Research Article Multicriteria Decision-Making Approach ...downloads.hindawi.com/journals/tswj/2014/868515.pdf · So the concept of hesitant interval-valued intuitionistic fuzzy sets

Research ArticleMulticriteria Decision-Making Approach with HesitantInterval-Valued Intuitionistic Fuzzy Sets

Juan-juan Peng12 Jian-qiang Wang1 Jing Wang1 and Xiao-hong Chen1

1 School of Business Central South University Changsha 410083 China2 School of Economics and Management Hubei University of Automotive Technology Shiyan 442002 China

Correspondence should be addressed to Jian-qiang Wang jqwangqqcom

Received 24 August 2013 Accepted 24 December 2013 Published 27 March 2014

Academic Editors X-l Luo J Mula W Szeto and T Tuma

Copyright copy 2014 Juan-juan Peng et alThis 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

The definition of hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs) is developed based on interval-valued intuitionisticfuzzy sets (IVIFSs) and hesitant fuzzy sets (HFSs)Then some operations on HIVIFSs are introduced in detail and their propertiesare further discussed In addition some hesitant interval-valued intuitionistic fuzzy number aggregation operators based on t-conorms and t-norms are proposed which can be used to aggregate decision-makersrsquo information inmulticriteria decision-making(MCDM) problems Some valuable proposals of these operators are studied In particular based on algebraic and Einstein t-conorms and t-norms some hesitant interval-valued intuitionistic fuzzy algebraic aggregation operators and Einstein aggregationoperators can be obtained respectively Furthermore an approach of MCDM problems based on the proposed aggregationoperators is given using hesitant interval-valued intuitionistic fuzzy information Finally an illustrative example is provided todemonstrate the applicability and effectiveness of the developed approach and the study is supported by a sensitivity analysis anda comparison analysis

1 Introduction

Since fuzzy sets were proposed by Zadeh [1] the studieson multicriteria decision-making (MCDM) problems havemade great progress Further fuzzy sets were generalized tointuitionistic fuzzy sets (IFSs) by Atanassov [2 3] where eachelement in an IFS has a membership degree and a nonmem-bership degree between 0 and 1 respectivelyThenAtanassovand Gargov [4] proposed the notion of interval-valuedintuitionistic fuzzy sets (IVIFSs) which are the extension ofIFSs where the membership degree and nonmembershipdegree of an element in an IVIFS are respectively representedby intervals in [0 1] rather than crisp values between 0and 1 In recent years many researchers have studied thetheory of IVIFSs and applied it to various fields [5ndash8] Forinstance Atanassov [9] introduced the operators of IVIFSsLee [10] proposed a method for ranking interval-valued

intuitionistic fuzzy numbers (IVIFNs) for fuzzy decision-making problems Lee [11] provided an enhanced MCDMmethod of machine design schemes under the interval-valued intuitionistic fuzzy environment Li [12] proposed aTOPSIS based nonlinear-programming method for MCDMproblems with IVIFSs Park et al [13] extended the TOPSISmethod to solve group MCDM problems in interval-valuedintuitionistic fuzzy environment in which all the preferenceinformation provided by decision-makers is presented asIVIFNs Chen et al [14] developed an approach to tacklegroup MCDM problems in the context of IVIFSs Nayagamand Sivaraman [15] introduced a method for ranking IVIFSsand compared it to other methods by means of numericalexamples Chen et al [16] presented a MCDMmethod basedon the proposed interval-valued intuitionistic fuzzy weightedaverage (IVIFWA) operator Meng et al [17] developedan induced generalized interval-valued intuitionistic fuzzy

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 868515 22 pageshttpdxdoiorg1011552014868515

2 The Scientific World Journal

hybrid Shapley averaging (GIVIFHSA) operator and appliedit to MCDM problems

Hesitant fuzzy sets (HFSs) another extension of tradi-tional fuzzy sets provide a useful reference for our studyunder hesitant fuzzy environment HFSs were first intro-duced by Torra and Narukawa [18] and they permit themembership degrees of an element to be a set of severalpossible values between 0 and 1 HFSs are highly useful inhandling the situations where people have hesitancy in pro-viding their preferences over objects in the decision-makingprocess Some aggregation operators of HFSs were studiedand applied to decision-making problems [19ndash21] Then thecorrelation coefficients of HFSs the distance measures andcorrelation measures of HFSs were discussed [22ndash24] basedon which Peng et al [25] presented a generalized hesitantfuzzy synergetic weighted distance measure Zhang and Wei[26] developed the E-VIKOR method and TOPSIS methodto solve MCDM problems with hesitant fuzzy informationZhang [27] developed a wide range of hesitant fuzzy poweraggregation operators for hesitant fuzzy information Chen etal [28] generalized the concept of HFSs to hesitant interval-valued fuzzy sets (HIVFSs) in which themembership degreesof an element to a given set are not exactly defined butdenoted by several possible interval values Wei [29] definedHIVFSs and some hesitant interval-valued fuzzy aggregationoperators Wei and Zhao [30] developed some Einsteinoperations on HIVFSs and the induced hesitant interval-valued fuzzy Einstein aggregation (HIVFEA) operators andapplied them to MCDM problems Zhu et al [31] defineddual HFSs (DHFSs) in terms of two functions that returntwo sets ofmembership degrees and nonmembership degreesrather than crisp numbers in HFSs If the idea of dual HFSsis used from a new perspective then another extension ofHFSs may be defined in terms of one function that theelement of HFSs returns a set of IFSs which are called hes-itant intuitionistic fuzzy sets (HIFSs) But decision-makersusually cannot estimate criteria values of alternatives withexact numerical values when the information is not knownpreciselyTherefore interval values in fuzzy sets can representit better than specific numbers such as interval-valued fuzzysets (IVFSs) and IVIFSs Furthermore although the theoriesof IVIFSs and HFSs have been developed and generalizedthey cannot deal with all sorts of uncertainties in differentreal problems For example when we ask the opinion of anexpert about a certain statement he or she may answer thatthe possibility that the statement is true is [01 02] and thatthe statement is false is [04 05] or the possibility that thestatement is true is [05 06] and that the statement is false is[03 05]This issue is beyond the scope of IVFSs and IVIFSsTherefore some new theories are required

So the concept of hesitant interval-valued intuitionisticfuzzy sets (HIVIFSs) is developed in this paper Comparingto the existing fuzzy sets mentioned above HIVIFSs are anew extension of HFSs which support a more flexible andsimpler approach when decision-makers provide their deci-sion information in a hesitant interval-valued intuitionisticfuzzy environment Furthermore IVIFSsHFSsHIVFSs andHIFSs are all the special cases of HIVIFSs

In this paper HFSs are extended based on IVIFSsHIVIFSs are defined and their properties and applicationsare also discussed Thus the rest of this paper is organized asfollows In Section 2 the definitions and properties of IVIFSsand HFSs are briefly reviewed In Section 3 the notion ofHIVIFSs is proposed and the operations and properties ofHIVIFSs based on 119905-conorms and 119905-norms are discussed InSection 4 some hesitant interval-valued intuitionistic fuzzynumber aggregation operators are developed and applied toMCDMproblems Section 5 gives an example to illustrate theapplication of the developedmethod Finally the conclusionsare drawn in Section 6

2 Preliminaries

In this section some basic concepts and definitions related toHIVIFSs are introduced including interval numbers IVIFSsand HFSs These will be utilized in the subsequent analysis

21 Interval Numbers and Their Operations

Definition 1 (see [32ndash34]) Let 119886 = [119886119871 119886119880] = 119909 | 119886

119871le 119909 le

119886119880 then 119886 is called an interval number In particular if 0 le

119886119871le 119909 le 119886

119880 then 119886 is reduced to a positive interval numberConsider any two interval fuzzy numbers 119886 = [119886

119871 119886119880]

and = [119887119871 119887119880] and their operations are defined as follows

(1) 119886 = hArr 119886119871= 119887119871 119886119880= 119887119880

(2) 119886 + = [119886119871+ 119887119871 119886119880+ 119887119880]

(3) 119886 minus = [119886119871minus 119887119880 119886119880minus 119887119871]

(4) 119886 times = [min119886119871119887119871 119886119871119887119880 119886119880119887119871 119886119880119887119880max119886119871119887119871 119886119871119887119880 119886119880119887119871 119886119880119887119880]

(5) 119896119886 = [119896119886119871 119896119886119880] 119896 gt 0

22 IVIFSs Atanassov first proposed IFSs being enlarge-ment and development of Zadehrsquos fuzzy sets IFSs contain thedegree of nonmembership which makes it possible for us tomodel unknown informationThe definition of IVIFSs givenby Atanassov and Gargov [4] is shown as follows

Definition 2 (see [4]) Let 119863[0 1] be the set of all closedsubintervals of the interval [0 1] Let 119883 be a given set and119883 = An IVIFS in 119883 is an expression given by 119860 =

⟨119909 120583119860(119909) ]119860(119909)⟩ | 119909 isin 119883 where 120583

119860 119883 rarr 119863[0 1] ]

119860rarr

119863[0 1] with the condition 0 lt sup119909120583119860(119909) + sup

119909]119860(119909) le

1 The intervals 120583119860(119909) and ]

119860(119909) denote the degree of

belongingness and nonbelongingness of the element 119909 to theset 119860 respectively Thus for each 119909 isin 119883 120583

119860(119909) and ]

119860(119909)

are closed intervals whose lower and upper boundaries aredenoted by 120583119871

119860(119909) 120583119880

119860(119909) and ]119871

119860(119909) ]119880119860(119909) respectively and

then

119860 = ⟨119909 [120583119871

119860(119909) 120583

119880

119860(119909)] []119871

119860(119909) ]119880

119860(119909)]⟩ | 119909 isin 119883 (1)

where 0 lt 120583119880

119860(119909) + ]119880

119860(119909) le 1 120583119871

119860(119909) ge 0 ]119871

119860(119909) ge 0 For each

element 119909 the hesitancy degree can be calculated as followsΠ119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) = [1 minus 120583

119880

119860(119909) minus ]119880

119860(119909) 1 minus 120583

119871

119860(119909) minus

The Scientific World Journal 3

]119871119860(119909)] The set of all IVIFSs in 119883 is denoted by IVIFS(119883)

An interval-valued intuitionistic fuzzy number (IVIFN) isdenoted by 119860 = ([119886 119887] [119888 119889]) and the degree of hesitance isdenoted by [119890 119891] = [1 minus 119886 minus 119889 1 minus 119886 minus 119888] for convenience

Definition 3 (see [16]) Let 119894= ⟨[119886119894 119887119894] [119888119894 119889119894]⟩ (1 le 119894 le 119899)

be a collection of IVIFNs and let 119908119894(1 le 119894 le 119899) be the crisp

values where 119894= ⟨[119886119894 119887119894] [119888119894 119889119894]⟩ = [[119886

119894 119887119894] [1 minus 119889

119894 1 minus 119888

119894]]

0 le 119886119894le 119887119894le 1 0 le 119888

119894le 119889119894le 1 0 le 119887

119894+ 119889119894le 1 and

1 le 119894 le 119899 and then the interval-valued intuitionistic fuzzyweighted average operator can be defined as follows

IVIFWA119908(1 2

119899)

=sum119899

119894=1[[119886119894 119887119894] [1 minus 119889

119894 1 minus 119888119894]] times 119908

119894

sum119899

119894=1119908119894

= [[sum119899

119894=1119886119894119908119894

sum119899

119894=1119908119894

sum119899

119894=1119887119894119908119894

sum119899

119894=1119908119894

]

[sum119899

119894=1(1 minus 119889

119894) 119908119894

sum119899

119894=1119908119894

sum119899

119894=1(1 minus 119888119894) 119908119894

sum119899

119894=1119908119894

]]

= [[119886 ] [119888 119889]]

(2)

where IVIFWA119908(1 2

119899) = [[119886 ] [119888 119889]] = ⟨[119886 ]

[1 minus 119889 1 minus 119888]⟩ is an interval-valued intuitionistic fuzzy value119886 119888 and 119889 are calculated by the Karnik-Mendel algorithms[35]

Example 4 Let 1

= ⟨[03 06] [01 02]⟩ and 2

=

⟨[04 06] [01 03]⟩ be two IVIFNs and 1199081= 03 119908

2= 05

According to (2)

IVIFWA119908(1 2)

= [[03 times 03 + 04 times 05

03 + 0506 times 03 + 06 times 06

03 + 05]

[(1 minus 02) times 03 + (1 minus 03) times 05

03 + 05

(1 minus 01) times 03 + (1 minus 01) times 05

03 + 05]]

= [[03625 06750] [07375 09000]]

= ⟨[03625 06750] [1 minus 09000 1 minus 07375]⟩

= ⟨[03625 06750] [01000 02625]⟩

(3)

Definition 5 (see [36]) Let = ⟨[119886 119887] [119888 119889]⟩ be an IVIFNand then an accuracy function 119871() can be defined as follows

119871 () =119886 + 119887 minus 119889 (1 minus 119887) minus 119888 (1 minus 119886)

2 (4)

where 119871() isin [minus1 1] and 1 le 119894 le 119899

Definition 6 (see [36]) Let 1and 2be two IVIFNs and then

the following comparison method must exist

(1) If 119871(1) gt 119871(

2) then

1gt 2

(2) If 119871(1) = 119871(

2) then

1= 2

Example 7 Let 1= ⟨[04 06] [01 02]⟩ and

2= ⟨[05

06] [02 03]⟩ be two IVIFNs According to (4) 119871(1) =

(04 + 06 minus 02 times (1 minus 06) minus 01 times (1 minus 04))2 = 043 and119871(2) = 044 119871(

2) gt 119871(

1) can be obtained so the optimal

one(s) is 2

Definition 8 (see [37ndash39]) A function 119879 [0 1] times [0 1] rarr

[0 1] is called 119905-norm if it satisfies the following conditions

(1) for all 119909 isin [0 1] 119879(1 119909) = 119909(2) for all 119909 119910 isin [0 1] 119879(119909 119910) = 119879(119910 119909)(3) for all 119909 119910 119911 isin [0 1] 119879(119909 119879(119910 119911)) = 119879(119879(119909 119910) 119911)(4) if 119909 le 119909

1015840 119910 le 119910

1015840 then 119879(119909 119910) le 119879(1199091015840 1199101015840)


[0 1] is called 119905-conorm if it satisfies the following conditions


1015840 119910 le 119910

1015840 then 119878(119909 119910) le 119878(1199091015840 1199101015840)

There are some well-known Archimedean 119905-conorms and 119905-norms [39 40]

(1) Let 119896(119905) = minus In 119905 119897(119905) = minus In(1 minus 119905) 119896minus1(119905) = 119890minus119905

119897minus1(119905) = 1 minus 119890

minus119905 and then algebraic 119905-conorms and 119905-norms are obtained as follows119879(119909 119910) = 119909119910 119878(119909 119910) =1 minus (1 minus 119909)(1 minus 119910)

(2) Let 119896(119905) = In((2 minus 119905)119905) 119897(119905) = In((2 minus (1 minus 119905))(1 minus

119905)) 119896minus1(119905) = 2(119890119905+ 1) 119897minus1(119905) = 1 minus (2(119890

119905+ 1)) and

then Einstein 119905-conorms and 119905-norms are obtained asfollows 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) 119878(119909 119910) =(119909 + 119910)(1 + 119909119910)

(3) Let 119896(119905) = In((120574 minus (1 minus 120574)119905)119905) 120574 gt 0 119897(119905) = In((120574 minus(1 minus 120574)(1 minus 119905))(1 minus 119905)) 119896minus1(119905) = 120574(119890

119905+ 120574 minus 1) 119897minus1(119905) =

1minus(120574(119890119905+120574minus1)) and thenHamacher 119905-conorms and

119905-norms are obtained as follows

119879 (119909 119910) =119909119910

120574 + (1 minus 120574) (119909 + 119910 minus 119909119910) 120574 gt 0

119878 (119909 119910) =119909 + 119910 minus 119909119910 minus (1 minus 120574) 119909119910

1 minus (1 minus 120574) 119909119910 120574 gt 0

(5)

Based on the Archimedean 119905-conorms and 119905-normssome operations of IVIFSs are discussed as follows

Definition 10 Let = ⟨[119886 119887] [119888 119889]⟩ 1= ⟨[1198861 1198871] [1198881 1198891]⟩

2= ⟨[1198862 1198872] [1198882 1198892]⟩ be three IVIFNs 120582 ge 0 and then their

operations could be defined as follows [19 41ndash43]

(1) 120582 = ⟨[119896minus1(120582119896(119886)) 119896

minus1(120582119896(119887))] [119897

minus1(120582119897(119888))

119897minus1(120582119897(119889))]⟩

(2) 120582 = ⟨[119897minus1(120582119897(119886)) 119897

minus1(120582119897(119887))] [119896

minus1(120582119896(119888))

119896minus1(120582119896(119889))]⟩ 120582 gt 0

(3) 1oplus 2

= ⟨[119897minus1(119897(1198861) + 119897(119886

2)) 119897minus1(119897(1198871) + 119897(119887

2))]

[119896minus1(119896(1198881) + 119896(119888

2)) 119896minus1(119896(1198891) + 119896(119889

2))]⟩


(4) 119886 otimes = ⟨[119896minus1(119896(1198861) + 119896(119886

2)) 119896minus1(119896(1198871) + 119896(119887

2))]

[119897minus1(119897(1198881) + 119897(119888

2)) 119897minus1(119897(1198891) + 119897(119889

2))]⟩

Here 119897(119905) = 119896(1 minus 119905) and 119896 [0 1] rarr [0infin) is a strictlydecreasing function

23 HFSs

Definition 11 (see [44]) Let119883 be a universal set and aHFS on119883 is in terms of a function that when applied to119883will returna subset of [0 1] which can be represented as follows

119864 = ⟨119909 ℎ119864(119909)⟩ | 119909 isin 119883 (6)

where ℎ119864(119909) is a set of values in [0 1] denoting the possible

membership degrees of the element 119909 isin 119883 to the set 119864 ℎ119864(119909)

is called a hesitant fuzzy element (HFE) [23] and 119867 is theset of all HFEs It is noteworthy that if 119883 contains only oneelement then 119864 is called a hesitant fuzzy number (HFN)briefly denoted by 119864 = ℎ

119864(119909) The set of all hesitant fuzzy

numbers is represented as HFNSTorra [44] defined some operations on HFNs and Xia

and Xu [19 22] defined some new operations on HFNs andthe score function

Definition 12 (see [43]) Let ℎ ℎ1 and ℎ

2be three HFNs 120582 ge

0 and then four operations are defined as follows

(1) ℎ120582 = ⋃120574isinℎ

119896minus1(120582119896(120574))

(2) 120582ℎ = ⋃120574isinℎ

119897minus1(120582119897(120574))

(3) ℎ1oplus ℎ2= ⋃1205741isinℎ11205742isinℎ2

119897minus1(119897(1205741) + 119897(120574

2))

(4) ℎ1otimes ℎ2= ⋃1205741isinℎ11205742isinℎ2

119896minus1(119896(1205741) + 119896(120574

2))


Definition 13 (see [19]) Let ℎ isin HFNs and 119904(ℎ) =

(1ℎ)sum120574isinℎ

120574 is called the score function of ℎ where ℎ isthe number of elements in ℎ For two HFNs ℎ

1and ℎ

2 if

119904(ℎ1) gt 119904(ℎ

2) then ℎ

1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2

Example 14 Let ℎ1= 03 05 06 ℎ

2= 04 07 be two

HFNs According to Definition 13 119904(ℎ1) = (13)times (03+05+

06) = 04667 119904(ℎ2) = 055 119904(ℎ

2) gt 119904(ℎ

1) so ℎ

2gt ℎ1

Furthermore Torra and Narukawa [18 44] proposed anaggregation principle for HFEs

Definition 15 (see [18 44]) Let 119864 = ℎ1 ℎ2 ℎ

119899 be a set of

119899 HFEs let 120599 be a function on 119864 and let 120599 [0 1]119899 rarr [0 1]and then

120599119864= ⋃120574isinℎ1timesℎ2timessdotsdotsdottimesℎ

119899

120599 (120574) (7)

3 HIVIFSs and Their Operations

HFSs are the extension of traditional fuzzy sets and theirmembership degree of an element is a set of several possiblevalues between 0 and 1 In some cases decision-makers

usually cannot estimate criteria values of alternatives with anexact numerical value when the information is not preciselyknown Therefore interval values in fuzzy sets can representit better than specific numbers such as IVFSs and IVIFSsFurthermore IVIFSs could describe the object being ldquoneitherthis nor thatrdquo and the membership degree and nonmember-ship degree of IVIFSs are interval values respectively Thusprecise numerical values in HFSs can be replaced by IVIFSswhich are more flexible in the real world and this is what thissection will solve

Definition 16 Assume that 119883 is a finite universal set AHIVIFS 119860 in119883 is an object in the following form

119864 = ⟨119909119867119864 (119909)⟩ | 119909 isin 119883 (8)

where 119867119864(119909) is a finite set of values in IVIFSs denoting the

possiblemembership degrees andnonmembership degrees ofthe element 119909 isin 119883 to the set 119864

Based on the definition given above

119867119864 (119909) =

119899(119867119864(119909))

⋃119894=1

⟨[120583119871

119864119894

(119909) 120583119880

119864119894

(119909)] []119871119864119894

(119909) ]119880119864119894

(119909)]⟩

(9)

where 0 le 120583119871

1198641

(119909) le 120583119880

1198641

(119909) le 120583119871

1198642

(119909) le 120583119880

1198642

(119909) le sdot sdot sdot

120583119871

119899(119867119864(119909))

(119909) le 120583119880

119899(119867119864(119909))

(119909) le 1 0 le 120583119880

119864119894

(119909) + ]119880119864119894

(119909) le 1120583119871

119864119894

(119909) ge 0 ]119871119864119894

(119909) ge 0 and 119899(119867119864(119909)) ge 1 Actually HIVIFSs

have several possible membership degrees taking the formof IVIFSs instead of FSs in HFSs If 119899(119867

119864(119909)) = 1 then

the HIVIFS is reduced to an IVIFS if 120583119871119864119894

(119909) = 120583119880

119864119894

(119909) (119894 =

1 2 119899(119867119864(119909))) and ]119871

119864119894

(119909) = ]119880119864119894

(119909) = 0 (119894 = 1 2

119899(119867119864(119909))) then the HIVIFS is reduced to a HFS if 120583119871

119864119894

(119909) =

120583119880

119864119894

(119909) (119894 = 1 2 119899(119867119864(119909))) or ]119871

119864119894

(119909) = ]119880119864119894

(119909) (119894 =

1 2 119899(119867119864(119909))) then the HIVIFS is reduced to a HIVFS

if 120583119871119864119894

(119909) = 120583119880

119864119894

(119909) (119894 = 1 2 119899(119867119864(119909))) and ]119871

119864119894

(119909) =

]119880119864119894

(119909) (119894 = 1 2 119899(119867119864(119909))) then the HIVIFS is reduced

to a HIFS Furthermore 119867119864(119909) is called a hesitant interval-

valued intuitionistic fuzzy element (HIVIFE) and 119864 is theset of all HIVIFEs In particular if 119883 has only one element⟨119909119867119864(119909)⟩ is called a hesitant interval-valued intuitionistic

fuzzy number (HIVIFN) briefly denoted by

119867119864=

119899(119867119864)

⋃119894=1

⟨[119886119894 119887119894] [119888119894 119889119894]⟩ (10)

The set of all HIVIFNs is denoted by HIVIFNS

Definition 17 Let119860 isin HIVIFS(119883)119860 = ⟨119909119867119860(119909)⟩ | 119909 isin 119883

and for all 119909 isin 119883 Π119860(119909) = ⋃

119899(119867119860(119909))

119894=1[1minus120583

119880

119860119894

(119909)minus]119880119860119894

(119909) 1minus

120583119871

119860119894

(119909) minus ]119871119860119894

(119909)] ThenΠ119860(119909) is called the hesitant interval-

valued intuitionistic index of 119909

Example 18 Let 119883 = 1199091 1199092 and let 119860 = ⟨119909

1 ⟨[03 04]

[01 02]⟩ ⟨04 02⟩⟩ ⟨1199092 ⟨[05 06][02 04]⟩⟩ be a

HIVIFS and then Π119860(1199091) = [04 06] 04 Π

119860(1199092) = [0

03] Thus Π119860(119909) = ⟨119909

1 [04 06] 04⟩ ⟨119909

2 [0 03]⟩


The operations of HIVIFNs are defined as follows

Definition 19 Let1198671= ⋃119899(1198671)

1198941=1

⟨[1198861198941

1198871198941

] [1198881198941

1198891198941

]⟩ and1198672=

⋃119899(1198672)

1198942=1

⟨[1198861198942

1198871198942

] [1198881198942

1198891198942

]⟩ be two HIVIFNs 120582 ge 0 and fouroperations are defined as follows

(1) 1205821198671= ⋃119899(1198671)

1198941=1

⟨[119897minus1(120582119897(1198861198941

)) 119897minus1(120582119897(1198871198941

))][119896minus1(120582119896(1198881198941

)) 119896minus1(120582119896(119889

1198941

))]⟩

(2) (1198671)120582= ⋃119899(1198671)

119894=1⟨[119896minus1(120582119896(1198861198941

)) 119896minus1(120582119896(1198871198941

))][119897minus1(120582119897(1198881198941

)) 119897minus1(120582119897(1198891198941

))]⟩

(3) 1198671oplus1198672= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119897minus1(119897(1198861198941

)+119897(1198861198942

)) 119897minus1(119897(1198871198941

)+

119897(1198871198942

))] [119896minus1(119896(1198881198941

) + 119896(1198881198942

)) 119896minus1(119896(1198891198941

) + 119896(1198891198942

))]⟩

(4) 1198671otimes 1198672

= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119896minus1(119896(1198861198941

) + 119896(1198861198942

))

119896minus1(119896(1198871198941

) + 119896(1198871198942

))] [119897minus1(119897(1198881198941

) + 119897(1198881198942

)) 119897minus1(119897(1198891198941

) +

119897(1198891198942

))]⟩


Example 20 Let1198671= ⟨[01 03] [02 04]⟩ ⟨[02 03] [03

04]⟩ and1198672= ⟨[03 04] [02 03]⟩ be two HIVIFNs and

119896(119909) = minus In119909 119896minus1(119909) = 119890minus119909 119897(119909) = minus In(1 minus 119909) 119897minus1(119909) =

1 minus 119890minus119909 and 120582 = 2 The following can be calculated

(1) 21198671

= ⟨[1 minus 119890minus2(minus log(1minus01))

1 minus 119890minus2(minus log(1minus03))

]

[119890minus2(minus log 02)

119890minus2(minus log 04)

]⟩ ⟨[1 minus 119890minus2(minus log(1minus02))

1 minus

119890minus2(minus log(1minus03))

] [119890minus2(minus log 03)


]⟩ = ⟨[019

051] [004 016]⟩ ⟨[036 051] [009016]⟩

(2) (1198671)2= ⟨[001 009] [036 064]⟩ ⟨[004 009]

[051 064]⟩

(3) 1198671oplus 1198672= ⟨[037 058] [004 012]⟩ ⟨[044 058]

[006 012]⟩

(4) 1198671otimes 1198672= ⟨[003 012] [036 058]⟩ ⟨[006 012]

[044 058]⟩

Theorem 21 Let1198671 1198672 1198673isin 119867119868119881119868119865119873119878 120582 120582

1 1205822gt 0 and

then

(1) 1198671oplus 1198672= 1198672oplus 1198671

(2) 1198671otimes 1198672= 1198672otimes 1198671

(3) 1205821198671oplus 1205821198672= 120582(119867

1oplus 1198672)

(4) (1198671)120582otimes (1198672)120582= (1198671otimes 1198672)120582

(5) (1198671oplus 1198672) oplus 1198673= 1198671oplus (1198672oplus 1198673)

(6) (1198671otimes 1198672) otimes 1198673= 1198671otimes (1198672otimes 1198673)

(7) ((1198671)1205821)1205822 = (119867

1)12058211205822

Proof According to Definition 19 it is clear that (1) (2) (5)and (6) are obvious (3) (4) and (7)will be proved as follows

(3) 1205821198671oplus 1205821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(120582119897 (119886

1198941

) + 120582119897 (1198861198942

))

119897minus1(120582119897 (119887

1198941

) + 120582119897 (1198871198942

))]

[119896minus1(120582119896 (119888

1198941

) + 120582119896 (1198881198942

))

119896minus1(120582119896 (119889

1198941

) + 120582119896 (1198891198942

))]⟩

=

119899(1198671)

⋃119894=1

119899(1198672)

⋃119895=1

⟨[119897minus1(120582 (119897 (119886

1198941

) + 119897 (1198861198942

)))

119897minus1(120582 (119897 (119887

1198941

) + 119897 (1198871198942

)))]

[119896minus1(120582 (119896 (119888

1198941

) + 119896 (1198881198942

)))

119896minus1(120582 (119896 (119889

1198941

) + 119896 (1198891198942

)))]⟩

= 120582 (1198671oplus 1198672)

(4) (1198671)120582otimes (1198672)120582

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582119896 (119886

1198941

) + 120582119896 (1198861198942

))

119896minus1(120582119896 (119887

1198941

) + 120582119896 (1198871198942

))]

[119897minus1(120582119897 (1198881198941

) + 120582119897 (1198881198942

))

119897minus1(120582119897 (119889

1198941

) + 120582119897 (1198891198942

))]⟩

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582 (119896 (119886

1198941

) + 119896 (1198861198942

)))

119896minus1(120582 (119896 (119887

1198941

) + 119896 (1198871198942

)))]

[119897minus1(120582 (119897 (119888

1198941

) + 119897 (1198881198942

)))

119897minus1(120582 (119897 (119889

1198941

) + 119897 (1198891198942

)))]⟩

= (1198671otimes 1198672)120582

(7) ((1198671)1205821)1205822

=

119899(119867)

⋃1198941=1

⟨[119896minus1(1205822119896 (119896minus1(1205821119896 (1198861198941

))))

119896minus1(1205822119896 (119896minus1(1205821119896 (1198871198941

))))]

[119897minus1(1205822119897 (119897minus1(1205821119897 (1198881198941

))))


119897minus1(1205822119897 (119897minus1(1205821119897 (1198891198941

))))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058221205821119896 (1198861198941

))

119896minus1(12058221205821119896 (1198871198941

))]

[119897minus1(12058221205821119897 (1198881198941

))

119897minus1(12058221205821119897 (1198891198941

))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058211205822119896 (1198861198941

))

119896minus1(12058211205822119896 (1198871198941

))]

[119897minus1(12058211205822119897 (1198881198941

))

119897minus1(12058211205822119897 (1198891198941

))]⟩

= (1198671)12058211205822

(11)The proof is completed

Based on Definitions 5 6 and 13 the ranking method forHIVIFNs is defined as follows

Definition 22 Let 119867 isin HIVIFNs 119878(119867) = (1119867)sum120574isin119867

120574 iscalled the score function of 119867 where 119867 is the number ofthe interval-valued intuitionistic fuzzy values in 119867 For twoHIVIFNs 119867

1and 119867

2 if 119878(119867

1) gt 119878(119867

2) then 119867

1gt 1198672 if

119878(1198671) = 119878(119867

2) then119867

1= 1198672

Note that 119878(1198671) and 119878(119867

2) could be compared by utilizing

Definitions 5 and 6

Example 23 Let1198671= ⟨[03 04] [01 02]⟩ ⟨[03 05] [02


then

119878 (1198671) =

1

2times ⟨[03 + 03 04 + 05] [01 + 02 02 + 04]⟩

= ⟨[030 045] [015 030]⟩

119878 (1198672) = ⟨[03 04] [02 03]⟩

(12)According to Definitions 5 and 6

119871 (119878 (1198671))

=030 + 045 minus 030 times (1 minus 045) minus 015 times (1 minus 030)

2

= 024

119871 (119878 (1198672))


2

= 019

(13)

Hence 119878(1198671) gt 119878(119867

2) which indicates that 119867

1is preferred

to1198672

4 HIVIFN Aggregation Operators and TheirApplications in MCDM Problems

In this section HIVIFN aggregation operators are proposedand some properties of these operators are discussed Inparticular some hesitant interval-valued intuitionistic fuzzyalgebraic aggregation operators are proposed based on alge-braic 119905-conorms and 119905-norms Then how to utilize theseoperators to MCDM problems is discussed as well

41 HIVIFN Aggregation Operators

Definition 24 Let 119867119895(119895 = 1 2 119899) be a collection of

HIVIFNs and HIVIFNWA HIVIFNS119899 rarr HIVIFNS andthen

HIVIFNWA119908(1198671 1198672 119867

119899) =

119899

⨁119895=1

119908119895119867119895 (14)

The HIVIFNWA operator is called the HIVIFN weightedaveraging operator of dimension 119899 where 119908 = (119908

1 1199082

119908119899) is the weight vector of 119867

119895(119895 = 1 2 119899) with 119908

119895ge

0 (119895 = 1 2 119899) and sum119899119895=1

119908119895= 1

Theorem 25 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2

119899) be a collection of HIVIFNs and let119908 = (1199081 1199082 119908

119899)

be the weight vector of 119867119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 =

1 2 119899) and sum119899119895=1

119908119895= 1 Then the aggregated result using

the HIVIFNWA operator is also a HIVIFN and

119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(15)

Proof By using mathematical induction on 119899 we have thefollowing


(1) For 119899 = 2 since

11990811198671=

119899(1198671)

⋃1198941=1

⟨[119897minus1(1199081119897 (1198861198941

)) 119897minus1(1199081119897 (1198871198941

))]

[119896minus1(1199081119896 (1198881198941

)) 119896minus1(1199081119896 (1198891198941

))]⟩

11990821198672=

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199082119897 (1198861198942

)) 119897minus1(1199082119897 (1198871198942

))]

[119896minus1(1199082119896 (1198881198942

)) 119896minus1(1199082119896 (1198891198942

))]⟩

(16)

the following can be obtained

HIVIFNWA119908(1198671 1198672)

= 11990811198671oplus 11990821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

))]⟩

(17)

(2) If (15) holds for 119899 = 119896 then

HIVIFNWA119908(1198671 1198672 119867

119896)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119896119897 (119886119894119896

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119896119897 (119887119894119896

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119896119896 (119888119894119896

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119896119896 (119889119894119896

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[

[

119897minus1(

119896

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119896

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(18)

When 119899 = 119896 + 1 in terms of (1) and (4) in Definition 19

HIVIFNWA119908(1198671 1198672 119867

119896 119867119896+1

)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(119897 (119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

)

+ sdot sdot sdot + 119908119896119897 (119886119894119896

)))

+ 119908119896+1

119897 (119886119894119896+1

))

119897minus1(119897 (119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

)

+ sdot sdot sdot + 119908119896119897 (119887119894119896

)))

+ 119908119896+1

119897 (120583119887119894119896+1

))]

[119896minus1(119896 (119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

)

+ sdot sdot sdot + 119908119896119896 (119888119894119896

)))

+ 119908119896+1

119896 (119888119894119896+1

))

119896minus1(119896 (119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot + 119908119896119896 (119889119894119896

)))

+ 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot +

119908119896119897 (119886119894119896

) + 119908119896+1

119897 (119886119894119896+1

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942


119908119896119897 (119887119894119896

) + 119908119896+1

119897 (119887119894119896+1

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942


119908119896119896 (119888119894119896

) + 119908119896+1

119896 (119888119894119896+1

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942


119908119896119896 (119889119894119896

) + 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[

[

119897minus1(

119896+1

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896+1

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896+1

sum119895=1

119908119895119896 (119888119894119895

))


119896minus1(

119896+1

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(19)

that is (15) holds for 119899 = 119896+1 thus (15) holds for all 119899 Then

HIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119899119897 (119886119894119899

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119899119897 (119887119894119899

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119899119896 (119888119894119899

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119899119896 (119889119894119899

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(20)


HIVIFNs HIVIFNWG HIVIFNS119899 rarr HIVIFNS and then

HIVIFNWG119908(1198671 1198672 119867

119899) =

119899

⨂119895=1

(119867119895)119908119895

(21)

The HIVIFNWG operator is called the HIVIFN weightedgeometric operator of dimension 119899 and119908 = (119908

1 1199082 119908

119899)

is the weight vector of 119867119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 =

1 2 119899) and sum119899119895=1

119908119895= 1

Similarly the following theorems can be obtained

Theorem 27 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)

be the weight vector of119860119895(119895 = 1 2 119899) with119908

119895ge 0 (119895 = 1

2 119899) andsum119899119895=1

119908119895= 1Then the aggregated result using the

HIVIFNWG operator is also a HIVIFN and

HIVIFNWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119896=1

⟨[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119886119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119887119894119895

))]

]

[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119888119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119889119894119895

))]

]

⟩

(22)


HIVIFNsHIVIFNWAAHIVIFNS119899 rarr HIVIFNS and then

HIVIFNWAA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)2

)

12

(23)

The HIVIFNWAA operator is called the HIVIFN weightedarithmetic averaging operator of dimension 119899 where 119908 =

(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 29 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1


the HIVIFNWAA operator is also a HIVIFN and

119867119868119881119868119865119873119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119886

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119888

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119889

119894119895

))))))]

]

⟩

(24)



HIVIFNsHIVIFNWAGHIVIFNS119899 rarr HIVIFNS and then

HIVIFNWAG119908(1198671 1198672 119867

119899) =

1

2(

119899

⨂119895=1

(2119867119895)119908119895

)

(25)

The HIVIFNWAG operator is called the HIVIFN weightedarithmetic geometric operator of dimension 119899 where 119908 =

(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 31 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =

1 2 119899) be a collection of HIVIFNs and let 119908 =

(1199081 1199082 119908

119899) be the weight vector of 119860

119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 Then the

aggregated result using the HIVIFNWAG operator is also aHIVIFN and

119867119868119881119868119865119873119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119886

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119888

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119889

119894119895

))))))]

]

⟩

(26)


HIVIFNsGHIVIFNWAHIVIFNS119899 rarr HIVIFNS and then

GHIVIFNWA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)120582

)

1120582

(27)

TheGHIVIFNWAoperator is called the generalizedHIVIFNweighted averaging operator of dimension 119899 where 119908 =

(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1

the GHIVIFNWA operator is reduced to the HIVIFNWAoperator If 120582 = 2 the GHIVIFNWA operator is reduced tothe HIVIFNWAA operator

Theorem 33 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 120582 gt 0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Then

the aggregated result using the GHIVIFNWA operator is also aHIVIFN and

119866119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

(28)


HIVIFNs GHIVIFNWG HIVIFNS119899 rarr HIVIFNS andthen

GHIVIFNWG119908(1198671 1198672 119867

119899) =

1

120582(

119899

⨂119895=1

(120582119867119895)119908119895

)

(29)

TheGHIVIFNWGoperator is called the generalizedHIVIFNweighted geometric operator of dimension 119899 where 119908 =

(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1

the GHIVIFNWG operator is reduced to the HIVIFNWGoperator If 120582 = 2 then theGHIVIFNWGoperator is reducedto the HIVIFNWAG operator


Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)

be the weight vector of 119860119895(119895 = 1 2 119899) with 120582 gt 0

119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1Then the aggregated

result using the GHIVIFNWG operator is also a HIVIFN and

119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)

Note that Theorems 27ndash35 could be proved by using themathematical induction method and are omitted here

Based on these hesitant interval-valued intuitionisticfuzzy aggregation operators it is easy to obtain the followingproperties

Property 1 (idempotency) Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899) be a collection of HIVIFNs If

119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)

Proof According to Theorem 33 and 119867119895

= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894

119889119894]⟩ = 119867Similarly GHIVIFNWG(119867

1 1198672 119867

119899) = 119867

Property 2 (commutativity) Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899) be a collection of HIVIFNs and

let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899) be anypermutation of119867

119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)

is any permutation of119867119895

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) + sdot sdot sdot + 119908119899119897 (119896minus1(120582119896 (119886

119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) + sdot sdot sdot + 119908119899119897 (119896minus1(120582119896 (

119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩

= GHIVIFNWA119908(1 2

119899)

(35)


Similarly GHIVIFNWG(1 2

119899) =

GHIVIFNWG(1198671 1198672 119867

119899)

Property 3 (boundary) Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩

(119895 = 1 2 119899) be a collection of HIVIFNs and then

119867minusle GHIVIFAWA (119867

1 1198672 119867

119899) le 119867

+

119867minusle GHIVIFAWG (119867

1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])

Proof The process is omitted here

42 HIVIFN Algebraic Aggregation Operators and HIV-IFN Einstein Aggregation Operators Obviously different 119905-conorms and 119905-norms may lead to different aggregationoperators In the following HIVIFN algebraic aggregationoperators and Einstein aggregation operators are presentedbased on algebraic norms and Einstein norms

Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2

119899) be a collection of HIVIFNs let 119908 = (1199081 1199082 119908

119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890

minus119909 119897(119909) = minus ln(1minus119909) 119897minus1(119909) = 1minus119890

minus119909 119879(119909 119910) = 119909119910

and 119878(119909 119910) = 1 minus ((1 minus 119909)(1 minus 119910)) be algebraic 119905-conorm and119905-norm Then some HIVIFN algebraic aggregation operatorscould be obtained as follows

(1) Hesitant interval-valued intuitionistic fuzzy numberalgebraic weighted averaging operator is as follows

HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)

(2) Hesitant interval-valued intuitionistic fuzzy numberalgebraic weighted geometric operator is as follows

HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)

(3) Hesitant interval-valued intuitionistic fuzzy num-ber algebraic weighted arithmetic averaging operator is asfollows

119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)

(4) Hesitant interval-valued intuitionistic fuzzy number alge-braic weighted arithmetic geometric operator is as follows

HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)

(5) Generalized hesitant interval-valued intuitionisticfuzzy number algebraic weighted averaging operator is asfollows


GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)

(6) Generalized hesitant interval-valued intuitionistic fuzzynumber algebraic weighted geometric operator is as follows

GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)

In particular if 120582 = 1 then (41) is reduced to (37) and (42) isreduced to (38) if 120582 = 2 then (41) is reduced to (39) and (42)is reduced to (40)

Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02


let 119908 = (04 06) be the weight of them and 120582 = 1 2 5According toTheorem 36 the following can be calculated

(1) HIVIFNAWA119908(1198671 1198672) = ⟨[1 minus (1 minus 02)

04times

(1 minus 01)06 1minus(1 minus 03)

04times(1 minus 03)

06] [01

04times0206 0204times

0406] [1 minus (1 minus 04)

04times (1 minus 01)

06 1 minus (1 minus 05)

04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)

According to Definitions 24 6 and 8 Consider the following

119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus

HIVIFNAWA119908(1198671 1198672) gt HIVIFNAWG

119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)

Consider 119871(119878(HIVIFNAWAA119908(1198671 1198672))) = 01031

119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus

HIVIFNAWAA119908(1198671 1198672) gt HIVIFNAWAG

119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)


According to Definitions 22 5 and 6

119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)

Consider 119871(119878(GHIVIFNAWA119908(1198671 1198672))) = 01404

119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus

GHIVIFNAWA119908(1198671 1198672) gt GHIVIFNAWG

119908(1198671 1198672)

(51)

In the three cases listed above the aggregation resultsby using the GHIVIFNAWA operator are greater than theaggregation results by utilizing the GHIVIFNAWG operator

Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =

1 2 119899) be a collection of HIVIFNs and let 119908 = (1199081 1199082

119908119899)119879 be the weight vector of 119860

119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus

119909)119909) and 119896minus1(119909) = 2(119890119909+1) 119897(119909) = ln((2minus(1minus119909))(1minus119909))

119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))

and 119878(119909 119910) = (119909 + 119910)(1 + 119909119910) be the Einstein 119905-conorm and119905-norm respectively Then some HIVIFN Einstein aggregationoperators could be obtained as follows

(1) Hesitant interval-valued intuitionistic fuzzy numberEinstein weighted averaging operator is as follows

119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)

(2) Hesitant interval-valued intuitionistic fuzzy numberEinstein weighted geometric operator is as follows

119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)

(3) Hesitant interval-valued intuitionistic fuzzy numberEinstein weighted arithmetic averaging operator is as follows

119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)

(4) Hesitant interval-valued intuitionistic fuzzy numberEinstein weighted arithmetic geometric operator is as follows


119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)

(5) Generalized hesitant interval-valued intuitionisticfuzzy number Einstein weighted averaging operator is asfollows

119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)

(6) Generalized hesitant interval-valued intuitionisticfuzzy number Einstein weighted geometric operator is asfollows


119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)


43 The MCDM Approach Based on the GHIVIFNWA andGHIVIFNWG Operators Let 119860 = 119886

1 1198862 119886119898 be a finite

set of alternatives and let 119862 = 1198881 1198882 119888119899 be a finite set of

criteria whose criteria weight vector is 119908 = (1199081 1199082 119908

119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899

be the hesitant interval-valued intuitionistic fuzzy decisionmatrix where

119894119895is a criterion value denoted by HIVIFNs

The characteristics of the alternatives 119886119894(119894 = 1 2 119898) with

respect to the attributes 119888119895(119895 = 1 2 119899) can be denoted

by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895

]⟩ In the following wepropose one approach to rank and select the most desirablealternative(s) The procedure of this approach is shown asfollows

Step 1 Aggregate the HIVIFNs 119894119895(119894 = 1 2 119899 119895 =

1 2 119899) of the alternative 119886119894(119894 = 1 2 119898)

Utilize the GHIVIFNWA or GHIVIFNWG operator toobtain the overall values 119910

119894for the alternatives 119886

119894(119894 =

1 2 119898) respectively that is

119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)

or119910119894= GHIVIFNWG

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)


Step 2 Calculate the score values According toDefinition 22 the score of overall values 119878(119910

119894) (119894 = 1 2 119898)

could be calculated

Step 3 Rank the preference order of all alternatives 119886119894(119894 =

1 2 119898) 119871(119878(119910119894)) (119894 = 1 2 119898) could be obtained

according to Definition 5 The greater the value of 119871(119878(119910119894))

is the better the alternative 119886119894(119894 = 1 2 119898) will be

Step 4 Select the optimal one(s)

5 Illustrative Example

In this section the proposed approach and one existingmethod are utilized to evaluate four companies with hesitantinterval-valued intuitionistic fuzzy information

The enterprisersquos board of directors intends to find anautomobile company and establish a foundation for deeperand more extensive cooperation with it in the following fiveyears Suppose there are four possible projects 119886

119894(119894 = 1 2 3 4)

to be evaluated It is necessary to compare these companiesand rank them in terms of their importance Four criteriasuggested by the Balanced Scorecard methodology couldbe taken into account (it should be noted that all of them

are of the maximization type) 1198881 economy 119888

2 comfort 119888

3

design and 1198884 safety And suppose that the weight vector of

the criteria is 119908 = (02 03 015 035) The decision-makersare required to provide their evaluation of the company 119886

119894

under the criterion 119888119895(119894 = 1 2 3 4 119895 = 1 2 3 4) The

hesitant interval-valued intuitionistic fuzzy decision matrix119877 = (

119894119895)4times4

is shown in Table 1 where 119894119895(119894 = 1 2 3 4

119895 = 1 2 3 4) are in the form of HIVIFNs

51 Illustration of the Proposed Approach In order to get theoptimal alternative(s) the following steps are involved



For the convenience of analysis and computation weuse hesitant interval-valued intuitionistic fuzzy algebraicaggregation operators to fuse the attribute values which arerepresented in the form of HIVIFNs in MCDM problemsLet 119896(119909) = minus ln(119909) and let 119896minus1(119909) = 119890

minus119909 119897(119909) = minus ln(1 minus

119909) 119897minus1(119909) = 1 minus 119890

minus119909 and 119879(119909 119910) = 119909119910 and 119878(119909 119910) = 1 minus

((1 minus 119909)(1 minus 119910)) be algebraic 119905-conorm and 119905-normThen theGHIVIFNWA or GHIVIFNWG operators are respectivelyreduced to the GHIVIFNAWA or GHIVIFNAWG operatorsand (62) and (63) are reduced to the following expression thatis

119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)

Let 120582 = 2 and according to the formula listed above theoverall HIVIFNs 119910

119894of the alternatives 119886

119894(119894 = 1 2 3 4) could

be obtained and shown in Table 2

Step 2 Based on Definition 22 and Table 2 the score valuesof overall HIVIFNs 119878(119910

119894) (119894 = 1 2 3 4) can be obtained and

shown in Table 3


Table 1 Hesitant interval-valued intuitionistic fuzzy decision matrix 119877 = (119894119895)4times4

1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩

Table 2 The overall HIVIFNs of alternatives

120582 = 2 GHIVIFAWA (HIVIFAWA) GHIVIFNWG (HIVIFAWG)

1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩

Step 3 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-

dance with the scores 119878(119910119894) (119894 = 1 2 3 4) of the aggregated

hesitant interval-valued intuitionistic fuzzy values by usingDefinitions 5 and 6 From Table 3 the following results canbe obtained

Case 1 The GHIVIFAWA operator is as follows

119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)

So the final ranking of alternatives is 1198864≻ 1198862≻ 1198863≻ 1198861

Case 2 The GHIVIFAWG operator is as follows

119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)


Step 4 Select the best one(s) In Step 3 if the GHIVIFNAWAoperator is utilized then the optimal alternative is 119886

2while the

worst alternative is 1198861 if the GHIVIFNAWGoperator is used

then the optimal alternative is 1198864while the worst alternative

is 1198861

52 Sensitivity Analysis In Step 1 two aggregation operatorscan be used and the sensitivity analysis will be conducted inthese following cases

(1)The hesitant interval-valued intuitionistic fuzzy alge-braic aggregation operators in Step 1 are illustrated as follows

In order to investigate the influence of 120582 on the rankingof alternatives different 120582 are utilizedThe ranking results areshown in Tables 4 and 5

From Tables 4 and 5 the GHIVIFNAWA and GHIV-IFNAWG operators have produced different rankings ofthe alternatives However for each operator the rankingsobtained are consistent as 120582 changes Moreover 119886

4or 1198862is

always the optimal one while the worst one is always 1198861

(2) The hesitant interval-valued intuitionistic fuzzy Ein-stein aggregation operators in Step 1 are illustrated as follows

Let 119896(119909) = ln((2 minus 119909)119909) and let 119896minus1(119909) = 2(119890119909+

1) 119897(119909) = ln((2 minus (1 minus 119909))(1 minus 119909)) 119897minus1(119909) = 1 minus (2(119890119909+

1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =

(119909 + 119910)(1 + 119909119910) be the Einstein 119905-conorm and 119905-normrespectively Then the GHIVIFNWA and GHIVIFNWGoperators are respectively reduced to the GHIVIFNEWAand GHIVIFNEWG operators According to (58) and (60)the following results could be obtained and shown in Tables6 and 7

From Tables 6 and 7 the GHIVIFNEWA and GHIV-IFNEWG operators have produced different rankings of thealternatives Furthermore for each operator the aggrega-tion parameter 120582 also leads to different aggregation resultsbut the final rankings of alternatives are the same as theparameter changes What is more regardless of using theGHIVIFNEWA and GHIVIFNEWG operators is that 119886

4or

1198862is always the optimal one while the worst one is always 119886

1

It can be concluded from the sensitivity analysis thatdifferent 119905-conorms and 119905-norms could lead to differentaggregation results However the rankings using each opera-tor are consistent

53 Comparison Analysis Based on the same decision-making problem if the method of Chen et al [16] isemployed HIVIFNs are transformed to IVIFNs by using thescore function firstly and then IVIFNs could be aggregated bythe interval-valued intuitionistic fuzzy weighted aggregationoperators proposed by Chen et al [16]


Table 3 The score values of overall HIVIFNs

GHIVIFAWA 120582 = 2 (HIVIFAWA) GHIVIFAWG 120582 = 2 (HIVIFAWG)119878(1199101) ⟨[04651 05135] [02052 02545]⟩ 119878(119910

1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩

Table 4 Rankings obtained using the GHIVIFNAWA operator

120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861

Based on Definition 3 and sum4

119894=1119908119894= 1 the interval-

valued intuitionistic fuzzy weighted average values of allalternatives could be obtained as follows

IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07

2times 03 + 05 times 015 + 04 times 035

05 times 02 +04 + 07

2times 03 + 06 times 015 + 04 times 035]

[(1 minus 03) times 02 + (1 minus02 + 03

2) times 03 + (1 minus 04)

times 015 + (1 minus 02) times 035 (1 minus 02) times 02

+ (1 minus01 + 03

2) times 03 + (1 minus 03) times 015

+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)

According to Definitions 5 and 6

119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863

≻ 1198861and the best optimal one is

1198864 The ranking here is the same as the result using the

GHIVIFNAWA and GHIVIFNAWA operatorsAccording to the calculation results although the existing

method can produce the same result as the proposedmethodthe method being compared has a problem that how to trans-form HIVIFNs to IVIFNs in the first step could avoid infor-mation loss in the process of transformation By contrastthe proposed approach based on different 119905-conorms and 119905-norms can be used to deal with different relationships amongthe aggregated arguments could handle MCDM problemsin a flexible and objective manner under hesitant interval-valued intuitionistic fuzzy environment and can providemore choices for decision-makers Additionally different 119905-conorms and 119905-norms and aggregation operators could bechosen in the practical decision-making process At the sametime different results may be produced which reflectedthe preferences of decision-makers Therefore the developedapproach can produce better results than the existingmethod


Table 5 Rankings obtained using the GHIVIFNAWG operator

120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861

Table 6 Rankings obtained using the GHIVIFNEWA operator

120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861

Table 7 Rankings obtained using the GHIVIFNEWG operator

120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion

HFSs are the extension of traditional fuzzy sets and theirmembership degree of an element is a set of several possiblevalues between 0 and 1 IVIFSs can describe the fuzzyconcept ldquoneither this nor thatrdquo and the membership degreesand nonmembership degrees of IVIFSs are not only realnumbers but interval values respectively Precise numericalvalues in HFSs can be replaced by IVIFSs which providemore preference information for decision-makers In thispaper the definition of HIVIFSs was developed and appliedto the MCDM problems in which the evaluation valuesof alternatives on criteria were expressed with HIVIFNsFurthermore based on 119905-conorms and 119905-norms some aggre-gation operators namely the HIVIFNWA and HIVIFNWGHIVIFNWAA and HIVIFNWGA and GHIVIFNWA andGHIVIFNWG operators were proposed respectively Theirproperties were discussed in detail as well In particular thecorresponding hesitant interval-valued intuitionistic fuzzyalgebraic aggregation operators based on algebraic 119905-conormand 119905-norm and hesitant interval-valued intuitionistic fuzzyEinstein aggregation operators based on Einstein 119905-conormand 119905-normwere presented In addition different aggregationoperators were utilized to fuse the hesitant interval-valuedintuitionistic fuzzy information to get the overall HIVIFNs of

alternatives and the ranking of all given alternatives At lastthe example was presented to illustrate the fuzzy decision-making process and the sensitivity analysis and comparisonanalysis were conducted to enrich the paper The prominentfeature of the proposed method is that it could providea useful and flexible way to efficiently facilitate decision-makers under a hesitant interval-valued intuitionistic fuzzyenvironment and the related calculations are simple Henceit has enriched and developed the theories and methodsof MCDM problems and also has provided a new idea forsolvingMCDMproblems In the future research the distanceand similarity measure of HIVIFSs will be studied to solveMCDM problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work issupported by the National Natural Science Foundation of


China (nos 71271218 and 71221061) the Research Projectof Education of Hubei (no Q20122302) and the ScienceFoundation for Doctors of Hubei University of AutomotiveTechnology (no BK201405)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash356 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionSofia 1983

[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[4] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[5] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[6] Z S Xu and J Chen ldquoAn approach to group decision makingbased on interval-valued intuitionistic judgment matricesrdquoSystem Engineering Theory and Practice vol 27 no 4 pp 126ndash133 2007

[7] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009

[8] J Wu H B Huang and Q W Cao ldquoResearch on AHP withinterval-valued intuitionistic fuzzy sets and its application inmulti-criteria decision making problemsrdquo Applied Mathemat-ical Modelling vol 37 no 24 pp 9898ndash9906 2013

[9] K T Atanassov ldquoOperators over interval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 64 no 2 pp 159ndash1741994

[10] W Lee ldquoA novel method for ranking interval-valued intuition-istic fuzzy numbers and its application to decision makingrdquoin Proceedings of the International Conference on IntelligentHuman-Machine Systems and Cybernetics (IHMSC rsquo09) vol 2pp 282ndash285 Zhejiang China August 2009

[11] W Lee ldquoAn enhanced multicriteria decision-making methodof machine design schemes under interval-valued intuitionisticfuzzy environmentrdquo in Proceedings of the 10th IEEE Interna-tional Conference on Computer-Aided Industrial Design andConceptual Design (CD rsquo2009) pp 721ndash725 Wenzhou ChinaNovember 2009

[12] D F Li ldquoTOPSIS-based nonlinear-programming methodologyfor multiattribute decision making with interval-valued intu-itionistic fuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol18 no 2 pp 299ndash311 2010

[13] JH Park I Y Park Y CKwun andXG Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[14] T Y Chen H P Wang and Y Y Lu ldquoA multicriteria groupdecision-making approach based on interval-valued intuition-istic fuzzy sets a comparative perspectiverdquo Expert Systems withApplications vol 38 no 6 pp 7647ndash7658 2011

[15] V L G Nayagam and G Sivaraman ldquoRanking of interval-valued intuitionistic fuzzy setsrdquoApplied SoftComputing Journalvol 11 no 4 pp 3368ndash3372 2011

[16] S M Chen LW Lee H C Liu and SW Yang ldquoMultiattributedecision making based on interval-valued intuitionistic fuzzy

valuesrdquo Expert Systems with Applications vol 39 no 12 pp10343ndash10351 2012

[17] F Y Meng C Q Tan and Q Zhang ldquoThe induced generalizedinterval-valued intuitionistic fuzzy hybrid Shapley averagingoperator and its application in decision makingrdquo Knowledge-Based Systems vol 42 pp 9ndash19 2013

[18] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju Island Republic of Korea August2009

[19] MMXia andZ S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[20] B Zhu Z S Xu and M M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

[21] G W Wei ldquoHesitant fuzzy prioritized operators and theirapplication to multiple attribute decision makingrdquo Knowledge-Based Systems vol 31 pp 176ndash182 2012

[22] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[23] Z S Xu and M M Xia ldquoOn distance and correlation measuresof hesitant fuzzy informationrdquo International Journal of Intelli-gent Systems vol 26 no 5 pp 410ndash425 2011

[24] N Chen Z S Xu and M M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modeling vol 37 no 4 pp 2197ndash22112013

[25] DH Peng C Y Gao and Z F Gao ldquoGeneralized hesitant fuzzysynergetic weighted distance measures and their applicationto multiple criteria decision-makingrdquo Applied MathematicalModeling vol 37 no 8 pp 5837ndash5850 2013

[26] N Zhang and G W Wei ldquoExtension of VIKOR method fordecision making problem based on hesitant fuzzy setrdquo AppliedMathematical Modeling vol 37 no 7 pp 4938ndash4947 2013

[27] Z M Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013

[28] N Chen Z S Xu and M M Xia ldquoInterval-valued hesitantpreference relation relations and their applications to groupdecision makingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013

[29] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013

[30] G W Wei and X F Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent andFuzzy Systems vol 24 no 4 pp 789ndash803 2013

[31] B Zhu Z S Xu and M M Xia ldquoDual hesitant fuzzy setsrdquoJournal of Applied Mathematics vol 2012 Article ID 879629 13pages 2012

[32] A Sengupta and T K Pal ldquoOn comparing interval numbersrdquoEuropean Journal of Operational Research vol 127 no 1 pp 28ndash43 2000

[33] P Y Chen ldquoAn interval estimation for the number of signalsrdquoSignal Processing vol 85 no 8 pp 1623ndash1633 2005

[34] Z S Xu ldquoDependent uncertain ordered weighted aggregationoperatorsrdquo Information Fusion vol 9 no 2 pp 310ndash316 2008


[35] N N Karnik and J M Mendel ldquoCentroid of a type-2 fuzzy setrdquoInformation Sciences vol 132 no 1ndash4 pp 195ndash220 2001

[36] V L GNayagam SMuralikrishnan andG Sivaraman ldquoMulti-criteria decision-making method based on interval-valuedintuitionistic fuzzy setsrdquo Expert Systems with Applications vol38 no 3 pp 1464ndash1467 2011

[37] B Schweizer and A Sklar Probabilistic Metric Spaces ElsevierNew York NY USA 1983

[38] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[39] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[40] E P Klement and R Mesiar Logical Algebraic Analytic andProbabilistic Aspects of Triangular Norms Elsevier AmsterdamThe Netherlands 2005

[41] Z S Xu and Z D Sun ldquoPriority method for a kind of multi-attribute decision-making problemsrdquo Journal of ManagementSciences in China vol 3 no 5 pp 35ndash39 2002

[42] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011

[43] M M Xia Research on fuzzy decision information aggregationtechniques and measures [Dissertation] Southeast UniversityNanjing China 2012

[44] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Algebra

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Decision SciencesAdvances in

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Stochastic AnalysisInternational Journal of


hybrid Shapley averaging (GIVIFHSA) operator and appliedit to MCDM problems

Hesitant fuzzy sets (HFSs) another extension of tradi-tional fuzzy sets provide a useful reference for our studyunder hesitant fuzzy environment HFSs were first intro-duced by Torra and Narukawa [18] and they permit themembership degrees of an element to be a set of severalpossible values between 0 and 1 HFSs are highly useful inhandling the situations where people have hesitancy in pro-viding their preferences over objects in the decision-makingprocess Some aggregation operators of HFSs were studiedand applied to decision-making problems [19ndash21] Then thecorrelation coefficients of HFSs the distance measures andcorrelation measures of HFSs were discussed [22ndash24] basedon which Peng et al [25] presented a generalized hesitantfuzzy synergetic weighted distance measure Zhang and Wei[26] developed the E-VIKOR method and TOPSIS methodto solve MCDM problems with hesitant fuzzy informationZhang [27] developed a wide range of hesitant fuzzy poweraggregation operators for hesitant fuzzy information Chen etal [28] generalized the concept of HFSs to hesitant interval-valued fuzzy sets (HIVFSs) in which themembership degreesof an element to a given set are not exactly defined butdenoted by several possible interval values Wei [29] definedHIVFSs and some hesitant interval-valued fuzzy aggregationoperators Wei and Zhao [30] developed some Einsteinoperations on HIVFSs and the induced hesitant interval-valued fuzzy Einstein aggregation (HIVFEA) operators andapplied them to MCDM problems Zhu et al [31] defineddual HFSs (DHFSs) in terms of two functions that returntwo sets ofmembership degrees and nonmembership degreesrather than crisp numbers in HFSs If the idea of dual HFSsis used from a new perspective then another extension ofHFSs may be defined in terms of one function that theelement of HFSs returns a set of IFSs which are called hes-itant intuitionistic fuzzy sets (HIFSs) But decision-makersusually cannot estimate criteria values of alternatives withexact numerical values when the information is not knownpreciselyTherefore interval values in fuzzy sets can representit better than specific numbers such as interval-valued fuzzysets (IVFSs) and IVIFSs Furthermore although the theoriesof IVIFSs and HFSs have been developed and generalizedthey cannot deal with all sorts of uncertainties in differentreal problems For example when we ask the opinion of anexpert about a certain statement he or she may answer thatthe possibility that the statement is true is [01 02] and thatthe statement is false is [04 05] or the possibility that thestatement is true is [05 06] and that the statement is false is[03 05]This issue is beyond the scope of IVFSs and IVIFSsTherefore some new theories are required

So the concept of hesitant interval-valued intuitionisticfuzzy sets (HIVIFSs) is developed in this paper Comparingto the existing fuzzy sets mentioned above HIVIFSs are anew extension of HFSs which support a more flexible andsimpler approach when decision-makers provide their deci-sion information in a hesitant interval-valued intuitionisticfuzzy environment Furthermore IVIFSsHFSsHIVFSs andHIFSs are all the special cases of HIVIFSs

In this paper HFSs are extended based on IVIFSsHIVIFSs are defined and their properties and applicationsare also discussed Thus the rest of this paper is organized asfollows In Section 2 the definitions and properties of IVIFSsand HFSs are briefly reviewed In Section 3 the notion ofHIVIFSs is proposed and the operations and properties ofHIVIFSs based on 119905-conorms and 119905-norms are discussed InSection 4 some hesitant interval-valued intuitionistic fuzzynumber aggregation operators are developed and applied toMCDMproblems Section 5 gives an example to illustrate theapplication of the developedmethod Finally the conclusionsare drawn in Section 6

2 Preliminaries

In this section some basic concepts and definitions related toHIVIFSs are introduced including interval numbers IVIFSsand HFSs These will be utilized in the subsequent analysis

21 Interval Numbers and Their Operations

Definition 1 (see [32ndash34]) Let 119886 = [119886119871 119886119880] = 119909 | 119886

119871le 119909 le

119886119880 then 119886 is called an interval number In particular if 0 le

119886119871le 119909 le 119886

119880 then 119886 is reduced to a positive interval numberConsider any two interval fuzzy numbers 119886 = [119886

119871 119886119880]

and = [119887119871 119887119880] and their operations are defined as follows

(1) 119886 = hArr 119886119871= 119887119871 119886119880= 119887119880

(2) 119886 + = [119886119871+ 119887119871 119886119880+ 119887119880]

(3) 119886 minus = [119886119871minus 119887119880 119886119880minus 119887119871]

(4) 119886 times = [min119886119871119887119871 119886119871119887119880 119886119880119887119871 119886119880119887119880max119886119871119887119871 119886119871119887119880 119886119880119887119871 119886119880119887119880]

(5) 119896119886 = [119896119886119871 119896119886119880] 119896 gt 0

22 IVIFSs Atanassov first proposed IFSs being enlarge-ment and development of Zadehrsquos fuzzy sets IFSs contain thedegree of nonmembership which makes it possible for us tomodel unknown informationThe definition of IVIFSs givenby Atanassov and Gargov [4] is shown as follows

Definition 2 (see [4]) Let 119863[0 1] be the set of all closedsubintervals of the interval [0 1] Let 119883 be a given set and119883 = An IVIFS in 119883 is an expression given by 119860 =

⟨119909 120583119860(119909) ]119860(119909)⟩ | 119909 isin 119883 where 120583

119860 119883 rarr 119863[0 1] ]

119860rarr

119863[0 1] with the condition 0 lt sup119909120583119860(119909) + sup

119909]119860(119909) le

1 The intervals 120583119860(119909) and ]

119860(119909) denote the degree of

belongingness and nonbelongingness of the element 119909 to theset 119860 respectively Thus for each 119909 isin 119883 120583

119860(119909) and ]

119860(119909)

are closed intervals whose lower and upper boundaries aredenoted by 120583119871

119860(119909) 120583119880

119860(119909) and ]119871

119860(119909) ]119880119860(119909) respectively and

then

119860 = ⟨119909 [120583119871

119860(119909) 120583

119880

119860(119909)] []119871

119860(119909) ]119880

119860(119909)]⟩ | 119909 isin 119883 (1)

where 0 lt 120583119880

119860(119909) + ]119880

119860(119909) le 1 120583119871

119860(119909) ge 0 ]119871

119860(119909) ge 0 For each

element 119909 the hesitancy degree can be calculated as followsΠ119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) = [1 minus 120583

119880

119860(119909) minus ]119880

119860(119909) 1 minus 120583

119871

119860(119909) minus






values where 119894= ⟨[119886119894 119887119894] [119888119894 119889119894]⟩ = [[119886

119894 119887119894] [1 minus 119889

119894 1 minus 119888

119894]]

0 le 119886119894le 119887119894le 1 0 le 119888

119894le 119889119894le 1 0 le 119887

119894+ 119889119894le 1 and


IVIFWA119908(1 2

119899)

=sum119899

119894=1[[119886119894 119887119894] [1 minus 119889

119894 1 minus 119888119894]] times 119908

119894

sum119899

119894=1119908119894

= [[sum119899

119894=1119886119894119908119894

sum119899

119894=1119908119894

sum119899

119894=1119887119894119908119894

sum119899

119894=1119908119894

]

[sum119899

119894=1(1 minus 119889

119894) 119908119894

sum119899

119894=1119908119894

sum119899

119894=1(1 minus 119888119894) 119908119894

sum119899

119894=1119908119894

]]

= [[119886 ] [119888 119889]]

(2)


119899) = [[119886 ] [119888 119889]] = ⟨[119886 ]


Example 4 Let 1

= ⟨[03 06] [01 02]⟩ and 2

=

⟨[04 06] [01 03]⟩ be two IVIFNs and 1199081= 03 119908

2= 05

According to (2)

IVIFWA119908(1 2)

= [[03 times 03 + 04 times 05

03 + 0506 times 03 + 06 times 06

03 + 05]


03 + 05


03 + 05]]

= [[03625 06750] [07375 09000]]

= ⟨[03625 06750] [1 minus 09000 1 minus 07375]⟩

= ⟨[03625 06750] [01000 02625]⟩

(3)



2 (4)




(1) If 119871(1) gt 119871(

2) then

1gt 2

(2) If 119871(1) = 119871(

2) then

1= 2

Example 7 Let 1= ⟨[04 06] [01 02]⟩ and

2= ⟨[05



2) gt 119871(


one(s) is 2




1015840 119910 le 119910

1015840 then 119879(119909 119910) le 119879(1199091015840 1199101015840)




1015840 119910 le 119910

1015840 then 119878(119909 119910) le 119878(1199091015840 1199101015840)



119897minus1(119905) = 1 minus 119890




119905+ 1)) and



119905+ 120574 minus 1) 119897minus1(119905) =



119879 (119909 119910) =119909119910

120574 + (1 minus 120574) (119909 + 119910 minus 119909119910) 120574 gt 0


1 minus (1 minus 120574) 119909119910 120574 gt 0

(5)


Definition 10 Let = ⟨[119886 119887] [119888 119889]⟩ 1= ⟨[1198861 1198871] [1198881 1198891]⟩



(1) 120582 = ⟨[119896minus1(120582119896(119886)) 119896

minus1(120582119896(119887))] [119897

minus1(120582119897(119888))

119897minus1(120582119897(119889))]⟩

(2) 120582 = ⟨[119897minus1(120582119897(119886)) 119897

minus1(120582119897(119887))] [119896

minus1(120582119896(119888))

119896minus1(120582119896(119889))]⟩ 120582 gt 0

(3) 1oplus 2

= ⟨[119897minus1(119897(1198861) + 119897(119886

2)) 119897minus1(119897(1198871) + 119897(119887

2))]

[119896minus1(119896(1198881) + 119896(119888

2)) 119896minus1(119896(1198891) + 119896(119889

2))]⟩


(4) 119886 otimes = ⟨[119896minus1(119896(1198861) + 119896(119886

2)) 119896minus1(119896(1198871) + 119896(119887

2))]

[119897minus1(119897(1198881) + 119897(119888

2)) 119897minus1(119897(1198891) + 119897(119889

2))]⟩


23 HFSs


119864 = ⟨119909 ℎ119864(119909)⟩ | 119909 isin 119883 (6)










(1) ℎ120582 = ⋃120574isinℎ

119896minus1(120582119896(120574))

(2) 120582ℎ = ⋃120574isinℎ

119897minus1(120582119897(120574))


119897minus1(119897(1205741) + 119897(120574

2))


119896minus1(119896(1205741) + 119896(120574

2))





1and ℎ

2 if

119904(ℎ1) gt 119904(ℎ

2) then ℎ

1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2


2= 04 07 be two


06) = 04667 119904(ℎ2) = 055 119904(ℎ

2) gt 119904(ℎ

1) so ℎ

2gt ℎ1



119899 be a set of



119899

120599 (120574) (7)





119864 = ⟨119909119867119864 (119909)⟩ | 119909 isin 119883 (8)




119867119864 (119909) =

119899(119867119864(119909))

⋃119894=1

⟨[120583119871

119864119894

(119909) 120583119880

119864119894

(119909)] []119871119864119894

(119909) ]119880119864119894

(119909)]⟩

(9)

where 0 le 120583119871

1198641

(119909) le 120583119880

1198641

(119909) le 120583119871

1198642

(119909) le 120583119880

1198642


120583119871

119899(119867119864(119909))

(119909) le 120583119880

119899(119867119864(119909))

(119909) le 1 0 le 120583119880

119864119894

(119909) + ]119880119864119894

(119909) le 1120583119871

119864119894

(119909) ge 0 ]119871119864119894



119864(119909)) = 1 then


(119909) = 120583119880

119864119894

(119909) (119894 =

1 2 119899(119867119864(119909))) and ]119871

119864119894

(119909) = ]119880119864119894

(119909) = 0 (119894 = 1 2


119864119894

(119909) =

120583119880

119864119894

(119909) (119894 = 1 2 119899(119867119864(119909))) or ]119871

119864119894

(119909) = ]119880119864119894

(119909) (119894 =


if 120583119871119864119894

(119909) = 120583119880

119864119894

(119909) (119894 = 1 2 119899(119867119864(119909))) and ]119871

119864119894

(119909) =

]119880119864119894





119867119864=

119899(119867119864)

⋃119894=1

⟨[119886119894 119887119894] [119888119894 119889119894]⟩ (10)



and for all 119909 isin 119883 Π119860(119909) = ⋃

119899(119867119860(119909))

119894=1[1minus120583

119880

119860119894

(119909)minus]119880119860119894

(119909) 1minus

120583119871

119860119894

(119909) minus ]119871119860119894




1 ⟨[03 04]

[01 02]⟩ ⟨04 02⟩⟩ ⟨1199092 ⟨[05 06][02 04]⟩⟩ be a


119860(1199092) = [0

03] Thus Π119860(119909) = ⟨119909

1 [04 06] 04⟩ ⟨119909

2 [0 03]⟩



Definition 19 Let1198671= ⋃119899(1198671)

1198941=1

⟨[1198861198941

1198871198941

] [1198881198941

1198891198941

]⟩ and1198672=

⋃119899(1198672)

1198942=1

⟨[1198861198942

1198871198942

] [1198881198942

1198891198942


(1) 1205821198671= ⋃119899(1198671)

1198941=1

⟨[119897minus1(120582119897(1198861198941

)) 119897minus1(120582119897(1198871198941

))][119896minus1(120582119896(1198881198941

)) 119896minus1(120582119896(119889

1198941

))]⟩

(2) (1198671)120582= ⋃119899(1198671)

119894=1⟨[119896minus1(120582119896(1198861198941

)) 119896minus1(120582119896(1198871198941

))][119897minus1(120582119897(1198881198941

)) 119897minus1(120582119897(1198891198941

))]⟩

(3) 1198671oplus1198672= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119897minus1(119897(1198861198941

)+119897(1198861198942

)) 119897minus1(119897(1198871198941

)+

119897(1198871198942

))] [119896minus1(119896(1198881198941

) + 119896(1198881198942

)) 119896minus1(119896(1198891198941

) + 119896(1198891198942

))]⟩

(4) 1198671otimes 1198672

= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119896minus1(119896(1198861198941

) + 119896(1198861198942

))

119896minus1(119896(1198871198941

) + 119896(1198871198942

))] [119897minus1(119897(1198881198941

) + 119897(1198881198942

)) 119897minus1(119897(1198891198941

) +

119897(1198891198942

))]⟩


Example 20 Let1198671= ⟨[01 03] [02 04]⟩ ⟨[02 03] [03




(1) 21198671



]




1 minus




]⟩ = ⟨[019

051] [004 016]⟩ ⟨[036 051] [009016]⟩

(2) (1198671)2= ⟨[001 009] [036 064]⟩ ⟨[004 009]

[051 064]⟩

(3) 1198671oplus 1198672= ⟨[037 058] [004 012]⟩ ⟨[044 058]

[006 012]⟩

(4) 1198671otimes 1198672= ⟨[003 012] [036 058]⟩ ⟨[006 012]

[044 058]⟩

Theorem 21 Let1198671 1198672 1198673isin 119867119868119881119868119865119873119878 120582 120582

1 1205822gt 0 and

then

(1) 1198671oplus 1198672= 1198672oplus 1198671

(2) 1198671otimes 1198672= 1198672otimes 1198671

(3) 1205821198671oplus 1205821198672= 120582(119867

1oplus 1198672)

(4) (1198671)120582otimes (1198672)120582= (1198671otimes 1198672)120582



(7) ((1198671)1205821)1205822 = (119867

1)12058211205822


(3) 1205821198671oplus 1205821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(120582119897 (119886

1198941

) + 120582119897 (1198861198942

))

119897minus1(120582119897 (119887

1198941

) + 120582119897 (1198871198942

))]

[119896minus1(120582119896 (119888

1198941

) + 120582119896 (1198881198942

))

119896minus1(120582119896 (119889

1198941

) + 120582119896 (1198891198942

))]⟩

=

119899(1198671)

⋃119894=1

119899(1198672)

⋃119895=1

⟨[119897minus1(120582 (119897 (119886

1198941

) + 119897 (1198861198942

)))

119897minus1(120582 (119897 (119887

1198941

) + 119897 (1198871198942

)))]

[119896minus1(120582 (119896 (119888

1198941

) + 119896 (1198881198942

)))

119896minus1(120582 (119896 (119889

1198941

) + 119896 (1198891198942

)))]⟩

= 120582 (1198671oplus 1198672)

(4) (1198671)120582otimes (1198672)120582

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582119896 (119886

1198941

) + 120582119896 (1198861198942

))

119896minus1(120582119896 (119887

1198941

) + 120582119896 (1198871198942

))]

[119897minus1(120582119897 (1198881198941

) + 120582119897 (1198881198942

))

119897minus1(120582119897 (119889

1198941

) + 120582119897 (1198891198942

))]⟩

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582 (119896 (119886

1198941

) + 119896 (1198861198942

)))

119896minus1(120582 (119896 (119887

1198941

) + 119896 (1198871198942

)))]

[119897minus1(120582 (119897 (119888

1198941

) + 119897 (1198881198942

)))

119897minus1(120582 (119897 (119889

1198941

) + 119897 (1198891198942

)))]⟩

= (1198671otimes 1198672)120582

(7) ((1198671)1205821)1205822

=

119899(119867)

⋃1198941=1

⟨[119896minus1(1205822119896 (119896minus1(1205821119896 (1198861198941

))))

119896minus1(1205822119896 (119896minus1(1205821119896 (1198871198941

))))]

[119897minus1(1205822119897 (119897minus1(1205821119897 (1198881198941

))))


119897minus1(1205822119897 (119897minus1(1205821119897 (1198891198941

))))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058221205821119896 (1198861198941

))

119896minus1(12058221205821119896 (1198871198941

))]

[119897minus1(12058221205821119897 (1198881198941

))

119897minus1(12058221205821119897 (1198891198941

))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058211205822119896 (1198861198941

))

119896minus1(12058211205822119896 (1198871198941

))]

[119897minus1(12058211205822119897 (1198881198941

))

119897minus1(12058211205822119897 (1198891198941

))]⟩

= (1198671)12058211205822





1and 119867

2 if 119878(119867

1) gt 119878(119867

2) then 119867

1gt 1198672 if

119878(1198671) = 119878(119867

2) then119867

1= 1198672

Note that 119878(1198671) and 119878(119867


Definitions 5 and 6

Example 23 Let1198671= ⟨[03 04] [01 02]⟩ ⟨[03 05] [02


then

119878 (1198671) =

1

2times ⟨[03 + 03 04 + 05] [01 + 02 02 + 04]⟩

= ⟨[030 045] [015 030]⟩

119878 (1198672) = ⟨[03 04] [02 03]⟩


119871 (119878 (1198671))


2

= 024

119871 (119878 (1198672))


2

= 019

(13)

Hence 119878(1198671) gt 119878(119867


1is preferred

to1198672






HIVIFNWA119908(1198671 1198672 119867

119899) =

119899

⨁119895=1

119908119895119867119895 (14)


1 1199082


119895(119895 = 1 2 119899) with 119908

119895ge

0 (119895 = 1 2 119899) and sum119899119895=1

119908119895= 1

Theorem 25 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(15)



(1) For 119899 = 2 since

11990811198671=

119899(1198671)

⋃1198941=1

⟨[119897minus1(1199081119897 (1198861198941

)) 119897minus1(1199081119897 (1198871198941

))]

[119896minus1(1199081119896 (1198881198941

)) 119896minus1(1199081119896 (1198891198941

))]⟩

11990821198672=

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199082119897 (1198861198942

)) 119897minus1(1199082119897 (1198871198942

))]

[119896minus1(1199082119896 (1198881198942

)) 119896minus1(1199082119896 (1198891198942

))]⟩

(16)


HIVIFNWA119908(1198671 1198672)

= 11990811198671oplus 11990821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

))]⟩

(17)

(2) If (15) holds for 119899 = 119896 then

HIVIFNWA119908(1198671 1198672 119867

119896)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119896119897 (119886119894119896

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119896119897 (119887119894119896

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119896119896 (119888119894119896

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119896119896 (119889119894119896

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[

[

119897minus1(

119896

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119896

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(18)


HIVIFNWA119908(1198671 1198672 119867

119896 119867119896+1

)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(119897 (119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

)

+ sdot sdot sdot + 119908119896119897 (119886119894119896

)))

+ 119908119896+1

119897 (119886119894119896+1

))

119897minus1(119897 (119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

)

+ sdot sdot sdot + 119908119896119897 (119887119894119896

)))

+ 119908119896+1

119897 (120583119887119894119896+1

))]

[119896minus1(119896 (119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

)

+ sdot sdot sdot + 119908119896119896 (119888119894119896

)))

+ 119908119896+1

119896 (119888119894119896+1

))

119896minus1(119896 (119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot + 119908119896119896 (119889119894119896

)))

+ 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942


119908119896119897 (119886119894119896

) + 119908119896+1

119897 (119886119894119896+1

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942


119908119896119897 (119887119894119896

) + 119908119896+1

119897 (119887119894119896+1

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942


119908119896119896 (119888119894119896

) + 119908119896+1

119896 (119888119894119896+1

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942


119908119896119896 (119889119894119896

) + 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[

[

119897minus1(

119896+1

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896+1

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896+1

sum119895=1

119908119895119896 (119888119894119895

))


119896minus1(

119896+1

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(19)


HIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119899119897 (119886119894119899

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119899119897 (119887119894119899

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119899119896 (119888119894119899

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119899119896 (119889119894119899

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(20)



HIVIFNWG119908(1198671 1198672 119867

119899) =

119899

⨂119895=1

(119867119895)119908119895

(21)


1 1199082 119908

119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1

119908119895= 1


Theorem 27 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 = 1

2 119899) andsum119899119895=1



HIVIFNWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119896=1

⟨[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119886119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119887119894119895

))]

]

[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119888119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119889119894119895

))]

]

⟩

(22)



HIVIFNWAA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)2

)

12

(23)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 29 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119886

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119888

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119889

119894119895

))))))]

]

⟩

(24)




HIVIFNWAG119908(1198671 1198672 119867

119899) =

1

2(

119899

⨂119895=1

(2119867119895)119908119895

)

(25)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 31 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 Then the


119867119868119881119868119865119873119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119886

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119888

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119889

119894119895

))))))]

]

⟩

(26)



GHIVIFNWA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)120582

)

1120582

(27)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1


Theorem 33 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 120582 gt 0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Then


119866119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

(28)



GHIVIFNWG119908(1198671 1198672 119867

119899) =

1

120582(

119899

⨂119895=1

(120582119867119895)119908119895

)

(29)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1



Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119908119895ge 0 (119895 = 1 2 119899) andsum119899



119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)




= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)


= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














values where 119894= ⟨[119886119894 119887119894] [119888119894 119889119894]⟩ = [[119886

119894 119887119894] [1 minus 119889

119894 1 minus 119888

119894]]

0 le 119886119894le 119887119894le 1 0 le 119888

119894le 119889119894le 1 0 le 119887

119894+ 119889119894le 1 and


IVIFWA119908(1 2

119899)

=sum119899

119894=1[[119886119894 119887119894] [1 minus 119889

119894 1 minus 119888119894]] times 119908

119894

sum119899

119894=1119908119894

= [[sum119899

119894=1119886119894119908119894

sum119899

119894=1119908119894

sum119899

119894=1119887119894119908119894

sum119899

119894=1119908119894

]

[sum119899

119894=1(1 minus 119889

119894) 119908119894

sum119899

119894=1119908119894

sum119899

119894=1(1 minus 119888119894) 119908119894

sum119899

119894=1119908119894

]]

= [[119886 ] [119888 119889]]

(2)


119899) = [[119886 ] [119888 119889]] = ⟨[119886 ]


Example 4 Let 1

= ⟨[03 06] [01 02]⟩ and 2

=

⟨[04 06] [01 03]⟩ be two IVIFNs and 1199081= 03 119908

2= 05

According to (2)

IVIFWA119908(1 2)

= [[03 times 03 + 04 times 05

03 + 0506 times 03 + 06 times 06

03 + 05]


03 + 05


03 + 05]]

= [[03625 06750] [07375 09000]]

= ⟨[03625 06750] [1 minus 09000 1 minus 07375]⟩

= ⟨[03625 06750] [01000 02625]⟩

(3)



2 (4)




(1) If 119871(1) gt 119871(

2) then

1gt 2

(2) If 119871(1) = 119871(

2) then

1= 2

Example 7 Let 1= ⟨[04 06] [01 02]⟩ and

2= ⟨[05



2) gt 119871(


one(s) is 2




1015840 119910 le 119910

1015840 then 119879(119909 119910) le 119879(1199091015840 1199101015840)




1015840 119910 le 119910

1015840 then 119878(119909 119910) le 119878(1199091015840 1199101015840)



119897minus1(119905) = 1 minus 119890




119905+ 1)) and



119905+ 120574 minus 1) 119897minus1(119905) =



119879 (119909 119910) =119909119910

120574 + (1 minus 120574) (119909 + 119910 minus 119909119910) 120574 gt 0


1 minus (1 minus 120574) 119909119910 120574 gt 0

(5)


Definition 10 Let = ⟨[119886 119887] [119888 119889]⟩ 1= ⟨[1198861 1198871] [1198881 1198891]⟩



(1) 120582 = ⟨[119896minus1(120582119896(119886)) 119896

minus1(120582119896(119887))] [119897

minus1(120582119897(119888))

119897minus1(120582119897(119889))]⟩

(2) 120582 = ⟨[119897minus1(120582119897(119886)) 119897

minus1(120582119897(119887))] [119896

minus1(120582119896(119888))

119896minus1(120582119896(119889))]⟩ 120582 gt 0

(3) 1oplus 2

= ⟨[119897minus1(119897(1198861) + 119897(119886

2)) 119897minus1(119897(1198871) + 119897(119887

2))]

[119896minus1(119896(1198881) + 119896(119888

2)) 119896minus1(119896(1198891) + 119896(119889

2))]⟩


(4) 119886 otimes = ⟨[119896minus1(119896(1198861) + 119896(119886

2)) 119896minus1(119896(1198871) + 119896(119887

2))]

[119897minus1(119897(1198881) + 119897(119888

2)) 119897minus1(119897(1198891) + 119897(119889

2))]⟩


23 HFSs


119864 = ⟨119909 ℎ119864(119909)⟩ | 119909 isin 119883 (6)










(1) ℎ120582 = ⋃120574isinℎ

119896minus1(120582119896(120574))

(2) 120582ℎ = ⋃120574isinℎ

119897minus1(120582119897(120574))


119897minus1(119897(1205741) + 119897(120574

2))


119896minus1(119896(1205741) + 119896(120574

2))





1and ℎ

2 if

119904(ℎ1) gt 119904(ℎ

2) then ℎ

1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2


2= 04 07 be two


06) = 04667 119904(ℎ2) = 055 119904(ℎ

2) gt 119904(ℎ

1) so ℎ

2gt ℎ1



119899 be a set of



119899

120599 (120574) (7)





119864 = ⟨119909119867119864 (119909)⟩ | 119909 isin 119883 (8)




119867119864 (119909) =

119899(119867119864(119909))

⋃119894=1

⟨[120583119871

119864119894

(119909) 120583119880

119864119894

(119909)] []119871119864119894

(119909) ]119880119864119894

(119909)]⟩

(9)

where 0 le 120583119871

1198641

(119909) le 120583119880

1198641

(119909) le 120583119871

1198642

(119909) le 120583119880

1198642


120583119871

119899(119867119864(119909))

(119909) le 120583119880

119899(119867119864(119909))

(119909) le 1 0 le 120583119880

119864119894

(119909) + ]119880119864119894

(119909) le 1120583119871

119864119894

(119909) ge 0 ]119871119864119894



119864(119909)) = 1 then


(119909) = 120583119880

119864119894

(119909) (119894 =

1 2 119899(119867119864(119909))) and ]119871

119864119894

(119909) = ]119880119864119894

(119909) = 0 (119894 = 1 2


119864119894

(119909) =

120583119880

119864119894

(119909) (119894 = 1 2 119899(119867119864(119909))) or ]119871

119864119894

(119909) = ]119880119864119894

(119909) (119894 =


if 120583119871119864119894

(119909) = 120583119880

119864119894

(119909) (119894 = 1 2 119899(119867119864(119909))) and ]119871

119864119894

(119909) =

]119880119864119894





119867119864=

119899(119867119864)

⋃119894=1

⟨[119886119894 119887119894] [119888119894 119889119894]⟩ (10)



and for all 119909 isin 119883 Π119860(119909) = ⋃

119899(119867119860(119909))

119894=1[1minus120583

119880

119860119894

(119909)minus]119880119860119894

(119909) 1minus

120583119871

119860119894

(119909) minus ]119871119860119894




1 ⟨[03 04]

[01 02]⟩ ⟨04 02⟩⟩ ⟨1199092 ⟨[05 06][02 04]⟩⟩ be a


119860(1199092) = [0

03] Thus Π119860(119909) = ⟨119909

1 [04 06] 04⟩ ⟨119909

2 [0 03]⟩



Definition 19 Let1198671= ⋃119899(1198671)

1198941=1

⟨[1198861198941

1198871198941

] [1198881198941

1198891198941

]⟩ and1198672=

⋃119899(1198672)

1198942=1

⟨[1198861198942

1198871198942

] [1198881198942

1198891198942


(1) 1205821198671= ⋃119899(1198671)

1198941=1

⟨[119897minus1(120582119897(1198861198941

)) 119897minus1(120582119897(1198871198941

))][119896minus1(120582119896(1198881198941

)) 119896minus1(120582119896(119889

1198941

))]⟩

(2) (1198671)120582= ⋃119899(1198671)

119894=1⟨[119896minus1(120582119896(1198861198941

)) 119896minus1(120582119896(1198871198941

))][119897minus1(120582119897(1198881198941

)) 119897minus1(120582119897(1198891198941

))]⟩

(3) 1198671oplus1198672= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119897minus1(119897(1198861198941

)+119897(1198861198942

)) 119897minus1(119897(1198871198941

)+

119897(1198871198942

))] [119896minus1(119896(1198881198941

) + 119896(1198881198942

)) 119896minus1(119896(1198891198941

) + 119896(1198891198942

))]⟩

(4) 1198671otimes 1198672

= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119896minus1(119896(1198861198941

) + 119896(1198861198942

))

119896minus1(119896(1198871198941

) + 119896(1198871198942

))] [119897minus1(119897(1198881198941

) + 119897(1198881198942

)) 119897minus1(119897(1198891198941

) +

119897(1198891198942

))]⟩


Example 20 Let1198671= ⟨[01 03] [02 04]⟩ ⟨[02 03] [03




(1) 21198671



]




1 minus




]⟩ = ⟨[019

051] [004 016]⟩ ⟨[036 051] [009016]⟩

(2) (1198671)2= ⟨[001 009] [036 064]⟩ ⟨[004 009]

[051 064]⟩

(3) 1198671oplus 1198672= ⟨[037 058] [004 012]⟩ ⟨[044 058]

[006 012]⟩

(4) 1198671otimes 1198672= ⟨[003 012] [036 058]⟩ ⟨[006 012]

[044 058]⟩

Theorem 21 Let1198671 1198672 1198673isin 119867119868119881119868119865119873119878 120582 120582

1 1205822gt 0 and

then

(1) 1198671oplus 1198672= 1198672oplus 1198671

(2) 1198671otimes 1198672= 1198672otimes 1198671

(3) 1205821198671oplus 1205821198672= 120582(119867

1oplus 1198672)

(4) (1198671)120582otimes (1198672)120582= (1198671otimes 1198672)120582



(7) ((1198671)1205821)1205822 = (119867

1)12058211205822


(3) 1205821198671oplus 1205821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(120582119897 (119886

1198941

) + 120582119897 (1198861198942

))

119897minus1(120582119897 (119887

1198941

) + 120582119897 (1198871198942

))]

[119896minus1(120582119896 (119888

1198941

) + 120582119896 (1198881198942

))

119896minus1(120582119896 (119889

1198941

) + 120582119896 (1198891198942

))]⟩

=

119899(1198671)

⋃119894=1

119899(1198672)

⋃119895=1

⟨[119897minus1(120582 (119897 (119886

1198941

) + 119897 (1198861198942

)))

119897minus1(120582 (119897 (119887

1198941

) + 119897 (1198871198942

)))]

[119896minus1(120582 (119896 (119888

1198941

) + 119896 (1198881198942

)))

119896minus1(120582 (119896 (119889

1198941

) + 119896 (1198891198942

)))]⟩

= 120582 (1198671oplus 1198672)

(4) (1198671)120582otimes (1198672)120582

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582119896 (119886

1198941

) + 120582119896 (1198861198942

))

119896minus1(120582119896 (119887

1198941

) + 120582119896 (1198871198942

))]

[119897minus1(120582119897 (1198881198941

) + 120582119897 (1198881198942

))

119897minus1(120582119897 (119889

1198941

) + 120582119897 (1198891198942

))]⟩

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582 (119896 (119886

1198941

) + 119896 (1198861198942

)))

119896minus1(120582 (119896 (119887

1198941

) + 119896 (1198871198942

)))]

[119897minus1(120582 (119897 (119888

1198941

) + 119897 (1198881198942

)))

119897minus1(120582 (119897 (119889

1198941

) + 119897 (1198891198942

)))]⟩

= (1198671otimes 1198672)120582

(7) ((1198671)1205821)1205822

=

119899(119867)

⋃1198941=1

⟨[119896minus1(1205822119896 (119896minus1(1205821119896 (1198861198941

))))

119896minus1(1205822119896 (119896minus1(1205821119896 (1198871198941

))))]

[119897minus1(1205822119897 (119897minus1(1205821119897 (1198881198941

))))


119897minus1(1205822119897 (119897minus1(1205821119897 (1198891198941

))))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058221205821119896 (1198861198941

))

119896minus1(12058221205821119896 (1198871198941

))]

[119897minus1(12058221205821119897 (1198881198941

))

119897minus1(12058221205821119897 (1198891198941

))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058211205822119896 (1198861198941

))

119896minus1(12058211205822119896 (1198871198941

))]

[119897minus1(12058211205822119897 (1198881198941

))

119897minus1(12058211205822119897 (1198891198941

))]⟩

= (1198671)12058211205822





1and 119867

2 if 119878(119867

1) gt 119878(119867

2) then 119867

1gt 1198672 if

119878(1198671) = 119878(119867

2) then119867

1= 1198672

Note that 119878(1198671) and 119878(119867


Definitions 5 and 6

Example 23 Let1198671= ⟨[03 04] [01 02]⟩ ⟨[03 05] [02


then

119878 (1198671) =

1

2times ⟨[03 + 03 04 + 05] [01 + 02 02 + 04]⟩

= ⟨[030 045] [015 030]⟩

119878 (1198672) = ⟨[03 04] [02 03]⟩


119871 (119878 (1198671))


2

= 024

119871 (119878 (1198672))


2

= 019

(13)

Hence 119878(1198671) gt 119878(119867


1is preferred

to1198672






HIVIFNWA119908(1198671 1198672 119867

119899) =

119899

⨁119895=1

119908119895119867119895 (14)


1 1199082


119895(119895 = 1 2 119899) with 119908

119895ge

0 (119895 = 1 2 119899) and sum119899119895=1

119908119895= 1

Theorem 25 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(15)



(1) For 119899 = 2 since

11990811198671=

119899(1198671)

⋃1198941=1

⟨[119897minus1(1199081119897 (1198861198941

)) 119897minus1(1199081119897 (1198871198941

))]

[119896minus1(1199081119896 (1198881198941

)) 119896minus1(1199081119896 (1198891198941

))]⟩

11990821198672=

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199082119897 (1198861198942

)) 119897minus1(1199082119897 (1198871198942

))]

[119896minus1(1199082119896 (1198881198942

)) 119896minus1(1199082119896 (1198891198942

))]⟩

(16)


HIVIFNWA119908(1198671 1198672)

= 11990811198671oplus 11990821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

))]⟩

(17)

(2) If (15) holds for 119899 = 119896 then

HIVIFNWA119908(1198671 1198672 119867

119896)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119896119897 (119886119894119896

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119896119897 (119887119894119896

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119896119896 (119888119894119896

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119896119896 (119889119894119896

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[

[

119897minus1(

119896

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119896

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(18)


HIVIFNWA119908(1198671 1198672 119867

119896 119867119896+1

)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(119897 (119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

)

+ sdot sdot sdot + 119908119896119897 (119886119894119896

)))

+ 119908119896+1

119897 (119886119894119896+1

))

119897minus1(119897 (119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

)

+ sdot sdot sdot + 119908119896119897 (119887119894119896

)))

+ 119908119896+1

119897 (120583119887119894119896+1

))]

[119896minus1(119896 (119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

)

+ sdot sdot sdot + 119908119896119896 (119888119894119896

)))

+ 119908119896+1

119896 (119888119894119896+1

))

119896minus1(119896 (119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot + 119908119896119896 (119889119894119896

)))

+ 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942


119908119896119897 (119886119894119896

) + 119908119896+1

119897 (119886119894119896+1

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942


119908119896119897 (119887119894119896

) + 119908119896+1

119897 (119887119894119896+1

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942


119908119896119896 (119888119894119896

) + 119908119896+1

119896 (119888119894119896+1

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942


119908119896119896 (119889119894119896

) + 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[

[

119897minus1(

119896+1

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896+1

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896+1

sum119895=1

119908119895119896 (119888119894119895

))


119896minus1(

119896+1

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(19)


HIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119899119897 (119886119894119899

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119899119897 (119887119894119899

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119899119896 (119888119894119899

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119899119896 (119889119894119899

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(20)



HIVIFNWG119908(1198671 1198672 119867

119899) =

119899

⨂119895=1

(119867119895)119908119895

(21)


1 1199082 119908

119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1

119908119895= 1


Theorem 27 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 = 1

2 119899) andsum119899119895=1



HIVIFNWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119896=1

⟨[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119886119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119887119894119895

))]

]

[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119888119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119889119894119895

))]

]

⟩

(22)



HIVIFNWAA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)2

)

12

(23)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 29 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119886

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119888

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119889

119894119895

))))))]

]

⟩

(24)




HIVIFNWAG119908(1198671 1198672 119867

119899) =

1

2(

119899

⨂119895=1

(2119867119895)119908119895

)

(25)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 31 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 Then the


119867119868119881119868119865119873119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119886

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119888

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119889

119894119895

))))))]

]

⟩

(26)



GHIVIFNWA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)120582

)

1120582

(27)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1


Theorem 33 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 120582 gt 0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Then


119866119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

(28)



GHIVIFNWG119908(1198671 1198672 119867

119899) =

1

120582(

119899

⨂119895=1

(120582119867119895)119908119895

)

(29)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1



Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119908119895ge 0 (119895 = 1 2 119899) andsum119899



119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)




= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)


= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(4) 119886 otimes = ⟨[119896minus1(119896(1198861) + 119896(119886

2)) 119896minus1(119896(1198871) + 119896(119887

2))]

[119897minus1(119897(1198881) + 119897(119888

2)) 119897minus1(119897(1198891) + 119897(119889

2))]⟩


23 HFSs


119864 = ⟨119909 ℎ119864(119909)⟩ | 119909 isin 119883 (6)










(1) ℎ120582 = ⋃120574isinℎ

119896minus1(120582119896(120574))

(2) 120582ℎ = ⋃120574isinℎ

119897minus1(120582119897(120574))


119897minus1(119897(1205741) + 119897(120574

2))


119896minus1(119896(1205741) + 119896(120574

2))





1and ℎ

2 if

119904(ℎ1) gt 119904(ℎ

2) then ℎ

1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2


2= 04 07 be two


06) = 04667 119904(ℎ2) = 055 119904(ℎ

2) gt 119904(ℎ

1) so ℎ

2gt ℎ1



119899 be a set of



119899

120599 (120574) (7)





119864 = ⟨119909119867119864 (119909)⟩ | 119909 isin 119883 (8)




119867119864 (119909) =

119899(119867119864(119909))

⋃119894=1

⟨[120583119871

119864119894

(119909) 120583119880

119864119894

(119909)] []119871119864119894

(119909) ]119880119864119894

(119909)]⟩

(9)

where 0 le 120583119871

1198641

(119909) le 120583119880

1198641

(119909) le 120583119871

1198642

(119909) le 120583119880

1198642


120583119871

119899(119867119864(119909))

(119909) le 120583119880

119899(119867119864(119909))

(119909) le 1 0 le 120583119880

119864119894

(119909) + ]119880119864119894

(119909) le 1120583119871

119864119894

(119909) ge 0 ]119871119864119894



119864(119909)) = 1 then


(119909) = 120583119880

119864119894

(119909) (119894 =

1 2 119899(119867119864(119909))) and ]119871

119864119894

(119909) = ]119880119864119894

(119909) = 0 (119894 = 1 2


119864119894

(119909) =

120583119880

119864119894

(119909) (119894 = 1 2 119899(119867119864(119909))) or ]119871

119864119894

(119909) = ]119880119864119894

(119909) (119894 =


if 120583119871119864119894

(119909) = 120583119880

119864119894

(119909) (119894 = 1 2 119899(119867119864(119909))) and ]119871

119864119894

(119909) =

]119880119864119894





119867119864=

119899(119867119864)

⋃119894=1

⟨[119886119894 119887119894] [119888119894 119889119894]⟩ (10)



and for all 119909 isin 119883 Π119860(119909) = ⋃

119899(119867119860(119909))

119894=1[1minus120583

119880

119860119894

(119909)minus]119880119860119894

(119909) 1minus

120583119871

119860119894

(119909) minus ]119871119860119894




1 ⟨[03 04]

[01 02]⟩ ⟨04 02⟩⟩ ⟨1199092 ⟨[05 06][02 04]⟩⟩ be a


119860(1199092) = [0

03] Thus Π119860(119909) = ⟨119909

1 [04 06] 04⟩ ⟨119909

2 [0 03]⟩



Definition 19 Let1198671= ⋃119899(1198671)

1198941=1

⟨[1198861198941

1198871198941

] [1198881198941

1198891198941

]⟩ and1198672=

⋃119899(1198672)

1198942=1

⟨[1198861198942

1198871198942

] [1198881198942

1198891198942


(1) 1205821198671= ⋃119899(1198671)

1198941=1

⟨[119897minus1(120582119897(1198861198941

)) 119897minus1(120582119897(1198871198941

))][119896minus1(120582119896(1198881198941

)) 119896minus1(120582119896(119889

1198941

))]⟩

(2) (1198671)120582= ⋃119899(1198671)

119894=1⟨[119896minus1(120582119896(1198861198941

)) 119896minus1(120582119896(1198871198941

))][119897minus1(120582119897(1198881198941

)) 119897minus1(120582119897(1198891198941

))]⟩

(3) 1198671oplus1198672= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119897minus1(119897(1198861198941

)+119897(1198861198942

)) 119897minus1(119897(1198871198941

)+

119897(1198871198942

))] [119896minus1(119896(1198881198941

) + 119896(1198881198942

)) 119896minus1(119896(1198891198941

) + 119896(1198891198942

))]⟩

(4) 1198671otimes 1198672

= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119896minus1(119896(1198861198941

) + 119896(1198861198942

))

119896minus1(119896(1198871198941

) + 119896(1198871198942

))] [119897minus1(119897(1198881198941

) + 119897(1198881198942

)) 119897minus1(119897(1198891198941

) +

119897(1198891198942

))]⟩


Example 20 Let1198671= ⟨[01 03] [02 04]⟩ ⟨[02 03] [03




(1) 21198671



]




1 minus




]⟩ = ⟨[019

051] [004 016]⟩ ⟨[036 051] [009016]⟩

(2) (1198671)2= ⟨[001 009] [036 064]⟩ ⟨[004 009]

[051 064]⟩

(3) 1198671oplus 1198672= ⟨[037 058] [004 012]⟩ ⟨[044 058]

[006 012]⟩

(4) 1198671otimes 1198672= ⟨[003 012] [036 058]⟩ ⟨[006 012]

[044 058]⟩

Theorem 21 Let1198671 1198672 1198673isin 119867119868119881119868119865119873119878 120582 120582

1 1205822gt 0 and

then

(1) 1198671oplus 1198672= 1198672oplus 1198671

(2) 1198671otimes 1198672= 1198672otimes 1198671

(3) 1205821198671oplus 1205821198672= 120582(119867

1oplus 1198672)

(4) (1198671)120582otimes (1198672)120582= (1198671otimes 1198672)120582



(7) ((1198671)1205821)1205822 = (119867

1)12058211205822


(3) 1205821198671oplus 1205821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(120582119897 (119886

1198941

) + 120582119897 (1198861198942

))

119897minus1(120582119897 (119887

1198941

) + 120582119897 (1198871198942

))]

[119896minus1(120582119896 (119888

1198941

) + 120582119896 (1198881198942

))

119896minus1(120582119896 (119889

1198941

) + 120582119896 (1198891198942

))]⟩

=

119899(1198671)

⋃119894=1

119899(1198672)

⋃119895=1

⟨[119897minus1(120582 (119897 (119886

1198941

) + 119897 (1198861198942

)))

119897minus1(120582 (119897 (119887

1198941

) + 119897 (1198871198942

)))]

[119896minus1(120582 (119896 (119888

1198941

) + 119896 (1198881198942

)))

119896minus1(120582 (119896 (119889

1198941

) + 119896 (1198891198942

)))]⟩

= 120582 (1198671oplus 1198672)

(4) (1198671)120582otimes (1198672)120582

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582119896 (119886

1198941

) + 120582119896 (1198861198942

))

119896minus1(120582119896 (119887

1198941

) + 120582119896 (1198871198942

))]

[119897minus1(120582119897 (1198881198941

) + 120582119897 (1198881198942

))

119897minus1(120582119897 (119889

1198941

) + 120582119897 (1198891198942

))]⟩

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582 (119896 (119886

1198941

) + 119896 (1198861198942

)))

119896minus1(120582 (119896 (119887

1198941

) + 119896 (1198871198942

)))]

[119897minus1(120582 (119897 (119888

1198941

) + 119897 (1198881198942

)))

119897minus1(120582 (119897 (119889

1198941

) + 119897 (1198891198942

)))]⟩

= (1198671otimes 1198672)120582

(7) ((1198671)1205821)1205822

=

119899(119867)

⋃1198941=1

⟨[119896minus1(1205822119896 (119896minus1(1205821119896 (1198861198941

))))

119896minus1(1205822119896 (119896minus1(1205821119896 (1198871198941

))))]

[119897minus1(1205822119897 (119897minus1(1205821119897 (1198881198941

))))


119897minus1(1205822119897 (119897minus1(1205821119897 (1198891198941

))))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058221205821119896 (1198861198941

))

119896minus1(12058221205821119896 (1198871198941

))]

[119897minus1(12058221205821119897 (1198881198941

))

119897minus1(12058221205821119897 (1198891198941

))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058211205822119896 (1198861198941

))

119896minus1(12058211205822119896 (1198871198941

))]

[119897minus1(12058211205822119897 (1198881198941

))

119897minus1(12058211205822119897 (1198891198941

))]⟩

= (1198671)12058211205822





1and 119867

2 if 119878(119867

1) gt 119878(119867

2) then 119867

1gt 1198672 if

119878(1198671) = 119878(119867

2) then119867

1= 1198672

Note that 119878(1198671) and 119878(119867


Definitions 5 and 6

Example 23 Let1198671= ⟨[03 04] [01 02]⟩ ⟨[03 05] [02


then

119878 (1198671) =

1

2times ⟨[03 + 03 04 + 05] [01 + 02 02 + 04]⟩

= ⟨[030 045] [015 030]⟩

119878 (1198672) = ⟨[03 04] [02 03]⟩


119871 (119878 (1198671))


2

= 024

119871 (119878 (1198672))


2

= 019

(13)

Hence 119878(1198671) gt 119878(119867


1is preferred

to1198672






HIVIFNWA119908(1198671 1198672 119867

119899) =

119899

⨁119895=1

119908119895119867119895 (14)


1 1199082


119895(119895 = 1 2 119899) with 119908

119895ge

0 (119895 = 1 2 119899) and sum119899119895=1

119908119895= 1

Theorem 25 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(15)



(1) For 119899 = 2 since

11990811198671=

119899(1198671)

⋃1198941=1

⟨[119897minus1(1199081119897 (1198861198941

)) 119897minus1(1199081119897 (1198871198941

))]

[119896minus1(1199081119896 (1198881198941

)) 119896minus1(1199081119896 (1198891198941

))]⟩

11990821198672=

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199082119897 (1198861198942

)) 119897minus1(1199082119897 (1198871198942

))]

[119896minus1(1199082119896 (1198881198942

)) 119896minus1(1199082119896 (1198891198942

))]⟩

(16)


HIVIFNWA119908(1198671 1198672)

= 11990811198671oplus 11990821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

))]⟩

(17)

(2) If (15) holds for 119899 = 119896 then

HIVIFNWA119908(1198671 1198672 119867

119896)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119896119897 (119886119894119896

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119896119897 (119887119894119896

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119896119896 (119888119894119896

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119896119896 (119889119894119896

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[

[

119897minus1(

119896

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119896

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(18)


HIVIFNWA119908(1198671 1198672 119867

119896 119867119896+1

)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(119897 (119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

)

+ sdot sdot sdot + 119908119896119897 (119886119894119896

)))

+ 119908119896+1

119897 (119886119894119896+1

))

119897minus1(119897 (119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

)

+ sdot sdot sdot + 119908119896119897 (119887119894119896

)))

+ 119908119896+1

119897 (120583119887119894119896+1

))]

[119896minus1(119896 (119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

)

+ sdot sdot sdot + 119908119896119896 (119888119894119896

)))

+ 119908119896+1

119896 (119888119894119896+1

))

119896minus1(119896 (119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot + 119908119896119896 (119889119894119896

)))

+ 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942


119908119896119897 (119886119894119896

) + 119908119896+1

119897 (119886119894119896+1

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942


119908119896119897 (119887119894119896

) + 119908119896+1

119897 (119887119894119896+1

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942


119908119896119896 (119888119894119896

) + 119908119896+1

119896 (119888119894119896+1

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942


119908119896119896 (119889119894119896

) + 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[

[

119897minus1(

119896+1

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896+1

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896+1

sum119895=1

119908119895119896 (119888119894119895

))


119896minus1(

119896+1

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(19)


HIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119899119897 (119886119894119899

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119899119897 (119887119894119899

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119899119896 (119888119894119899

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119899119896 (119889119894119899

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(20)



HIVIFNWG119908(1198671 1198672 119867

119899) =

119899

⨂119895=1

(119867119895)119908119895

(21)


1 1199082 119908

119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1

119908119895= 1


Theorem 27 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 = 1

2 119899) andsum119899119895=1



HIVIFNWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119896=1

⟨[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119886119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119887119894119895

))]

]

[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119888119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119889119894119895

))]

]

⟩

(22)



HIVIFNWAA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)2

)

12

(23)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 29 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119886

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119888

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119889

119894119895

))))))]

]

⟩

(24)




HIVIFNWAG119908(1198671 1198672 119867

119899) =

1

2(

119899

⨂119895=1

(2119867119895)119908119895

)

(25)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 31 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 Then the


119867119868119881119868119865119873119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119886

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119888

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119889

119894119895

))))))]

]

⟩

(26)



GHIVIFNWA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)120582

)

1120582

(27)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1


Theorem 33 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 120582 gt 0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Then


119866119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

(28)



GHIVIFNWG119908(1198671 1198672 119867

119899) =

1

120582(

119899

⨂119895=1

(120582119867119895)119908119895

)

(29)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1



Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119908119895ge 0 (119895 = 1 2 119899) andsum119899



119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)




= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)


= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Definition 19 Let1198671= ⋃119899(1198671)

1198941=1

⟨[1198861198941

1198871198941

] [1198881198941

1198891198941

]⟩ and1198672=

⋃119899(1198672)

1198942=1

⟨[1198861198942

1198871198942

] [1198881198942

1198891198942


(1) 1205821198671= ⋃119899(1198671)

1198941=1

⟨[119897minus1(120582119897(1198861198941

)) 119897minus1(120582119897(1198871198941

))][119896minus1(120582119896(1198881198941

)) 119896minus1(120582119896(119889

1198941

))]⟩

(2) (1198671)120582= ⋃119899(1198671)

119894=1⟨[119896minus1(120582119896(1198861198941

)) 119896minus1(120582119896(1198871198941

))][119897minus1(120582119897(1198881198941

)) 119897minus1(120582119897(1198891198941

))]⟩

(3) 1198671oplus1198672= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119897minus1(119897(1198861198941

)+119897(1198861198942

)) 119897minus1(119897(1198871198941

)+

119897(1198871198942

))] [119896minus1(119896(1198881198941

) + 119896(1198881198942

)) 119896minus1(119896(1198891198941

) + 119896(1198891198942

))]⟩

(4) 1198671otimes 1198672

= ⋃119899(1198671)

1198941=1

⋃119899(1198672)

1198942=1

⟨[119896minus1(119896(1198861198941

) + 119896(1198861198942

))

119896minus1(119896(1198871198941

) + 119896(1198871198942

))] [119897minus1(119897(1198881198941

) + 119897(1198881198942

)) 119897minus1(119897(1198891198941

) +

119897(1198891198942

))]⟩


Example 20 Let1198671= ⟨[01 03] [02 04]⟩ ⟨[02 03] [03




(1) 21198671



]




1 minus




]⟩ = ⟨[019

051] [004 016]⟩ ⟨[036 051] [009016]⟩

(2) (1198671)2= ⟨[001 009] [036 064]⟩ ⟨[004 009]

[051 064]⟩

(3) 1198671oplus 1198672= ⟨[037 058] [004 012]⟩ ⟨[044 058]

[006 012]⟩

(4) 1198671otimes 1198672= ⟨[003 012] [036 058]⟩ ⟨[006 012]

[044 058]⟩

Theorem 21 Let1198671 1198672 1198673isin 119867119868119881119868119865119873119878 120582 120582

1 1205822gt 0 and

then

(1) 1198671oplus 1198672= 1198672oplus 1198671

(2) 1198671otimes 1198672= 1198672otimes 1198671

(3) 1205821198671oplus 1205821198672= 120582(119867

1oplus 1198672)

(4) (1198671)120582otimes (1198672)120582= (1198671otimes 1198672)120582



(7) ((1198671)1205821)1205822 = (119867

1)12058211205822


(3) 1205821198671oplus 1205821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(120582119897 (119886

1198941

) + 120582119897 (1198861198942

))

119897minus1(120582119897 (119887

1198941

) + 120582119897 (1198871198942

))]

[119896minus1(120582119896 (119888

1198941

) + 120582119896 (1198881198942

))

119896minus1(120582119896 (119889

1198941

) + 120582119896 (1198891198942

))]⟩

=

119899(1198671)

⋃119894=1

119899(1198672)

⋃119895=1

⟨[119897minus1(120582 (119897 (119886

1198941

) + 119897 (1198861198942

)))

119897minus1(120582 (119897 (119887

1198941

) + 119897 (1198871198942

)))]

[119896minus1(120582 (119896 (119888

1198941

) + 119896 (1198881198942

)))

119896minus1(120582 (119896 (119889

1198941

) + 119896 (1198891198942

)))]⟩

= 120582 (1198671oplus 1198672)

(4) (1198671)120582otimes (1198672)120582

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582119896 (119886

1198941

) + 120582119896 (1198861198942

))

119896minus1(120582119896 (119887

1198941

) + 120582119896 (1198871198942

))]

[119897minus1(120582119897 (1198881198941

) + 120582119897 (1198881198942

))

119897minus1(120582119897 (119889

1198941

) + 120582119897 (1198891198942

))]⟩

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119896minus1(120582 (119896 (119886

1198941

) + 119896 (1198861198942

)))

119896minus1(120582 (119896 (119887

1198941

) + 119896 (1198871198942

)))]

[119897minus1(120582 (119897 (119888

1198941

) + 119897 (1198881198942

)))

119897minus1(120582 (119897 (119889

1198941

) + 119897 (1198891198942

)))]⟩

= (1198671otimes 1198672)120582

(7) ((1198671)1205821)1205822

=

119899(119867)

⋃1198941=1

⟨[119896minus1(1205822119896 (119896minus1(1205821119896 (1198861198941

))))

119896minus1(1205822119896 (119896minus1(1205821119896 (1198871198941

))))]

[119897minus1(1205822119897 (119897minus1(1205821119897 (1198881198941

))))


119897minus1(1205822119897 (119897minus1(1205821119897 (1198891198941

))))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058221205821119896 (1198861198941

))

119896minus1(12058221205821119896 (1198871198941

))]

[119897minus1(12058221205821119897 (1198881198941

))

119897minus1(12058221205821119897 (1198891198941

))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058211205822119896 (1198861198941

))

119896minus1(12058211205822119896 (1198871198941

))]

[119897minus1(12058211205822119897 (1198881198941

))

119897minus1(12058211205822119897 (1198891198941

))]⟩

= (1198671)12058211205822





1and 119867

2 if 119878(119867

1) gt 119878(119867

2) then 119867

1gt 1198672 if

119878(1198671) = 119878(119867

2) then119867

1= 1198672

Note that 119878(1198671) and 119878(119867


Definitions 5 and 6

Example 23 Let1198671= ⟨[03 04] [01 02]⟩ ⟨[03 05] [02


then

119878 (1198671) =

1

2times ⟨[03 + 03 04 + 05] [01 + 02 02 + 04]⟩

= ⟨[030 045] [015 030]⟩

119878 (1198672) = ⟨[03 04] [02 03]⟩


119871 (119878 (1198671))


2

= 024

119871 (119878 (1198672))


2

= 019

(13)

Hence 119878(1198671) gt 119878(119867


1is preferred

to1198672






HIVIFNWA119908(1198671 1198672 119867

119899) =

119899

⨁119895=1

119908119895119867119895 (14)


1 1199082


119895(119895 = 1 2 119899) with 119908

119895ge

0 (119895 = 1 2 119899) and sum119899119895=1

119908119895= 1

Theorem 25 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(15)



(1) For 119899 = 2 since

11990811198671=

119899(1198671)

⋃1198941=1

⟨[119897minus1(1199081119897 (1198861198941

)) 119897minus1(1199081119897 (1198871198941

))]

[119896minus1(1199081119896 (1198881198941

)) 119896minus1(1199081119896 (1198891198941

))]⟩

11990821198672=

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199082119897 (1198861198942

)) 119897minus1(1199082119897 (1198871198942

))]

[119896minus1(1199082119896 (1198881198942

)) 119896minus1(1199082119896 (1198891198942

))]⟩

(16)


HIVIFNWA119908(1198671 1198672)

= 11990811198671oplus 11990821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

))]⟩

(17)

(2) If (15) holds for 119899 = 119896 then

HIVIFNWA119908(1198671 1198672 119867

119896)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119896119897 (119886119894119896

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119896119897 (119887119894119896

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119896119896 (119888119894119896

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119896119896 (119889119894119896

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[

[

119897minus1(

119896

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119896

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(18)


HIVIFNWA119908(1198671 1198672 119867

119896 119867119896+1

)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(119897 (119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

)

+ sdot sdot sdot + 119908119896119897 (119886119894119896

)))

+ 119908119896+1

119897 (119886119894119896+1

))

119897minus1(119897 (119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

)

+ sdot sdot sdot + 119908119896119897 (119887119894119896

)))

+ 119908119896+1

119897 (120583119887119894119896+1

))]

[119896minus1(119896 (119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

)

+ sdot sdot sdot + 119908119896119896 (119888119894119896

)))

+ 119908119896+1

119896 (119888119894119896+1

))

119896minus1(119896 (119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot + 119908119896119896 (119889119894119896

)))

+ 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942


119908119896119897 (119886119894119896

) + 119908119896+1

119897 (119886119894119896+1

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942


119908119896119897 (119887119894119896

) + 119908119896+1

119897 (119887119894119896+1

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942


119908119896119896 (119888119894119896

) + 119908119896+1

119896 (119888119894119896+1

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942


119908119896119896 (119889119894119896

) + 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[

[

119897minus1(

119896+1

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896+1

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896+1

sum119895=1

119908119895119896 (119888119894119895

))


119896minus1(

119896+1

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(19)


HIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119899119897 (119886119894119899

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119899119897 (119887119894119899

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119899119896 (119888119894119899

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119899119896 (119889119894119899

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(20)



HIVIFNWG119908(1198671 1198672 119867

119899) =

119899

⨂119895=1

(119867119895)119908119895

(21)


1 1199082 119908

119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1

119908119895= 1


Theorem 27 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 = 1

2 119899) andsum119899119895=1



HIVIFNWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119896=1

⟨[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119886119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119887119894119895

))]

]

[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119888119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119889119894119895

))]

]

⟩

(22)



HIVIFNWAA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)2

)

12

(23)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 29 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119886

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119888

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119889

119894119895

))))))]

]

⟩

(24)




HIVIFNWAG119908(1198671 1198672 119867

119899) =

1

2(

119899

⨂119895=1

(2119867119895)119908119895

)

(25)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 31 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 Then the


119867119868119881119868119865119873119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119886

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119888

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119889

119894119895

))))))]

]

⟩

(26)



GHIVIFNWA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)120582

)

1120582

(27)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1


Theorem 33 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 120582 gt 0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Then


119866119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

(28)



GHIVIFNWG119908(1198671 1198672 119867

119899) =

1

120582(

119899

⨂119895=1

(120582119867119895)119908119895

)

(29)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1



Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119908119895ge 0 (119895 = 1 2 119899) andsum119899



119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)




= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)


= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119897minus1(1205822119897 (119897minus1(1205821119897 (1198891198941

))))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058221205821119896 (1198861198941

))

119896minus1(12058221205821119896 (1198871198941

))]

[119897minus1(12058221205821119897 (1198881198941

))

119897minus1(12058221205821119897 (1198891198941

))]⟩

=

119899(119867)

⋃119894=1

⟨[119896minus1(12058211205822119896 (1198861198941

))

119896minus1(12058211205822119896 (1198871198941

))]

[119897minus1(12058211205822119897 (1198881198941

))

119897minus1(12058211205822119897 (1198891198941

))]⟩

= (1198671)12058211205822





1and 119867

2 if 119878(119867

1) gt 119878(119867

2) then 119867

1gt 1198672 if

119878(1198671) = 119878(119867

2) then119867

1= 1198672

Note that 119878(1198671) and 119878(119867


Definitions 5 and 6

Example 23 Let1198671= ⟨[03 04] [01 02]⟩ ⟨[03 05] [02


then

119878 (1198671) =

1

2times ⟨[03 + 03 04 + 05] [01 + 02 02 + 04]⟩

= ⟨[030 045] [015 030]⟩

119878 (1198672) = ⟨[03 04] [02 03]⟩


119871 (119878 (1198671))


2

= 024

119871 (119878 (1198672))


2

= 019

(13)

Hence 119878(1198671) gt 119878(119867


1is preferred

to1198672






HIVIFNWA119908(1198671 1198672 119867

119899) =

119899

⨁119895=1

119908119895119867119895 (14)


1 1199082


119895(119895 = 1 2 119899) with 119908

119895ge

0 (119895 = 1 2 119899) and sum119899119895=1

119908119895= 1

Theorem 25 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(15)



(1) For 119899 = 2 since

11990811198671=

119899(1198671)

⋃1198941=1

⟨[119897minus1(1199081119897 (1198861198941

)) 119897minus1(1199081119897 (1198871198941

))]

[119896minus1(1199081119896 (1198881198941

)) 119896minus1(1199081119896 (1198891198941

))]⟩

11990821198672=

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199082119897 (1198861198942

)) 119897minus1(1199082119897 (1198871198942

))]

[119896minus1(1199082119896 (1198881198942

)) 119896minus1(1199082119896 (1198891198942

))]⟩

(16)


HIVIFNWA119908(1198671 1198672)

= 11990811198671oplus 11990821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

))]⟩

(17)

(2) If (15) holds for 119899 = 119896 then

HIVIFNWA119908(1198671 1198672 119867

119896)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119896119897 (119886119894119896

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119896119897 (119887119894119896

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119896119896 (119888119894119896

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119896119896 (119889119894119896

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[

[

119897minus1(

119896

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119896

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(18)


HIVIFNWA119908(1198671 1198672 119867

119896 119867119896+1

)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(119897 (119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

)

+ sdot sdot sdot + 119908119896119897 (119886119894119896

)))

+ 119908119896+1

119897 (119886119894119896+1

))

119897minus1(119897 (119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

)

+ sdot sdot sdot + 119908119896119897 (119887119894119896

)))

+ 119908119896+1

119897 (120583119887119894119896+1

))]

[119896minus1(119896 (119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

)

+ sdot sdot sdot + 119908119896119896 (119888119894119896

)))

+ 119908119896+1

119896 (119888119894119896+1

))

119896minus1(119896 (119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot + 119908119896119896 (119889119894119896

)))

+ 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942


119908119896119897 (119886119894119896

) + 119908119896+1

119897 (119886119894119896+1

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942


119908119896119897 (119887119894119896

) + 119908119896+1

119897 (119887119894119896+1

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942


119908119896119896 (119888119894119896

) + 119908119896+1

119896 (119888119894119896+1

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942


119908119896119896 (119889119894119896

) + 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[

[

119897minus1(

119896+1

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896+1

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896+1

sum119895=1

119908119895119896 (119888119894119895

))


119896minus1(

119896+1

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(19)


HIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119899119897 (119886119894119899

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119899119897 (119887119894119899

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119899119896 (119888119894119899

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119899119896 (119889119894119899

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(20)



HIVIFNWG119908(1198671 1198672 119867

119899) =

119899

⨂119895=1

(119867119895)119908119895

(21)


1 1199082 119908

119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1

119908119895= 1


Theorem 27 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 = 1

2 119899) andsum119899119895=1



HIVIFNWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119896=1

⟨[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119886119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119887119894119895

))]

]

[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119888119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119889119894119895

))]

]

⟩

(22)



HIVIFNWAA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)2

)

12

(23)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 29 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119886

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119888

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119889

119894119895

))))))]

]

⟩

(24)




HIVIFNWAG119908(1198671 1198672 119867

119899) =

1

2(

119899

⨂119895=1

(2119867119895)119908119895

)

(25)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 31 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 Then the


119867119868119881119868119865119873119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119886

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119888

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119889

119894119895

))))))]

]

⟩

(26)



GHIVIFNWA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)120582

)

1120582

(27)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1


Theorem 33 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 120582 gt 0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Then


119866119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

(28)



GHIVIFNWG119908(1198671 1198672 119867

119899) =

1

120582(

119899

⨂119895=1

(120582119867119895)119908119895

)

(29)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1



Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119908119895ge 0 (119895 = 1 2 119899) andsum119899



119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)




= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)


= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(1) For 119899 = 2 since

11990811198671=

119899(1198671)

⋃1198941=1

⟨[119897minus1(1199081119897 (1198861198941

)) 119897minus1(1199081119897 (1198871198941

))]

[119896minus1(1199081119896 (1198881198941

)) 119896minus1(1199081119896 (1198891198941

))]⟩

11990821198672=

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199082119897 (1198861198942

)) 119897minus1(1199082119897 (1198871198942

))]

[119896minus1(1199082119896 (1198881198942

)) 119896minus1(1199082119896 (1198891198942

))]⟩

(16)


HIVIFNWA119908(1198671 1198672)

= 11990811198671oplus 11990821198672

=

119899(1198671)

⋃1198941=1

119899(1198672)

⋃1198942=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

))]⟩

(17)

(2) If (15) holds for 119899 = 119896 then

HIVIFNWA119908(1198671 1198672 119867

119896)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119896119897 (119886119894119896

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119896119897 (119887119894119896

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119896119896 (119888119894119896

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119896119896 (119889119894119896

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

⟨[

[

119897minus1(

119896

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119896

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(18)


HIVIFNWA119908(1198671 1198672 119867

119896 119867119896+1

)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896)

⋃119894119896=1

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(119897 (119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

)

+ sdot sdot sdot + 119908119896119897 (119886119894119896

)))

+ 119908119896+1

119897 (119886119894119896+1

))

119897minus1(119897 (119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

)

+ sdot sdot sdot + 119908119896119897 (119887119894119896

)))

+ 119908119896+1

119897 (120583119887119894119896+1

))]

[119896minus1(119896 (119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

)

+ sdot sdot sdot + 119908119896119896 (119888119894119896

)))

+ 119908119896+1

119896 (119888119894119896+1

))

119896minus1(119896 (119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot + 119908119896119896 (119889119894119896

)))

+ 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942


119908119896119897 (119886119894119896

) + 119908119896+1

119897 (119886119894119896+1

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942


119908119896119897 (119887119894119896

) + 119908119896+1

119897 (119887119894119896+1

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942


119908119896119896 (119888119894119896

) + 119908119896+1

119896 (119888119894119896+1

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942


119908119896119896 (119889119894119896

) + 119908119896+1

119896 (119889119894119896+1

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119896+1)

⋃119894119896+1=1

⟨[

[

119897minus1(

119896+1

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119896+1

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119896+1

sum119895=1

119908119895119896 (119888119894119895

))


119896minus1(

119896+1

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(19)


HIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119899119897 (119886119894119899

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119899119897 (119887119894119899

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119899119896 (119888119894119899

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119899119896 (119889119894119899

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(20)



HIVIFNWG119908(1198671 1198672 119867

119899) =

119899

⨂119895=1

(119867119895)119908119895

(21)


1 1199082 119908

119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1

119908119895= 1


Theorem 27 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 = 1

2 119899) andsum119899119895=1



HIVIFNWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119896=1

⟨[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119886119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119887119894119895

))]

]

[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119888119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119889119894119895

))]

]

⟩

(22)



HIVIFNWAA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)2

)

12

(23)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 29 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119886

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119888

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119889

119894119895

))))))]

]

⟩

(24)




HIVIFNWAG119908(1198671 1198672 119867

119899) =

1

2(

119899

⨂119895=1

(2119867119895)119908119895

)

(25)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 31 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 Then the


119867119868119881119868119865119873119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119886

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119888

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119889

119894119895

))))))]

]

⟩

(26)



GHIVIFNWA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)120582

)

1120582

(27)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1


Theorem 33 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 120582 gt 0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Then


119866119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

(28)



GHIVIFNWG119908(1198671 1198672 119867

119899) =

1

120582(

119899

⨂119895=1

(120582119867119895)119908119895

)

(29)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1



Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119908119895ge 0 (119895 = 1 2 119899) andsum119899



119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)




= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)


= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119896minus1(

119896+1

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(19)


HIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[119897minus1(1199081119897 (1198861198941

) + 1199082119897 (1198861198942

) + sdot sdot sdot 119908119899119897 (119886119894119899

))

119897minus1(1199081119897 (1198871198941

) + 1199082119897 (1198871198942

) + sdot sdot sdot + 119908119899119897 (119887119894119899

))]

[119896minus1(1199081119896 (1198881198941

) + 1199082119896 (1198881198942

) + sdot sdot sdot 119908119899119896 (119888119894119899

))

119896minus1(1199081119896 (1198891198941

) + 1199082119896 (1198891198942

)

+ sdot sdot sdot 119908119899119896 (119889119894119899

))]⟩

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119886119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119887119894119895

))]

]

[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119888119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119889119894119895

))]

]

⟩

(20)



HIVIFNWG119908(1198671 1198672 119867

119899) =

119899

⨂119895=1

(119867119895)119908119895

(21)


1 1199082 119908

119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1

119908119895= 1


Theorem 27 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 = 1

2 119899) andsum119899119895=1



HIVIFNWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119896=1

⟨[

[

119896minus1(

119899

sum119895=1

119908119895119896 (119886119894119895

))

119896minus1(

119899

sum119895=1

119908119895119896 (119887119894119895

))]

]

[

[

119897minus1(

119899

sum119895=1

119908119895119897 (119888119894119895

))

119897minus1(

119899

sum119895=1

119908119895119897 (119889119894119895

))]

]

⟩

(22)



HIVIFNWAA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)2

)

12

(23)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 29 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119895ge 0 (119895 =

1 2 119899) and sum119899119895=1



119867119868119881119868119865119873119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119886

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119888

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119889

119894119895

))))))]

]

⟩

(24)




HIVIFNWAG119908(1198671 1198672 119867

119899) =

1

2(

119899

⨂119895=1

(2119867119895)119908119895

)

(25)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 31 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 Then the


119867119868119881119868119865119873119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119886

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119888

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119889

119894119895

))))))]

]

⟩

(26)



GHIVIFNWA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)120582

)

1120582

(27)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1


Theorem 33 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 120582 gt 0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Then


119866119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

(28)



GHIVIFNWG119908(1198671 1198672 119867

119899) =

1

120582(

119899

⨂119895=1

(120582119867119895)119908119895

)

(29)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1



Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119908119895ge 0 (119895 = 1 2 119899) andsum119899



119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)




= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)


= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












HIVIFNWAG119908(1198671 1198672 119867

119899) =

1

2(

119899

⨂119895=1

(2119867119895)119908119895

)

(25)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 31 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 Then the


119867119868119881119868119865119873119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119886

119894119895

)))))) 119897minus1(1

2119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(2119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119888

119894119895

)))))) 119896minus1(1

2119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(2119896 (119889

119894119895

))))))]

]

⟩

(26)



GHIVIFNWA119908(1198671 1198672 119867

119899) = (

119899

⨁119895=1

119908119895(119867119895)120582

)

1120582

(27)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1


Theorem 33 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 120582 gt 0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Then


119866119867119868119881119868119865119873119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

(28)



GHIVIFNWG119908(1198671 1198672 119867

119899) =

1

120582(

119899

⨂119895=1

(120582119867119895)119908119895

)

(29)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895gt 0 (119895 = 1 2 119899) and sum

119899

119895=1119908119895= 1 If 120582 = 1



Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119908119895ge 0 (119895 = 1 2 119899) andsum119899



119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)




= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)


= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Theorem 35 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)


119908119895ge 0 (119895 = 1 2 119899) andsum119899



119866119867119868119881119868119865119873119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119894119895

))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (V119871

119867119895

119889119894119895

))))))]

]

⟩

(30)




= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


119867119895= 119867 = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899 then

GHIVIFNWA (1198671 1198672 119867

119899) = 119867

GHIVIFNWG (1198671 1198672 119867

119899) = 119867

(31)


= 119867 =

⋃119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894 119889119894]⟩ for all 119894 = 1 2 119899

GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894119895

)))))) 119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119894119895

)))))) 119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(119867)

⋃1198941=1

sdot sdot sdot

119899(119867)

⋃119894=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894))))))]

]

⟩

(32)

Since sum119899119895=1

119908119895= 1

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119897minus1(119897 (119896minus1(120582119896 (119886

119894))))))

= 119896minus1(1

120582119896 (119896minus1(120582119896 (119886

119894))))

= 119896minus1(1

120582(120582119896 (119886

119894))) = 119896

minus1(1

120582120582119896 (119886119894)) = 119896

minus1(119896 (119886119894)) = 119886

119894

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119894)))))) = 119887

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888119894)))))) = 119888

119894

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119894)))))) = 119889

119894

(33)


Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Hence GHIVIFNWA119908(1198671 1198672 119867

119899) = ⋃

119899(119867)

119894=1⟨[119886119894 119887119894] [119888119894


1 1198672 119867

119899) = 119867


= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

]

[119888119894119895

119889119894119895


let 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895


119895 and then

GHIVIFNWA (1 2

119899)

= GHIVIFNWA (1198671 1198672 119867

119899)

GHIVIFNWG (1 2

119899)

= GHIVIFNWG (1198671 1198672 119867

119899)

(34)

Proof Since 119895= ⋃119899(119895)

119894119895=1

⟨[119886119894119895

119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2 119899)


GHIVIFNWA119908(1198671 1198672 119867

119899)

=

119899(1)

⋃1198941=1

119899(1198672)

⋃1198942=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) +

119899

sum119895=2

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

=

119899(1)

⋃1198941=1

119899(2)

⋃1198942=1

119899(1198673)

⋃1198943=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942

))) +

119899

sum119895=2

119908119895119897 (119896minus1(120582119896 (119886

119894119895

))))))

119896minus1(1

120582119896(119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942

))) +

119899

sum119895=3

119908119895119897 (119896minus1(120582119896 (119887

119894119895

))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119888

119894119895

))))))

119897minus1(1

120582119897(119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942

))) +

119899

sum119895=3

119908119895119896 (119897minus1(120582119897 (119889

119894119895

))))))]

]

⟩

= sdot sdot sdot =

119899(1)

⋃1198941=1

sdot sdot sdot

119899(119899)

⋃119894119899=1

⟨[119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (119886

1198941

))) + 1199082119897 (119896minus1(120582119896 (119886

1198942


119894119899

))))))

119896minus1(1

120582119896 (119897minus1(1199081119897 (119896minus1(120582119896 (

1198941

))) + 1199082119897 (119896minus1(120582119896 (

1198942


119894119899

))))))]

[119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (1198881198941

))) + 1199082119896 (119897minus1(120582119897 (1198881198942


))))))

119897minus1(1

120582119897 (119896minus1(1199081119896 (119897minus1(120582119897 (119889

1198941

))) + 1199082119896 (119897minus1(120582119897 (119889

1198942


119894119899

))))))]⟩


119899)

(35)



119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899) =

GHIVIFNWG(1198671 1198672 119867

119899)


119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩



1 1198672 119867

119899) le 119867

+


1 1198672 119867

119899) le 119867

+

(36)

where119867minus = ([0 0] [1 1]) and119867+ = ([1 1] [0 0])



Theorem 36 Let 119867119895= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 = 1 2


119899)119879


119908119895ge 0 (119895 = 1 2 119899) andsum119899

119895=1119908119895= 1 119896(119909) = minus ln(119909) and

119896minus1(119909) = 119890


minus119909 119879(119909 119910) = 119909119910



HIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus

119899

prod119895=1

(1 minus 119886119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119887119894119895

)119908119895

]

]

[

[

119899

prod119895=1

(119888119894119895

)119908119895

119899

prod119895=1

(119889119894119895

)119908119895

]

]

⟩

(37)


HIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119899

prod119895=1

(119886119894119895

)119908119895

119899

prod119895=1

(119887119894119895

)119908119895

]

]

[

[

1 minus

119899

prod119895=1

(1 minus 119888119894119895

)119908119895

1 minus

119899

prod119895=1

(1 minus 119889119894119895

)119908119895

]

]

⟩

(38)


119867119868119881119868119865119873119860119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (119886119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119887119894119895

)2

)119908119895

)

12

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)2

)119908119895

)

12

]

]

⟩

(39)


HIVIFNAWAG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)2

)119908119895

)

12

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)2

)119908119895

)

12

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)2

)119908119895

)

12

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)2

)119908119895

)

12

]

]

⟩

(40)



GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










GHIVIFNAWA119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

(1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119888119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(41)


GHIVIFNAWG119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119886119894119895

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod119895=1

(1 minus (1 minus 119887119894119895

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod119895=1

(1 minus (119888119894119895

)120582

)

119908119895

)

1120582

(1 minus

119899

prod119895=1

(1 minus (119889119894119895

)120582

)

119908119895

)

1120582

]

]

⟩

(42)


Example 37 Let1198671= ⟨[02 03] [01 02]⟩ ⟨[04 05] [02




04times


04times(1 minus 03)

06] [01


0406] [1 minus (1 minus 04)



04times

(1 minus 03)06] [02

04times 0206 0304

times 0406]⟩ = ⟨[01414

03000] [01516 03031]⟩ ⟨[02347 03881] [02000

03565]⟩ Consider

HIVIFNAWG119908(1198671 1198672)

= ⟨[01320 03000] [01614 03268]⟩

⟨[01741 03680] [02000 03618]⟩

(43)


119878 (HIVIFNAWA119908(1198671 1198672))

= ⟨[01881 03441] [01758 03298]⟩

119878 (HIVIFNAWG119908(1198671 1198672))

= ⟨[01531 03340] [01807 03443]⟩

(44)

119871(119878(HIVIFNAWA119908(1198671 1198672))) =

00866 119871(119878(HIVIFNAWG119908(1198671 1198672))) = 00524 Thus


119908(1198671 1198672) (45)

(2) HIVIFNAWAA119908(1198671 1198672) =

⟨[01487 03000] [01508 02989]⟩ ⟨[02701 03972][02000 03554]⟩

HIVIFNAWAG119908(1198671 1198672)

= ⟨[01313 03000] [01677 03375]⟩

⟨[01686 03637] [02000 03642]⟩

(46)


119878 (HIVIFNAWAA119908(1198671 1198672))

= ⟨[02094 03486] [01754 03272]⟩

119878 (HIVIFNAWAG119908(1198671 1198672))

= ⟨[01500 03319] [01839 03509]⟩

(47)


119871(119878(HIVIFNAWAG119908(1198671 1198672))) = 00456 Thus


119908(1198671 1198672)

(48)

(3) GHIVIFNAWA119908(1198671 1198672) = ⟨[01680 03000]

[01481 02847]⟩ ⟨[03333 04262] [02000 03512]⟩

GHIVIFNAWG119908(1198671 1198672)

= ⟨[01292 03000] [01813 03628]⟩

⟨[01540 03502] [02000 03720]⟩

(49)



119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878 (GHIVIFNAWA119908(1198671 1198672))

= ⟨[02507 03631] [01741 03180]⟩

119878 (GHIVIFNAWG119908(1198671 1198672))

= ⟨[01416 03251] [01907 03674]⟩

(50)


119871(119878(GHIVIFNAWG119908(1198671 1198672))) = 00459 Thus


119908(1198671 1198672)

(51)


Theorem 38 Let 119867119895

= ⋃119899(119867119895)

119894119895=1

⟨[119886119894119895

119887119894119895

] [119888119894119895

119889119894119895

]⟩ (119895 =



119895(119895 = 1 2 119899) with 120582 gt

0 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 119896(119909) = ln((2 minus


119897minus1(119909) = 1 minus (2(119890

119909+ 1))119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910))



119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119886

119894119895

) (1 minus 119886119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119887

119894119895

) (1 minus 119887119894119895

))119908119895

+ 1

]]

]

[[

[

2

prod119899

119895=1((2 minus 119888

119894119895

) 119888119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119889

119894119895

) 119889119894119895

)119908119895

+ 1

]]

]

(52)


119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

prod119899

119895=1((2 minus 119886

119894119895

) 119886119894119895

)119908119895

+ 1

2

prod119899

119895=1((2 minus 119887

119894119895

) 119887119894119895

)119908119895

+ 1

]]

]

[[

[

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119888

119894119895

) (1 minus 119888119894119895

))119908119895

+ 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

minus 1

prod119899

119895=1((1 + 119889

119894119895

) (1 minus 119889119894119895

))119908119895

+ 1

]]

]

(53)


119867119868119881119868119865119873119864119882119860119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

(54)

where

120572119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

1 minus 119886119894119895

) 120573119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

1 minus 119887119894119895

)

120583119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

119888119894119895

) ]119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

119889119894119895

)

(55)



119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119867119868119881119868119865119873119864119882119860119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))12

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))12

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))12

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))12

+ 1

]]

]

(56)

where

120583119894119895

= (1198862

119894119895

minus 119886119894119895

+ 1

119886119894119895

) ]119894119895

= (1198872

119894119895

minus 119887119894119895

+ 1

119887119894119895

)

120572119894119895

= (1198882

119894119895

minus 119888119894119895

+ 1

1 minus 119888119894119895

) 120573119894119895

= (1198892

119894119895

minus 119889119894119895

+ 1

1 minus 119889119894119895

)

(57)


119866119867119868119881119868119865119873119864119882119860119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(58)

where

120572119894119895

= (

1 + (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119886119894119895

) 119886119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119887119894119895

) 119887119894119895

)120582

+ 1))

)

120583119894119895

= (

1 + (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

1 minus (2 (((1 + 119888119894119895

) (1 minus 119888119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

1 minus (2 (((1 + 119889119894119895

) (1 minus 119889119894119895

))120582

+ 1))

)

(59)



119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119866119867119868119881119868119865119873119864119882119866119908(1198671 1198672 119867

119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

[[

[

1 minus2

((prod119899

119895=1120583119908119895

119894119895

+ 3) (prod119899

119895=1120583119908119895

119894119895

minus 1))1120582

+ 1

1 minus2

((prod119899

119895=1]119908119895

119894119895

+ 3) (prod119899

119895=1]119908119895

119894119895

minus 1))1120582

+ 1

]]

]

[[

[

2

((prod119899

119895=1120572119908119895

119894119895

+ 3) (prod119899

119895=1120572119908119895

119894119895

minus 1))1120582

+ 1

2

((prod119899

119895=1120573119908119895

119894119895

+ 3) (prod119899

119895=1120573119908119895

119894119895

minus 1))1120582

+ 1

]]

]

(60)

where

120583119894119895

= (

1 + (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

1 minus (2 (((1 + 119886119894119895

) (1 minus 119886119894119895

))120582

+ 1))

)

]119894119895

= (

1 + (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

1 minus (2 (((1 + 119887119894119895

) (1 minus 119887119894119895

))120582

+ 1))

)

120572119894119895

= (

1 + (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119888119894119895

) 119888119894119895

)120582

+ 1))

)

120573119894119895

= (

1 + (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

1 minus (2 (((2 minus 119889119894119895

) 119889119894119895

)120582

+ 1))

)

(61)



1 1198862 119886119898 be a finite



119899)

where 119908119895ge 0 (119895 = 1 2 119899) sum119899

119895=1119908119895= 1 Let 119877 = (

119894119895)119898times119899





by 119894119895= ⋃119899(119894119895)

119903=1⟨[119886119903

119894119895

119887119903

119894119895

] [119888119903

119894119895

119889119903

119894119895






119894(119894 =


119910119894= GHIVIFNWA

119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119886

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119887

119903

119894119895))))))]

]

[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119888

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119889

119903

119894119895))))))]

]

⟩

(62)


119908(1198941 1198942

119894119899)

=

119899(1198671)

⋃1198941=1

sdot sdot sdot

119899(119867119899)

⋃119894119899=1

⟨[

[

119897minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119886

119903

119894119895)))))) 119897

minus1(1

120582119897(119896minus1(

119899

sum119895=1

119908119895119896 (119897minus1(120582119897 (119887

119903

119894119895))))))]

]

[

[

119896minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119888

119903

119894119895)))))) 119896

minus1(1

120582119896(119897minus1(

119899

sum119895=1

119908119895119897 (119896minus1(120582119896 (119889

119903

119894119895))))))]

]

⟩

(63)



119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894) (119894 = 1 2 119898)

could be calculated









119894(119894 = 1 2 3 4)



2 comfort 119888

3



119894



119894119895)4times4








119909) 119897minus1(119909) = 1 minus 119890



119910119894= GHIVIFNAWA (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

(1 minus

4

prod119895=1

(1 minus (119886119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119888119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(64)

or

119910119894= GHIVIFAWG (

1198941 1198942 1198943 1198944)

=

119899(1198941)

⋃1198941=1

sdot sdot sdot

119899(1198944)

⋃1198944=1

⟨[

[

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119886119896

119894119895)120582

)119908119895

)

1120582

1 minus (1 minus

4

prod119895=1

(1 minus (1 minus 119887119896

119894119895)120582

)119908119895

)

1120582

]

]

[

[

(1 minus

4

prod119895=1

(1 minus (119888119896

119894119895)120582

)119908119895

)

1120582

(1 minus

4

prod119895=1

(1 minus (119889119896

119894119895)120582

)119908119895

)

1120582

]

]

⟩

(119894 = 1 2 3 4)

(65)



119894(119894 = 1 2 3 4) could




shown in Table 3



1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198881

1198882 119888

31198884

1198861

⟨[04 05] [02 03]⟩ ⟨[01 02] [03 04]⟩ ⟨07 03⟩ ⟨[05 06] [03 04]⟩ ⟨04 02⟩

1198862

⟨[07 08] [01 02]⟩ ⟨[03 04] [02 03]⟩ ⟨[06 07] [01 03]⟩ ⟨[05 06] [02 04]⟩ ⟨[06 07] [01 03]⟩

1198863

⟨[04 05] [03 04]⟩ ⟨[04 05] [02 03]⟩ ⟨[07 08] [01 02]⟩ ⟨[06 07] [01 03]⟩

1198864

⟨[05 06] [02 03]⟩ ⟨06 03⟩⟨[02 03] [01 02]⟩⟨[05 06] [02 03]⟩

⟨[07 08] [01 02]⟩



1199101

⟨[03925 04601] [01712 02389]⟩⟨[05376 05668] [02391 02700]⟩

⟨[03373 04415] [01971 02628]⟩⟨[04788 05150] [02507 02896]⟩

1199102

⟨[05513 06545] [01359 02874]⟩⟨[06113 07126] [01107 02874]⟩

⟨[04732 05731] [01537 03020]⟩⟨[05990 06980] [01207 03020]⟩

1199103 ⟨[05398 06427] [01517 02975]⟩ ⟨[04921 05911] [01884 03118]⟩

1199104

⟨[05929 06696] [01576 02440]⟩⟨[06125 06905] [01751 02594]⟩

⟨[05880 06539] [02123 02698]⟩⟨[04938 05348] [02016 02556]⟩





119871 (119878 (1199101)) = 03725 119871 (119878 (119910

2)) = 05612

119871 (119878 (1199103)) = 05032 119871 (119878 (119910

4)) = 05681

(66)



119871 (119878 (1199101)) = 03049 119871 (119878 (119910

2)) = 04990

119871 (119878 (1199103)) = 04300 119871 (119878 (119910

4)) = 04669

(67)



2while the



is 1198861





4or 1198862is





1)) 119879(119909 119910) = 119909119910(1 + (1 minus 119909)(1 minus 119910)) and 119878(119909 119910) =



4or


1






1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1) ⟨[04081 04783] [02239 02762]⟩

119878(1199102) ⟨[05813 06836] [01233 02874]⟩ 119878(119910

2) ⟨[05361 06356] [01372 03020]⟩

119878(1199103) ⟨[05398 06427] [01517 02975]⟩ 119878(119910

3) ⟨[04921 05911] [01884 03118]⟩

119878(1199104) ⟨[06027 06801] [01664 02517]⟩ 119878(119910

4) ⟨[05409 05944] [02070 02627]⟩


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03627 05529 04927 05581 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03725 05612 05032 05681 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 03914 05836 05363 05940 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04493 06105 05804 06263 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 04949 06443 06287 06624 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05152 06642 06537 06788 1198864≻ 1198862≻ 1198863≻ 1198861


119894=1119908119894= 1 the interval-


IVIFWA119908(11988611 11988612 11988613 11988614)

=sum4

119894=1[[1198861119894 1198871119894] [1 minus 119889

1119894 1 minus 1198881119894]] times 119908

119894

sum119899

119894=1119908119894

= [[

4

sum119894=1

1198861119894119908119894

4

sum119894=1

1198871119894119908119894] [

119899

sum119894=1

(1 minus 1198891119894) 119908119894

119899

sum119894=1

(1 minus 1198881119894) 119908119894]]

= [[04 times 02 +03 + 07


05 times 02 +04 + 07



2) times 03 + (1 minus 04)


+ (1 minus01 + 03


+ (1 minus 02) times 035]]

= [[0445 0495] [0735 0785]]

= ⟨[0445 0495] [1 minus 0785 1 minus 0735]⟩

= ⟨[0445 0495] [0215 0265]⟩

IVIFWA119908(11988621 11988622 11988623 11988624)

= [[0560 0660] [0635 087]]

= ⟨[0560 0660] [013 0365]⟩

IVIFWA119908(11988631 11988632 11988633 11988634)

= [[0515 0615] [0695 0830]]

= ⟨[0515 0615] [0170 0305]⟩

IVIFWA119908(11988641 11988642 11988643 11988644)

= [[0578 0648] [0743 0813]]

= ⟨[0578 0648] [0187 0257]⟩

(68)


119871 (IVIFWA119908(11988611 11988612 11988613 11988614)) = 0343

119871 (IVIFWA119908(11988621 11988622 11988623 11988624)) = 0519

119871 (IVIFWA119908(11988631 11988632 11988633 11988634)) = 0486

119871 (IVIFWA119908(11988641 11988642 11988643 11988644)) = 0528

(69)

So 1198864

≻ 1198862

≻ 1198863







120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03291 05148 04329 04482 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03049 04990 04300 04669 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02819 04517 03804 04076 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02428 03995 02877 03338 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 02071 03523 02824 02993 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01858 03372 02608 02699 1198862≻ 1198864≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 04143 05478 04864 05515 119886

4≻ 1198862≻ 1198863≻ 1198861

120582 = 2 03708 05586 04994 05641 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 5 02752 05955 05559 06075 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 10 04751 06317 06113 06505 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 20 05146 06669 06571 06812 1198864≻ 1198862≻ 1198863≻ 1198861

120582 = 30 05295 06825 06759 06925 1198864≻ 1198862≻ 1198863≻ 1198861


120582 1198861

1198862

1198863

1198864

Rankings120582 = 1 03342 05213 04551 05127 119886

2≻ 1198864≻ 1198863≻ 1198861

120582 = 2 03156 04967 04093 04274 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 5 02669 04315 03622 03760 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 10 02262 03767 03048 03184 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 20 01951 03414 02905 02986 1198862≻ 1198864≻ 1198863≻ 1198861

120582 = 30 01745 03034 02578 02602 1198862≻ 1198864≻ 1198863≻ 1198861

6 Conclusion





Acknowledgments




References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Multicriteria Decision-Making Approach ...downloads.hindawi.com/journals/tswj/2014/868515.pdf · So the concept of hesitant interval-valued intuitionistic fuzzy sets

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