Research ArticleThe Interval-Valued Trapezoidal Approximation ofInterval-Valued Fuzzy Numbers and Its Application in FuzzyRisk Analysis
Zengtai Gong1 and Shexiang Hai12
1 College of Mathematics and Statistics Northwest Normal University Lanzhou 730070 China2 School of Science Lanzhou University of Technology Lanzhou 730050 China
Correspondence should be addressed to Zengtai Gong zt-gong163com
Received 27 February 2014 Revised 27 May 2014 Accepted 9 July 2014 Published 12 August 2014
Academic Editor Francisco J Marcellan
Copyright copy 2014 Z Gong and S Hai This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Taking into account that interval-valued fuzzy numbers can provide more flexibility to represent the imprecise information andinterval-valued trapezoidal fuzzy numbers are widely used in practice this paper devotes to seek an approximation operator thatproduces an interval-valued trapezoidal fuzzy number which is the nearest one to the given interval-valued fuzzy number and theapproximation operator preserves the core of the original interval-valued fuzzy number with respect to the weighted distance Asan application we use the interval-valued trapezoidal approximation to handle fuzzy risk analysis problems which overcome thedrawback of existing fuzzy risk analysis methods
1 Introduction
The theory of fuzzy set proposed by Zadeh [1] has receiveda great deal of attention due to its capability of handlinguncertainty Uncertainty exists almost everywhere except inthe most idealized situations it is not only an inevitable andubiquitous phenomenon but also a fundamental scientificprinciple As a generalization of an ordinary Zadehrsquos fuzzyset the notion of interval-valued fuzzy sets was suggestedfor the first time by Gorzalczany [2] and Turksen [3] It wasintroduced to alleviate some drawbacks of fuzzy set theoryand has been applied to the fields of approximate inferencesignal transmission and control and so forth
In 1998 Wang and Li [4] defined interval-valued fuzzynumbers and gave their extended operations In practiceinterval-valued trapezoidal fuzzy numbers are widely usedin decision making risk analysis sensitivity analysis andother fields [5ndash7] In this paper we are interested in approxi-mating interval-valued fuzzy numbers by means of interval-valued trapezoidal fuzzy numbers to simplify calculationsThe interval-valued trapezoidal approximationmust preservesome parameters of the given interval-valued fuzzy numbersuch as 120572-level set invariance translation invariance scale
invariance identity nearness criterion ranking invarianceand continuity Considering that the core (120572-level set where120572 = 1) of an interval-valued fuzzy number is an importantparameter in practical problems we use the Karush-Kuhn-TucherTheorem to investigate the interval-valued trapezoidalapproximation of an interval-valued fuzzy number whichpreserves its core
The plan of this paper goes as follows Section 2 containssome basic notations of interval-valued fuzzy numbers andthe 120572-level set of interval-valued fuzzy numbers is presentedwhich differs from [8] Some results related to interval-valued fuzzy numbers are investigated these results will befrequently referred to in the subsequent sections Section 3is devoted to seek an approximation operator 119879 IF(119877) rarrIF119879(119877) that produces an interval-valued trapezoidal fuzzynumber which is the nearest one to the given interval-valuedfuzzy number among all interval-valued trapezoidal fuzzynumbers and it preserves the core of the original interval-valued fuzzy number with respect to the weighted distance119863119868 In Section 4 some properties of the approximation
operator such as translation invariance scale invarianceidentity nearness criterion ranking invariance and distanceproperty are discussed As an application we also use the
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 254853 22 pageshttpdxdoiorg1011552014254853
2 Journal of Applied Mathematics
approximation operator to handle fuzzy risk analysis prob-lems which provides us with a useful way to deal with fuzzyrisk analysis problems in Section 5
2 Preliminaries
21 Fuzzy Numbers In 1972 Chang and Zadeh [9] intro-duced the conception of fuzzy numbers with the considera-tion of the properties of probability functions Since then thetheory of fuzzy numbers and its applications have expansivelybeen developed in data analysis artificial intelligence anddecision making This section will remind us of the basicnotations of fuzzy numbers and give readers a better under-standing of the paper
Definition 1 (see [11ndash13]) A fuzzy number119860 is a subset of thereal line 119877 with the membership function 120583 119877 rarr [0 1]
such that the following holds
(i) 119860 is normal that is there is an 1199090isin 119877with 120583(119909
0) = 1
(ii) 119860 is fuzzy convex that is 120583(120582119909 + (1 minus 120582)119910) ge
min120583(119909) 120583(119910) for any 119909 119910 isin 119877 and 120582 isin [0 1](iii) 120583 is upper semicontinuous that is 120583minus1([120572 1]) is
closed for any 120572 isin [0 1](iv) The support of 120583 is bounded that is the closure of
119909 isin 119877 120583(119909) gt 0 is bounded
We denote by 119865(119877) the set of all fuzzy numbers on 119877Let 119860 isin 119865(119877) whose membership function 120583(119909) can
generally be defined as [14]
120583 (119909) =
119897119860 (119909) 119886 le 119909 lt 119887
1 119887 le 119909 le 119888
119903119860 (119909) 119888 lt 119909 le 119889
0 otherwise
(1)
where 119886 119887 119888 119889 isin 119877 119897119860 [119886 119887) rarr [0 1] is a nondecreas-
ing upper semicontinuous function such that 119897119860(119886) = 0
119897119860(119887) = 1 119903
119860 (119888 119889] rarr [0 1] is a nonincreasing upper
semicontinuous function satisfying 119903119860(119888) = 1 119903
119860(119889) = 0 119897
119860
and 119903119860are called the left and the right side of 119860 respectively
For any 120572 isin (0 1] the 120572-level set of a fuzzy number 119860 isa crisp set defined as [15]
119860120572= 119909 isin 119877 120583 (119909) ge 120572 (2)
The support or 0-level set 1198600of a fuzzy number is defined as
1198600= 119909 isin 119877 120583(119909) ge 0 (3)
It is well known that every 120572-level set of a fuzzy number 119860 isa closed interval denoted as
119860120572= [119860minus (120572) 119860+ (
120572)] (4)
where119860minus (120572) = inf 119909 isin 119877 120583 (119909) ge 120572
119860+ (120572) = sup 119909 isin 119877 120583 (119909) ge 120572
(5)
It is obvious that 119860minus(120572) and 119860
+(120572) are the inverse functions
of 119897119860and 119903119860 respectively
An often used fuzzy number is the trapezoidal fuzzynumber which is completely characterized by four realnumbers 119905
1le 1199052le 1199053le 1199054 denoted by 119879 = 119905
1 1199052 1199053 1199054
and with the membership function
120583 (119909) =
119909 minus 1199051
1199052minus 1199051
1199051le 119909 lt 119905
2
1 1199052le 119909 le 119905
3
1199054minus 119909
1199054minus 1199053
1199053lt 119909 le 119905
4
0 otherwise
(6)
Wewrite 119865119879(119877) as the family of all trapezoidal fuzzy numberson 119877
22 Interval-Valued Fuzzy Numbers This section is devotedto review basic concept of interval-valued fuzzy numberswhich will be used extensively throughout this paper
Let 119868 be a closed unit interval that is 119868 = [0 1] and [119868] =119886 = [119886
minus 119886+] 119886minusle 119886+ 119886minus 119886+isin 119868
Definition 2 (see [16]) Let 119883 be an ordinary nonempty setThen the mapping 119860 119883 rarr [119868] is called an interval-valuedfuzzy set on119883 All interval-valued fuzzy sets on119883 are denotedby IF(119883)
An interval-valued fuzzy set 119860 defined on119883 is given by
119860 = (119909 [119860119871(119909) 119860
119880(119909)]) 119909 isin 119883 (7)
where 0 le 119860119871(119909) le 119860119880(119909) le 1 The interval-valued fuzzy set119860 can be represented by an interval 119860(119909) = [119860119871(x) 119860119880(119909)]and the ordinary fuzzy sets 119860119871 119883 rarr 119868 and 119860119880 119883 rarr 119868
are called a lower and an upper fuzzy set of 119860 respectively
Definition 3 (see [17]) If an interval-valued fuzzy set 119860(119909) =[119860119871(119909) 119860
119880(119909)] satisfies the following conditions
(i) 119860 is normal that is there is an 1199090isin 119877 with 119860(119909
0) =
[1 1]
(ii) 119860 is convex that is 119860119871(120582119909 + (1 minus 120582)119910) ge
min(119860119871(119909) 119860119871(119910)) and 119860119880(120582119909 + (1 minus 120582)119910) ge
min(119860119880(119909) 119860119880(119910)) for any 119909 119910 isin 119877 and 120582 isin [0 1]
(iii) 119860119871(119909) and 119860119880(119909) are upper semicontinuous
(iv) the support of 119860119871(119909) and 119860119880(119909) are bounded that isthe closure of 119909 isin 119877 119860119871(119909) gt 0 and 119909 isin 119877 119860119880(119909) gt 0 are bounded
then 119860 is called an interval-valued fuzzy number on 119877 Allinterval-valued fuzzy numbers on 119877 are denoted by IF(119877)
Journal of Applied Mathematics 3
For any 119860 = [119860119871 119860119880] isin IF(119877) the lower fuzzy number119860119871 and the upper fuzzy number 119860119880 can be represented as
119860119871(119909) =
119897119860119871 (119909) 119886
119871le 119909 lt 119887
119871
1 119887119871le 119909 le 119888
119871
119903119860119871 (119909) 119888
119871lt 119909 le 119889
119871
0 otherwise
(8)
119860119880(119909) =
119897119860119880 (119909) 119886
119880le 119909 lt 119887
119880
1 119887119880le 119909 le 119888
119880
119903119860119880 (119909) 119888
119880lt 119909 le 119889
119880
0 otherwise
(9)
respectively where 119886119871 119887119871 119888119871 119889119871 119886119880 119887119880 119888119880 119889119880 isin 119877 119897119860119871
[119886119871 119887119871) rarr [0 1] and 119897
119860119880 [119886
119880 b119880) rarr [0 1] are
nondecreasing upper semicontinuous functions such that119897119860119871(119886119871) = 0 119897
119860119871(119887119871) = 1 119897
119860119880(119886119880) = 0 and 119897
119860119880(119887119880) =
1 119903119860119871 (119888
119871 119889119871] rarr [0 1] and 119903
119860119880 (119888
119880 119889119880] rarr [0 1]
are nonincreasing upper semicontinuous functions fulfilling119903119860119871(119888119871) = 1 119903
119860119871(119889119871) = 0 119903
119860119880(119888119880) = 1 and 119903
119860119880(119889119880) = 0
If 119886119871 = 119886119880 119887119871 = 119887
119880 119888119871 = 119888119880 119889119871 = 119889
119880 119897119860119871(119909) =
119897119860119880(119909) and 119903
119860119871(119909) = 119903
119860119880(119909) that is 119860119871(119909) = 119860119880(119909) then
the interval-valued fuzzy number 119860 = [119860119871 119860119880] is a fuzzy
numberFor any 120572 isin [0 1] the 120572-level set of an interval-valued
fuzzy number 119860 is defined as
119860120572= (119909 119910) isin 119877
2 119860119871(119909) ge 120572 119860
119880(119910) ge 120572
= (119909 119910) isin 1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)]
119910 isin [119860119880
minus(120572) 119860
119880
+(120572)]
(10)
where 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and 119860119880
+(120572) are the inverse
functions of 119897119860119871 119903119860119871 119897119860119880 and 119903
119860119880 respectively If 119860119871 = 119860119880
then this definition coincides with (4) The core of 119860 ispresented as
core119860 = (119909 119910) isin 1198772 119909 isin [119860119871minus(1) 119860
119871
+(1)]
119910 isin [119860119880
minus(1) 119860
119880
+(1)]
(11)
Theorem 4 Let 119860119871 119860119880 isin 119865(119877) 119860 = [119860119871 119860119880] isin IF(119877) if andonly if 119860119880
minus(120572) le 119860
119871
minus(120572) 119860119880
+(120572) ge 119860
119871
+(120572) for any 120572 isin [0 1]
Proof If If 119909 isin [119886119880 119887119880) then there exist 1205721 1205722isin [0 1] such
that
1205721= 119897119860119871 (119909) 120572
2= 119897119860119880 (119909) (12)
Since 119860119880minus(120572) le 119860
119871
minus(120572) for any 120572 isin [0 1] this implies that
119909 = 119860119880
minus(1205722) le 119860119871
minus(1205722) ≜ 1199091015840 (13)
where 1199091015840 isin [119886119880 119887119871] By the monotonicity of 119897119860119871 we have
119897119860119871 (119909) le 119897119860
119871 (1199091015840) = 1205722= 119897119860119880 (119909) (14)
Similarly we can prove that 119903119860119880(119909) ge 119903
119860119871(119909) for any 119909 isin
(119888119880 119889119880] If 119909 isin [119887119880 119888119880] then 119860119880(119909) = 1 ge 119860119871(119909) Therefore
119860119871(119909) le 119860
119880(119909) for any 119909 isin [119886119880 119889119880] that is 119860 = [119860119871 119860119880] isin
IF(119877)Only if If 120572 isin [0 1] then there exist 119909
1isin [119886119880 119887119880] 1199092isin
[119886119871 119887119871] such that
1199091= 119860119880
minus(120572) 119909
2= 119860119871
minus(120572) (15)
Since 119897119860119880(119909) ge 119897
119860119871(119909) for any 119909 isin [119886119880 119887119880] this implies that
120572 = 119897119860119880 (1199091) ge 119897119860119871 (1199091) ≜ 1205721015840 (16)
where 1205721015840 isin [0 1] By the monotonicity of 119860119871minus we have
119860119871
minus(120572) ge 119860
119871
minus(1205721015840) = 1199091= 119860119880
minus(120572) (17)
Similarly we can prove that119860119880+(120572) ge 119860
119871
+(120572) for any120572 isin [0 1]
This concludes the proof
It is well known interval-valued fuzzy numbers with sim-plemembership functions are preferred in practice Howeveras a particular of interval-valued fuzzy numbers interval-valued trapezoidal fuzzy numbers could be wide appliedin real mathematical modeling Thus the properties of theinterval-valued trapezoidal fuzzy number are discussed asfollows
Definition 5 (see [6 18ndash20]) Let 119860 = [119860119871 119860119880] isin IF(119877) If
119860119871 119860119880isin 119865119879(119877) then 119860 is called an interval-valued trape-
zoidal fuzzy numberThe lower trapezoidal fuzzy number119860119871is expressed as
119860119871(119909) =
119909 minus 119905119871
1
119905119871
2minus 119905119871
1
119905119871
1le 119909 lt 119905
119871
2
1 119905119871
2le 119909 le 119905
119871
3
119905119871
4minus 119909
119905119871
4minus 119905119871
3
119905119871
3lt 119909 le 119905
119871
4
0 otherwise
(18)
and the upper trapezoidal fuzzy number 119860119880 is expressed as
119860119880(119909) =
119909 minus 119905119880
1
119905119880
2minus 119905119880
1
119905119880
1le 119909 lt 119905
119880
2
1 119905119880
2le 119909 le 119905
119880
3
119905119880
4minus 119909
119905119880
4minus 119905119880
3
119905119880
3lt 119909 le 119905
119880
4
0 otherwise
(19)
An interval-valued trapezoidal fuzzy number 119860 can berepresented as 119860 = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] The family
of all interval-valued trapezoidal fuzzy numbers on 119877 isdenoted as IF119879(119877)
Theorem 6 Let 119860119871 119860119880 isin 119865119879(119877) 119860 = [119860119871 119860119880] isin 119868119865119879(119877) ifand only if 119905119880
1le 119905119871
1 1199051198802le 119905119871
2 1199051198803ge 119905119871
3and 1199051198804ge 119905119871
4
4 Journal of Applied Mathematics
23 TheWeighted Distance of Interval-Valued Fuzzy NumbersIn 2007 Zeng and Li [21] introduced the weighted distance offuzzy numbers 119860 and 119861 as follows
1198892
119891(119860 119861) = int
1
0
119891 (120572) (119860minus (120572) minus 119861minus (
120572))2119889120572
+ int
1
0
119891 (120572) (119860+ (120572) minus 119861+ (
120572))2119889120572
(20)
where the function 119891(120572) is nonnegative and increasing on[0 1] with 119891(0) = 0 and int1
0119891(120572)119889120572 = 12 The function
119891(120572) is also called the weighting function The property ofmonotone increasing of function 119891(120572)means that the higherthe cut level the more important its weight in determiningthe distance of fuzzy numbers 119860 and 119861 Both conditions119891(0) = 0 and int1
0119891(120572)119889120572 = 12 ensure that the distance
defined by (20) is the extension of the ordinary distance in119877 defined by its absolute value That means this distancebecomes an absolute value in119877when a fuzzy number reducesto a real number In applications the function 119891(120572) can bechosen according to the actual situation
We will define the weighted distance of interval-valuedfuzzy numbers as follows It can be considered as a naturalextension of the weighted distance 119889
119891(119860 119861) of fuzzy num-
bers
Definition 7 Let 119860 119861 isin IF(119877) The weighted distance of 119860and 119861 is defined as
119863119868 (119860 119861) =
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus 119861
119871
minus(120572))
2
119889120572
+ int
1
0
119891(120572)(119860119871
+(120572) minus 119861
119871
+(120572))
2
119889120572)
12
+ (int
1
0
119891 (120572) (119860119880
minus(120572) minus 119861
119880
minus(120572))
2
119889120572
+ int
1
0
119891(120572)(119860119880
+(120572) minus 119861
119880
+(120572))
2
119889120572)
12
]
=
1
2
[119889119891(119860119871 119861119871) + 119889119891(119860119880 119861119880)]
(21)
If 119860119871 = 119860119880 and 119861119871 = 119861119880 then119863119868(119860 119861) = 119889
119891(119860 119861)
Property 1 Let119860 119861 isin IF(119877) Then119863119868(119860 119861) = 0 if and only if
119863(119860119871 119861119871) = 0 and119863(119860119880 119861119880) = 0
Theorem 8 (IF(119877) 119863119868) is a metric space
By the completeness of metric space (119865(119877) 119889119891) we can
obtain the following conclusion
Theorem 9 The metric space (IF(119877) 119863119868) is complete
24 The Ranking of Interval-Valued Fuzzy Numbers Theranking of fuzzy numbers was studied by many researchers
and itwas extended to interval-valued fuzzy numbers becauseof its attraction and applicabilityWewill propose a ranking ofinterval-valued fuzzy numbers which embodies the impor-tance of the core of interval-valued fuzzy numbers
Definition 10 Let 119860 119861 isin IF(119877) The ranking of 119860 119861 can bedefined by the following formula
119860 ⪰ 119861 lArrrArr 119860119871
minus(1) + 119860
119871
+(1) ge 119861
119871
minus(1) + 119861
119871
+(1)
119860119880
minus(1) + 119860
119880
+(1) ge 119861
119880
minus(1) + 119861
119880
+(1)
(22)
Example 11 Let
119860119871(119909) = 119861
119871(119909) =
1 minus (119909 minus 3)2 119909 isin [2 4]
0 otherwise
119860119880(119909) =
1 minus (119909 minus 3)2 119909 isin [2 3)
1 119909 isin [3 5]
minus119909 + 6 119909 isin (5 6]
0 otherwise
119861119880(119909) =
1 minus (119909 minus 3)2 119909 isin [2 3)
1 minus
1
9
(119909 minus 3)2 119909 isin [3 6]
0 otherwise
(23)
We obtain core119860 = (119909 119910) isin 1198772 119909 = 3 119910 isin [3 5] and
core119861 = (119909 119910) isin 1198772 119909 = 3 119910 = 3 By a direct calculationwe have 119860 ⪰ 119861
3 Weighted Interval-ValuedTrapezoidal Approximation
31 Criteria for Interval-Valued Trapezoidal ApproximationIf we want to approximate an interval-valued fuzzy numberby an interval-valued trapezoidal fuzzy number we mustuse an approximate operator 119879 IF(119877) rarr IF119879(119877) whichtransforms a family of all interval-valued fuzzy numbers 119860into a family of interval-valued trapezoidal fuzzy numbers119879(119860) that is 119879 119860 rarr 119879(119860) Since interval-valuedtrapezoidal approximation could also be performed in manyways we propose a number of criteria which the approxi-mation operator should possess at least one Reference [22]has given some criteria for the fuzzy number approximationsimilarly we give some criteria for interval-valued trapezoidalapproximation as follows
311 120572-Level Set Invariance An approximation operator 119879 is120572-level set invariant if
(119879 (119860))120572= 119860120572 (24)
Remark 12 For any two different levels 1205721and 120572
2(1205721= 1205722)
we obtain one and only one approximation operator which isinvariant both in 120572
1- and 120572
2-level set
Proof Let 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2 119909 isin
[119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] Then we
Journal of Applied Mathematics 5
can obtain one and only one interval-valued trapezoidal fuzzynumber 119879(119860) = [(119879(119860))119871 (119879(119860))119880] where
(119879 (A))119871120572= [
119860119871
minus(1205722) minus 119860119871
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
minus(1205721)
119860119871
+(1205722) minus 119860119871
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
+(1205721)]
(119879 (119860))119880
120572= [
119860119880
minus(1205722) minus 119860119880
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
minus(1205721)
119860119880
+(1205722) minus 119860119880
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
+(1205721)]
(25)
It is obvious that
(119879 (119860))1205721= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205721) 119860119871
+(1205721)]
119910 isin [119860119880
minus(1205721) 119860119880
+(1205721)]
(119879 (119860))1205722= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205722) 119860119871
+(1205722)]
119910 isin [119860119880
minus(1205722) 119860119880
+(1205722)]
(26)
Hence (119879(119860))1205721= 1198601205721and (119879(119860))
1205722= 1198601205722
312 Translation Invariance For 119860 isin IF(119877) and 119911 isin 119877 wedefine
119860 + 119911 = [(119860 + 119911)119871 (119860 + 119911)
119880] (27)
where (119860 + 119911)120572= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(120572) + 119911 119860
119871
+(120572) +
119911] 119910 isin [119860119880
minus(120572) + 119911 119860
119880
+(120572) + 119911] 120572 isin [0 1] that is
(119860 + 119911)119871
minus(120572) = 119860
119871
minus(120572) + 119911
(119860 + 119911)119871
+(120572) = 119860
119871
+(120572) + 119911
(28)
(119860 + 119911)119880
minus(120572) = 119860
119880
minus(120572) + 119911
(119860 + 119911)119880
+(120572) = 119860
119880
+(120572) + 119911
(29)
An approximation operator 119879 is invariant to translation if
119879 (119860 + 119911) = 119879 (119860) + 119911 119911 isin 119877 (30)
Translation invariance means that the relative position of theinterval-valued trapezoidal approximation remains constantwhen the membership function is moved to the left or to theright
313 Scale Invariance For 119860 isin IF(119877) and 120582 isin 119877 0 wedefine
120582119860 = [120582119860119871 120582119860119880] (31)
When 120582 gt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
minus(120572)
120582119860119871
+(120572)] 119910 isin [120582119860
119880
minus(120572) 120582119860
119880
+(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
minus(120572)
(120582119860)119871
+(120572) = 120582119860
119871
+(120572)
(120582119860)119880
minus(120572) = 120582119860
119880
minus(120572)
(120582119860)119880
+(120572) = 120582119860
119880
+(120572)
(32)
When 120582 lt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
+(120572)
120582119860119871
minus(120572)] 119910 isin [120582119860
119880
+(120572) 120582119860
119880
minus(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
+(120572)
(120582119860)119871
+(120572) = 120582119860
119871
minus(120572)
(33)
(120582119860)119880
minus(120572) = 120582119860
119880
+(120572)
(120582119860)119880
+(120572) = 120582119860
119880
minus(120572)
(34)
We say that an approximation operator 119879 is scale invariant if
119879 (120582119860) = 120582119879 (119860) 120582 isin 119877 0 (35)
314 Identity This criterion states that the interval-valuedtrapezoidal approximation of an interval-valued trapezoidalfuzzy number is equivalent to that number that is if 119860 isin
IF119879(119877) then
119879 (119860) = 119860 (36)
315 Nearness Criterion An approximation operator 119879 ful-fills the nearness criterion if for any interval-valued fuzzynumber119860 its output value119879(119860) is the nearest interval-valuedtrapezoidal fuzzy number to 119860 with respect to the weighteddistance 119863
119868defined by (21) In other words for any 119861 isin
IF119879(119877) we have
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (37)
Remark 13 We can verify that IF(119877) is closed and convex so119879(119860) exists and is unique
316 Ranking Invariance A reasonable approximation oper-ator should preserve the accepted ranking We say that anapproximation operator 119879 is ranking invariant if for any119860 119861 isin IF(119877)
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (38)
317 Continuity Let 119860 119861 isin IF(119877) An approximationoperator 119879 is continuous if for any 120576 gt 0 there is 120575 gt 0 when119863119868(119860 119861) lt 120575 we have
119863119868 (119879 (119860) 119879 (119861)) lt 120576 (39)
The continuity constraint means that if two interval-valuedfuzzy numbers are close then their interval-valued trape-zoidal approximations also should be close
6 Journal of Applied Mathematics
32 Interval-Valued Trapezoidal Approximation Based on theWeighted Distance In this section we are looking for anapproximation operator 119879 IF(119877) rarr IF119879(119877) whichproduces an interval-valued trapezoidal fuzzy number thatis the nearest one to the given interval-valued fuzzy numberand preserves its core with respect to the weighted distance119863119868defined by (21)
Lemma 14 Let 119860 isin 119865(119877) 119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1]
If function 119891(120572) is nonnegative and increasing on [0 1] with119891(0) = 0 and int1
0119891(120572)119889120572 = 12 then we have
(i)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860minus (
1) minus 119860minus (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
le 119860minus (1) (40)
(ii)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (41)
Proof (i) See [23] the proof of Theorem 31(ii) Since 119860
+(120572) is a nonincreasing function we have
119860+(120572) ge 119860
+(1) for any 120572 isin [0 1] By 119891(120572) ge 0 we can prove
that
(1 minus 120572)119860+ (120572) 119891 (120572) ge (1 minus 120572)119860+ (
1) 119891 (120572)
= [(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572)
(42)
According to the monotonicity of integration we have
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
ge int
1
0
[(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572) 119889120572
(43)
That is
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
minus 119860+ (1) int
1
0
120572 (1 minus 120572) 119891 (120572) 119889120572
ge 119860+ (1) int
1
0
(120572 minus 1)2119891 (120572) 119889120572
(44)
Because int10(120572 minus 1)
2119891(120572)119889120572 gt 0 it follows that
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (45)
Theorem 15 (see [24]) Let 119891 1198921 1198922 119892
119898 119877119899rarr 119877 be
convex and differentiable functions Then 119909 solves the convexprogramming problem
min 119891 (119909)
119904119905 119892119894(119909) le 119887119894
119894 isin 1 2 119898
(46)
if and only if there exist 120583119894 119894 isin 1 2 119898 such that
(i) nabla119891(119909) + Σ119898119894=1120583119894nabla119892119894(119909) = 0
(ii) 119892119894(119909) minus 119887
119894le 0
(iii) 120583119894ge 0
(iv) 120583119894(119887119894minus 119892119894(119909)) = 0
Suppose that 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2
119909 isin [119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] We
will try to find an interval-valued trapezoidal fuzzy number119879(119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] which is the nearest
interval-valued trapezoidal fuzzy number of 119860 and preservesits core with respect to the weighted distance 119863
119868 Thus we have
to find such real numbers 1199051198711 1199051198712 1199051198713 1199051198714 1199051198801 1199051198802 1199051198803and 1199051198804that
minimize
119863119868 (119860 119879 (119860))
=
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus (119905
119871
1+ (119905119871
2minus 119905119871
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119905119871
3) 120572))
2
119889120572)
12
+ (int
1
0
f (120572) (119860119880minus(120572) minus (119905
119880
1+ (119905119880
2minus 119905119880
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119905119880
3) 120572))
2
119889120572)
12
]
(47)
with respect to condition core119860 = core119879(119860) that is
119905119871
2= 119860119871
minus(1) 119905
119871
3= 119860119871
+(1)
119905119880
2= 119860119880
minus(1) 119905
119880
3= 119860119880
+(1)
(48)
It follows that
119905119871
2le 119905119871
3 119905
119880
2le 119905119880
3 (49)
Making use of Theorem 4 we have
119905119880
2le 119905119871
2 119905
119871
3le 119905119880
3 (50)
Journal of Applied Mathematics 7
Using (47) and (50) together with Theorem 6 we only need tominimize the function
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) [119860119871
minus(120572) minus (119905
119871
1+ (119860119871
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119860119871
+(1)) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
minus(120572) minus (119905
119880
1+ (119860119880
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119860119880
+(1)) 120572)]
2
119889120572
(51)
subject to
119905119880
1minus 119905119871
1le 0 119905
119871
4minus 119905119880
4le 0 (52)
After simple calculations we obtain
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
4)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
4)
2
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 sdot 119905
119871
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 sdot 119905
119871
4
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 sdot 119905
119880
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 sdot 119905
119880
4
+ int
1
0
119891 (120572) (119860119871
minus(120572) minus 120572 sdot 119860
119871
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119871
+(120572) minus 120572 sdot 119860
119871
+(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
minus(120572) minus 120572 sdot 119860
119880
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
+(120572) minus 120572 sdot 119860
119880
+(1))
2
119889120572
(53)
subject to
119905119880
1minus 119905119871
1le 0
119905119871
4minus 119905119880
4le 0
(54)
We present the main result of the paper as follows
Theorem 16 Let 119860 = [119860119871 119860119880] isin IF(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] is the nearest
interval-valued trapezoidal fuzzy number to 119860 and preservesits core with respect to the weighted distance 119863
119868 Consider the
following(i) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(55)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(56)
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(57)
(ii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(58)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(59)
8 Journal of Applied Mathematics
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(60)
(iii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
(61)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(62)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(63)
(iv) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(64)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(65)
Proof Because the function 119865 in (53) and conditions (54)satisfy the hypothesis of convexity and differentiability inTheorem 15 after some simple calculations conditions (i)ndash(iv) inTheorem 15 with respect to the minimization problem(53) in conditions (54) can be shown as follows
2119905119871
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 minus 1205831
= 0
(66)
2119905119871
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 + 1205832
= 0
(67)
2119905119880
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 + 1205831
= 0
(68)
Journal of Applied Mathematics 9
2119905119880
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 minus 1205832
= 0
(69)
1205831(119905119880
1minus 119905119871
1) = 0 (70)
1205832(119905119871
4minus 119905119880
4) = 0 (71)
1205831ge 0 (72)
1205832ge 0 (73)
119905119880
1minus 119905119871
1le 0 (74)
119905119871
4minus 119905119880
4le 0 (75)
(i) In the case 1205831gt 0 and 120583
2= 0 the solution of the system
(66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(76)
Firstly we have from (55) that 1205831gt 0 and it follows from
(56) that
119905119880
4minus 119905119871
4=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(77)
Then conditions (72) (73) (74) and (75) are verified
Secondly combing with (48) (55) and Lemma 14 (i) wecan prove that
119905119880
2minus 119905119880
1
= 119860119880
minus(1) + ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(78)
And on the basis of (50) we have
119905119871
2minus 119905119871
1ge 119905119880
2minus 119905119880
1gt 0 (79)
By making use of (48) and Lemma 14 (ii) we get
119905119880
4minus 119905119880
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119880
+(1) ge 0
119905119871
4minus 119905119871
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119871
+(1) ge 0
(80)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(ii) In the case 120583
1gt 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
approximation operator to handle fuzzy risk analysis prob-lems which provides us with a useful way to deal with fuzzyrisk analysis problems in Section 5
2 Preliminaries
21 Fuzzy Numbers In 1972 Chang and Zadeh [9] intro-duced the conception of fuzzy numbers with the considera-tion of the properties of probability functions Since then thetheory of fuzzy numbers and its applications have expansivelybeen developed in data analysis artificial intelligence anddecision making This section will remind us of the basicnotations of fuzzy numbers and give readers a better under-standing of the paper
Definition 1 (see [11ndash13]) A fuzzy number119860 is a subset of thereal line 119877 with the membership function 120583 119877 rarr [0 1]
such that the following holds
(i) 119860 is normal that is there is an 1199090isin 119877with 120583(119909
0) = 1
(ii) 119860 is fuzzy convex that is 120583(120582119909 + (1 minus 120582)119910) ge
min120583(119909) 120583(119910) for any 119909 119910 isin 119877 and 120582 isin [0 1](iii) 120583 is upper semicontinuous that is 120583minus1([120572 1]) is
closed for any 120572 isin [0 1](iv) The support of 120583 is bounded that is the closure of
119909 isin 119877 120583(119909) gt 0 is bounded
We denote by 119865(119877) the set of all fuzzy numbers on 119877Let 119860 isin 119865(119877) whose membership function 120583(119909) can
generally be defined as [14]
120583 (119909) =
119897119860 (119909) 119886 le 119909 lt 119887
1 119887 le 119909 le 119888
119903119860 (119909) 119888 lt 119909 le 119889
0 otherwise
(1)
where 119886 119887 119888 119889 isin 119877 119897119860 [119886 119887) rarr [0 1] is a nondecreas-
ing upper semicontinuous function such that 119897119860(119886) = 0
119897119860(119887) = 1 119903
119860 (119888 119889] rarr [0 1] is a nonincreasing upper
semicontinuous function satisfying 119903119860(119888) = 1 119903
119860(119889) = 0 119897
119860
and 119903119860are called the left and the right side of 119860 respectively
For any 120572 isin (0 1] the 120572-level set of a fuzzy number 119860 isa crisp set defined as [15]
119860120572= 119909 isin 119877 120583 (119909) ge 120572 (2)
The support or 0-level set 1198600of a fuzzy number is defined as
1198600= 119909 isin 119877 120583(119909) ge 0 (3)
It is well known that every 120572-level set of a fuzzy number 119860 isa closed interval denoted as
119860120572= [119860minus (120572) 119860+ (
120572)] (4)
where119860minus (120572) = inf 119909 isin 119877 120583 (119909) ge 120572
119860+ (120572) = sup 119909 isin 119877 120583 (119909) ge 120572
(5)
It is obvious that 119860minus(120572) and 119860
+(120572) are the inverse functions
of 119897119860and 119903119860 respectively
An often used fuzzy number is the trapezoidal fuzzynumber which is completely characterized by four realnumbers 119905
1le 1199052le 1199053le 1199054 denoted by 119879 = 119905
1 1199052 1199053 1199054
and with the membership function
120583 (119909) =
119909 minus 1199051
1199052minus 1199051
1199051le 119909 lt 119905
2
1 1199052le 119909 le 119905
3
1199054minus 119909
1199054minus 1199053
1199053lt 119909 le 119905
4
0 otherwise
(6)
Wewrite 119865119879(119877) as the family of all trapezoidal fuzzy numberson 119877
22 Interval-Valued Fuzzy Numbers This section is devotedto review basic concept of interval-valued fuzzy numberswhich will be used extensively throughout this paper
Let 119868 be a closed unit interval that is 119868 = [0 1] and [119868] =119886 = [119886
minus 119886+] 119886minusle 119886+ 119886minus 119886+isin 119868
Definition 2 (see [16]) Let 119883 be an ordinary nonempty setThen the mapping 119860 119883 rarr [119868] is called an interval-valuedfuzzy set on119883 All interval-valued fuzzy sets on119883 are denotedby IF(119883)
An interval-valued fuzzy set 119860 defined on119883 is given by
119860 = (119909 [119860119871(119909) 119860
119880(119909)]) 119909 isin 119883 (7)
where 0 le 119860119871(119909) le 119860119880(119909) le 1 The interval-valued fuzzy set119860 can be represented by an interval 119860(119909) = [119860119871(x) 119860119880(119909)]and the ordinary fuzzy sets 119860119871 119883 rarr 119868 and 119860119880 119883 rarr 119868
are called a lower and an upper fuzzy set of 119860 respectively
Definition 3 (see [17]) If an interval-valued fuzzy set 119860(119909) =[119860119871(119909) 119860
119880(119909)] satisfies the following conditions
(i) 119860 is normal that is there is an 1199090isin 119877 with 119860(119909
0) =
[1 1]
(ii) 119860 is convex that is 119860119871(120582119909 + (1 minus 120582)119910) ge
min(119860119871(119909) 119860119871(119910)) and 119860119880(120582119909 + (1 minus 120582)119910) ge
min(119860119880(119909) 119860119880(119910)) for any 119909 119910 isin 119877 and 120582 isin [0 1]
(iii) 119860119871(119909) and 119860119880(119909) are upper semicontinuous
(iv) the support of 119860119871(119909) and 119860119880(119909) are bounded that isthe closure of 119909 isin 119877 119860119871(119909) gt 0 and 119909 isin 119877 119860119880(119909) gt 0 are bounded
then 119860 is called an interval-valued fuzzy number on 119877 Allinterval-valued fuzzy numbers on 119877 are denoted by IF(119877)
Journal of Applied Mathematics 3
For any 119860 = [119860119871 119860119880] isin IF(119877) the lower fuzzy number119860119871 and the upper fuzzy number 119860119880 can be represented as
119860119871(119909) =
119897119860119871 (119909) 119886
119871le 119909 lt 119887
119871
1 119887119871le 119909 le 119888
119871
119903119860119871 (119909) 119888
119871lt 119909 le 119889
119871
0 otherwise
(8)
119860119880(119909) =
119897119860119880 (119909) 119886
119880le 119909 lt 119887
119880
1 119887119880le 119909 le 119888
119880
119903119860119880 (119909) 119888
119880lt 119909 le 119889
119880
0 otherwise
(9)
respectively where 119886119871 119887119871 119888119871 119889119871 119886119880 119887119880 119888119880 119889119880 isin 119877 119897119860119871
[119886119871 119887119871) rarr [0 1] and 119897
119860119880 [119886
119880 b119880) rarr [0 1] are
nondecreasing upper semicontinuous functions such that119897119860119871(119886119871) = 0 119897
119860119871(119887119871) = 1 119897
119860119880(119886119880) = 0 and 119897
119860119880(119887119880) =
1 119903119860119871 (119888
119871 119889119871] rarr [0 1] and 119903
119860119880 (119888
119880 119889119880] rarr [0 1]
are nonincreasing upper semicontinuous functions fulfilling119903119860119871(119888119871) = 1 119903
119860119871(119889119871) = 0 119903
119860119880(119888119880) = 1 and 119903
119860119880(119889119880) = 0
If 119886119871 = 119886119880 119887119871 = 119887
119880 119888119871 = 119888119880 119889119871 = 119889
119880 119897119860119871(119909) =
119897119860119880(119909) and 119903
119860119871(119909) = 119903
119860119880(119909) that is 119860119871(119909) = 119860119880(119909) then
the interval-valued fuzzy number 119860 = [119860119871 119860119880] is a fuzzy
numberFor any 120572 isin [0 1] the 120572-level set of an interval-valued
fuzzy number 119860 is defined as
119860120572= (119909 119910) isin 119877
2 119860119871(119909) ge 120572 119860
119880(119910) ge 120572
= (119909 119910) isin 1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)]
119910 isin [119860119880
minus(120572) 119860
119880
+(120572)]
(10)
where 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and 119860119880
+(120572) are the inverse
functions of 119897119860119871 119903119860119871 119897119860119880 and 119903
119860119880 respectively If 119860119871 = 119860119880
then this definition coincides with (4) The core of 119860 ispresented as
core119860 = (119909 119910) isin 1198772 119909 isin [119860119871minus(1) 119860
119871
+(1)]
119910 isin [119860119880
minus(1) 119860
119880
+(1)]
(11)
Theorem 4 Let 119860119871 119860119880 isin 119865(119877) 119860 = [119860119871 119860119880] isin IF(119877) if andonly if 119860119880
minus(120572) le 119860
119871
minus(120572) 119860119880
+(120572) ge 119860
119871
+(120572) for any 120572 isin [0 1]
Proof If If 119909 isin [119886119880 119887119880) then there exist 1205721 1205722isin [0 1] such
that
1205721= 119897119860119871 (119909) 120572
2= 119897119860119880 (119909) (12)
Since 119860119880minus(120572) le 119860
119871
minus(120572) for any 120572 isin [0 1] this implies that
119909 = 119860119880
minus(1205722) le 119860119871
minus(1205722) ≜ 1199091015840 (13)
where 1199091015840 isin [119886119880 119887119871] By the monotonicity of 119897119860119871 we have
119897119860119871 (119909) le 119897119860
119871 (1199091015840) = 1205722= 119897119860119880 (119909) (14)
Similarly we can prove that 119903119860119880(119909) ge 119903
119860119871(119909) for any 119909 isin
(119888119880 119889119880] If 119909 isin [119887119880 119888119880] then 119860119880(119909) = 1 ge 119860119871(119909) Therefore
119860119871(119909) le 119860
119880(119909) for any 119909 isin [119886119880 119889119880] that is 119860 = [119860119871 119860119880] isin
IF(119877)Only if If 120572 isin [0 1] then there exist 119909
1isin [119886119880 119887119880] 1199092isin
[119886119871 119887119871] such that
1199091= 119860119880
minus(120572) 119909
2= 119860119871
minus(120572) (15)
Since 119897119860119880(119909) ge 119897
119860119871(119909) for any 119909 isin [119886119880 119887119880] this implies that
120572 = 119897119860119880 (1199091) ge 119897119860119871 (1199091) ≜ 1205721015840 (16)
where 1205721015840 isin [0 1] By the monotonicity of 119860119871minus we have
119860119871
minus(120572) ge 119860
119871
minus(1205721015840) = 1199091= 119860119880
minus(120572) (17)
Similarly we can prove that119860119880+(120572) ge 119860
119871
+(120572) for any120572 isin [0 1]
This concludes the proof
It is well known interval-valued fuzzy numbers with sim-plemembership functions are preferred in practice Howeveras a particular of interval-valued fuzzy numbers interval-valued trapezoidal fuzzy numbers could be wide appliedin real mathematical modeling Thus the properties of theinterval-valued trapezoidal fuzzy number are discussed asfollows
Definition 5 (see [6 18ndash20]) Let 119860 = [119860119871 119860119880] isin IF(119877) If
119860119871 119860119880isin 119865119879(119877) then 119860 is called an interval-valued trape-
zoidal fuzzy numberThe lower trapezoidal fuzzy number119860119871is expressed as
119860119871(119909) =
119909 minus 119905119871
1
119905119871
2minus 119905119871
1
119905119871
1le 119909 lt 119905
119871
2
1 119905119871
2le 119909 le 119905
119871
3
119905119871
4minus 119909
119905119871
4minus 119905119871
3
119905119871
3lt 119909 le 119905
119871
4
0 otherwise
(18)
and the upper trapezoidal fuzzy number 119860119880 is expressed as
119860119880(119909) =
119909 minus 119905119880
1
119905119880
2minus 119905119880
1
119905119880
1le 119909 lt 119905
119880
2
1 119905119880
2le 119909 le 119905
119880
3
119905119880
4minus 119909
119905119880
4minus 119905119880
3
119905119880
3lt 119909 le 119905
119880
4
0 otherwise
(19)
An interval-valued trapezoidal fuzzy number 119860 can berepresented as 119860 = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] The family
of all interval-valued trapezoidal fuzzy numbers on 119877 isdenoted as IF119879(119877)
Theorem 6 Let 119860119871 119860119880 isin 119865119879(119877) 119860 = [119860119871 119860119880] isin 119868119865119879(119877) ifand only if 119905119880
1le 119905119871
1 1199051198802le 119905119871
2 1199051198803ge 119905119871
3and 1199051198804ge 119905119871
4
4 Journal of Applied Mathematics
23 TheWeighted Distance of Interval-Valued Fuzzy NumbersIn 2007 Zeng and Li [21] introduced the weighted distance offuzzy numbers 119860 and 119861 as follows
1198892
119891(119860 119861) = int
1
0
119891 (120572) (119860minus (120572) minus 119861minus (
120572))2119889120572
+ int
1
0
119891 (120572) (119860+ (120572) minus 119861+ (
120572))2119889120572
(20)
where the function 119891(120572) is nonnegative and increasing on[0 1] with 119891(0) = 0 and int1
0119891(120572)119889120572 = 12 The function
119891(120572) is also called the weighting function The property ofmonotone increasing of function 119891(120572)means that the higherthe cut level the more important its weight in determiningthe distance of fuzzy numbers 119860 and 119861 Both conditions119891(0) = 0 and int1
0119891(120572)119889120572 = 12 ensure that the distance
defined by (20) is the extension of the ordinary distance in119877 defined by its absolute value That means this distancebecomes an absolute value in119877when a fuzzy number reducesto a real number In applications the function 119891(120572) can bechosen according to the actual situation
We will define the weighted distance of interval-valuedfuzzy numbers as follows It can be considered as a naturalextension of the weighted distance 119889
119891(119860 119861) of fuzzy num-
bers
Definition 7 Let 119860 119861 isin IF(119877) The weighted distance of 119860and 119861 is defined as
119863119868 (119860 119861) =
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus 119861
119871
minus(120572))
2
119889120572
+ int
1
0
119891(120572)(119860119871
+(120572) minus 119861
119871
+(120572))
2
119889120572)
12
+ (int
1
0
119891 (120572) (119860119880
minus(120572) minus 119861
119880
minus(120572))
2
119889120572
+ int
1
0
119891(120572)(119860119880
+(120572) minus 119861
119880
+(120572))
2
119889120572)
12
]
=
1
2
[119889119891(119860119871 119861119871) + 119889119891(119860119880 119861119880)]
(21)
If 119860119871 = 119860119880 and 119861119871 = 119861119880 then119863119868(119860 119861) = 119889
119891(119860 119861)
Property 1 Let119860 119861 isin IF(119877) Then119863119868(119860 119861) = 0 if and only if
119863(119860119871 119861119871) = 0 and119863(119860119880 119861119880) = 0
Theorem 8 (IF(119877) 119863119868) is a metric space
By the completeness of metric space (119865(119877) 119889119891) we can
obtain the following conclusion
Theorem 9 The metric space (IF(119877) 119863119868) is complete
24 The Ranking of Interval-Valued Fuzzy Numbers Theranking of fuzzy numbers was studied by many researchers
and itwas extended to interval-valued fuzzy numbers becauseof its attraction and applicabilityWewill propose a ranking ofinterval-valued fuzzy numbers which embodies the impor-tance of the core of interval-valued fuzzy numbers
Definition 10 Let 119860 119861 isin IF(119877) The ranking of 119860 119861 can bedefined by the following formula
119860 ⪰ 119861 lArrrArr 119860119871
minus(1) + 119860
119871
+(1) ge 119861
119871
minus(1) + 119861
119871
+(1)
119860119880
minus(1) + 119860
119880
+(1) ge 119861
119880
minus(1) + 119861
119880
+(1)
(22)
Example 11 Let
119860119871(119909) = 119861
119871(119909) =
1 minus (119909 minus 3)2 119909 isin [2 4]
0 otherwise
119860119880(119909) =
1 minus (119909 minus 3)2 119909 isin [2 3)
1 119909 isin [3 5]
minus119909 + 6 119909 isin (5 6]
0 otherwise
119861119880(119909) =
1 minus (119909 minus 3)2 119909 isin [2 3)
1 minus
1
9
(119909 minus 3)2 119909 isin [3 6]
0 otherwise
(23)
We obtain core119860 = (119909 119910) isin 1198772 119909 = 3 119910 isin [3 5] and
core119861 = (119909 119910) isin 1198772 119909 = 3 119910 = 3 By a direct calculationwe have 119860 ⪰ 119861
3 Weighted Interval-ValuedTrapezoidal Approximation
31 Criteria for Interval-Valued Trapezoidal ApproximationIf we want to approximate an interval-valued fuzzy numberby an interval-valued trapezoidal fuzzy number we mustuse an approximate operator 119879 IF(119877) rarr IF119879(119877) whichtransforms a family of all interval-valued fuzzy numbers 119860into a family of interval-valued trapezoidal fuzzy numbers119879(119860) that is 119879 119860 rarr 119879(119860) Since interval-valuedtrapezoidal approximation could also be performed in manyways we propose a number of criteria which the approxi-mation operator should possess at least one Reference [22]has given some criteria for the fuzzy number approximationsimilarly we give some criteria for interval-valued trapezoidalapproximation as follows
311 120572-Level Set Invariance An approximation operator 119879 is120572-level set invariant if
(119879 (119860))120572= 119860120572 (24)
Remark 12 For any two different levels 1205721and 120572
2(1205721= 1205722)
we obtain one and only one approximation operator which isinvariant both in 120572
1- and 120572
2-level set
Proof Let 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2 119909 isin
[119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] Then we
Journal of Applied Mathematics 5
can obtain one and only one interval-valued trapezoidal fuzzynumber 119879(119860) = [(119879(119860))119871 (119879(119860))119880] where
(119879 (A))119871120572= [
119860119871
minus(1205722) minus 119860119871
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
minus(1205721)
119860119871
+(1205722) minus 119860119871
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
+(1205721)]
(119879 (119860))119880
120572= [
119860119880
minus(1205722) minus 119860119880
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
minus(1205721)
119860119880
+(1205722) minus 119860119880
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
+(1205721)]
(25)
It is obvious that
(119879 (119860))1205721= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205721) 119860119871
+(1205721)]
119910 isin [119860119880
minus(1205721) 119860119880
+(1205721)]
(119879 (119860))1205722= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205722) 119860119871
+(1205722)]
119910 isin [119860119880
minus(1205722) 119860119880
+(1205722)]
(26)
Hence (119879(119860))1205721= 1198601205721and (119879(119860))
1205722= 1198601205722
312 Translation Invariance For 119860 isin IF(119877) and 119911 isin 119877 wedefine
119860 + 119911 = [(119860 + 119911)119871 (119860 + 119911)
119880] (27)
where (119860 + 119911)120572= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(120572) + 119911 119860
119871
+(120572) +
119911] 119910 isin [119860119880
minus(120572) + 119911 119860
119880
+(120572) + 119911] 120572 isin [0 1] that is
(119860 + 119911)119871
minus(120572) = 119860
119871
minus(120572) + 119911
(119860 + 119911)119871
+(120572) = 119860
119871
+(120572) + 119911
(28)
(119860 + 119911)119880
minus(120572) = 119860
119880
minus(120572) + 119911
(119860 + 119911)119880
+(120572) = 119860
119880
+(120572) + 119911
(29)
An approximation operator 119879 is invariant to translation if
119879 (119860 + 119911) = 119879 (119860) + 119911 119911 isin 119877 (30)
Translation invariance means that the relative position of theinterval-valued trapezoidal approximation remains constantwhen the membership function is moved to the left or to theright
313 Scale Invariance For 119860 isin IF(119877) and 120582 isin 119877 0 wedefine
120582119860 = [120582119860119871 120582119860119880] (31)
When 120582 gt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
minus(120572)
120582119860119871
+(120572)] 119910 isin [120582119860
119880
minus(120572) 120582119860
119880
+(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
minus(120572)
(120582119860)119871
+(120572) = 120582119860
119871
+(120572)
(120582119860)119880
minus(120572) = 120582119860
119880
minus(120572)
(120582119860)119880
+(120572) = 120582119860
119880
+(120572)
(32)
When 120582 lt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
+(120572)
120582119860119871
minus(120572)] 119910 isin [120582119860
119880
+(120572) 120582119860
119880
minus(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
+(120572)
(120582119860)119871
+(120572) = 120582119860
119871
minus(120572)
(33)
(120582119860)119880
minus(120572) = 120582119860
119880
+(120572)
(120582119860)119880
+(120572) = 120582119860
119880
minus(120572)
(34)
We say that an approximation operator 119879 is scale invariant if
119879 (120582119860) = 120582119879 (119860) 120582 isin 119877 0 (35)
314 Identity This criterion states that the interval-valuedtrapezoidal approximation of an interval-valued trapezoidalfuzzy number is equivalent to that number that is if 119860 isin
IF119879(119877) then
119879 (119860) = 119860 (36)
315 Nearness Criterion An approximation operator 119879 ful-fills the nearness criterion if for any interval-valued fuzzynumber119860 its output value119879(119860) is the nearest interval-valuedtrapezoidal fuzzy number to 119860 with respect to the weighteddistance 119863
119868defined by (21) In other words for any 119861 isin
IF119879(119877) we have
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (37)
Remark 13 We can verify that IF(119877) is closed and convex so119879(119860) exists and is unique
316 Ranking Invariance A reasonable approximation oper-ator should preserve the accepted ranking We say that anapproximation operator 119879 is ranking invariant if for any119860 119861 isin IF(119877)
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (38)
317 Continuity Let 119860 119861 isin IF(119877) An approximationoperator 119879 is continuous if for any 120576 gt 0 there is 120575 gt 0 when119863119868(119860 119861) lt 120575 we have
119863119868 (119879 (119860) 119879 (119861)) lt 120576 (39)
The continuity constraint means that if two interval-valuedfuzzy numbers are close then their interval-valued trape-zoidal approximations also should be close
6 Journal of Applied Mathematics
32 Interval-Valued Trapezoidal Approximation Based on theWeighted Distance In this section we are looking for anapproximation operator 119879 IF(119877) rarr IF119879(119877) whichproduces an interval-valued trapezoidal fuzzy number thatis the nearest one to the given interval-valued fuzzy numberand preserves its core with respect to the weighted distance119863119868defined by (21)
Lemma 14 Let 119860 isin 119865(119877) 119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1]
If function 119891(120572) is nonnegative and increasing on [0 1] with119891(0) = 0 and int1
0119891(120572)119889120572 = 12 then we have
(i)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860minus (
1) minus 119860minus (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
le 119860minus (1) (40)
(ii)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (41)
Proof (i) See [23] the proof of Theorem 31(ii) Since 119860
+(120572) is a nonincreasing function we have
119860+(120572) ge 119860
+(1) for any 120572 isin [0 1] By 119891(120572) ge 0 we can prove
that
(1 minus 120572)119860+ (120572) 119891 (120572) ge (1 minus 120572)119860+ (
1) 119891 (120572)
= [(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572)
(42)
According to the monotonicity of integration we have
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
ge int
1
0
[(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572) 119889120572
(43)
That is
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
minus 119860+ (1) int
1
0
120572 (1 minus 120572) 119891 (120572) 119889120572
ge 119860+ (1) int
1
0
(120572 minus 1)2119891 (120572) 119889120572
(44)
Because int10(120572 minus 1)
2119891(120572)119889120572 gt 0 it follows that
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (45)
Theorem 15 (see [24]) Let 119891 1198921 1198922 119892
119898 119877119899rarr 119877 be
convex and differentiable functions Then 119909 solves the convexprogramming problem
min 119891 (119909)
119904119905 119892119894(119909) le 119887119894
119894 isin 1 2 119898
(46)
if and only if there exist 120583119894 119894 isin 1 2 119898 such that
(i) nabla119891(119909) + Σ119898119894=1120583119894nabla119892119894(119909) = 0
(ii) 119892119894(119909) minus 119887
119894le 0
(iii) 120583119894ge 0
(iv) 120583119894(119887119894minus 119892119894(119909)) = 0
Suppose that 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2
119909 isin [119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] We
will try to find an interval-valued trapezoidal fuzzy number119879(119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] which is the nearest
interval-valued trapezoidal fuzzy number of 119860 and preservesits core with respect to the weighted distance 119863
119868 Thus we have
to find such real numbers 1199051198711 1199051198712 1199051198713 1199051198714 1199051198801 1199051198802 1199051198803and 1199051198804that
minimize
119863119868 (119860 119879 (119860))
=
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus (119905
119871
1+ (119905119871
2minus 119905119871
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119905119871
3) 120572))
2
119889120572)
12
+ (int
1
0
f (120572) (119860119880minus(120572) minus (119905
119880
1+ (119905119880
2minus 119905119880
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119905119880
3) 120572))
2
119889120572)
12
]
(47)
with respect to condition core119860 = core119879(119860) that is
119905119871
2= 119860119871
minus(1) 119905
119871
3= 119860119871
+(1)
119905119880
2= 119860119880
minus(1) 119905
119880
3= 119860119880
+(1)
(48)
It follows that
119905119871
2le 119905119871
3 119905
119880
2le 119905119880
3 (49)
Making use of Theorem 4 we have
119905119880
2le 119905119871
2 119905
119871
3le 119905119880
3 (50)
Journal of Applied Mathematics 7
Using (47) and (50) together with Theorem 6 we only need tominimize the function
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) [119860119871
minus(120572) minus (119905
119871
1+ (119860119871
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119860119871
+(1)) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
minus(120572) minus (119905
119880
1+ (119860119880
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119860119880
+(1)) 120572)]
2
119889120572
(51)
subject to
119905119880
1minus 119905119871
1le 0 119905
119871
4minus 119905119880
4le 0 (52)
After simple calculations we obtain
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
4)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
4)
2
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 sdot 119905
119871
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 sdot 119905
119871
4
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 sdot 119905
119880
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 sdot 119905
119880
4
+ int
1
0
119891 (120572) (119860119871
minus(120572) minus 120572 sdot 119860
119871
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119871
+(120572) minus 120572 sdot 119860
119871
+(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
minus(120572) minus 120572 sdot 119860
119880
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
+(120572) minus 120572 sdot 119860
119880
+(1))
2
119889120572
(53)
subject to
119905119880
1minus 119905119871
1le 0
119905119871
4minus 119905119880
4le 0
(54)
We present the main result of the paper as follows
Theorem 16 Let 119860 = [119860119871 119860119880] isin IF(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] is the nearest
interval-valued trapezoidal fuzzy number to 119860 and preservesits core with respect to the weighted distance 119863
119868 Consider the
following(i) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(55)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(56)
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(57)
(ii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(58)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(59)
8 Journal of Applied Mathematics
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(60)
(iii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
(61)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(62)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(63)
(iv) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(64)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(65)
Proof Because the function 119865 in (53) and conditions (54)satisfy the hypothesis of convexity and differentiability inTheorem 15 after some simple calculations conditions (i)ndash(iv) inTheorem 15 with respect to the minimization problem(53) in conditions (54) can be shown as follows
2119905119871
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 minus 1205831
= 0
(66)
2119905119871
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 + 1205832
= 0
(67)
2119905119880
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 + 1205831
= 0
(68)
Journal of Applied Mathematics 9
2119905119880
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 minus 1205832
= 0
(69)
1205831(119905119880
1minus 119905119871
1) = 0 (70)
1205832(119905119871
4minus 119905119880
4) = 0 (71)
1205831ge 0 (72)
1205832ge 0 (73)
119905119880
1minus 119905119871
1le 0 (74)
119905119871
4minus 119905119880
4le 0 (75)
(i) In the case 1205831gt 0 and 120583
2= 0 the solution of the system
(66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(76)
Firstly we have from (55) that 1205831gt 0 and it follows from
(56) that
119905119880
4minus 119905119871
4=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(77)
Then conditions (72) (73) (74) and (75) are verified
Secondly combing with (48) (55) and Lemma 14 (i) wecan prove that
119905119880
2minus 119905119880
1
= 119860119880
minus(1) + ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(78)
And on the basis of (50) we have
119905119871
2minus 119905119871
1ge 119905119880
2minus 119905119880
1gt 0 (79)
By making use of (48) and Lemma 14 (ii) we get
119905119880
4minus 119905119880
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119880
+(1) ge 0
119905119871
4minus 119905119871
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119871
+(1) ge 0
(80)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(ii) In the case 120583
1gt 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
For any 119860 = [119860119871 119860119880] isin IF(119877) the lower fuzzy number119860119871 and the upper fuzzy number 119860119880 can be represented as
119860119871(119909) =
119897119860119871 (119909) 119886
119871le 119909 lt 119887
119871
1 119887119871le 119909 le 119888
119871
119903119860119871 (119909) 119888
119871lt 119909 le 119889
119871
0 otherwise
(8)
119860119880(119909) =
119897119860119880 (119909) 119886
119880le 119909 lt 119887
119880
1 119887119880le 119909 le 119888
119880
119903119860119880 (119909) 119888
119880lt 119909 le 119889
119880
0 otherwise
(9)
respectively where 119886119871 119887119871 119888119871 119889119871 119886119880 119887119880 119888119880 119889119880 isin 119877 119897119860119871
[119886119871 119887119871) rarr [0 1] and 119897
119860119880 [119886
119880 b119880) rarr [0 1] are
nondecreasing upper semicontinuous functions such that119897119860119871(119886119871) = 0 119897
119860119871(119887119871) = 1 119897
119860119880(119886119880) = 0 and 119897
119860119880(119887119880) =
1 119903119860119871 (119888
119871 119889119871] rarr [0 1] and 119903
119860119880 (119888
119880 119889119880] rarr [0 1]
are nonincreasing upper semicontinuous functions fulfilling119903119860119871(119888119871) = 1 119903
119860119871(119889119871) = 0 119903
119860119880(119888119880) = 1 and 119903
119860119880(119889119880) = 0
If 119886119871 = 119886119880 119887119871 = 119887
119880 119888119871 = 119888119880 119889119871 = 119889
119880 119897119860119871(119909) =
119897119860119880(119909) and 119903
119860119871(119909) = 119903
119860119880(119909) that is 119860119871(119909) = 119860119880(119909) then
the interval-valued fuzzy number 119860 = [119860119871 119860119880] is a fuzzy
numberFor any 120572 isin [0 1] the 120572-level set of an interval-valued
fuzzy number 119860 is defined as
119860120572= (119909 119910) isin 119877
2 119860119871(119909) ge 120572 119860
119880(119910) ge 120572
= (119909 119910) isin 1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)]
119910 isin [119860119880
minus(120572) 119860
119880
+(120572)]
(10)
where 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and 119860119880
+(120572) are the inverse
functions of 119897119860119871 119903119860119871 119897119860119880 and 119903
119860119880 respectively If 119860119871 = 119860119880
then this definition coincides with (4) The core of 119860 ispresented as
core119860 = (119909 119910) isin 1198772 119909 isin [119860119871minus(1) 119860
119871
+(1)]
119910 isin [119860119880
minus(1) 119860
119880
+(1)]
(11)
Theorem 4 Let 119860119871 119860119880 isin 119865(119877) 119860 = [119860119871 119860119880] isin IF(119877) if andonly if 119860119880
minus(120572) le 119860
119871
minus(120572) 119860119880
+(120572) ge 119860
119871
+(120572) for any 120572 isin [0 1]
Proof If If 119909 isin [119886119880 119887119880) then there exist 1205721 1205722isin [0 1] such
that
1205721= 119897119860119871 (119909) 120572
2= 119897119860119880 (119909) (12)
Since 119860119880minus(120572) le 119860
119871
minus(120572) for any 120572 isin [0 1] this implies that
119909 = 119860119880
minus(1205722) le 119860119871
minus(1205722) ≜ 1199091015840 (13)
where 1199091015840 isin [119886119880 119887119871] By the monotonicity of 119897119860119871 we have
119897119860119871 (119909) le 119897119860
119871 (1199091015840) = 1205722= 119897119860119880 (119909) (14)
Similarly we can prove that 119903119860119880(119909) ge 119903
119860119871(119909) for any 119909 isin
(119888119880 119889119880] If 119909 isin [119887119880 119888119880] then 119860119880(119909) = 1 ge 119860119871(119909) Therefore
119860119871(119909) le 119860
119880(119909) for any 119909 isin [119886119880 119889119880] that is 119860 = [119860119871 119860119880] isin
IF(119877)Only if If 120572 isin [0 1] then there exist 119909
1isin [119886119880 119887119880] 1199092isin
[119886119871 119887119871] such that
1199091= 119860119880
minus(120572) 119909
2= 119860119871
minus(120572) (15)
Since 119897119860119880(119909) ge 119897
119860119871(119909) for any 119909 isin [119886119880 119887119880] this implies that
120572 = 119897119860119880 (1199091) ge 119897119860119871 (1199091) ≜ 1205721015840 (16)
where 1205721015840 isin [0 1] By the monotonicity of 119860119871minus we have
119860119871
minus(120572) ge 119860
119871
minus(1205721015840) = 1199091= 119860119880
minus(120572) (17)
Similarly we can prove that119860119880+(120572) ge 119860
119871
+(120572) for any120572 isin [0 1]
This concludes the proof
It is well known interval-valued fuzzy numbers with sim-plemembership functions are preferred in practice Howeveras a particular of interval-valued fuzzy numbers interval-valued trapezoidal fuzzy numbers could be wide appliedin real mathematical modeling Thus the properties of theinterval-valued trapezoidal fuzzy number are discussed asfollows
Definition 5 (see [6 18ndash20]) Let 119860 = [119860119871 119860119880] isin IF(119877) If
119860119871 119860119880isin 119865119879(119877) then 119860 is called an interval-valued trape-
zoidal fuzzy numberThe lower trapezoidal fuzzy number119860119871is expressed as
119860119871(119909) =
119909 minus 119905119871
1
119905119871
2minus 119905119871
1
119905119871
1le 119909 lt 119905
119871
2
1 119905119871
2le 119909 le 119905
119871
3
119905119871
4minus 119909
119905119871
4minus 119905119871
3
119905119871
3lt 119909 le 119905
119871
4
0 otherwise
(18)
and the upper trapezoidal fuzzy number 119860119880 is expressed as
119860119880(119909) =
119909 minus 119905119880
1
119905119880
2minus 119905119880
1
119905119880
1le 119909 lt 119905
119880
2
1 119905119880
2le 119909 le 119905
119880
3
119905119880
4minus 119909
119905119880
4minus 119905119880
3
119905119880
3lt 119909 le 119905
119880
4
0 otherwise
(19)
An interval-valued trapezoidal fuzzy number 119860 can berepresented as 119860 = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] The family
of all interval-valued trapezoidal fuzzy numbers on 119877 isdenoted as IF119879(119877)
Theorem 6 Let 119860119871 119860119880 isin 119865119879(119877) 119860 = [119860119871 119860119880] isin 119868119865119879(119877) ifand only if 119905119880
1le 119905119871
1 1199051198802le 119905119871
2 1199051198803ge 119905119871
3and 1199051198804ge 119905119871
4
4 Journal of Applied Mathematics
23 TheWeighted Distance of Interval-Valued Fuzzy NumbersIn 2007 Zeng and Li [21] introduced the weighted distance offuzzy numbers 119860 and 119861 as follows
1198892
119891(119860 119861) = int
1
0
119891 (120572) (119860minus (120572) minus 119861minus (
120572))2119889120572
+ int
1
0
119891 (120572) (119860+ (120572) minus 119861+ (
120572))2119889120572
(20)
where the function 119891(120572) is nonnegative and increasing on[0 1] with 119891(0) = 0 and int1
0119891(120572)119889120572 = 12 The function
119891(120572) is also called the weighting function The property ofmonotone increasing of function 119891(120572)means that the higherthe cut level the more important its weight in determiningthe distance of fuzzy numbers 119860 and 119861 Both conditions119891(0) = 0 and int1
0119891(120572)119889120572 = 12 ensure that the distance
defined by (20) is the extension of the ordinary distance in119877 defined by its absolute value That means this distancebecomes an absolute value in119877when a fuzzy number reducesto a real number In applications the function 119891(120572) can bechosen according to the actual situation
We will define the weighted distance of interval-valuedfuzzy numbers as follows It can be considered as a naturalextension of the weighted distance 119889
119891(119860 119861) of fuzzy num-
bers
Definition 7 Let 119860 119861 isin IF(119877) The weighted distance of 119860and 119861 is defined as
119863119868 (119860 119861) =
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus 119861
119871
minus(120572))
2
119889120572
+ int
1
0
119891(120572)(119860119871
+(120572) minus 119861
119871
+(120572))
2
119889120572)
12
+ (int
1
0
119891 (120572) (119860119880
minus(120572) minus 119861
119880
minus(120572))
2
119889120572
+ int
1
0
119891(120572)(119860119880
+(120572) minus 119861
119880
+(120572))
2
119889120572)
12
]
=
1
2
[119889119891(119860119871 119861119871) + 119889119891(119860119880 119861119880)]
(21)
If 119860119871 = 119860119880 and 119861119871 = 119861119880 then119863119868(119860 119861) = 119889
119891(119860 119861)
Property 1 Let119860 119861 isin IF(119877) Then119863119868(119860 119861) = 0 if and only if
119863(119860119871 119861119871) = 0 and119863(119860119880 119861119880) = 0
Theorem 8 (IF(119877) 119863119868) is a metric space
By the completeness of metric space (119865(119877) 119889119891) we can
obtain the following conclusion
Theorem 9 The metric space (IF(119877) 119863119868) is complete
24 The Ranking of Interval-Valued Fuzzy Numbers Theranking of fuzzy numbers was studied by many researchers
and itwas extended to interval-valued fuzzy numbers becauseof its attraction and applicabilityWewill propose a ranking ofinterval-valued fuzzy numbers which embodies the impor-tance of the core of interval-valued fuzzy numbers
Definition 10 Let 119860 119861 isin IF(119877) The ranking of 119860 119861 can bedefined by the following formula
119860 ⪰ 119861 lArrrArr 119860119871
minus(1) + 119860
119871
+(1) ge 119861
119871
minus(1) + 119861
119871
+(1)
119860119880
minus(1) + 119860
119880
+(1) ge 119861
119880
minus(1) + 119861
119880
+(1)
(22)
Example 11 Let
119860119871(119909) = 119861
119871(119909) =
1 minus (119909 minus 3)2 119909 isin [2 4]
0 otherwise
119860119880(119909) =
1 minus (119909 minus 3)2 119909 isin [2 3)
1 119909 isin [3 5]
minus119909 + 6 119909 isin (5 6]
0 otherwise
119861119880(119909) =
1 minus (119909 minus 3)2 119909 isin [2 3)
1 minus
1
9
(119909 minus 3)2 119909 isin [3 6]
0 otherwise
(23)
We obtain core119860 = (119909 119910) isin 1198772 119909 = 3 119910 isin [3 5] and
core119861 = (119909 119910) isin 1198772 119909 = 3 119910 = 3 By a direct calculationwe have 119860 ⪰ 119861
3 Weighted Interval-ValuedTrapezoidal Approximation
31 Criteria for Interval-Valued Trapezoidal ApproximationIf we want to approximate an interval-valued fuzzy numberby an interval-valued trapezoidal fuzzy number we mustuse an approximate operator 119879 IF(119877) rarr IF119879(119877) whichtransforms a family of all interval-valued fuzzy numbers 119860into a family of interval-valued trapezoidal fuzzy numbers119879(119860) that is 119879 119860 rarr 119879(119860) Since interval-valuedtrapezoidal approximation could also be performed in manyways we propose a number of criteria which the approxi-mation operator should possess at least one Reference [22]has given some criteria for the fuzzy number approximationsimilarly we give some criteria for interval-valued trapezoidalapproximation as follows
311 120572-Level Set Invariance An approximation operator 119879 is120572-level set invariant if
(119879 (119860))120572= 119860120572 (24)
Remark 12 For any two different levels 1205721and 120572
2(1205721= 1205722)
we obtain one and only one approximation operator which isinvariant both in 120572
1- and 120572
2-level set
Proof Let 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2 119909 isin
[119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] Then we
Journal of Applied Mathematics 5
can obtain one and only one interval-valued trapezoidal fuzzynumber 119879(119860) = [(119879(119860))119871 (119879(119860))119880] where
(119879 (A))119871120572= [
119860119871
minus(1205722) minus 119860119871
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
minus(1205721)
119860119871
+(1205722) minus 119860119871
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
+(1205721)]
(119879 (119860))119880
120572= [
119860119880
minus(1205722) minus 119860119880
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
minus(1205721)
119860119880
+(1205722) minus 119860119880
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
+(1205721)]
(25)
It is obvious that
(119879 (119860))1205721= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205721) 119860119871
+(1205721)]
119910 isin [119860119880
minus(1205721) 119860119880
+(1205721)]
(119879 (119860))1205722= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205722) 119860119871
+(1205722)]
119910 isin [119860119880
minus(1205722) 119860119880
+(1205722)]
(26)
Hence (119879(119860))1205721= 1198601205721and (119879(119860))
1205722= 1198601205722
312 Translation Invariance For 119860 isin IF(119877) and 119911 isin 119877 wedefine
119860 + 119911 = [(119860 + 119911)119871 (119860 + 119911)
119880] (27)
where (119860 + 119911)120572= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(120572) + 119911 119860
119871
+(120572) +
119911] 119910 isin [119860119880
minus(120572) + 119911 119860
119880
+(120572) + 119911] 120572 isin [0 1] that is
(119860 + 119911)119871
minus(120572) = 119860
119871
minus(120572) + 119911
(119860 + 119911)119871
+(120572) = 119860
119871
+(120572) + 119911
(28)
(119860 + 119911)119880
minus(120572) = 119860
119880
minus(120572) + 119911
(119860 + 119911)119880
+(120572) = 119860
119880
+(120572) + 119911
(29)
An approximation operator 119879 is invariant to translation if
119879 (119860 + 119911) = 119879 (119860) + 119911 119911 isin 119877 (30)
Translation invariance means that the relative position of theinterval-valued trapezoidal approximation remains constantwhen the membership function is moved to the left or to theright
313 Scale Invariance For 119860 isin IF(119877) and 120582 isin 119877 0 wedefine
120582119860 = [120582119860119871 120582119860119880] (31)
When 120582 gt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
minus(120572)
120582119860119871
+(120572)] 119910 isin [120582119860
119880
minus(120572) 120582119860
119880
+(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
minus(120572)
(120582119860)119871
+(120572) = 120582119860
119871
+(120572)
(120582119860)119880
minus(120572) = 120582119860
119880
minus(120572)
(120582119860)119880
+(120572) = 120582119860
119880
+(120572)
(32)
When 120582 lt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
+(120572)
120582119860119871
minus(120572)] 119910 isin [120582119860
119880
+(120572) 120582119860
119880
minus(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
+(120572)
(120582119860)119871
+(120572) = 120582119860
119871
minus(120572)
(33)
(120582119860)119880
minus(120572) = 120582119860
119880
+(120572)
(120582119860)119880
+(120572) = 120582119860
119880
minus(120572)
(34)
We say that an approximation operator 119879 is scale invariant if
119879 (120582119860) = 120582119879 (119860) 120582 isin 119877 0 (35)
314 Identity This criterion states that the interval-valuedtrapezoidal approximation of an interval-valued trapezoidalfuzzy number is equivalent to that number that is if 119860 isin
IF119879(119877) then
119879 (119860) = 119860 (36)
315 Nearness Criterion An approximation operator 119879 ful-fills the nearness criterion if for any interval-valued fuzzynumber119860 its output value119879(119860) is the nearest interval-valuedtrapezoidal fuzzy number to 119860 with respect to the weighteddistance 119863
119868defined by (21) In other words for any 119861 isin
IF119879(119877) we have
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (37)
Remark 13 We can verify that IF(119877) is closed and convex so119879(119860) exists and is unique
316 Ranking Invariance A reasonable approximation oper-ator should preserve the accepted ranking We say that anapproximation operator 119879 is ranking invariant if for any119860 119861 isin IF(119877)
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (38)
317 Continuity Let 119860 119861 isin IF(119877) An approximationoperator 119879 is continuous if for any 120576 gt 0 there is 120575 gt 0 when119863119868(119860 119861) lt 120575 we have
119863119868 (119879 (119860) 119879 (119861)) lt 120576 (39)
The continuity constraint means that if two interval-valuedfuzzy numbers are close then their interval-valued trape-zoidal approximations also should be close
6 Journal of Applied Mathematics
32 Interval-Valued Trapezoidal Approximation Based on theWeighted Distance In this section we are looking for anapproximation operator 119879 IF(119877) rarr IF119879(119877) whichproduces an interval-valued trapezoidal fuzzy number thatis the nearest one to the given interval-valued fuzzy numberand preserves its core with respect to the weighted distance119863119868defined by (21)
Lemma 14 Let 119860 isin 119865(119877) 119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1]
If function 119891(120572) is nonnegative and increasing on [0 1] with119891(0) = 0 and int1
0119891(120572)119889120572 = 12 then we have
(i)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860minus (
1) minus 119860minus (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
le 119860minus (1) (40)
(ii)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (41)
Proof (i) See [23] the proof of Theorem 31(ii) Since 119860
+(120572) is a nonincreasing function we have
119860+(120572) ge 119860
+(1) for any 120572 isin [0 1] By 119891(120572) ge 0 we can prove
that
(1 minus 120572)119860+ (120572) 119891 (120572) ge (1 minus 120572)119860+ (
1) 119891 (120572)
= [(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572)
(42)
According to the monotonicity of integration we have
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
ge int
1
0
[(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572) 119889120572
(43)
That is
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
minus 119860+ (1) int
1
0
120572 (1 minus 120572) 119891 (120572) 119889120572
ge 119860+ (1) int
1
0
(120572 minus 1)2119891 (120572) 119889120572
(44)
Because int10(120572 minus 1)
2119891(120572)119889120572 gt 0 it follows that
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (45)
Theorem 15 (see [24]) Let 119891 1198921 1198922 119892
119898 119877119899rarr 119877 be
convex and differentiable functions Then 119909 solves the convexprogramming problem
min 119891 (119909)
119904119905 119892119894(119909) le 119887119894
119894 isin 1 2 119898
(46)
if and only if there exist 120583119894 119894 isin 1 2 119898 such that
(i) nabla119891(119909) + Σ119898119894=1120583119894nabla119892119894(119909) = 0
(ii) 119892119894(119909) minus 119887
119894le 0
(iii) 120583119894ge 0
(iv) 120583119894(119887119894minus 119892119894(119909)) = 0
Suppose that 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2
119909 isin [119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] We
will try to find an interval-valued trapezoidal fuzzy number119879(119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] which is the nearest
interval-valued trapezoidal fuzzy number of 119860 and preservesits core with respect to the weighted distance 119863
119868 Thus we have
to find such real numbers 1199051198711 1199051198712 1199051198713 1199051198714 1199051198801 1199051198802 1199051198803and 1199051198804that
minimize
119863119868 (119860 119879 (119860))
=
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus (119905
119871
1+ (119905119871
2minus 119905119871
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119905119871
3) 120572))
2
119889120572)
12
+ (int
1
0
f (120572) (119860119880minus(120572) minus (119905
119880
1+ (119905119880
2minus 119905119880
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119905119880
3) 120572))
2
119889120572)
12
]
(47)
with respect to condition core119860 = core119879(119860) that is
119905119871
2= 119860119871
minus(1) 119905
119871
3= 119860119871
+(1)
119905119880
2= 119860119880
minus(1) 119905
119880
3= 119860119880
+(1)
(48)
It follows that
119905119871
2le 119905119871
3 119905
119880
2le 119905119880
3 (49)
Making use of Theorem 4 we have
119905119880
2le 119905119871
2 119905
119871
3le 119905119880
3 (50)
Journal of Applied Mathematics 7
Using (47) and (50) together with Theorem 6 we only need tominimize the function
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) [119860119871
minus(120572) minus (119905
119871
1+ (119860119871
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119860119871
+(1)) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
minus(120572) minus (119905
119880
1+ (119860119880
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119860119880
+(1)) 120572)]
2
119889120572
(51)
subject to
119905119880
1minus 119905119871
1le 0 119905
119871
4minus 119905119880
4le 0 (52)
After simple calculations we obtain
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
4)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
4)
2
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 sdot 119905
119871
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 sdot 119905
119871
4
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 sdot 119905
119880
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 sdot 119905
119880
4
+ int
1
0
119891 (120572) (119860119871
minus(120572) minus 120572 sdot 119860
119871
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119871
+(120572) minus 120572 sdot 119860
119871
+(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
minus(120572) minus 120572 sdot 119860
119880
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
+(120572) minus 120572 sdot 119860
119880
+(1))
2
119889120572
(53)
subject to
119905119880
1minus 119905119871
1le 0
119905119871
4minus 119905119880
4le 0
(54)
We present the main result of the paper as follows
Theorem 16 Let 119860 = [119860119871 119860119880] isin IF(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] is the nearest
interval-valued trapezoidal fuzzy number to 119860 and preservesits core with respect to the weighted distance 119863
119868 Consider the
following(i) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(55)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(56)
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(57)
(ii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(58)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(59)
8 Journal of Applied Mathematics
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(60)
(iii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
(61)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(62)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(63)
(iv) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(64)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(65)
Proof Because the function 119865 in (53) and conditions (54)satisfy the hypothesis of convexity and differentiability inTheorem 15 after some simple calculations conditions (i)ndash(iv) inTheorem 15 with respect to the minimization problem(53) in conditions (54) can be shown as follows
2119905119871
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 minus 1205831
= 0
(66)
2119905119871
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 + 1205832
= 0
(67)
2119905119880
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 + 1205831
= 0
(68)
Journal of Applied Mathematics 9
2119905119880
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 minus 1205832
= 0
(69)
1205831(119905119880
1minus 119905119871
1) = 0 (70)
1205832(119905119871
4minus 119905119880
4) = 0 (71)
1205831ge 0 (72)
1205832ge 0 (73)
119905119880
1minus 119905119871
1le 0 (74)
119905119871
4minus 119905119880
4le 0 (75)
(i) In the case 1205831gt 0 and 120583
2= 0 the solution of the system
(66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(76)
Firstly we have from (55) that 1205831gt 0 and it follows from
(56) that
119905119880
4minus 119905119871
4=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(77)
Then conditions (72) (73) (74) and (75) are verified
Secondly combing with (48) (55) and Lemma 14 (i) wecan prove that
119905119880
2minus 119905119880
1
= 119860119880
minus(1) + ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(78)
And on the basis of (50) we have
119905119871
2minus 119905119871
1ge 119905119880
2minus 119905119880
1gt 0 (79)
By making use of (48) and Lemma 14 (ii) we get
119905119880
4minus 119905119880
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119880
+(1) ge 0
119905119871
4minus 119905119871
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119871
+(1) ge 0
(80)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(ii) In the case 120583
1gt 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
23 TheWeighted Distance of Interval-Valued Fuzzy NumbersIn 2007 Zeng and Li [21] introduced the weighted distance offuzzy numbers 119860 and 119861 as follows
1198892
119891(119860 119861) = int
1
0
119891 (120572) (119860minus (120572) minus 119861minus (
120572))2119889120572
+ int
1
0
119891 (120572) (119860+ (120572) minus 119861+ (
120572))2119889120572
(20)
where the function 119891(120572) is nonnegative and increasing on[0 1] with 119891(0) = 0 and int1
0119891(120572)119889120572 = 12 The function
119891(120572) is also called the weighting function The property ofmonotone increasing of function 119891(120572)means that the higherthe cut level the more important its weight in determiningthe distance of fuzzy numbers 119860 and 119861 Both conditions119891(0) = 0 and int1
0119891(120572)119889120572 = 12 ensure that the distance
defined by (20) is the extension of the ordinary distance in119877 defined by its absolute value That means this distancebecomes an absolute value in119877when a fuzzy number reducesto a real number In applications the function 119891(120572) can bechosen according to the actual situation
We will define the weighted distance of interval-valuedfuzzy numbers as follows It can be considered as a naturalextension of the weighted distance 119889
119891(119860 119861) of fuzzy num-
bers
Definition 7 Let 119860 119861 isin IF(119877) The weighted distance of 119860and 119861 is defined as
119863119868 (119860 119861) =
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus 119861
119871
minus(120572))
2
119889120572
+ int
1
0
119891(120572)(119860119871
+(120572) minus 119861
119871
+(120572))
2
119889120572)
12
+ (int
1
0
119891 (120572) (119860119880
minus(120572) minus 119861
119880
minus(120572))
2
119889120572
+ int
1
0
119891(120572)(119860119880
+(120572) minus 119861
119880
+(120572))
2
119889120572)
12
]
=
1
2
[119889119891(119860119871 119861119871) + 119889119891(119860119880 119861119880)]
(21)
If 119860119871 = 119860119880 and 119861119871 = 119861119880 then119863119868(119860 119861) = 119889
119891(119860 119861)
Property 1 Let119860 119861 isin IF(119877) Then119863119868(119860 119861) = 0 if and only if
119863(119860119871 119861119871) = 0 and119863(119860119880 119861119880) = 0
Theorem 8 (IF(119877) 119863119868) is a metric space
By the completeness of metric space (119865(119877) 119889119891) we can
obtain the following conclusion
Theorem 9 The metric space (IF(119877) 119863119868) is complete
24 The Ranking of Interval-Valued Fuzzy Numbers Theranking of fuzzy numbers was studied by many researchers
and itwas extended to interval-valued fuzzy numbers becauseof its attraction and applicabilityWewill propose a ranking ofinterval-valued fuzzy numbers which embodies the impor-tance of the core of interval-valued fuzzy numbers
Definition 10 Let 119860 119861 isin IF(119877) The ranking of 119860 119861 can bedefined by the following formula
119860 ⪰ 119861 lArrrArr 119860119871
minus(1) + 119860
119871
+(1) ge 119861
119871
minus(1) + 119861
119871
+(1)
119860119880
minus(1) + 119860
119880
+(1) ge 119861
119880
minus(1) + 119861
119880
+(1)
(22)
Example 11 Let
119860119871(119909) = 119861
119871(119909) =
1 minus (119909 minus 3)2 119909 isin [2 4]
0 otherwise
119860119880(119909) =
1 minus (119909 minus 3)2 119909 isin [2 3)
1 119909 isin [3 5]
minus119909 + 6 119909 isin (5 6]
0 otherwise
119861119880(119909) =
1 minus (119909 minus 3)2 119909 isin [2 3)
1 minus
1
9
(119909 minus 3)2 119909 isin [3 6]
0 otherwise
(23)
We obtain core119860 = (119909 119910) isin 1198772 119909 = 3 119910 isin [3 5] and
core119861 = (119909 119910) isin 1198772 119909 = 3 119910 = 3 By a direct calculationwe have 119860 ⪰ 119861
3 Weighted Interval-ValuedTrapezoidal Approximation
31 Criteria for Interval-Valued Trapezoidal ApproximationIf we want to approximate an interval-valued fuzzy numberby an interval-valued trapezoidal fuzzy number we mustuse an approximate operator 119879 IF(119877) rarr IF119879(119877) whichtransforms a family of all interval-valued fuzzy numbers 119860into a family of interval-valued trapezoidal fuzzy numbers119879(119860) that is 119879 119860 rarr 119879(119860) Since interval-valuedtrapezoidal approximation could also be performed in manyways we propose a number of criteria which the approxi-mation operator should possess at least one Reference [22]has given some criteria for the fuzzy number approximationsimilarly we give some criteria for interval-valued trapezoidalapproximation as follows
311 120572-Level Set Invariance An approximation operator 119879 is120572-level set invariant if
(119879 (119860))120572= 119860120572 (24)
Remark 12 For any two different levels 1205721and 120572
2(1205721= 1205722)
we obtain one and only one approximation operator which isinvariant both in 120572
1- and 120572
2-level set
Proof Let 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2 119909 isin
[119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] Then we
Journal of Applied Mathematics 5
can obtain one and only one interval-valued trapezoidal fuzzynumber 119879(119860) = [(119879(119860))119871 (119879(119860))119880] where
(119879 (A))119871120572= [
119860119871
minus(1205722) minus 119860119871
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
minus(1205721)
119860119871
+(1205722) minus 119860119871
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
+(1205721)]
(119879 (119860))119880
120572= [
119860119880
minus(1205722) minus 119860119880
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
minus(1205721)
119860119880
+(1205722) minus 119860119880
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
+(1205721)]
(25)
It is obvious that
(119879 (119860))1205721= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205721) 119860119871
+(1205721)]
119910 isin [119860119880
minus(1205721) 119860119880
+(1205721)]
(119879 (119860))1205722= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205722) 119860119871
+(1205722)]
119910 isin [119860119880
minus(1205722) 119860119880
+(1205722)]
(26)
Hence (119879(119860))1205721= 1198601205721and (119879(119860))
1205722= 1198601205722
312 Translation Invariance For 119860 isin IF(119877) and 119911 isin 119877 wedefine
119860 + 119911 = [(119860 + 119911)119871 (119860 + 119911)
119880] (27)
where (119860 + 119911)120572= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(120572) + 119911 119860
119871
+(120572) +
119911] 119910 isin [119860119880
minus(120572) + 119911 119860
119880
+(120572) + 119911] 120572 isin [0 1] that is
(119860 + 119911)119871
minus(120572) = 119860
119871
minus(120572) + 119911
(119860 + 119911)119871
+(120572) = 119860
119871
+(120572) + 119911
(28)
(119860 + 119911)119880
minus(120572) = 119860
119880
minus(120572) + 119911
(119860 + 119911)119880
+(120572) = 119860
119880
+(120572) + 119911
(29)
An approximation operator 119879 is invariant to translation if
119879 (119860 + 119911) = 119879 (119860) + 119911 119911 isin 119877 (30)
Translation invariance means that the relative position of theinterval-valued trapezoidal approximation remains constantwhen the membership function is moved to the left or to theright
313 Scale Invariance For 119860 isin IF(119877) and 120582 isin 119877 0 wedefine
120582119860 = [120582119860119871 120582119860119880] (31)
When 120582 gt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
minus(120572)
120582119860119871
+(120572)] 119910 isin [120582119860
119880
minus(120572) 120582119860
119880
+(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
minus(120572)
(120582119860)119871
+(120572) = 120582119860
119871
+(120572)
(120582119860)119880
minus(120572) = 120582119860
119880
minus(120572)
(120582119860)119880
+(120572) = 120582119860
119880
+(120572)
(32)
When 120582 lt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
+(120572)
120582119860119871
minus(120572)] 119910 isin [120582119860
119880
+(120572) 120582119860
119880
minus(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
+(120572)
(120582119860)119871
+(120572) = 120582119860
119871
minus(120572)
(33)
(120582119860)119880
minus(120572) = 120582119860
119880
+(120572)
(120582119860)119880
+(120572) = 120582119860
119880
minus(120572)
(34)
We say that an approximation operator 119879 is scale invariant if
119879 (120582119860) = 120582119879 (119860) 120582 isin 119877 0 (35)
314 Identity This criterion states that the interval-valuedtrapezoidal approximation of an interval-valued trapezoidalfuzzy number is equivalent to that number that is if 119860 isin
IF119879(119877) then
119879 (119860) = 119860 (36)
315 Nearness Criterion An approximation operator 119879 ful-fills the nearness criterion if for any interval-valued fuzzynumber119860 its output value119879(119860) is the nearest interval-valuedtrapezoidal fuzzy number to 119860 with respect to the weighteddistance 119863
119868defined by (21) In other words for any 119861 isin
IF119879(119877) we have
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (37)
Remark 13 We can verify that IF(119877) is closed and convex so119879(119860) exists and is unique
316 Ranking Invariance A reasonable approximation oper-ator should preserve the accepted ranking We say that anapproximation operator 119879 is ranking invariant if for any119860 119861 isin IF(119877)
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (38)
317 Continuity Let 119860 119861 isin IF(119877) An approximationoperator 119879 is continuous if for any 120576 gt 0 there is 120575 gt 0 when119863119868(119860 119861) lt 120575 we have
119863119868 (119879 (119860) 119879 (119861)) lt 120576 (39)
The continuity constraint means that if two interval-valuedfuzzy numbers are close then their interval-valued trape-zoidal approximations also should be close
6 Journal of Applied Mathematics
32 Interval-Valued Trapezoidal Approximation Based on theWeighted Distance In this section we are looking for anapproximation operator 119879 IF(119877) rarr IF119879(119877) whichproduces an interval-valued trapezoidal fuzzy number thatis the nearest one to the given interval-valued fuzzy numberand preserves its core with respect to the weighted distance119863119868defined by (21)
Lemma 14 Let 119860 isin 119865(119877) 119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1]
If function 119891(120572) is nonnegative and increasing on [0 1] with119891(0) = 0 and int1
0119891(120572)119889120572 = 12 then we have
(i)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860minus (
1) minus 119860minus (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
le 119860minus (1) (40)
(ii)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (41)
Proof (i) See [23] the proof of Theorem 31(ii) Since 119860
+(120572) is a nonincreasing function we have
119860+(120572) ge 119860
+(1) for any 120572 isin [0 1] By 119891(120572) ge 0 we can prove
that
(1 minus 120572)119860+ (120572) 119891 (120572) ge (1 minus 120572)119860+ (
1) 119891 (120572)
= [(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572)
(42)
According to the monotonicity of integration we have
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
ge int
1
0
[(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572) 119889120572
(43)
That is
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
minus 119860+ (1) int
1
0
120572 (1 minus 120572) 119891 (120572) 119889120572
ge 119860+ (1) int
1
0
(120572 minus 1)2119891 (120572) 119889120572
(44)
Because int10(120572 minus 1)
2119891(120572)119889120572 gt 0 it follows that
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (45)
Theorem 15 (see [24]) Let 119891 1198921 1198922 119892
119898 119877119899rarr 119877 be
convex and differentiable functions Then 119909 solves the convexprogramming problem
min 119891 (119909)
119904119905 119892119894(119909) le 119887119894
119894 isin 1 2 119898
(46)
if and only if there exist 120583119894 119894 isin 1 2 119898 such that
(i) nabla119891(119909) + Σ119898119894=1120583119894nabla119892119894(119909) = 0
(ii) 119892119894(119909) minus 119887
119894le 0
(iii) 120583119894ge 0
(iv) 120583119894(119887119894minus 119892119894(119909)) = 0
Suppose that 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2
119909 isin [119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] We
will try to find an interval-valued trapezoidal fuzzy number119879(119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] which is the nearest
interval-valued trapezoidal fuzzy number of 119860 and preservesits core with respect to the weighted distance 119863
119868 Thus we have
to find such real numbers 1199051198711 1199051198712 1199051198713 1199051198714 1199051198801 1199051198802 1199051198803and 1199051198804that
minimize
119863119868 (119860 119879 (119860))
=
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus (119905
119871
1+ (119905119871
2minus 119905119871
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119905119871
3) 120572))
2
119889120572)
12
+ (int
1
0
f (120572) (119860119880minus(120572) minus (119905
119880
1+ (119905119880
2minus 119905119880
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119905119880
3) 120572))
2
119889120572)
12
]
(47)
with respect to condition core119860 = core119879(119860) that is
119905119871
2= 119860119871
minus(1) 119905
119871
3= 119860119871
+(1)
119905119880
2= 119860119880
minus(1) 119905
119880
3= 119860119880
+(1)
(48)
It follows that
119905119871
2le 119905119871
3 119905
119880
2le 119905119880
3 (49)
Making use of Theorem 4 we have
119905119880
2le 119905119871
2 119905
119871
3le 119905119880
3 (50)
Journal of Applied Mathematics 7
Using (47) and (50) together with Theorem 6 we only need tominimize the function
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) [119860119871
minus(120572) minus (119905
119871
1+ (119860119871
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119860119871
+(1)) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
minus(120572) minus (119905
119880
1+ (119860119880
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119860119880
+(1)) 120572)]
2
119889120572
(51)
subject to
119905119880
1minus 119905119871
1le 0 119905
119871
4minus 119905119880
4le 0 (52)
After simple calculations we obtain
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
4)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
4)
2
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 sdot 119905
119871
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 sdot 119905
119871
4
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 sdot 119905
119880
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 sdot 119905
119880
4
+ int
1
0
119891 (120572) (119860119871
minus(120572) minus 120572 sdot 119860
119871
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119871
+(120572) minus 120572 sdot 119860
119871
+(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
minus(120572) minus 120572 sdot 119860
119880
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
+(120572) minus 120572 sdot 119860
119880
+(1))
2
119889120572
(53)
subject to
119905119880
1minus 119905119871
1le 0
119905119871
4minus 119905119880
4le 0
(54)
We present the main result of the paper as follows
Theorem 16 Let 119860 = [119860119871 119860119880] isin IF(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] is the nearest
interval-valued trapezoidal fuzzy number to 119860 and preservesits core with respect to the weighted distance 119863
119868 Consider the
following(i) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(55)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(56)
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(57)
(ii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(58)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(59)
8 Journal of Applied Mathematics
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(60)
(iii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
(61)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(62)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(63)
(iv) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(64)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(65)
Proof Because the function 119865 in (53) and conditions (54)satisfy the hypothesis of convexity and differentiability inTheorem 15 after some simple calculations conditions (i)ndash(iv) inTheorem 15 with respect to the minimization problem(53) in conditions (54) can be shown as follows
2119905119871
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 minus 1205831
= 0
(66)
2119905119871
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 + 1205832
= 0
(67)
2119905119880
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 + 1205831
= 0
(68)
Journal of Applied Mathematics 9
2119905119880
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 minus 1205832
= 0
(69)
1205831(119905119880
1minus 119905119871
1) = 0 (70)
1205832(119905119871
4minus 119905119880
4) = 0 (71)
1205831ge 0 (72)
1205832ge 0 (73)
119905119880
1minus 119905119871
1le 0 (74)
119905119871
4minus 119905119880
4le 0 (75)
(i) In the case 1205831gt 0 and 120583
2= 0 the solution of the system
(66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(76)
Firstly we have from (55) that 1205831gt 0 and it follows from
(56) that
119905119880
4minus 119905119871
4=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(77)
Then conditions (72) (73) (74) and (75) are verified
Secondly combing with (48) (55) and Lemma 14 (i) wecan prove that
119905119880
2minus 119905119880
1
= 119860119880
minus(1) + ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(78)
And on the basis of (50) we have
119905119871
2minus 119905119871
1ge 119905119880
2minus 119905119880
1gt 0 (79)
By making use of (48) and Lemma 14 (ii) we get
119905119880
4minus 119905119880
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119880
+(1) ge 0
119905119871
4minus 119905119871
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119871
+(1) ge 0
(80)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(ii) In the case 120583
1gt 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
can obtain one and only one interval-valued trapezoidal fuzzynumber 119879(119860) = [(119879(119860))119871 (119879(119860))119880] where
(119879 (A))119871120572= [
119860119871
minus(1205722) minus 119860119871
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
minus(1205721)
119860119871
+(1205722) minus 119860119871
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119871
+(1205721)]
(119879 (119860))119880
120572= [
119860119880
minus(1205722) minus 119860119880
minus(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
minus(1205721)
119860119880
+(1205722) minus 119860119880
+(1205721)
1205722minus 1205721
(120572 minus 1205721) + 119860119880
+(1205721)]
(25)
It is obvious that
(119879 (119860))1205721= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205721) 119860119871
+(1205721)]
119910 isin [119860119880
minus(1205721) 119860119880
+(1205721)]
(119879 (119860))1205722= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(1205722) 119860119871
+(1205722)]
119910 isin [119860119880
minus(1205722) 119860119880
+(1205722)]
(26)
Hence (119879(119860))1205721= 1198601205721and (119879(119860))
1205722= 1198601205722
312 Translation Invariance For 119860 isin IF(119877) and 119911 isin 119877 wedefine
119860 + 119911 = [(119860 + 119911)119871 (119860 + 119911)
119880] (27)
where (119860 + 119911)120572= (119909 119910) isin 119877
2 119909 isin [119860
119871
minus(120572) + 119911 119860
119871
+(120572) +
119911] 119910 isin [119860119880
minus(120572) + 119911 119860
119880
+(120572) + 119911] 120572 isin [0 1] that is
(119860 + 119911)119871
minus(120572) = 119860
119871
minus(120572) + 119911
(119860 + 119911)119871
+(120572) = 119860
119871
+(120572) + 119911
(28)
(119860 + 119911)119880
minus(120572) = 119860
119880
minus(120572) + 119911
(119860 + 119911)119880
+(120572) = 119860
119880
+(120572) + 119911
(29)
An approximation operator 119879 is invariant to translation if
119879 (119860 + 119911) = 119879 (119860) + 119911 119911 isin 119877 (30)
Translation invariance means that the relative position of theinterval-valued trapezoidal approximation remains constantwhen the membership function is moved to the left or to theright
313 Scale Invariance For 119860 isin IF(119877) and 120582 isin 119877 0 wedefine
120582119860 = [120582119860119871 120582119860119880] (31)
When 120582 gt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
minus(120572)
120582119860119871
+(120572)] 119910 isin [120582119860
119880
minus(120572) 120582119860
119880
+(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
minus(120572)
(120582119860)119871
+(120572) = 120582119860
119871
+(120572)
(120582119860)119880
minus(120572) = 120582119860
119880
minus(120572)
(120582119860)119880
+(120572) = 120582119860
119880
+(120572)
(32)
When 120582 lt 0 (120582119860)120572= (119909 119910) isin 119877
2 119909 isin [120582119860
119871
+(120572)
120582119860119871
minus(120572)] 119910 isin [120582119860
119880
+(120572) 120582119860
119880
minus(120572)] 120572 isin [0 1] that is
(120582119860)119871
minus(120572) = 120582119860
119871
+(120572)
(120582119860)119871
+(120572) = 120582119860
119871
minus(120572)
(33)
(120582119860)119880
minus(120572) = 120582119860
119880
+(120572)
(120582119860)119880
+(120572) = 120582119860
119880
minus(120572)
(34)
We say that an approximation operator 119879 is scale invariant if
119879 (120582119860) = 120582119879 (119860) 120582 isin 119877 0 (35)
314 Identity This criterion states that the interval-valuedtrapezoidal approximation of an interval-valued trapezoidalfuzzy number is equivalent to that number that is if 119860 isin
IF119879(119877) then
119879 (119860) = 119860 (36)
315 Nearness Criterion An approximation operator 119879 ful-fills the nearness criterion if for any interval-valued fuzzynumber119860 its output value119879(119860) is the nearest interval-valuedtrapezoidal fuzzy number to 119860 with respect to the weighteddistance 119863
119868defined by (21) In other words for any 119861 isin
IF119879(119877) we have
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (37)
Remark 13 We can verify that IF(119877) is closed and convex so119879(119860) exists and is unique
316 Ranking Invariance A reasonable approximation oper-ator should preserve the accepted ranking We say that anapproximation operator 119879 is ranking invariant if for any119860 119861 isin IF(119877)
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (38)
317 Continuity Let 119860 119861 isin IF(119877) An approximationoperator 119879 is continuous if for any 120576 gt 0 there is 120575 gt 0 when119863119868(119860 119861) lt 120575 we have
119863119868 (119879 (119860) 119879 (119861)) lt 120576 (39)
The continuity constraint means that if two interval-valuedfuzzy numbers are close then their interval-valued trape-zoidal approximations also should be close
6 Journal of Applied Mathematics
32 Interval-Valued Trapezoidal Approximation Based on theWeighted Distance In this section we are looking for anapproximation operator 119879 IF(119877) rarr IF119879(119877) whichproduces an interval-valued trapezoidal fuzzy number thatis the nearest one to the given interval-valued fuzzy numberand preserves its core with respect to the weighted distance119863119868defined by (21)
Lemma 14 Let 119860 isin 119865(119877) 119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1]
If function 119891(120572) is nonnegative and increasing on [0 1] with119891(0) = 0 and int1
0119891(120572)119889120572 = 12 then we have
(i)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860minus (
1) minus 119860minus (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
le 119860minus (1) (40)
(ii)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (41)
Proof (i) See [23] the proof of Theorem 31(ii) Since 119860
+(120572) is a nonincreasing function we have
119860+(120572) ge 119860
+(1) for any 120572 isin [0 1] By 119891(120572) ge 0 we can prove
that
(1 minus 120572)119860+ (120572) 119891 (120572) ge (1 minus 120572)119860+ (
1) 119891 (120572)
= [(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572)
(42)
According to the monotonicity of integration we have
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
ge int
1
0
[(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572) 119889120572
(43)
That is
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
minus 119860+ (1) int
1
0
120572 (1 minus 120572) 119891 (120572) 119889120572
ge 119860+ (1) int
1
0
(120572 minus 1)2119891 (120572) 119889120572
(44)
Because int10(120572 minus 1)
2119891(120572)119889120572 gt 0 it follows that
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (45)
Theorem 15 (see [24]) Let 119891 1198921 1198922 119892
119898 119877119899rarr 119877 be
convex and differentiable functions Then 119909 solves the convexprogramming problem
min 119891 (119909)
119904119905 119892119894(119909) le 119887119894
119894 isin 1 2 119898
(46)
if and only if there exist 120583119894 119894 isin 1 2 119898 such that
(i) nabla119891(119909) + Σ119898119894=1120583119894nabla119892119894(119909) = 0
(ii) 119892119894(119909) minus 119887
119894le 0
(iii) 120583119894ge 0
(iv) 120583119894(119887119894minus 119892119894(119909)) = 0
Suppose that 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2
119909 isin [119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] We
will try to find an interval-valued trapezoidal fuzzy number119879(119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] which is the nearest
interval-valued trapezoidal fuzzy number of 119860 and preservesits core with respect to the weighted distance 119863
119868 Thus we have
to find such real numbers 1199051198711 1199051198712 1199051198713 1199051198714 1199051198801 1199051198802 1199051198803and 1199051198804that
minimize
119863119868 (119860 119879 (119860))
=
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus (119905
119871
1+ (119905119871
2minus 119905119871
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119905119871
3) 120572))
2
119889120572)
12
+ (int
1
0
f (120572) (119860119880minus(120572) minus (119905
119880
1+ (119905119880
2minus 119905119880
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119905119880
3) 120572))
2
119889120572)
12
]
(47)
with respect to condition core119860 = core119879(119860) that is
119905119871
2= 119860119871
minus(1) 119905
119871
3= 119860119871
+(1)
119905119880
2= 119860119880
minus(1) 119905
119880
3= 119860119880
+(1)
(48)
It follows that
119905119871
2le 119905119871
3 119905
119880
2le 119905119880
3 (49)
Making use of Theorem 4 we have
119905119880
2le 119905119871
2 119905
119871
3le 119905119880
3 (50)
Journal of Applied Mathematics 7
Using (47) and (50) together with Theorem 6 we only need tominimize the function
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) [119860119871
minus(120572) minus (119905
119871
1+ (119860119871
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119860119871
+(1)) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
minus(120572) minus (119905
119880
1+ (119860119880
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119860119880
+(1)) 120572)]
2
119889120572
(51)
subject to
119905119880
1minus 119905119871
1le 0 119905
119871
4minus 119905119880
4le 0 (52)
After simple calculations we obtain
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
4)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
4)
2
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 sdot 119905
119871
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 sdot 119905
119871
4
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 sdot 119905
119880
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 sdot 119905
119880
4
+ int
1
0
119891 (120572) (119860119871
minus(120572) minus 120572 sdot 119860
119871
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119871
+(120572) minus 120572 sdot 119860
119871
+(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
minus(120572) minus 120572 sdot 119860
119880
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
+(120572) minus 120572 sdot 119860
119880
+(1))
2
119889120572
(53)
subject to
119905119880
1minus 119905119871
1le 0
119905119871
4minus 119905119880
4le 0
(54)
We present the main result of the paper as follows
Theorem 16 Let 119860 = [119860119871 119860119880] isin IF(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] is the nearest
interval-valued trapezoidal fuzzy number to 119860 and preservesits core with respect to the weighted distance 119863
119868 Consider the
following(i) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(55)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(56)
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(57)
(ii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(58)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(59)
8 Journal of Applied Mathematics
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(60)
(iii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
(61)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(62)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(63)
(iv) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(64)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(65)
Proof Because the function 119865 in (53) and conditions (54)satisfy the hypothesis of convexity and differentiability inTheorem 15 after some simple calculations conditions (i)ndash(iv) inTheorem 15 with respect to the minimization problem(53) in conditions (54) can be shown as follows
2119905119871
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 minus 1205831
= 0
(66)
2119905119871
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 + 1205832
= 0
(67)
2119905119880
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 + 1205831
= 0
(68)
Journal of Applied Mathematics 9
2119905119880
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 minus 1205832
= 0
(69)
1205831(119905119880
1minus 119905119871
1) = 0 (70)
1205832(119905119871
4minus 119905119880
4) = 0 (71)
1205831ge 0 (72)
1205832ge 0 (73)
119905119880
1minus 119905119871
1le 0 (74)
119905119871
4minus 119905119880
4le 0 (75)
(i) In the case 1205831gt 0 and 120583
2= 0 the solution of the system
(66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(76)
Firstly we have from (55) that 1205831gt 0 and it follows from
(56) that
119905119880
4minus 119905119871
4=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(77)
Then conditions (72) (73) (74) and (75) are verified
Secondly combing with (48) (55) and Lemma 14 (i) wecan prove that
119905119880
2minus 119905119880
1
= 119860119880
minus(1) + ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(78)
And on the basis of (50) we have
119905119871
2minus 119905119871
1ge 119905119880
2minus 119905119880
1gt 0 (79)
By making use of (48) and Lemma 14 (ii) we get
119905119880
4minus 119905119880
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119880
+(1) ge 0
119905119871
4minus 119905119871
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119871
+(1) ge 0
(80)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(ii) In the case 120583
1gt 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
32 Interval-Valued Trapezoidal Approximation Based on theWeighted Distance In this section we are looking for anapproximation operator 119879 IF(119877) rarr IF119879(119877) whichproduces an interval-valued trapezoidal fuzzy number thatis the nearest one to the given interval-valued fuzzy numberand preserves its core with respect to the weighted distance119863119868defined by (21)
Lemma 14 Let 119860 isin 119865(119877) 119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1]
If function 119891(120572) is nonnegative and increasing on [0 1] with119891(0) = 0 and int1
0119891(120572)119889120572 = 12 then we have
(i)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860minus (
1) minus 119860minus (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
le 119860minus (1) (40)
(ii)
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (41)
Proof (i) See [23] the proof of Theorem 31(ii) Since 119860
+(120572) is a nonincreasing function we have
119860+(120572) ge 119860
+(1) for any 120572 isin [0 1] By 119891(120572) ge 0 we can prove
that
(1 minus 120572)119860+ (120572) 119891 (120572) ge (1 minus 120572)119860+ (
1) 119891 (120572)
= [(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572)
(42)
According to the monotonicity of integration we have
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
ge int
1
0
[(120572 minus 1)2minus 120572 (120572 minus 1)]119860+ (
1) 119891 (120572) 119889120572
(43)
That is
int
1
0
(1 minus 120572)119860+ (120572) 119891 (120572) 119889120572
minus 119860+ (1) int
1
0
120572 (1 minus 120572) 119891 (120572) 119889120572
ge 119860+ (1) int
1
0
(120572 minus 1)2119891 (120572) 119889120572
(44)
Because int10(120572 minus 1)
2119891(120572)119889120572 gt 0 it follows that
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860+ (
1) minus 119860+ (120572)] 119889120572
int
1
0(120572 minus 1)
2119891 (120572) 119889120572
ge 119860+ (1) (45)
Theorem 15 (see [24]) Let 119891 1198921 1198922 119892
119898 119877119899rarr 119877 be
convex and differentiable functions Then 119909 solves the convexprogramming problem
min 119891 (119909)
119904119905 119892119894(119909) le 119887119894
119894 isin 1 2 119898
(46)
if and only if there exist 120583119894 119894 isin 1 2 119898 such that
(i) nabla119891(119909) + Σ119898119894=1120583119894nabla119892119894(119909) = 0
(ii) 119892119894(119909) minus 119887
119894le 0
(iii) 120583119894ge 0
(iv) 120583119894(119887119894minus 119892119894(119909)) = 0
Suppose that 119860 = [119860119871 119860119880] isin IF(119877) 119860120572= (119909 119910) isin 119877
2
119909 isin [119860119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin [0 1] We
will try to find an interval-valued trapezoidal fuzzy number119879(119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] which is the nearest
interval-valued trapezoidal fuzzy number of 119860 and preservesits core with respect to the weighted distance 119863
119868 Thus we have
to find such real numbers 1199051198711 1199051198712 1199051198713 1199051198714 1199051198801 1199051198802 1199051198803and 1199051198804that
minimize
119863119868 (119860 119879 (119860))
=
1
2
[(int
1
0
119891 (120572) (119860119871
minus(120572) minus (119905
119871
1+ (119905119871
2minus 119905119871
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119905119871
3) 120572))
2
119889120572)
12
+ (int
1
0
f (120572) (119860119880minus(120572) minus (119905
119880
1+ (119905119880
2minus 119905119880
1) 120572))
2
119889120572
+int
1
0
119891(120572)(119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119905119880
3) 120572))
2
119889120572)
12
]
(47)
with respect to condition core119860 = core119879(119860) that is
119905119871
2= 119860119871
minus(1) 119905
119871
3= 119860119871
+(1)
119905119880
2= 119860119880
minus(1) 119905
119880
3= 119860119880
+(1)
(48)
It follows that
119905119871
2le 119905119871
3 119905
119880
2le 119905119880
3 (49)
Making use of Theorem 4 we have
119905119880
2le 119905119871
2 119905
119871
3le 119905119880
3 (50)
Journal of Applied Mathematics 7
Using (47) and (50) together with Theorem 6 we only need tominimize the function
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) [119860119871
minus(120572) minus (119905
119871
1+ (119860119871
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119860119871
+(1)) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
minus(120572) minus (119905
119880
1+ (119860119880
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119860119880
+(1)) 120572)]
2
119889120572
(51)
subject to
119905119880
1minus 119905119871
1le 0 119905
119871
4minus 119905119880
4le 0 (52)
After simple calculations we obtain
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
4)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
4)
2
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 sdot 119905
119871
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 sdot 119905
119871
4
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 sdot 119905
119880
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 sdot 119905
119880
4
+ int
1
0
119891 (120572) (119860119871
minus(120572) minus 120572 sdot 119860
119871
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119871
+(120572) minus 120572 sdot 119860
119871
+(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
minus(120572) minus 120572 sdot 119860
119880
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
+(120572) minus 120572 sdot 119860
119880
+(1))
2
119889120572
(53)
subject to
119905119880
1minus 119905119871
1le 0
119905119871
4minus 119905119880
4le 0
(54)
We present the main result of the paper as follows
Theorem 16 Let 119860 = [119860119871 119860119880] isin IF(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] is the nearest
interval-valued trapezoidal fuzzy number to 119860 and preservesits core with respect to the weighted distance 119863
119868 Consider the
following(i) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(55)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(56)
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(57)
(ii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(58)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(59)
8 Journal of Applied Mathematics
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(60)
(iii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
(61)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(62)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(63)
(iv) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(64)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(65)
Proof Because the function 119865 in (53) and conditions (54)satisfy the hypothesis of convexity and differentiability inTheorem 15 after some simple calculations conditions (i)ndash(iv) inTheorem 15 with respect to the minimization problem(53) in conditions (54) can be shown as follows
2119905119871
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 minus 1205831
= 0
(66)
2119905119871
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 + 1205832
= 0
(67)
2119905119880
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 + 1205831
= 0
(68)
Journal of Applied Mathematics 9
2119905119880
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 minus 1205832
= 0
(69)
1205831(119905119880
1minus 119905119871
1) = 0 (70)
1205832(119905119871
4minus 119905119880
4) = 0 (71)
1205831ge 0 (72)
1205832ge 0 (73)
119905119880
1minus 119905119871
1le 0 (74)
119905119871
4minus 119905119880
4le 0 (75)
(i) In the case 1205831gt 0 and 120583
2= 0 the solution of the system
(66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(76)
Firstly we have from (55) that 1205831gt 0 and it follows from
(56) that
119905119880
4minus 119905119871
4=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(77)
Then conditions (72) (73) (74) and (75) are verified
Secondly combing with (48) (55) and Lemma 14 (i) wecan prove that
119905119880
2minus 119905119880
1
= 119860119880
minus(1) + ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(78)
And on the basis of (50) we have
119905119871
2minus 119905119871
1ge 119905119880
2minus 119905119880
1gt 0 (79)
By making use of (48) and Lemma 14 (ii) we get
119905119880
4minus 119905119880
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119880
+(1) ge 0
119905119871
4minus 119905119871
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119871
+(1) ge 0
(80)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(ii) In the case 120583
1gt 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
Using (47) and (50) together with Theorem 6 we only need tominimize the function
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) [119860119871
minus(120572) minus (119905
119871
1+ (119860119871
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119871
+(120572) minus (119905
119871
4minus (119905119871
4minus 119860119871
+(1)) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
minus(120572) minus (119905
119880
1+ (119860119880
minus(1) minus 119905
119871
1) 120572)]
2
119889120572
+ int
1
0
119891 (120572) [119860119880
+(120572) minus (119905
119880
4minus (119905119880
4minus 119860119880
+(1)) 120572)]
2
119889120572
(51)
subject to
119905119880
1minus 119905119871
1le 0 119905
119871
4minus 119905119880
4le 0 (52)
After simple calculations we obtain
119865 (119905119871
1 119905119871
4 119905119880
1 119905119880
4)
= int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119871
4)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
1)
2
+ int
1
0
119891 (120572) (1 minus 120572)2119889120572 sdot (119905
119880
4)
2
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 sdot 119905
119871
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 sdot 119905
119871
4
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 sdot 119905
119880
1
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 sdot 119905
119880
4
+ int
1
0
119891 (120572) (119860119871
minus(120572) minus 120572 sdot 119860
119871
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119871
+(120572) minus 120572 sdot 119860
119871
+(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
minus(120572) minus 120572 sdot 119860
119880
minus(1))
2
119889120572
+ int
1
0
119891 (120572) (119860119880
+(120572) minus 120572 sdot 119860
119880
+(1))
2
119889120572
(53)
subject to
119905119880
1minus 119905119871
1le 0
119905119871
4minus 119905119880
4le 0
(54)
We present the main result of the paper as follows
Theorem 16 Let 119860 = [119860119871 119860119880] isin IF(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)] is the nearest
interval-valued trapezoidal fuzzy number to 119860 and preservesits core with respect to the weighted distance 119863
119868 Consider the
following(i) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(55)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(56)
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(57)
(ii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
(58)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(59)
8 Journal of Applied Mathematics
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(60)
(iii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
(61)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(62)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(63)
(iv) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(64)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(65)
Proof Because the function 119865 in (53) and conditions (54)satisfy the hypothesis of convexity and differentiability inTheorem 15 after some simple calculations conditions (i)ndash(iv) inTheorem 15 with respect to the minimization problem(53) in conditions (54) can be shown as follows
2119905119871
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 minus 1205831
= 0
(66)
2119905119871
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 + 1205832
= 0
(67)
2119905119880
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 + 1205831
= 0
(68)
Journal of Applied Mathematics 9
2119905119880
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 minus 1205832
= 0
(69)
1205831(119905119880
1minus 119905119871
1) = 0 (70)
1205832(119905119871
4minus 119905119880
4) = 0 (71)
1205831ge 0 (72)
1205832ge 0 (73)
119905119880
1minus 119905119871
1le 0 (74)
119905119871
4minus 119905119880
4le 0 (75)
(i) In the case 1205831gt 0 and 120583
2= 0 the solution of the system
(66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(76)
Firstly we have from (55) that 1205831gt 0 and it follows from
(56) that
119905119880
4minus 119905119871
4=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(77)
Then conditions (72) (73) (74) and (75) are verified
Secondly combing with (48) (55) and Lemma 14 (i) wecan prove that
119905119880
2minus 119905119880
1
= 119860119880
minus(1) + ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(78)
And on the basis of (50) we have
119905119871
2minus 119905119871
1ge 119905119880
2minus 119905119880
1gt 0 (79)
By making use of (48) and Lemma 14 (ii) we get
119905119880
4minus 119905119880
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119880
+(1) ge 0
119905119871
4minus 119905119871
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119871
+(1) ge 0
(80)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(ii) In the case 120583
1gt 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
then we have
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(60)
(iii) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
(61)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
(62)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
(63)
(iv) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(64)
then we have
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(65)
Proof Because the function 119865 in (53) and conditions (54)satisfy the hypothesis of convexity and differentiability inTheorem 15 after some simple calculations conditions (i)ndash(iv) inTheorem 15 with respect to the minimization problem(53) in conditions (54) can be shown as follows
2119905119871
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 minus 1205831
= 0
(66)
2119905119871
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 + 1205832
= 0
(67)
2119905119880
1int
1
0
119891 (120572) (1 minus 120572)2119889120572
+ 2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572 + 1205831
= 0
(68)
Journal of Applied Mathematics 9
2119905119880
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 minus 1205832
= 0
(69)
1205831(119905119880
1minus 119905119871
1) = 0 (70)
1205832(119905119871
4minus 119905119880
4) = 0 (71)
1205831ge 0 (72)
1205832ge 0 (73)
119905119880
1minus 119905119871
1le 0 (74)
119905119871
4minus 119905119880
4le 0 (75)
(i) In the case 1205831gt 0 and 120583
2= 0 the solution of the system
(66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(76)
Firstly we have from (55) that 1205831gt 0 and it follows from
(56) that
119905119880
4minus 119905119871
4=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(77)
Then conditions (72) (73) (74) and (75) are verified
Secondly combing with (48) (55) and Lemma 14 (i) wecan prove that
119905119880
2minus 119905119880
1
= 119860119880
minus(1) + ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(78)
And on the basis of (50) we have
119905119871
2minus 119905119871
1ge 119905119880
2minus 119905119880
1gt 0 (79)
By making use of (48) and Lemma 14 (ii) we get
119905119880
4minus 119905119880
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119880
+(1) ge 0
119905119871
4minus 119905119871
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119871
+(1) ge 0
(80)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(ii) In the case 120583
1gt 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
2119905119880
4int
1
0
119891 (120572) (1 minus 120572)2119889120572
+2int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572 minus 1205832
= 0
(69)
1205831(119905119880
1minus 119905119871
1) = 0 (70)
1205832(119905119871
4minus 119905119880
4) = 0 (71)
1205831ge 0 (72)
1205832ge 0 (73)
119905119880
1minus 119905119871
1le 0 (74)
119905119871
4minus 119905119880
4le 0 (75)
(i) In the case 1205831gt 0 and 120583
2= 0 the solution of the system
(66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
(76)
Firstly we have from (55) that 1205831gt 0 and it follows from
(56) that
119905119880
4minus 119905119871
4=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(77)
Then conditions (72) (73) (74) and (75) are verified
Secondly combing with (48) (55) and Lemma 14 (i) wecan prove that
119905119880
2minus 119905119880
1
= 119860119880
minus(1) + ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)[120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(78)
And on the basis of (50) we have
119905119871
2minus 119905119871
1ge 119905119880
2minus 119905119880
1gt 0 (79)
By making use of (48) and Lemma 14 (ii) we get
119905119880
4minus 119905119880
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119880
+(1) ge 0
119905119871
4minus 119905119871
3= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus 119860119871
+(1) ge 0
(80)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(ii) In the case 120583
1gt 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= 119905119880
1= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
119905119871
4= 119905119880
4
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(81)We have from (58) and (59) that 120583
1gt 0 and 120583
2gt 0 Then
conditions (72) (73) (74) and (75) are verifiedFurthermore by making use of (48) (50) and (58)
similar to (i) we can prove that119905119871
2minus 119905119871
1ge 119905
U2minus 119905119880
1gt 0 (82)
According to (48) (59) and Lemma 14 (ii) we obtain
119905119880
4minus 119905119880
3
= minus119860119880
+(1) minus ((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
gt minus119860119880
+(1) minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(83)This implies that
119905119871
4minus 119905119871
3ge 119905119880
4minus 119905119880
3gt 0 (84)
It follows from (49) that (1199051198711 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are
two trapezoidal fuzzy numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued
trapezoidal approximation of 119860 in this case(iii) In the case 120583
1= 0 and 120583
2gt 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= 119905119880
4= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
1205831= 0
1205832= minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(85)
First we have from (62) that 1205832gt 0 Also it follows from
(61) we can prove that
119905119871
1minus 119905119880
1=
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= (int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572)
times (int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
ge 0
(86)
Then conditions (72) (73) (74) and (75) are verifiedSecondly combing with (48) and Lemma 14 (i) we have
119905119871
2minus 119905119871
1
= 119860119871
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
119905119880
2minus 119905119880
1
= 119860119880
minus(1) +
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
ge 0
(87)
According to (48) (50) (62) and the second result ofLemma 14 (ii) similar to (ii) we can prove that 119905119871
4minus 119905119871
3ge
119905119880
4minus 119905119880
3gt 0 It follows from (49) that (119905119871
1 119905119871
2 119905119871
3 119905119871
4) and
(119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy numbers
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 11
Therefore by Theorem 6 and (50) 119879(119860) = [(1199051198711 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IFT(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this case(iv) In the case 120583
1= 0 and 120583
2= 0 the solution of the
system (66)ndash(75) is
119905119871
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
1= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119871
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
119905119880
4= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
1205831= 0 120583
2= 0
(88)
By (64) similar to (i) and (iii) we have 1199051198711minus119905119880
1ge 0 and 119905119880
4minus119905119871
4ge
0 then conditions (72) (73) (74) and (75) are verifiedFurthermore similar to (i) and (iii) we can prove that
(119905119871
1 119905119871
2 119905119871
3 119905119871
4) and (119905119880
1 119905119880
2 119905119880
3 119905119880
4) are two trapezoidal fuzzy
numbersTherefore by Theorem 6 and (50) 119879(119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin IF119879(119877) is the nearest interval-valued trape-
zoidal approximation of 119860 in this caseFor any interval-valued fuzzy number we can apply one
and only one of the above situations (i)ndash(iv) to calculate theinterval-valued trapezoidal approximation of it We denote
Ω1= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
Ω2= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω3= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 gt 0
Ω4= 119860 = [119860
119871 119860119880] isin IF (119877)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 ge 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minusint
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 le 0
(89)
It is obvious that the cases (i)ndash(iv) cover the set of all interval-valued fuzzy numbers and Ω
1 Ω2 Ω3 and Ω
4are disjoint
sets So the approximation operator always gives an interval-valued trapezoidal fuzzy number
By the discussion of Theorem 16 we could find thenearest interval-valued trapezoidal fuzzy number for a giveninterval-valued fuzzy number Furthermore it preserves thecore of the given interval-valued fuzzy number with respectto the weighted distance119863
119868
Remark 17 If 119860 = [119860119871 119860119880] isin 119865(119877) that is 119860119871 = 119860119880 thenour conclusion is consisten with [23]
Corollary 18 Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) where 119903 gt 0 and
119860119871(119909) =
(
119909 minus 119886119871
1
119886119871
2minus 119886119871
1
)
119903
119886119871
1le 119909 lt 119886
119871
2
1 119886119871
2le 119909 le 119886
119871
3
(
119886119871
4minus 119909
119886119871
4minus 119886119871
3
)
119903
119886119871
3lt 119909 le 119886
119871
4
0 otherwise
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Applied Mathematics
119860119880(119909) =
(
119909 minus 119886119880
1
119886119880
2minus 119886119880
1
)
119903
119886119880
1le 119909 lt 119886
119880
2
1 119886119880
2le 119909 le 119886
119880
3
(
119886119880
4minus 119909
119886119880
4minus 119886119880
3
)
119903
119886119880
3lt 119909 le a119880
4
0 otherwise(90)
(i) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(91)
then
119905119871
1= 119905119880
1= minus
(minus61199032+ 5119903 + 1)(119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1)(119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(92)
(ii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(93)
then119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(94)
(iii) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) gt 0
(95)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= 119905119880
4
= minus
(minus61199032+ 5119903 + 1) (119886
119880
3+ 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4+ 119886119871
4)
2 (1 + 2119903) (1 + 3119903)
(96)
(iv) If 119891(120572) = 120572 and
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) ge 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) le 0
(97)
then
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
L3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
(98)
Proof Let 119860 = [119860119871 119860119880] = [(119886
119871
1 119886119871
2 119886119871
3 119886119871
4)119903
(119886119880
1 119886119880
2 119886119880
3 119886119880
4)119903] isin IF(119877) We have 119860119871
minus(120572) = 119886
119871
1+ (119886119871
2minus 119886119871
1) sdot
1205721119903 119860119880
minus(120572) = 119886
119880
1+(119886119880
2minus119886119880
1)sdot1205721119903 119860119871
+(120572) = 119886
119871
4minus(119886119871
4minus119886119871
3)sdot1205721119903
and 119860119880+(120572) = 119886
119880
4minus (119886119880
4minus 119886119880
3) sdot 1205721119903 It is obvious that
119860119871
minus(1) = 119886
119871
2 119860119880minus(1) = 119886
119880
2 119860119871+(1) = 119886
119871
3 and 119860119880
+(1) = 119886
119880
3 Then
by 119891(120572) = 120572 we can prove that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
=
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
12 (1 + 2119903) (1 + 3119903)
int
1
0
119891 (120572) (1 minus 120572)2119889120572 =
1
12
(99)
According to Theorem 16 the results can be easily obtained
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 13
Example 19 Let 119860 = [119860119871 119860119880] = [(2 4 5 7)
12
(2 3 6 8)12] isin IF(119877) and 119891(120572) = 120572 Since
(minus61199032+ 5119903 + 1) (119886
119880
2minus 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1minus 119886119871
1) = minus2 lt 0
(minus61199032+ 5119903 + 1) (119886
119880
3minus 119886119871
3) minus 2 (5119903 + 1) (119886
119880
4minus 119886119871
4) = minus5 lt 0
(100)
that is119860 = [119860119871 119860119880] satisfies condition (i) of Corollary 18 wehave119905119871
1= 119905119880
1
= minus
(minus61199032+ 5119903 + 1) (119886
119880
2+ 119886119871
2) minus 2 (5119903 + 1) (119886
119880
1+ 119886119871
1)
2 (1 + 2119903) (1 + 3119903)
=
7
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(101)
Therefore 119879(119860) = [(75 4 5 395) (75 3 6 445)] isin
IF119879(119877) is the nearest interval-valued trapezoidal fuzzy num-ber to 119860 which preserves the core of 119860
Remark 20 Let 119860 = [119860119871 119860119880] isin IF(119877) 119879(119860) = [119879(119860
119871)
119879(119860119880)] be not true in general
Example 21 Let 119860119871 = (2 4 5 7)12
isin 119865(119877) 119860119880 =
(2 3 6 8)12
isin 119865(119877) and 119891(120572) = 120572 Then according tocondition (iv) of Corollary 18 we have
119905119871
1= minus
(minus61199032+ 5119903 + 1) 119886
119871
2minus 2 (5119903 + 1) 119886
119871
1
(1 + 2119903) (1 + 3119903)
=
6
5
119905119880
1= minus
(minus61199032+ 5119903 + 1) 119886
119880
2minus 2 (5119903 + 1) 119886
119880
1
(1 + 2119903) (1 + 3119903)
=
8
5
119905119871
4= minus
(minus61199032+ 5119903 + 1) 119886
119871
3minus 2 (5119903 + 1) 119886
119871
4
(1 + 2119903) (1 + 3119903)
=
39
5
119905119880
4= minus
(minus61199032+ 5119903 + 1) 119886
119880
3minus 2 (5119903 + 1) 119886
119880
4
(1 + 2119903) (1 + 3119903)
=
44
5
(102)
Therefore 119879(119860119871) = (65 4 5 395) and 119879(119860119880) =
(85 3 6 445) Based on Example 19 we known that119879(119860) =
[119879(119860119871) 119879(119860
119880)] Further we have from Theorem 6 that
[119879(119860119871) 119879(119860
119880)] is not a trapezoidal interval-valued fuzzy
number
4 Properties of the Interval-ValuedTrapezoidal Approximation Operator
In this section we consider some properties of the approx-imation operator suggested in Section 32 With respect to
the criteria translation invariance scale invariance identitynearness criterion and ranking invariance we present thefollowing results
Theorem 22 The approximation operator 119879 119868119865(119877) rarr
119868119865119879(119877) which preserves the core of the initial interval-valued
fuzzy number has the following properties
(i) The operator 119879 is invariant to translations that isfor any 119860 isin 119868119865(119877) and 119911 isin 119877
119879 (119860 + 119911) = 119879 (119860) + 119911 (103)
(ii) The operator 119879 is scale invariance that is for any119860 isin 119868119865(119877) and 120582 isin 119877 0
119879 (120582119860) = 120582119879 (119860) (104)
(iii)The operator119879 fulfills the identity criterion that isfor any 119860 isin 119868119865119879(119877)
119879 (119860) = 119860 (105)
(iv) The operator 119879 fulfills the nearness criterion withrespect to the weighted distance119863
119868 that is
119863119868 (119860 119879 (119860)) le 119863119868 (
119860 119861) (106)
for every 119860 isin 119868119865(119877) and 119861 isin 119868119865119879(119877) such that core119861 =core119879(119860)
(v) The operator 119879 is core invariance that is for any119860 isin 119868119865(119877)
core119879 (119860) = core119860 (107)
(vi)The operator 119879 is ranking invariance that is
119860 ⪰ 119861 lArrrArr 119879 (119860) ⪰ 119879 (119861) (108)
for every 119860 119861 isin 119868119865(119877)
Proof If 119860 isin IF(119877) according to (28) and (48) we have
119905119871
2(119860 + 119911) = (119860 + 119911)
119871
minus(1) = 119860
119871
minus(1) + 119911 = 119905
119871
2(119860) + 119911 (109)
Similarly we can prove that
119905119871
3(119860 + 119911) = 119905
119871
3(119860) + 119911
119905119880
2(119860 + 119911) = 119905
119880
2(119860) + 119911
119905119880
3(119860 + 119911) = 119905
119880
3(119860) + 119911
(110)
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Journal of Applied Mathematics
Furthermore we have from (28) and (29) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
minus(1) minus (119860 + 119911)
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
minus(1) minus (119860 + 119911)
119880
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119871
+(1) minus (119860 + 119911)
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860 + 119911)119880
+(1) minus (119860 + 119911)
119880
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus119911int
1
0
119891 (120572) (1 minus 120572)2119889120572
(111)
Then one can easily prove that the interval-valued fuzzynumber 119860 satisfies one of conditions (i)ndash(iv) of Theorem 16if and only if the interval-valued fuzzy number 119860+119911 satisfiesthe same condition In any case ofTheorem 16 bymaking useof (111) we obtain
119905119871
119896(119860 + 119911) = 119905
119871
119896(119860) + 119911
119905119880
119896(119860 + 119911) = 119905
119880
119896(119860) + 119911
(112)
for every 119896 isin 1 4Therefore combine the above results with(109) and (110) and we have 119879(119860 + 119911) = 119879(119860) + 119911
(ii) Let 119860 isin IF(119877) If 120582 gt 0 combing with (32) similar to(i) we can prove that 119879(120582119860) = 120582119879(119860)
If 120582 lt 0 we have from (33) and (48) that
119905119871
2(120582119860) = (120582119860)
119871
minus(1) = 120582119860
119871
+(1) = 120582119905
119871
3(119860) (113)
Similarly we can prove that
119905119871
3(120582119860) = 120582119905
119871
2(119860) 119905
119880
2(120582119860) = 120582119905
119880
3(119860)
119905119880
3(120582119860) = 120582119905
119880
2(119860)
(114)
Furthermore it follows from (33) and (34) that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
minus(1) minus (120582119860)
119871
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
minus(1) minus (120582119860)
119880
minus(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119871
+(1) minus (120582119860)
119871
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (120582119860)119880
+(1) minus (120582119860)
119880
+(120572)] 119889120572
= 120582int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
(115)
Thus 120582119860 is in the case (i) of Theorem 16 if and only if 119860 isin the case (iii) of Theorem 16 Then making use of (115) andTheorem 16 we get
119905119871
1(120582119860) = 120582119905
119871
4(119860) 119905
119871
4(120582119860) = 120582119905
119871
1(119860)
119905119880
1(120582119860) = 120582119905
119880
4(119860) 119905
119880
4(120582119860) = 120582119905
119880
1(119860)
(116)
Therefore combine the above results with (113) and (114) andaccording to (33) and (34) we have
119879 (120582119860) = [(119905119871
1(120582119860) 119905
119871
2(120582119860) 119905
119871
3(120582119860) 119905
119871
4(120582119860))
(119905119880
1(120582119860) 119905
119880
2(120582119860) 119905
119880
3(120582119860) 119905
119880
4(120582119860))]
= [(120582119905119871
4 120582119905119871
3 120582119905119871
2 120582119905119871
1) (120582119905
119880
4 120582119905119880
3 120582119905119880
2 120582119905119880
1)]
= 120582 [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
= 120582119879 (119860)
(117)
Analogously 120582119860 is in the case (ii) of Theorem 16 if and onlyif 119860 is in the case (ii) of Theorem 16 120582119860 is in the case (iii) ofTheorem 16 if and only if119860 is in the case (i) ofTheorem 16120582119860is in the case (iv) of Theorem 16 if and only if 119860 is in the case(iv) ofTheorem 16 In each case 119905119871
119896(120582119860) = 120582119905
119871
5minus119896(119860) 119905119880119896(120582119860) =
120582119905119880
5minus119896(119860) for every 119896 isin 1 2 3 4 therefore 119879(120582119860) = 120582119879(119860)
(iii) If119860 isin IF119879(119877) then119860 is in the case (iv) ofTheorem 16and 119879(119860) = 119860
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 15
(iv) and (v) are the direct consequences of Theorem 16(vi) By (22) and Theorem 16 we can obtain the conclu-
sion
The continuity is considered the essential property foran approximation operator However the approximationoperator given by Theorem 16 is not continuous as thefollowing example proves
Example 23 (see [25]) Let us consider 119860 isin 119865(119877) sub IF(119877)119860120572= [119860minus(120572) 119860
+(120572)] 120572 isin [0 1] such that 119860
minus(1) lt 119860
+(1)
and the sequence of fuzzy numbers (119860119899)119899isin119873
is given by
(119860119899)minus(120572) = 119860minus (
120572) + 120572119899(119860+ (1) minus 119860minus (
1))
(119860119899)+(120572) = 119860+ (
120572)
120572 isin [0 1]
(118)
It is easy to check that the function (119860119899)minus(120572) is nondecreasing
and (119860119899)minus(1) = 119860
+(1) = (119860
119899)+(1) therefore 119860
119899is a fuzzy
number for any 119899 isin 119873 Then according to the weighteddistance 119889
119891defined by (20) we have
1198892
119891(119860119899 119860) = (119860
+ (1) minus 119860minus (
1))2int
1
0
119891 (120572) 1205722119899119889120572
le 119891 (1) (119860+ (1) minus 119860minus (
1))2int
1
0
1205722119899119889120572
=
119891 (1) (119860+ (1) minus 119860minus (
1))2
2119899 + 1
(119)
It is immediate that lim119899rarrinfin
119860119899= 119860 Now denote
119879 (119860) = (1199051 1199052 1199053 1199054)
119879 (119860119899) = (119905
1 (119899) 1199052 (
119899) 1199053 (119899) 1199054 (
119899))
119899 isin 119873
(120)
Because operator 119879 preserves the core of fuzzy number 119860 by(48) we have
lim119899rarrinfin
t2 (119899) = lim119899rarrinfin
(119860119899)minus(1) = 119860+ (
1) gt 119860minus (1) = 1199052
(121)
By seeing Lemma 3 [25] we cannot have lim119899rarrinfin
119879(119860119899) =
119879(119860)with respect to the weighted distance 119889119891 It follows from
Heinersquos criterion that 119879 is discontinuous
To overcome the handicap of discontinuity of the approxi-mation operator119879wepresent the following distance property
Lemma 24 Let 119879119899= [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899)) (119905
119880
1(119899)
119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] be a sequence of interval-valued trape-
zoidal fuzzy numbers If lim119899rarrinfin
119905119871
119894(119899) = 119905
119871
119894 lim119899rarrinfin
119905119880
119894(119899) =
119905119880
119894 119894 isin 1 2 3 4 then lim
119899rarrinfin119879119899= 119879 with respect
to the weighted distance 119863119868 where 119879 = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4)
(119905119880
1 119905119880
2 119905119880
3 119905119880
4)] isin 119868119865
119879(119877)
Proof It is similar to the proof of Lemma 2 in the paper [25]
Theorem 25 Let 119860 = [119860119871 119860119880] isin 119868119865(119877) 119860
120572= (119909 119910) isin
1198772 119909 isin [119860
119871
minus(120572) 119860
119871
+(120572)] 119910 isin [119860
119880
minus(120572) 119860
119880
+(120572)] 120572 isin
[0 1] and 119860119899= [119860
119871
119899 119860119880
119899](119899 isin 119873) be a sequence of
interval-valued fuzzy numbers where (119860119899)120572= (119909 119910) isin
1198772 119909 isin [(119860
119871
119899)minus(120572) (119860
119871
119899)+(120572)] 119910 isin [(119860
119880
119899)minus(120572) (119860
119880
119899)+(120572)]
120572 isin [0 1] If (119860119871119899)minus(120572) (119860
119871
119899)+(120572) (119860
119880
119899)minus(120572) and (119860119880
119899)+(120572)
are uniform convergent sequences to119860119871minus(120572) 119860
119871
+(120572) 119860
119880
minus(120572) and
119860119880
+(120572) respectively then
lim119899rarrinfin
119879 (119860119899) = 119879 (119860) (122)
with respect to the weighted distance119863119868
Proof We denote
119879 (119860) = [(119905119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))] 119899 isin 119873
(123)
Because (119860119871119899)minus(120572) (119860119871
119899)+(120572) (119860119880
119899)minus(120572) and (119860119880
119899)+(120572) are
uniform convergent sequences to 119860119871minus(120572) 119860119871
+(120572) 119860119880
minus(120572) and
119860119880
+(120572) respectively we have
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
(124)
and by (48) we obtain
lim119899rarrinfin
119905119871
2(119899) = lim
119899rarrinfin(119860119871
119899)minus(1) = 119860
119871
minus(1) = 119905
119871
2
lim119899rarrinfin
119905119880
2(119899) = lim
119899rarrinfin(119860119880
119899)minus(1) = 119860
119880
minus(1) = 119905
119880
2
lim119899rarrinfin
119905119871
3(119899) = lim
119899rarrinfin(119860119871
119899)+(1) = 119860
119871
+(1) = 119905
119871
3
lim119899rarrinfin
119905119880
3(119899) = lim
119899rarrinfin(119860119880
119899)+(1) = 119860
119880
+(1) = 119905
119880
3
(125)
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Journal of Applied Mathematics
Considering the following cases(i)119860 = [119860119871 119860119880] satisfies condition (i) ofTheorem 16 the
following situations are possible(ia) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(126)
then there exists119873 when 119899 gt 119873 119860119899satisfies condition (i) of
Theorem 16 We have from (124) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119880
+(1) minus 119860
119880
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119880
4
(127)
According to (125) and Lemma 24 we have lim119899rarrinfin
119879(119860119899) =
119879(119860) with respect to the weighted distance119863119868
(ib) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 lt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(128)
then there exists119873 when 119899 gt 119873119860119899satisfies condition (i) or
condition (ii) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(129)
In both two cases we can prove
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 17
= minus((int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
+int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
) = 119905119871
1
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
+ int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1)
minus (119860119880
119899)+(120572)] 119889120572)
times (2int
1
0
119891(120572)(1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(130)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ii) 119860 = [119860119871 119860119880] satisfies condition (ii) of Theorem 16
The proof is analogous with the proof of case (ia)(iii) 119860 = [119860119871 119860119880] satisfies condition (iii) of Theorem 16
The proof is analogous with the proof of case (i)(iv) 119860 = [119860119871 119860119880] satisfies condition (iv) of Theorem 16
the following situations are possible(iva) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(131)
the proof is analogous with the proof of (i119886)
(ivb) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(132)
then there exists 119873 when 119899 gt 119873 119860119899satisfies condition (i)
(ii) (iii) or (iv) of Theorem 16 and
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
= lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572
(133)
In either cases among (i) (ii) (iii) and (iv) it follows from(133) that
lim119899rarrinfin
119905119871
1(119899)
= minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572)
times (2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
1
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Journal of Applied Mathematics
lim119899rarrinfin
119905119880
1(119899) = lim
119899rarrinfin119905119871
1(119899) = 119905
119871
1= 119905119880
1
lim119899rarrinfin
119905119871
4(119899)
=minus lim119899rarrinfin
((int
1
0
119891 (120572) (1 minus 120572)[120572 sdot (119860119871
119899)+(1) minus (119860
119871
119899)+(120572)]119889120572
+ int
1
0
119891 (120572) (1 minus 120572)
times [120572 sdot (119860119880
119899)+(1) minus (119860
119880
119899)+(120572)] 119889120572)
times(2int
1
0
119891 (120572) (1 minus 120572)2119889120572)
minus1
)
= minus lim119899rarrinfin
int
1
0119891 (120572) (1 minus 120572) [120572 sdot (119860
119871
119899)+(1) minus (119860
119871
119899)+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= minus
int
1
0119891 (120572) (1 minus 120572) [120572 sdot 119860
119871
+(1) minus 119860
119871
+(120572)] 119889120572
int
1
0119891 (120572) (1 minus 120572)
2119889120572
= 119905119871
4
lim119899rarrinfin
119905119880
4(119899) = lim
119899rarrinfin119905119871
4(119899) = 119905
119871
4= 119905119880
4
(134)
Then according to (125) and Lemma 24 we havelim119899rarrinfin
119879(119860119899) = 119879(119860) with respect to the weighted
distance119863119868
(ivc) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 = 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 lt 0
(135)
the proof is analogous with the proof of (ib)(ivd) If
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572 gt 0
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572 = 0
(136)
the proof is analogous with the proof of (ib)
After we analyze all the cases the theorem is provenNext we will give an example to illustrate Theorem 25
Example 26 Let us consider interval-valued fuzzy number119860 = [119860
119871 119860119880] 119860120572= (119909 119910) isin 119877
2 119909 isin [119890
1205722
4 minus 120572] 119910 isin
[(12)1198901205722
5 minus 120572] 120572 isin [0 1] We will determine 119879(119860) with anerror less than 10minus2 with respect to the weighted distance119863
119868
Let 119860119899= [119860119871
119899 119860119880
119899] (119899 isin 119873) be a sequence of interval-
valued fuzzy numbers and
(119860119871
119899)minus(120572) = 1 + 120572
2+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
(119860119880
119899)minus(120572) =
1
2
(1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
)
(119860119871
119899)+(120572) = 4 minus 120572 (119860
119880
119899)+(120572) = 5 minus 120572
120572 isin [0 1]
(137)
From the Taylor formula we have
1198901205722
= 1 + 1205722+
1205724
2
+ sdot sdot sdot +
1205722119899
119899
+ int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 120572 isin [0 1]
(138)
Therefore for any 120572 isin [0 1] we can prove that
0 le 119860119871
minus(120572) minus (119860
119871
119899)minus(120572)
= int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905 le 119890int
1205722
0
(1205722minus 119905)
119899
119899
119889119905
= 119890 sdot
1205722119899+2
(119899 + 1)
le
119890
(119899 + 1)
(139)
0 le 119860119880
minus(120572) minus (119860
119880
119899)minus(120572)
=
1
2
int
1205722
0
(1205722minus 119905)
119899
119899
119890119905119889119905
le
119890
2
int
1205722
0
(1205722minus 119905)
119899
119899
119889119905 =
119890
2
sdot
1205722119899+2
(119899 + 1)
le
119890
2 (119899 + 1)
(140)
That is 119860 and 119860119899satisfy the hypothesis in Theorem 25
If 119891(120572) = 120572 then
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
minus(1) minus 119860
119880
minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
minus(1) minus 119860
119871
minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times [120572 (119860119880
minus(1) minus 119860
119871
minus(1)) minus (119860
119880
minus(120572) minus 119860
119871
minus(120572))] 119889120572
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 19
= int
1
0
119891 (120572) (1 minus 120572) [120572 (
119890
2
minus 119890) minus (
1
2
1198901205722
minus 1198901205722
)] 119889120572
=
119890
2
int
1
0
119891 (120572) (1 minus 120572) (1198901205722minus1minus 120572) 119889120572
gt 0
(141)
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119880
+(1) minus 119860
119880
+(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot 119860119871
+(1) minus 119860
119871
+(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 (119860119880
+(1) minus 119860
119871
+(1))
minus (119860119880
+(120572) minus 119860
119871
+(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) (120572 minus 1) 119889120572
lt 0
(142)
such that 119860 satisfies condition (iv) of Theorem 16 Further-more let
119866 (119899) = int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)] 119889120572
= int
1
0
119891 (120572) (1 minus 120572) [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
=
1
2
int
1
0
119891 (120572) (1 minus 120572) [(1 minus 120572) + (1205722minus 120572) +
1
2
(1205724minus 120572)
+ sdot sdot sdot +
1
119899
(1205722119899minus 120572)] 119889120572
(143)
It is obvious that 119866(119899) is decreasing and by (141) we have
lim119899rarrinfin
119866 (119899) = lim119899rarrinfin
int
1
0
119891 (120572) (1 minus 120572)
times [120572 ((119860119880
119899)minus(1) minus (119860
119871
119899)minus(1))
minus ((119860119880
119899)minus(120572) minus (119860
119871
119899)minus(120572))] 119889120572
= int
1
0
119891 (120572) (1 minus 120572)
times[120572 (119860119880
minus(1)minus 119860
119871
minus(1))minus(119860
119880
minus(120572) minus 119860
119871
minus(120572))]119889120572
gt 0
(144)
Therefore we can conclude that
int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119880
119899)minus(1) minus (119860
119880
119899)minus(120572)] 119889120572
minus int
1
0
119891 (120572) (1 minus 120572) [120572 sdot (119860119871
119899)minus(1) minus (119860
119871
119899)minus(120572)]119889120572 gt 0
119899 isin 119873
(145)It follows that 119860
119899satisfies condition (iv) of Theorem 16 We
denote119879 (119860) = [(119905
119871
1 119905119871
2 119905119871
3 119905119871
4) (119905119880
1 119905119880
2 119905119880
3 119905119880
4)]
119879 (119860119899) = [(119905
119871
1(119899) 119905119871
2(119899) 119905119871
3(119899) 119905119871
4(119899))
(119905119880
1(119899) 119905119880
2(119899) 119905119880
3(119899) 119905119880
4(119899))]
(119899 isin 119873)
(146)
UsingTheorem 16 (iv) together with (139) we have10038161003816100381610038161003816119905119871
1minus 119905119871
1(119899)
10038161003816100381610038161003816
= 12
100381610038161003816100381610038161003816100381610038161003816
int
1
0
120572 (1 minus 120572) [120572 ((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572))] 119889120572
100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
((119860119871
119899)minus(1) minus 119860
119871
minus(1))
minus 12int
1
0
120572 (1 minus 120572) ((119860119871
119899)minus(120572) minus 119860
119871
minus(120572)) 119889120572
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816(119860119871
119899)minus(1) minus 119860
119871
minus(1)
10038161003816100381610038161003816
+ 12int
1
0
120572 (1 minus 120572)
10038161003816100381610038161003816(119860119871
119899)minus(120572) minus 119860
119871
minus(120572)
10038161003816100381610038161003816119889120572
le
119890
(119899 + 1)
+
2119890
(119899 + 1)
=
3119890
(119899 + 1)
(147)
Similarly we can prove that10038161003816100381610038161003816119905119880
1minus 119905119880
1(119899)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
+
2119890
2 (119899 + 1)
=
3119890
2 (119899 + 1)
(148)
Combing (48) (139) and (140) we obtain10038161003816100381610038161003816119905119871
2minus 119905119871
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119871
minus(1) minus (119860
119871
119899)minus(1)
10038161003816100381610038161003816le
119890
(119899 + 1)
10038161003816100381610038161003816119905119880
2minus 119905119880
2(119899)
10038161003816100381610038161003816=
10038161003816100381610038161003816119860119880
minus(1) minus (119860
119880
119899)minus(1)
10038161003816100381610038161003816le
119890
2 (119899 + 1)
(149)
Therefore by making use of (147) (148) and (149) we get119863119868(119879 (119860) 119879 (119860119899
))
=
1
2
[(int
1
0
[120572 (119905119871
1+ (119905119871
2minus 119905119871
1) 120572)
minus(119905119871
1(119899) + (119905
119871
2(119899) minus 119905
119871
1(119899)) 120572)]
2
119889120572)
12
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
20 Journal of Applied Mathematics
+ (int
1
0
120572 [ (119905119880
1+ (119905119880
2minus 119905119880
1) 120572)
minus (119905119880
1(119899) + (119905
119880
2(119899) minus 119905
119880
1(119899)) 120572)]
2
119889120572)
12
]
=
1
2
[
[
radicint
1
0
120572((119905119871
1minus 119905119871
1(119899)) (1 minus 120572) + (119905
119871
2minus 119905119871
2(119899)) 120572)
2119889120572
+ radicint
1
0
120572((119905119880
1minus 119905119880
1(119899)) (1 minus 120572) + (119905
119880
2minus 119905119880
2(119899)) 120572)
2119889120572]
]
=
1
2
[ (
1
12
(119905119871
1minus 119905119871
1(119899))
2
+
1
6
(119905119871
1minus 119905119871
1(119899)) (119905
119871
2minus 119905119871
2(119899))
+
1
4
(119905119871
2minus 119905119871
2(119899))
2
)
12
+ (
1
12
(119905119880
1minus 119905119880
1(119899))
2
+
1
6
(119905119880
1minus 119905119880
1(119899)) (119905
119880
2minus 119905119880
2(119899))
+
1
4
(119905119880
2minus 119905119880
2(119899))
2
)
12
]
le
1
2
[radic1
6
(119905119871
1minus 119905119871
1(119899))2+
1
3
(119905119871
2minus 119905119871
2(119899))2
+radic1
6
(119905119880
1minus 119905119880
1(119899))2+
1
3
(119905119880
2minus 119905119880
2(119899))2]
lt
2119890
(119899 + 1)
(150)
It is obvious that for 119899 ge 5 we have119863119868(119879(119860) 119879(119860
119899)) lt 10
minus2For 119899 = 5 case (iv) inTheorem 16 is applicable to compute thenearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number 119860
5 and we obtain
119879 (1198605) = [(
21317
360360
163
60
3 4) (
21317
720720
163
120
4 5)]
(151)
Thenwe obtain an interval-valued trapezoidal approximation119879(1198605) with an error less than 10minus2
5 Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently a lot of methods have been presented for handlingfuzzy risk analysis problems However these researches didnot consider the risk analysis problems based on interval-valued fuzzy numbers Following we will use the approxima-tion operator presented in Section 32 to deal with fuzzy riskanalysis problems
Assume that there is a component 119860 consisting of 119899subcomponents 119860
1 1198602 119860
119899and each subcomponent is
evaluated by two evaluating items ldquoprobability of failurerdquo
Table 1 A 9-member linguistic term set (Schmucker 1984) [10]
Linguistic terms Trapezoidal fuzzy numbersAbsolutely low (0 0 0 0)Very low (0 0 002 007)Low (004 01 018 023)Fairly low (017 022 036 042)Medium (032 041 058 065)Fairly high (058 063 080 086)High (072 078 092 097)Very high (093 098 10 10)Absolutely high (10 10 10 10)
and ldquoseverity of lossrdquo We want to evaluate the probability offailure and severity of loss of component 119860 Assume that 119877
119894
denotes the probability of failure and 120596119894denotes the severity
of loss of the subcomponent 119860119894 respectively where 119877
119894and
120596119894are interval-valued fuzzy numbers and 1 le 119894 le 119899
The algorithm for dealing with fuzzy risk analysis is nowpresented as follows
Step 1 Use the fuzzy weighted mean method to integrate theevaluating values 119877
119894and 120596
119894of each subcomponent 119860
119894 where
1 le 119894 le 119899
Step 2 Transform interval-valued fuzzy numbers 119877119894and 120596
119894
into interval-valued trapezoidal fuzzy numbers 119879(119877119894) and
119879(120596119894) by means of the approximation operator 119879
Step 3 Use the interval-valued fuzzy number arithmeticoperations defined as [8] to calculate the probability of failure119877 of component 119860
119877 = [
119899
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
119899
sum
119894=1
119879 (120596119894)
= [(119903119871
1 119903119871
2 119903119871
3 119903119871
4) (119903119880
1 119903119880
2 119903119880
3 119903119880
4)]
(152)
Without a doubt 119877 is an interval-valued trapezoidal fuzzynumber
Step 4 Transform the interval-valued trapezoidal fuzzynumber 119877 into a standardized interval-valued trapezoidalfuzzy number 119877lowast
119877lowast= [(
119903119871
1
119896
119903119871
2
119896
119903119871
3
119896
119903119871
4
119896
) (
119903119880
1
119896
119903119880
2
119896
119903119880
3
119896
119903119880
4
119896
)] (153)
where 119896 = maxlceil|119903119871119895|rceil lceil|119903119880
119895|rceil 1 || denotes the absolute value
and lceilrceil denotes the upper bound and 1 le 119895 le 4
Step 5 Use the similarity measure of interval-valued fuzzynumbers introduced in [26] to calculate the similarity mea-sure of 119877lowast and each linguistic term shown in Table 1 Thelarger the similarity measure the higher the probability offailure of component 119860
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 21
Table 2 Evaluating values of the subcomponents 1198601 1198602 and 119860
3
Subcomponents 119860119894
Probability of failure 119877119894
Severity of loss 120596119894
1198601
[(03 05 08 10)2 (01 04 09 10)2] [(03 05 06 10)2 (01 04 09 10)2]1198602
[(04 08 08 10)2 (04 04 10 11)2] [(04 05 08 10)2 (04 04 10 11)2]1198603
[(03 07 08 10)2 (01 04 08 10)2] [(03 07 07 10)2 (01 04 08 10)2]
Table 3 Interval-valued trapezoidal approximation of 119877119894and 120596
119894
Subcomponents 119860119894
119879(119877119894) 119879(120596
119894)
1198601
[(
131
35
05 08
324
35
) (7435
04 09
337
35
)] [(
131
35
05 06
298
35
) (7435
04 09
337
35
)]
1198602
[(
192
35
08 08
324
35
) (1435
04 10
372
35
)] [(
153
35
05 08
324
35
) (1435
04 10
372
35
)]
1198603
[(
157
35
07 08
324
35
) (7435
04 08
324
35
)] [(
157
35
07 07
311
35
) (7435
04 08
324
35
)]
Example 27 Assume that the component 119860 consists of threesubcomponents 119860
1 1198602 and 119860
3 we evaluate the probability
of failure of the component 119860 There are some evaluatingvalues represented by interval-valued fuzzy numbers shownin Table 2 where 119877
119894denotes the probability of failure and 120596
119894
denotes the severity of loss of subcomponent 119860119894 and 1 le 119894 le
3
Step 1 Let 119891(120572) = 120572 According to Corollary 18 we obtaininterval-valued trapezoidal fuzzy numbers 119879(119877
119894) and 119879(120596
119894)
as shown in Table 3
Step 2 Calculate the probability of failure119877 of component119860By the interval-valued fuzzy number arithmetic operationsdefined as [8] we have
119877 = [
3
sum
119894=1
(119879 (119877119894) otimes 119879 (120596
119894))] ⊘
3
sum
119894=1
119879 (120596119894)
= [119879 (1198771) otimes 119879 (120596
1) oplus 119879 (119877
2) otimes 119879 (120596
2) oplus 119879 (119877
3) otimes 119879 (120596
3)]
⊘ [119879 (1205961) oplus 119879 (120596
2) oplus 119879 (120596
3)]
asymp [(0218 054 099 1959) (0085 018 204 3541)]
(154)
Step 3 Transform the interval-valued trapezoidal fuzzy num-ber 119877 into a standardized interval-valued trapezoidal fuzzynumber 119877lowast
119877lowast= [(00545 01350 02475 04898)
(00213 0045 05100 08853)]
(155)
Step 4 Calculate the similaritymeasure between the interval-valued trapezoidal fuzzy number 119877lowast and the linguistic termsshown in Table 1 we have
119878119865(119877lowast absolutely minus low) asymp 02797
119878119865(119877lowast very minus low) asymp 03131
119878119865(119877lowast low) asymp 04174
119878119865(119877lowast fairly minus low) asymp 04747
119878119865(119877lowastmedium) asymp 04748
119878119865(119877lowast fairly minus high) asymp 03445
119878119865(119877lowast high) asymp 02545
119878119865(119877lowast very minus high) asymp 01364
119878119865(119877lowast absolutely minus high) asymp 01166
(156)
It is obvious that 119878119865(119877lowastmedium) asymp 04748 is the largest
value therefore the interval-valued trapezoidal fuzzy num-ber 119877lowast is translated into the linguistic term ldquomediumrdquo Thatis the probability of failure of the component 119860 is medium
6 Conclusion
In this paper we use the 120572-level set of interval-valuedfuzzy numbers to investigate interval-valued trapezoidalapproximation of interval-valued fuzzy numbers and discusssome properties of the approximation operator includingtranslation invariance scale invariance identity nearness cri-terion and ranking invariance However Example 23 provesthat the approximation operator suggested in Section 32is not continuous Nevertheless Theorem 25 shows that theinterval-valued trapezoidal approximation has a relative goodbehavior As an application we use interval-valued trape-zoidal approximation to handle fuzzy risk analysis problemswhich provides us with a useful way to deal with fuzzy riskanalysis problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
22 Journal of Applied Mathematics
Acknowledgments
This work is supported by the National Natural ScienceFund of China (61262022) the Natural Scientific Fund ofGansu Province of China (1208RJZA251) and the Scien-tific Research Project of Northwest Normal University (noNWNU-KJCXGC-03-61)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] M BGorzalczany ldquoApproximate inferencewith interval-valuedfuzzy setsmdashan outlinerdquo in Proceedings of the Polish Symposiumon Interval and Fuzzy Mathematics pp 89ndash95 Poznan Poland1983
[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986
[4] G J Wang and X P Li ldquoThe applications of interval-valuedfuzzy numbers and interval-distribution numbersrdquo Fuzzy Setsand Systems vol 98 no 3 pp 331ndash335 1998
[5] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009
[6] B Farhadinia ldquoSensitivity analysis in interval-valued trape-zoidal fuzzy number linear programming problemsrdquo AppliedMathematical Modelling vol 38 no 1 pp 50ndash62 2014
[7] Z Y Xu S C Shang W B Qian andW H Shu ldquoA method forfuzzy risk analysis based on the new similarity of trapezoidalfuzzy numbersrdquo Expert Systems with Applications vol 37 no 3pp 1920ndash1927 2010
[8] D H Hong and S Lee ldquoSome algebraic properties and a dis-tance measure for interval-valued fuzzy numbersrdquo InformationSciences vol 148 no 1ndash4 pp 1ndash10 2002
[9] S S L Chang and L A Zadeh ldquoOn fuzzymapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics vol 2 no1 pp 30ndash34 1972
[10] K J Schmucker Fuzzy Sets Natural Language Computationsand Risk Analysis Computer Science Press 1984
[11] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[12] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[13] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[14] S Bodjanova ldquoMedian value and median interval of a fuzzynumberrdquo Information Sciences vol 172 no 1-2 pp 73ndash89 2005
[15] C XWu andMMa ldquoOn embedding problem of fuzzy numberspacemdash1rdquo Fuzzy Sets and Systems vol 44 no 1 pp 33ndash38 1991
[16] D H Hong ldquoA note on the correlation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol123 no 1 pp 89ndash92 2001
[17] G-J Wang and X-P Li ldquoCorrelation and information energyof interval-valued fuzzy numbersrdquo Fuzzy Sets and Systems vol103 no 1 pp 169ndash175 1999
[18] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquo
Computers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008
[19] S Chen and J Chen ldquoFuzzy risk analysis based on similaritymeasures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operatorsrdquo Expert Systems withApplications vol 36 no 3 pp 6309ndash6317 2009
[20] S-H Wei and S-M Chen ldquoFuzzy risk analysis based oninterval-valued fuzzy numbersrdquo Expert Systems with Applica-tions vol 36 no 2 part 1 pp 2285ndash2299 2009
[21] W Zeng and H Li ldquoWeighted triangular approximation offuzzy numbersrdquo International Journal of Approximate Reason-ing vol 46 no 1 pp 137ndash150 2007
[22] P Grzegorzewski and EMrowka ldquoTrapezoidal approximationsof fuzzy numbersrdquo Fuzzy Sets and Systems vol 153 no 1 pp115ndash135 2005
[23] S Abbasbandy and T Hajjari ldquoWeighted trapezoidal approx-imation-preserving cores of a fuzzy numberrdquo Computers ampMathematics with Applications vol 59 no 9 pp 3066ndash30772010
[24] R T Rockafellar Convex Analysis Princeton MathematicalSeries No 28 Princeton University Press Princeton NJ USA1970
[25] A I Ban and L C Coroianu ldquoDiscontinuity of the trapezoidalfuzzy number-valued operators preserving corerdquo Computers ampMathematics withApplications vol 62 no 8 pp 3103ndash3110 2011
[26] B Farhadinia and A I Ban ldquoDeveloping new similarity mea-sures of generalized intuitionistic fuzzy numbers and general-ized interval-valued fuzzy numbers from similarity measuresof generalized fuzzy numbersrdquo Mathematical and ComputerModelling vol 57 no 3-4 pp 812ndash825 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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