-
Research ArticleMultiple Attribute Decision Making Based
onHesitant Fuzzy Einstein Geometric Aggregation Operators
Xiaoqiang Zhou1,2 and Qingguo Li1
1 College of Mathematics and Econometrics, Hunan University,
Changsha 410082, China2 College of Mathematics, Hunan Institute of
Science and Technology, Yueyang 414006, China
Correspondence should be addressed to Qingguo Li;
[email protected]
Received 25 June 2013; Accepted 12 November 2013; Published 16
January 2014
Academic Editor: Yi-Chi Wang
Copyright © 2014 X. Zhou and Q. Li. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
We first define an accuracy function of hesitant fuzzy elements
(HFEs) and develop a new method to compare two HFEs. Then,based on
Einstein operators, we give some new operational laws on HFEs and
some desirable properties of these operations. Wealso develop
several new hesitant fuzzy aggregation operators, including the
hesitant fuzzy Einstein weighted geometric (HFEWG
𝜀)
operator and the hesitant fuzzy Einstein ordered weighted
geometric (HFEWG𝜀) operator, which are the extensions of the
weighted geometric operator and the ordered weighted geometric
(OWG) operator with hesitant fuzzy information,
respectively.Furthermore, we establish the connections between the
proposed and the existing hesitant fuzzy aggregation operators and
discussvarious properties of the proposed operators. Finally, we
apply the HFEWG
𝜀operator to solve the hesitant fuzzy decision making
problems.
1. Introduction
Atanassov [1, 2] introduced the concept of intuitionistic
fuzzyset (IFS) characterized by a membership function and a
non-membership function. It is more suitable to deal with
fuzzi-ness and uncertainty than the ordinary fuzzy set proposedby
Zadeh [3] characterized by one membership function.Information
aggregation is an important research topic inmany applications such
as fuzzy logic systems and multiat-tribute decisionmaking as
discussed byChen andHwang [4].Research on aggregation operators
with intuitionistic fuzzyinformation has received increasing
attention as shown inthe literature. Xu [5] developed some basic
arithmetic aggre-gation operators based on intuitionistic fuzzy
values (IFVs),such as the intuitionistic fuzzy weighted averaging
operatorand intuitionistic fuzzy ordered weighted averaging
operator,while Xu and Yager [6] presented some basic
geometricaggregation operators for aggregating IFVs, including
theintuitionistic fuzzy weighted geometric operator and
intu-itionistic fuzzy ordered weighted geometric operator. Basedon
these basic aggregation operators proposed in [6] and [5],
many generalized intuitionistic fuzzy aggregation operatorshave
been investigated [5–30]. Recently, Torra and Narukawa[31] and
Torra [32] proposed the hesitant fuzzy set (HFS),which is another
generalization form of fuzzy set. The char-acteristic of HFS is
that it allows membership degree to havea set of possible
values.Therefore, HFS is a very useful tool inthe situationswhere
there are somedifficulties in determiningthe membership of an
element to a set. Lately, research onaggregation methods and
multiple attribute decision makingtheories under hesitant fuzzy
environment is very active,and a lot of results have been obtained
for hesitant fuzzyinformation [33–43]. For example, Xia et al. [38]
developedsome confidence induced aggregation operators for
hesitantfuzzy information. Xia et al. [37] gave several series of
hesitantfuzzy aggregation operators with the help of
quasiarithmeticmeans. Wei [35] explored several hesitant fuzzy
prioritizedaggregation operators and applied them to hesitant
fuzzydecision making problems. Zhu et al. [43] investigated
thegeometric Bonferroni mean combining the Bonferroni meanand the
geometric mean under hesitant fuzzy environment.Xia and Xu [36]
presented some hesitant fuzzy operational
Hindawi Publishing CorporationJournal of Applied
MathematicsVolume 2014, Article ID 745617, 14
pageshttp://dx.doi.org/10.1155/2014/745617
-
2 Journal of Applied Mathematics
laws based on the relationship between the HFEs and theIFVs.
They also proposed a series of aggregation operators,such as
hesitant fuzzy weighted geometric (HFWG) operatorand hesitant fuzzy
ordered weighted geometric (HFOWG)operator. Furthermore, they
applied the proposed aggrega-tion operators to solve the multiple
attribute decisionmakingproblems.
Note that all aggregation operators introduced previouslyare
based on the algebraic product and algebraic sum of IFVs(orHFEs) to
carry out the combination process. However, thealgebraic operations
include algebraic product and algebraicsum, which are not the
unique operations that can be used toperform the intersection and
union.There aremany instancesof various t-norms and t-conorms
families which can bechosen tomodel the corresponding intersections
and unions,among which Einstein product and Einstein sum are
goodalternatives for they typically give the same smooth
approxi-mation as algebraic product and algebraic sum,
respectively.For intuitionistic fuzzy information,Wang and Liu [10,
11, 44]and Wei and Zhao [30] developed some new intuitionisticfuzzy
aggregation operators with the help of Einstein oper-ations. For
hesitant fuzzy information, however, it seems thatin the literature
there is little investigation on aggregationtechniques using the
Einstein operations to aggregate hesitantfuzzy information.
Therefore, it is necessary to develop somehesitant fuzzy
information aggregation operators based onEinstein operations.
The remainder of this paper is structured as follows.In Section
2, we briefly review some basic concepts andoperations related to
IFS andHFS. we also define an accuracyfunction of HFEs to
distinguish the two HFEs having thesame score values, based on
which we give the new com-parison laws on HFEs. In Section 3, we
present some newoperations for HFEs and discuss some basic
properties of theproposed operations. In Section 4, we develop some
novelhesitant fuzzy geometric aggregation operators with the helpof
Einstein operations, such as theHFEWG
𝜀operator and the
HFEOWG𝜀operator, and we further study various properties
of these operators. Section 5 gives an approach to solve
themultiple attribute hesitant fuzzy decision making problemsbased
on the HFEOWG
𝜀operator. Finally, Section 6 con-
cludes the paper.
2. Preliminaries
In this section, we briefly introduce Einstein operations
andsome notions of IFS and HFS. Meantime, we define anaccuracy
function of HFEs and redefine the comparison lawsbetween two
HFEs.
2.1. Einstein Operations. Since the appearance of fuzzy
settheory, the set theoretical operators have played an
importantrole and received more and more attention. It is well
knownthat the t-norms and t-conorms are the general
conceptsincluding all types of the specific operators, and they
satisfythe requirements of the conjunction and disjunction
opera-tors, respectively. There are various t-norms and
t-conormsfamilies that can be used to perform the corresponding
inter-sections and unions. Einstein sum ⊕
𝜀and Einstein product
⊗𝜀are examples of t-conorms and t-norms, respectively.They
are called Einstein operations and defined as [45]
𝑥⊗𝜀𝑦 =
𝑥 ⋅ 𝑦
1 + (1 − 𝑥) ⋅ (1 − 𝑦), 𝑥⊗
𝜀𝑦 =
𝑥 + 𝑦
1 + 𝑥 ⋅ 𝑦,
∀𝑥, 𝑦 ∈ [0, 1] .
(1)
2.2. Intuitionistic Fuzzy Set. Atanassov [1, 2] generalizedthe
concept of fuzzy set [3] and defined the concept ofintuitionistic
fuzzy set (IFS) as follows.
Definition 1. Let 𝑈 be fixed an IFS𝐴 on 𝑈 is given by;
𝐴 = {⟨𝑥, 𝜇𝐴(𝑥) , ]
𝐴(𝑥)⟩ | 𝑥 ∈ 𝑈} , (2)
where 𝜇𝐴
: 𝑈 → [0, 1] and ]𝐴
: 𝑈 → [0, 1], with thecondition 0 ≤ 𝜇
𝐴(𝑥) + ]
𝐴(𝑥) ≤ 1 for all 𝑥 ∈ 𝑈. Xu [5] called
𝑎 = (𝜇𝑎, ]𝑎) an IFV.
For IFVs, Wang and Liu [11] introduced some operationsas
follows.
Let 𝜆 > 0, 𝑎1= (𝜇𝑎1
, ]𝑎1
) and 𝑎2= (𝜇𝑎2
, ]𝑎2
) be two IFVs;then
(1) 𝑎1⊗𝜀𝑎2= (
𝜇𝑎1
+ 𝜇𝑎2
1 + 𝜇𝑎1
𝜇𝑎2
,]𝑎1
]𝑎2
1 + (1 − ]𝑎1
) (1 − ]𝑎2
))
(2) 𝑎1⊗𝜀𝑎2= (
𝜇𝑎1
𝜇𝑎2
1 + (1 − 𝜇𝑎1
) (1 − 𝜇𝑎2
),]𝑎1
+ ]𝑎2
1 + ]𝑎1
]𝑎2
)
(3) 𝑎∧𝜀𝜆
1= (
2]𝜆𝑎1
(2 − ]𝑎1
)𝜆
+ ]𝜆𝑎1
,(1 + 𝜇
𝑎1
)𝜆
− (1 − 𝜇𝑎1
)𝜆
(1 + 𝜇𝑎1
)𝜆
+ (1 − 𝜇𝑎1
)𝜆
) .
(3)
2.3. Hesitant Fuzzy Set. As another generalization of fuzzyset,
HFS was first introduced by Torra and Narukawa [31, 32].
Definition 2. Let𝑋 be a reference set; anHFS on𝑋 is in termsof a
function that when applied to𝑋 returns a subset of [0, 1].
To be easily understood, Xia and Xu use the
followingmathematical symbol to express the HFS:
𝐻 = {ℎ𝐻(𝑥)
𝑥| 𝑥 ∈ 𝑋} , (4)
where ℎ𝐻(𝑥) is a set of some values in [0, 1], denoting the
possible membership degrees of the element 𝑥 ∈ 𝑋 to the set𝐻.
For convenience, Xu and Xia [40] called ℎ
𝐻(𝑥) a hesitant
fuzzy element (HFE).Let ℎ be an HFE, ℎ− = min{𝛾 | 𝛾 ∈ ℎ}, and ℎ+
= max{𝛾 |
𝛾 ∈ ℎ}. Torra and Narukawa [31, 32] define the IFV 𝐴env(ℎ)as the
envelope of ℎ, where 𝐴env(ℎ) = (ℎ
−, 1 − ℎ+).
-
Journal of Applied Mathematics 3
Let𝛼 > 0, ℎ1and ℎ2be twoHFEs. Xia andXu [36] defined
some operations as follows:
(4) ℎ1⨁ℎ2= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{𝛾1+ 𝛾2− 𝛾1𝛾2}
(5) ℎ1⨂ℎ2= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{𝛾1𝛾2}
(6) 𝛼ℎ = ⋃𝛾∈ℎ
{𝛾𝛼
}
(7) ℎ𝛼
= ⋃𝛾∈ℎ
{1 − (1 − 𝛾)𝛼
} .
(5)
In [36], Xia and Xu defined the score function of anHFE ℎ to
compare the HFEs and gave the comparison laws.
Definition 3. Let ℎ be anHFE; 𝑠(ℎ) = (1/𝑛(ℎ))∑𝛾∈ℎ
𝛾 is calledthe score function of ℎ, where 𝑛(ℎ) is the number of
values ofℎ. For two HFEs ℎ
1and ℎ
2, if 𝑠(ℎ
1) > 𝑠(ℎ
2), then ℎ
1> ℎ2; if
𝑠(ℎ1) = 𝑠(ℎ
2), then ℎ
1= ℎ2.
From Definition 3, it can be seen that all HFEs areregarded as
the same if their score values are equal. In hesitantfuzzy decision
making process, however, we usually need tocompare two HFEs for
reordering or ranking. In the casewhere two HFEs have the same
score values, they can notbe distinguished by Definition 3.
Therefore, it is necessary todevelop a new method to overcome the
difficulty.
For an IFV, Hong and Choi [46] showed that the relationbetween
the score function and the accuracy function is sim-ilar to the
relation between mean and variance in statistics.From Definition 3,
we know that the score value of HFE ℎ isjust themean of the values
in ℎ.Motivated by the idea ofHongand Choi [46], we can define the
accuracy function of HFE ℎby using the variance of the values in
ℎ.
Definition 4. Let ℎ be an HFE; 𝑘(ℎ) = 1 −√(1/𝑛(ℎ))∑
𝛾∈ℎ(𝛾 − 𝑠(ℎ))
2 is called the accuracy function ofℎ, where 𝑛(ℎ) is the number
of values in ℎ and 𝑠(ℎ) is thescore function of ℎ.
It is well known that an efficient estimator is a measureof the
variance of an estimate’s sampling distribution instatistics: the
smaller the variance, the better the performanceof the estimator.
Motivated by this idea, it is meaningful andappropriate to
stipulate that the higher the accuracy degreeof HFE, the better the
HFE. Therefore, in the following, wedevelop a new method to compare
two HFEs, which is basedon the score function and the accuracy
function, defined asfollows.
Definition 5. Let ℎ1and ℎ
2be two HFEs and let 𝑠(⋅) and
𝑘(⋅) be the score function and accuracy function of
HFEs,respectively. Then
(1) if 𝑠(ℎ1) < 𝑠(ℎ
2), then ℎ
1is smaller than ℎ
2, denoted by
ℎ1≺ ℎ2;
(2) if 𝑠(ℎ1) = 𝑠(ℎ
2), then
(i) if 𝑘(ℎ1) < 𝑘(ℎ
2), then ℎ
1is smaller than ℎ
2,
denoted by ℎ1≺ ℎ2;
(ii) if 𝑘(ℎ1) = 𝑘(ℎ
2), then ℎ
1and ℎ
2represent
the same information, denoted by ℎ1≐ ℎ2. In
particular, if 𝛾1= 𝛾2for any 𝛾
1∈ ℎ1and 𝛾2∈ ℎ2,
then ℎ1is equal to ℎ
2, denoted by ℎ
1= ℎ2.
Example 6. Let ℎ1= {0.5}, ℎ
2= {0.1, 0.9}, ℎ
3= {0.3, 0.7},
ℎ4
= {0.1, 0.3, 0.7, 0.9}, ℎ5
= {0.2, 0.4, 0.6, 0.8}, and ℎ6
=
{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}; then 𝑠(ℎ1) =
𝑠(ℎ
2) =
𝑠(ℎ3) = 𝑠(ℎ
4) = 𝑠(ℎ
5) = 𝑠(ℎ
6) = 0.5, 𝑘(ℎ
1) = 1, 𝑘(ℎ
2) = 0.6,
𝑘(ℎ3) = 0.8, 𝑘(ℎ
4) = 0.6838, 𝑘(ℎ
5) = 0.7764, and 𝑘(ℎ
6) =
0.7418. By Definition 5, we have ℎ1≻ ℎ3≻ ℎ5≻ ℎ6≻ ℎ4≻ ℎ2.
3. Einstein Operations of Hesitant Fuzzy Sets
In this section, we will introduce the Einstein operationson
HFEs and analyze some desirable properties of
theseoperations.Motivated by the operational laws (1)–(3) on
IFVsand based on the interconnection between HFEs and IFVs,we give
some new operations of HFEs as follows.
Let 𝛼 > 0, ℎ, ℎ1, and ℎ
2be three HFEs; then
(8) ℎ1⊗𝜀ℎ2= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{𝛾1+ 𝛾2
1 + 𝛾1𝛾2
} ,
(9) ℎ1⊗𝜀ℎ2= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{𝛾1𝛾2
1 + (1 − 𝛾1) (1 − 𝛾
2)} ,
(10) ℎ∧𝜀𝛼
= ⋃𝛾∈ℎ
{2𝛾𝛼
(2 − 𝛾)𝛼
+ 𝛾𝛼} .
(6)
Proposition 7. Let 𝛼 > 0, 𝛼1> 0, 𝛼
2> 0, ℎ, ℎ
1and ℎ2be three
HFEs; then
(1) ℎ1⊗𝜀ℎ2= ℎ2⊗𝜀ℎ1,
(2) (ℎ1⊗𝜀ℎ2)⊗𝜀ℎ3= ℎ1⊗𝜀(ℎ2⊗𝜀ℎ3),
(3) (ℎ1⊗𝜀ℎ2)∧𝜀𝛼
= ℎ∧𝜀𝛼
1⊗𝜀ℎ∧𝜀𝛼
2,
(4) (ℎ∧𝜀𝛼1)∧𝜀𝛼2 = ℎ∧𝜀(𝛼1𝛼2);
(5) 𝐴 env(ℎ∧𝜀𝛼) = (𝐴 env(ℎ))
∧𝜀𝛼,
(6) 𝐴 env(ℎ1⊗𝜀ℎ2) = 𝐴 env(ℎ1)⊗𝜀𝐴 env(ℎ2).
Proof. (1) It is trivial.(2) By the operational law (9), we
have
(ℎ1⊗𝜀ℎ2) ⊗𝜀ℎ3
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,𝛾3∈ℎ3
{ ((𝛾1𝛾2/ (1 + (1 − 𝛾
1) (1 − 𝛾
2))) 𝛾3)
× (1 + (1 − (𝛾1𝛾2/ (1 + (1 − 𝛾
1) (1 − 𝛾
2))))
× (1 − 𝛾3))−1
}
-
4 Journal of Applied Mathematics
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,𝛾3∈ℎ3
{ (𝛾1𝛾2𝛾3) × (1 + (1 − 𝛾
1) (1 − 𝛾
2)
+ (1 − 𝛾1) (1 − 𝛾
3) + (1 − 𝛾
2)
× (1 − 𝛾3))−1
}
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,𝛾3∈ℎ3
{ (𝛾1(𝛾2𝛾3/ (1 + (1 − 𝛾
2) (1 − 𝛾
3))))
× (1 + (1 − 𝛾1)
× (1 − (𝛾2𝛾3/ (1 + (1 − 𝛾
2) (1 − 𝛾
3)))))−1
}
= ℎ1⊗𝜀(ℎ2⊗𝜀ℎ3) .
(7)
(3) Let ℎ = ℎ1⊗𝜀ℎ2; then ℎ = ℎ
1⊗𝜀ℎ2
=
⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{𝛾1𝛾2/(1 + (1 − 𝛾
1)(1 − 𝛾
2))}
(ℎ1⊗𝜀ℎ2)∧𝜀𝛼
= ℎ∧𝜀𝛼
= ⋃𝛾∈ℎ
{2𝛾𝛼
(2 − 𝛾)𝛼
+ 𝛾𝛼}
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{ (2(𝛾1𝛾2/ (1 + (1 − 𝛾
1) (1 − 𝛾
2)))𝛼
)
× ((2 − (𝛾1𝛾2/ (1 + (1 − 𝛾
1) (1 − 𝛾
2))))𝛼
+ (𝛾1𝛾2/ (1 + (1 − 𝛾
1) (1 − 𝛾
2)))𝛼
)−1
}
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{2(𝛾1𝛾2)𝛼
(4 − 2𝛾1− 2𝛾2+ 𝛾1𝛾2)𝛼
+ (𝛾1𝛾2)𝛼} ,
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{2(𝛾1𝛾2)𝛼
(2 − 𝛾1)𝛼
(2 − 𝛾2)𝛼
+ (𝛾1𝛾2)𝛼} .
(8)
Since ℎ∧𝜀𝛼1
= ⋃𝛾1∈ℎ{2𝛾𝛼1/((2 − 𝛾
1)𝛼
+ 𝛾𝛼1)} and ℎ∧𝜀𝛼
2=
⋃𝛾2∈ℎ{2𝛾𝛼
2/((2 − 𝛾
2)𝛼
+ 𝛾𝛼
2)}, then
ℎ∧𝜀𝛼
1⊗𝜀ℎ∧𝜀𝛼
2
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{ ((2𝛾𝛼
1/ ((2 − 𝛾
1)𝛼
+ 𝛾𝛼
1))
⋅ (2𝛾𝛼
2/ ((2 − 𝛾
2)𝛼
+ 𝛾𝛼
2)))
× (1 + (1 − (2𝛾𝛼
1/ ((2 − 𝛾
1)𝛼
+ 𝛾𝛼
1)))
× (1 − (2𝛾𝛼
2/ ((2 − 𝛾
2)𝛼
+ 𝛾𝛼
2))))−1
}
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{2(𝛾1𝛾2)𝛼
(2 − 𝛾1)𝛼
(2 − 𝛾2)𝛼
+ (𝛾1𝛾2)𝛼} .
(9)
Thus (ℎ1⊗𝜀ℎ2)∧𝜀𝛼
= ℎ∧𝜀𝛼
1⊗𝜀ℎ∧𝜀𝛼
2.
(4) Since ℎ∧𝜀𝛼1 = ⋃𝛾∈ℎ
{2𝛾𝛼1/((2 − 𝛾)𝛼1 + 𝛾𝛼1)}, then
(ℎ∧𝜀𝛼1)∧𝜀𝛼2
= ⋃𝛾∈ℎ
{ (2(2𝛾𝛼1/ ((2 − 𝛾)
𝛼1 + 𝛾𝛼1))𝛼2
)
× ((2 − (2𝛾𝛼1/ ((2 − 𝛾)𝛼1 + 𝛾𝛼1)))
𝛼2
+ (2𝛾𝛼1/ ((2 − 𝛾)𝛼1 + 𝛾𝛼1))
𝛼2
)−1
}
= ⋃𝛾∈ℎ
{2𝛾(𝛼1𝛼2)
(2 − 𝛾)(𝛼1𝛼2)
+ 𝛾(𝛼1𝛼2)}
= ℎ∧𝜀(𝛼1𝛼2).
(10)
(5) By the definition of the envelope of an HFE and theoperation
laws (3) and (10), we have
(𝐴env (ℎ))∧𝜀𝛼
= (ℎ−
, 1 − ℎ+
)∧𝜀𝛼
= (2(ℎ−)
𝛼
(2 − ℎ−)𝛼
+ (ℎ−)𝛼,[1 + (1 − ℎ+)]
𝛼
− [1 − (1 − ℎ+)]𝛼
[1 + (1 − ℎ+)]𝛼
+ [1 − (1 − ℎ+)]𝛼)
= (2(ℎ−)
𝛼
(2 − ℎ−)𝛼
+ (ℎ−)𝛼,(2 − ℎ+)
𝛼
− (ℎ+)𝛼
(2 − ℎ+)𝛼
+ (ℎ+)𝛼) .
𝐴env (ℎ∧𝜀𝛼
)
= 𝐴env (⋃𝛾∈ℎ
{2𝛾𝛼
(2 − 𝛾)𝛼
+ 𝛾𝛼})
= (2(ℎ−)
𝛼
(2 − ℎ−)𝛼
+ (ℎ−)𝛼, 1 −
2(ℎ+)𝛼
(2 − ℎ+)𝛼
+ (ℎ+)𝛼)
= (2(ℎ−)
𝛼
(2 − ℎ−)𝛼
+ (ℎ−)𝛼,(2 − ℎ+)
𝛼
− (ℎ+)𝛼
(2 − ℎ+)𝛼
+ (ℎ+)𝛼) .
(11)
Thus, 𝐴env(ℎ∧𝜀𝛼
) = (𝐴env(ℎ))∧𝜀𝛼.
(6) By the definition of the envelope of an HFE and theoperation
laws (2) and (9), we have
𝐴env (ℎ1) ⊗𝜀𝐴env (ℎ2)
= (ℎ−
1, 1 − ℎ
+
1) ⊗𝜀(ℎ−
2, 1 − ℎ
+
2)
= (ℎ−1ℎ−2
1 + (1 − ℎ−1) (1 − ℎ−
2),(1 − ℎ+
1) + (1 − ℎ+
2)
1 + (1 − ℎ+1) (1 − ℎ+
2))
-
Journal of Applied Mathematics 5
𝐴env (ℎ1⊗𝜀ℎ2)
= 𝐴env ( ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{𝛾1𝛾2
1 + (1 − 𝛾1) (1 − 𝛾
2)})
= (ℎ−1ℎ−2
1 + (1 − ℎ−1) (1 − ℎ−
2), 1 −
ℎ+1ℎ+2
1 + (1 − ℎ+1) (1 − ℎ+
2))
= (ℎ−1ℎ−2
1 + (1 − ℎ−1) (1 − ℎ−
2),(1 − ℎ+
1) + (1 − ℎ+
2)
1 + (1 − ℎ+1) (1 − ℎ+
2)) .
(12)
Thus, 𝐴env(ℎ1⊗𝜀ℎ2) = 𝐴env(ℎ1)⊗𝜀𝐴env(ℎ2).
Remark 8. Let 𝛼1> 0, 𝛼
2> 0, and ℎ be an HFE. It is worth
noting that ℎ∧𝜀𝛼1⊗𝜀ℎ∧𝜀𝛼2 ≐ ℎ∧𝜀(𝛼1+𝛼2) does not hold
necessarily
in general. To illustrate that, an example is given as
follows.
Example 9. Let ℎ = (0.3, 0.5), 𝛼1
= 𝛼2
= 1; thenℎ∧𝜀𝛼1⊗
𝜀ℎ∧𝜀𝛼2 = ℎ⊗
𝜀ℎ = ⋃
𝛾𝑖∈ℎ,𝛾𝑗∈ℎ,(𝑖,𝑗=1,2)
{𝛾𝑖𝛾𝑗/(1 + (1 −
𝛾𝑖)(1 − 𝛾
𝑗))} = (0.0604, 0.1111, 0.2), and ℎ∧𝜀(𝛼1+𝛼2) =
ℎ∧𝜀2 = ⋃𝛾∈ℎ
{2𝛾2/((2 − 𝛾)2
+ 𝛾2)} = (0.0604, 0.2). Clearly,𝑠(ℎ∧𝜀𝛼1⊗
𝜀ℎ∧𝜀𝛼2) = 0.1238 < 0.1302 = 𝑠(ℎ∧𝜀(𝛼1+𝛼2)). Thus
ℎ∧𝜀𝛼1⊗𝜀ℎ∧𝜀𝛼1 ≺ ℎ∧𝜀(𝛼1+𝛼2).
However, if the number of the values in ℎ is only one, thatis,
HFE ℎ is reduced to a fuzzy value, then the above resultholds.
Proposition 10. Let 𝛼1> 0, 𝛼
2> 0, and ℎ be an HFE, in
which the number of the values is only one, that is, ℎ =
{𝛾};then ℎ∧𝜀𝛼1⊗
𝜀ℎ∧𝜀𝛼2 = ℎ∧𝜀(𝛼1+𝛼2).
Proof. Since ℎ∧𝜀𝛼1 = ⋃𝛾∈ℎ
{2𝛾𝛼1/((2 − 𝛾)𝛼1 + 𝛾𝛼1)} and ℎ∧𝜀𝛼2 =
⋃𝛾∈ℎ
{2𝛾𝛼2/((2 − 𝛾)𝛼2 + 𝛾𝛼2)}, then
ℎ∧𝜀𝛼1⊗𝜀ℎ∧𝜀𝛼1
= ⋃𝛾∈ℎ
{ ((2𝛾𝛼1/ ((2 − 𝛾)
𝛼1 + 𝛾𝛼1))
⋅ (2𝛾𝛼2/ ((2 − 𝛾)
𝛼2 + 𝛾𝛼2)))
× (1 + (1 − (2𝛾𝛼1/ ((2 − 𝛾)
𝛼1 + 𝛾𝛼1)))
× (1 − (2𝛾𝛼2/ ((2 − 𝛾)
𝛼2 + 𝛾𝛼2))))−1
}
= ⋃𝛾∈ℎ
{ (2𝛾𝛼1 ⋅ 2𝛾𝛼2) × ([(2 − 𝛾)
𝛼1 + 𝛾𝛼1] ⋅ [(2 − 𝛾)
𝛼2 + 𝛾𝛼2]
+ [(2 − 𝛾)𝛼1 − 𝛾𝛼1]
⋅ [(2 − 𝛾)𝛼2 − 𝛾𝛼2])−1
}
= ⋃𝛾∈ℎ
{2𝛾𝛼1+𝛼2
(2 − 𝛾)𝛼1+𝛼2 + 𝛾𝛼1+𝛼2
}
= ℎ∧𝜀(𝛼1+𝛼2)
.
(13)
Proposition 10 shows that it is consistent with the result(iii)
in Theorem 2 in the literature [11].
4. Hesitant Fuzzy Einstein GeometricAggregation Operators
The weighted geometric operator [47] and the orderedweighted
geometric operator [48] are two of the most com-mon and basic
aggregation operators. Since their appearance,they have received
more and more attention. In this section,we extend them to
aggregate hesitant fuzzy information usingEinstein operations.
4.1. Hesitant Fuzzy Einstein Geometric Weighted
AggregationOperator. Based on the operational laws (5) and (7)
onHFEs,Xia and Xu [36] developed some hesitant fuzzy
aggregationoperators as listed below.
Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of HFEs; then.
(1) the hesitant fuzzy weighted geometric (HFWG) oper-ator
HFWG (ℎ1, ℎ2, . . . , ℎ
𝑛) =
𝑛
⨂𝑗=1
ℎ𝑗
𝜔𝑗
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛
{
{
{
𝑛
∏𝑗=1
𝛾𝑗
𝜔𝑗
}
}
}
,
(14)
where 𝜔 = (𝜔1, 𝜔2, . . . , 𝜔
𝑛)𝑇 is the weight vector of ℎ
𝑗(𝑗 =
1, 2, . . . , 𝑛) with 𝜔𝑗∈ [0, 1] and ∑𝑛
𝑗=1𝜔𝑗= 1.
(2) the hesitant fuzzy ordered weighted geometric(HFOWG)
operator
HFOWG (ℎ1, ℎ2, . . . , ℎ
𝑛)
=
𝑛
⨂𝑗=1
𝑤𝑗
ℎ𝜎(𝑗)
= ⋃𝛾𝜎(1)∈ℎ𝜎(1),𝛾𝜎(2)∈ℎ𝜎(2),...,𝛾𝜎(𝑛)∈ℎ𝜎(𝑛)
{
{
{
𝑛
∏𝑗=1
𝛾𝑤𝑗
𝜎(𝑗)
}
}
}
,
(15)
where 𝜎(1), 𝜎(2), . . . , 𝜎(𝑛) is a permutation of 1, 2, . . . ,
𝑛,such that ℎ
𝜎(𝑗−1)> ℎ𝜎(𝑗)
for all 𝑗 = 2, . . . , 𝑛 and 𝑤 =(𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 is aggregation-associated vector with 𝑤
𝑗∈
[0, 1] and ∑𝑛𝑗=1
𝑤𝑗= 1.
For convenience, let𝐻 be the set of all HFEs. Based on
theproposed Einstein operations onHFEs, we develop some
newaggregation operators for HFEs and discuss their
desirableproperties.
-
6 Journal of Applied Mathematics
Definition 11. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of
HFEs.
A hesitant fuzzy Einstein weighted geometric (HFEWG𝜀)
operator of dimension 𝑛 is a mapping HFEWG𝜀: 𝐻𝑛 → 𝐻
defined as follows:
HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛)
=
𝑛
⨂𝜀
𝑗=1
ℎ∧𝜀𝜔𝑗
𝑗
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛
{
{
{
2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
}
}
}
,
(16)
where 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 is the weight vector of ℎ
𝑗(𝑗 =
1, 2, . . . , 𝑛) and 𝑤𝑗> 0,∑𝑛
𝑗=1𝑤𝑗= 1. In particular, when 𝑤
𝑗=
1/𝑛, 𝑗 = 1, 2, . . . , 𝑛, the HFEWG𝜀operator is reduced to
the
hesitant fuzzy Einstein geometric (HFEG𝜀) operator:
HFEG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛)
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛
{
{
{
2∏𝑛
𝑗=1𝛾1/𝑛
𝑗
∏𝑛
𝑗=1(2 − 𝛾
𝑗)1/𝑛
+∏𝑛
𝑗=1𝛾1/𝑛
𝑗
}
}
}
.(17)
From Proposition 10, we easily get the following result.
Corollary 12. If all ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) are equal and
the
number of values in ℎ𝑗is only one, that is, ℎ
𝑗= ℎ = {𝛾} for
all 𝑗 = 1, 2, . . . , 𝑛, then
HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) = ℎ. (18)
Note that the HFEWG𝜀operator is not idempotent in
general; we give the following example to illustrate this
case.
Example 13. Let ℎ1
= ℎ2
= ℎ3
= ℎ = (0.3, 0.7), 𝑤 =(0.4, 0.25, 0.35)
𝑇; then HFEWG𝜀(ℎ1, ℎ2, ℎ3) = {0.3, 0.4137,
0.3782, 0.5126, 0.4323, 0.579, 0.5342, 0.7}. ByDefinition 3,
wehave 𝑠(HFEWG
𝜀(ℎ1, ℎ2, ℎ3)) = 0.4812 < 0.5 = 𝑠(ℎ). Hence
HFEWG𝜀(ℎ1, ℎ2, ℎ3) ≺ ℎ.
Lemma 14 (see [18, 49]). Let 𝛾𝑗> 0, 𝑤
𝑗> 0, 𝑗 = 1, 2, . . . , 𝑛,
and ∑𝑛𝑗=1
𝑤𝑗= 1. Then
𝑛
∏𝑗=1
𝛾𝑤𝑗
𝑗≤
𝑛
∑𝑗=1
𝑤𝑗𝛾𝑗 (19)
with equality if and only if 𝛾1= 𝛾2= ⋅ ⋅ ⋅ = 𝛾
𝑛.
Theorem 15. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of
HFEs
and 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 the weight vector of ℎ
𝑗(𝑗 =
1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛
𝑗=1𝑤𝑗= 1. Then
HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) ⪰ HFWG (ℎ
1, ℎ2, . . . , ℎ
𝑛) , (20)
where the equality holds if only if all ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛)
are
equal and the number of values in ℎ𝑗is only one.
Proof. For any 𝛾𝑗∈ ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛), by Lemma 14, we
have
𝑛
∏𝑗=1
(2 − 𝛾𝑗)𝑤𝑗
+
𝑛
∏𝑗=1
𝛾𝑤𝑗
𝑗≤
𝑛
∑𝑗=1
𝑤𝑗(2 − 𝛾
𝑗) +
𝑛
∑𝑗=1
𝑤𝑗𝛾𝑗= 2. (21)
Then
2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
≥
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗. (22)
It follows that 𝑠(⊗𝜀
𝑛
𝑗=1ℎ∧𝜀𝜔𝑗
𝑗) ≥ 𝑠(⊗
𝜀
𝑛
𝑗=1ℎ𝜔𝑗
𝑗), which completes
the proof of Theorem 15.
Theorem 15 tells us the result that the HFEWG𝜀operator
shows the decision maker’s more optimistic attitude than theHFWA
operator proposed by Xia and Xu [36] (i.e., (15)) inaggregation
process. To illustrate that, we give an exampleadopted from Example
1 in [36] as follows.
Example 16. Let ℎ1= (0.2, 0.3, 0.5), ℎ
2= (0.4, 0.6) be two
HFEs, and let 𝑤 = (0.7, 0.3)𝑇 be the weight vector of ℎ𝑗(𝑗 =
1, 2); then by Definition 11, we have
HFEWG𝜀(ℎ1, ℎ2) = ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{
{
{
2∏2
𝑗=1𝛾𝜔𝑗
𝑗
∏2
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏2
𝑗=1𝛾𝜔𝑗
𝑗
}
}
}
= {0.2482, 0.2856, 0.3276, 0.3744,
0.4683, 0.5288} .
(23)
However, Xia and Xu [36] used the HFWG operator toaggregate the
ℎ
𝑗(𝑗 = 1, 2) and got
HFEG (ℎ1, ℎ2)
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2
{
{
{
2
∏𝑗=1
𝛾𝑤𝑗
𝑗
}
}
}
= {0.2462, 0.2781, 0.3270, 0.3693, 0.4676, 0.5281} .
(24)
It is clear that 𝑠(HFEWG𝜀(ℎ1, ℎ2)) = 0.3722 > 0.3694 =
𝑠(HFEG(ℎ1, ℎ2)). Thus HFEWG
𝜀(ℎ1, ℎ2) ≻ HFEG(ℎ
1, ℎ2).
Based onDefinition 11 and the proposed operational laws,we can
obtain the following properties onHFEWG
𝜀operator.
Theorem 17. Let 𝛼 > 0, ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛), be a
collection of
HFEs and 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 the weight vector of ℎ
𝑗(𝑗 =
1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛
𝑖=1𝑤𝑗= 1. Then
HFEWG𝜀(ℎ∧𝜀𝛼
1, ℎ∧𝜀𝛼
2, . . . , ℎ
∧𝜀𝛼
𝑛)
= (HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛))∧𝜀𝛼
.
(25)
-
Journal of Applied Mathematics 7
Proof. Since ℎ∧𝜀𝛼𝑗
= ⋃𝛾∈ℎ𝑗
{2𝛾𝛼𝑗/((2 − 𝛾
𝑗)𝛼
+ 𝛾𝛼𝑗)} for all 𝑗 =
1, 2, . . . , 𝑛, by the definition of HFEWG𝜀, we have
HFEWG𝜀(ℎ∧𝜀𝛼
1, ℎ∧𝜀𝛼
2, . . . , ℎ
∧𝜀𝛼
𝑛)
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛
{
{
{
(2
𝑛
∏𝑗=1
(2𝛾𝛼
𝑗/ ((2 − 𝛾
𝑗)𝛼
+ 𝛾𝛼
𝑗))𝜔𝑗
)
× (
𝑛
∏𝑗=1
(2 − (2𝛾𝛼
𝑗/ ((2 − 𝛾
𝑗)𝛼
+ 𝛾𝛼
𝑗)))𝜔𝑗
+
𝑛
∏𝑗=1
(2𝛾𝛼
𝑗/ ((2 − 𝛾
𝑗)𝛼
+ 𝛾𝛼
𝑗))𝜔𝑗
)
−1
}
}
}
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛
{
{
{
2∏𝑛
𝑗=1𝛾𝛼𝜔𝑗
𝑗
∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝛼𝜔𝑗
+∏𝑛
𝑗=1𝛾𝛼𝜔𝑗
𝑗
}
}
}
.
(26)
Since HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) =
⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛
{2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗/(∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗 + ∏
𝑛
𝑗=1𝛾𝜔𝑗
𝑗)},
then
(HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛))∧𝜀𝛼
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛
{
{
{
(2(2
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗/(
𝑛
∏𝑗=1
(2 − 𝛾𝑗)𝜔𝑗
+
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗))
𝛼
)
× ((2 − (2
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗/(
𝑛
∏𝑗=1
(2 − 𝛾𝑗)𝜔𝑗
+
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗)))
𝛼
+ (2
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗/(
𝑛
∏𝑗=1
(2 − 𝛾𝑗)𝜔𝑗
+
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗))
𝛼
)
−1
}
}
}
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛
{
{
{
2(∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗)𝛼
(∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
)𝛼
+ (∏𝑛
𝑗=1𝛾𝛼𝜔𝑗
𝑗)𝛼
}
}
}
= ⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛
{
{
{
2∏𝑛
𝑗=1𝛾𝛼𝜔𝑗
𝑗
∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝛼𝜔𝑗
+∏𝑛
𝑗=1𝛾𝛼𝜔𝑗
𝑗
}
}
}
.
(27)
Theorem 18. Let ℎ be an𝐻𝐹𝐸, ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) a
collection
of HFEs, and 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 the weight vector of ℎ
𝑗
(𝑗 = 1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛
𝑖=1𝑤𝑗= 1. Then
HFEWG𝜀(ℎ1⊗𝜀ℎ, ℎ2⊗𝜀ℎ, . . . , ℎ
𝑛⊗𝜀ℎ)
= HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) ⊗𝜀ℎ.
(28)
Proof. By the definition of HFEWG𝜀and Einstein product
operator of HFEs, we have
HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) ⊗𝜀ℎ
= ⋃𝛾∈ℎ,𝛾
𝑗∈ℎ𝑗,𝑗=1,...,𝑛
{
{
{
(2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗/ (∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗)) ⋅ 𝛾
1 + (1 − (2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗/ (∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗))) (1 − 𝛾)
}
}
}
= ⋃𝛾∈ℎ,𝛾
𝑗∈ℎ𝑗,𝑗=1,...,𝑛
{
{
{
2𝛾∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
(2 − 𝛾)∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+ 𝛾∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
}
}
}
.
(29)
Since ℎ𝑗⊗𝜀ℎ = ⋃
𝛾𝑗∈ℎ𝑗,𝛾∈ℎ
{𝛾𝑗𝛾/(1 + (1 − 𝛾
𝑗)(1 − 𝛾))} for all 𝑗 =
1, 2, . . . , 𝑛, by the definition of HFEWG𝜀, we have
HFEWG𝜀(ℎ1⊗𝜀ℎ, ℎ2⊗𝜀ℎ, . . . , ℎ
𝑛⊕𝜀ℎ)
= ⋃𝛾∈ℎ,𝛾
𝑗∈ℎ𝑗,𝑗=1,...,𝑛
{
{
{
(2
𝑛
∏𝑗=1
(𝛾𝑗𝛾/ (1 + (1 − 𝛾
𝑗) (1 − 𝛾)))
𝜔𝑗
)
× (
𝑛
∏𝑗=1
(2 − (𝛾𝑗𝛾/ (1 + (1 − 𝛾
𝑗)
× (1 − 𝛾))))𝜔𝑗
+
𝑛
∏𝑗=1
(𝛾𝑗𝛾/ (1 + (1 − 𝛾
𝑗)
× (1 − 𝛾)))𝑤𝑗)
−1
}
}
}
= ⋃𝛾∈ℎ,𝛾
𝑗∈ℎ𝑗,𝑗=1,...,𝑛
{
{
{
2∏𝑛
𝑗=1(𝛾𝑗𝛾)𝜔𝑗
∏𝑛
𝑗=1((2 − 𝛾
𝑗) (2 − 𝛾))
𝜔𝑗
+∏𝑛
𝑗=1(𝛾𝑗𝛾)𝜔𝑗
}
}
}
-
8 Journal of Applied Mathematics
= ⋃𝛾∈ℎ,𝛾
𝑗∈ℎ𝑗,𝑗=1,...,𝑛
{
{
{
(2
𝑛
∏𝑗=1
𝛾𝜔𝑗 ⋅
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗)
× (
𝑛
∏𝑗=1
(2 − 𝛾)𝜔𝑗 ⋅
𝑛
∏𝑗=1
(2 − 𝛾𝑗)𝜔𝑗
+
𝑛
∏𝑗=1
𝛾𝜔𝑗
⋅
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗)
−1
}
}
}
= ⋃𝛾∈ℎ,𝛾
𝑗∈ℎ𝑗,𝑗=1,...,𝑛
{
{
{
(2𝛾∑𝑛
𝑗=1𝜔𝑗 ⋅
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗)
× ((2 − 𝛾)∑𝑛
𝑗=1𝜔𝑗 ⋅
𝑛
∏𝑗=1
(2 − 𝛾𝑗)𝜔𝑗
+ 𝛾∑𝑛
𝑗=1𝜔𝑗 ⋅
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗)
−1
}
}
}
= ⋃𝛾∈ℎ,𝛾
𝑗∈ℎ𝑗,𝑗=1,...,𝑛
{
{
{
2𝛾∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
(2 − 𝛾)∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+ 𝛾∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
}
}
}
.
(30)
Based onTheorems 17 and 18, the following property canbe
obtained easily.
Theorem 19. Let 𝛼 > 0, ℎ be an HFE, let ℎ𝑗(𝑗 = 1, 2, . . . ,
𝑛)
be a collection of HFEs, and let 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 be the
weight vector of ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) with 𝑤
𝑗∈ [0, 1] and
∑𝑛
𝑖=1𝑤𝑗= 1. Then
HFEWG𝜀(ℎ∧𝜀𝛼
1⊗𝜀ℎ, ℎ∧𝜀𝛼
1⊗𝜀ℎ, . . . , ℎ
∧𝜀𝛼
𝑛⊗𝜀ℎ)
= (HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) ⊗𝜀ℎ)∧𝜀𝛼
.
(31)
Theorem 20. Let ℎ𝑗and ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be two
collections
of HFEs and𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 the weight vector of ℎ
𝑗(𝑗 =
1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛
𝑖=1𝑤𝑗= 1. Then
HFEWG𝜀(ℎ1⊗𝜀ℎ
1, ℎ2⊗𝜀ℎ
2, . . . , ℎ
𝑛⊗𝜀ℎ
𝑛)
= HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) ⊗𝜀HFEWG
𝜀(ℎ
1, ℎ
2, . . . , ℎ
𝑛) .
(32)
Proof. By the definition of HFEWG𝜀and Einstein product
operator of HFEs, we have
HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) ⊗𝜀HFEWG
𝜀(ℎ
1, ℎ
2, . . . , ℎ
𝑛)
= ⋃
𝛾𝑗∈ℎ𝑗,𝛾
𝑗∈ℎ
𝑗,𝑗=1,...,𝑛
{
{
{
(2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗/ (∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗)) ⋅ (2∏
𝑛
𝑗=1𝛾𝑗
𝜔𝑗
/ (∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝑗
𝜔𝑗
))
1 + (1 − (2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗/ (∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗))) (1 − (2∏
𝑛
𝑗=1𝛾𝑗
𝜔𝑗/ (∏
𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝑗
𝜔𝑗)))
}
}
}
= ⋃
𝛾𝑗∈ℎ𝑗,𝛾
𝑗∈ℎ
𝑗,𝑗=1,...,𝑛
{
{
{
2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗⋅ ∏𝑛
𝑗=1𝛾𝑗
𝜔𝑗
∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
⋅ ∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗⋅ ∏𝑛
𝑗=1𝛾𝑗
𝜔𝑗
}
}
}
.
(33)
Since ℎ𝑗⊗𝜀ℎ𝑗= ⋃𝛾𝑗∈ℎ𝑗,𝛾
𝑗∈ℎ
𝑗
{𝛾𝑗𝛾𝑗/(1 + (1 − 𝛾
𝑗)(1 − 𝛾
𝑗))} for all
𝑗 = 1, 2, . . . , 𝑛, by the definition of HFEWG𝜀, we have
HFEWG𝜀(ℎ1⨂𝜀
ℎ
1, ℎ2⨂𝜀
ℎ
2, . . . , ℎ
𝑛⨂𝜀
ℎ
𝑛)
= ⋃
𝛾𝑗∈ℎ𝑗,𝛾
𝑗∈ℎ
𝑗,𝑗=1,...,𝑛
{
{
{
(2
𝑛
∏𝑗=1
(𝛾𝑗𝛾
𝑗/ (1 + (1 − 𝛾
𝑗) (1 − 𝛾
𝑗)))𝜔𝑗
)
× (
𝑛
∏𝑗=1
(2 − (𝛾𝑗𝛾
𝑗/ (1 + (1 − 𝛾
𝑗)
× (1 − 𝛾
𝑗))))𝜔𝑗
+
𝑛
∏𝑗=1
(𝛾𝑗𝛾
𝑗/ (1 + (1 − 𝛾
𝑗)
× (1 − 𝛾
𝑗)))𝜔𝑗
)
−1
}
}
}
= ⋃
𝛾𝑗∈ℎ𝑗,𝛾
𝑗∈ℎ
𝑗,𝑗=1,...,𝑛
{
{
{
(2
𝑛
∏𝑗=1
(𝛾𝑗𝛾
𝑗)𝜔𝑗
)
× (
𝑛
∏𝑗=1
[(2 − 𝛾𝑗) (2 − 𝛾
𝑗)]𝜔𝑗
+
𝑛
∏𝑗=1
(𝛾𝑗𝛾
𝑗)𝜔𝑗
)
−1
}
}
}
= ⋃
𝛾𝑗∈ℎ𝑗,𝛾
𝑗∈ℎ
𝑗,𝑗=1,...,𝑛
{
{
{
(2
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗⋅
𝑛
∏𝑗=1
𝛾
𝑗
𝜔𝑗
)
-
Journal of Applied Mathematics 9
× (
𝑛
∏𝑗=1
(2 − 𝛾𝑗)𝜔𝑗
⋅
𝑛
∏𝑗=1
(2 − 𝛾
𝑗)𝜔𝑗
+
𝑛
∏𝑗=1
𝛾𝜔𝑗
𝑗⋅
𝑛
∏𝑗=1
𝛾
𝑗
𝜔𝑗
)
−1
}
}
}
.
(34)
Theorem 21. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of
HFEs,
ℎ−min = min𝑗{ℎ−
𝑗| ℎ−𝑗= min{𝛾
𝑗∈ ℎ𝑗}}, and ℎ+max = max𝑗{ℎ
+
𝑗|
ℎ+𝑗= max{𝛾
𝑗∈ ℎ𝑗}}, and let 𝑤 = (𝑤
1, 𝑤2, . . . , 𝑤
𝑛)𝑇 be the
weight vector of ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) with 𝑤
𝑗∈ [0, 1] and
∑𝑛
𝑖=1𝑤𝑗= 1. Then
ℎ−
min ⪯ HFEWG𝜀 (ℎ1, ℎ2, . . . , ℎ𝑛) ⪯ ℎ+
max, (35)
where the equality holds if only if all ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛)
are
equal and the number of values in ℎ𝑗is only one.
Proof. Let 𝑓(𝑡) = (2 − 𝑡)/𝑡, 𝑡 ∈ [0, 1]. Then 𝑓(𝑡) = −2/𝑡2 <
0.Hence 𝑓(𝑡) is a decreasing function. Since ℎ−min ≤ ℎ
−
𝑗≤ 𝛾𝑗≤
ℎ+𝑗≤ ℎ+max for any 𝛾𝑗 ∈ ℎ𝑗 (𝑗 = 1, 2, . . . , 𝑛), then 𝑓(ℎ
+
max) ≤
𝑓(𝛾𝑗) ≤ 𝑓(ℎ−min); that is, (2 − ℎ
+
max)/ℎ+
max ≤ (2 − 𝛾𝑗)/𝛾𝑗 ≤
(2−ℎ−min)/ℎ−
min. Then for any 𝛾𝑗 ∈ ℎ𝑗 (𝑗 = 1, 2, . . . , 𝑛), we have
𝑛
∏𝑗=1
(2 − ℎ+maxℎ+max
)
𝑤𝑗
≤
𝑛
∏𝑗=1
(2 − 𝛾𝑗
𝛾𝑗
)
𝑤𝑗
≤
𝑛
∏𝑗=1
(1 − ℎ−min1 + ℎ−min
)
𝑤𝑗
⇐⇒ (2 − ℎ+maxℎ+max
)
∑𝑛
𝑗=1𝑤𝑗
≤
𝑛
∏𝑗=1
(2 − 𝛾𝑗
𝛾𝑗
)
𝑤𝑗
≤ (1 − ℎ−min1 + ℎ−min
)
∑𝑛
𝑗=1𝑤𝑗
⇐⇒ (2 − ℎ+maxℎ+max
)
≤
𝑛
∏𝑗=1
(2 − 𝛾𝑗
𝛾𝑗
)
𝑤𝑗
≤ (1 − ℎ−min1 + ℎ−min
) ⇐⇒2
ℎ+max
≤
𝑛
∏𝑗=1
(2 − 𝛾𝑗
𝛾𝑗
)
𝑤𝑗
+ 1 ≤2
ℎ−min⇐⇒
ℎ−
min2
≤1
∏𝑛
𝑗=1((2 − 𝛾
𝑗) /𝛾𝑗)𝑤𝑗
+ 1≤ (
ℎ+max2
) ⇐⇒ ℎ−
min
≤2
∏𝑛
𝑗=1((2 − 𝛾
𝑗) /𝛾𝑗)𝑤𝑗
+ 1≤ ℎ+
max ⇐⇒ ℎ−
min
≤2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
∏𝑛
𝑗=1(2 − 𝛾
𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝜔𝑗
𝑗
≤ ℎ+
max.
(36)
It follows that ℎ−min ≤ 𝑠(HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ𝑛)) ≤ ℎ+
max.Thus we have ℎ−min ⪯ HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ𝑛) ⪯ ℎ
+
max.
Remark 22. Let ℎ𝑗and ℎ
𝑗(𝑗 = 1, 2, . . . , 𝑛) be two collections
ofHFEs, and ℎ𝑗≺ ℎ
𝑗for all 𝑗; thenHFEWG
𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) ≺
HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) does not hold necessarily in general.
To illustrate that, an example is given as follows.
Example 23. Let ℎ1
= (0.45, 0.6), ℎ2
= (0.6, 0.7), ℎ3
=
(0.5, 0.6), ℎ1= (0.2, 0.9), ℎ
2= (0.45, 0.95), ℎ
3= (0.35, 0.8),
and 𝑤 = (0.5, 0.3, 0.2)𝑇; then HFEWG𝜀(ℎ1, ℎ2, ℎ3) = {0.5024,
0.5215, 0.5286, 0.5483, 0.5791, 0.6, 0.6077, 0.6291}
andHFEWG
𝜀(ℎ1, ℎ2, ℎ3) = {0.2778, 0.3372, 0.3835, 0.4595,
0.6088, 0.7099, 0.7833, 0.8947}. By Definition 3, we
have𝑠(HFEWG
𝜀(ℎ1, ℎ2, ℎ3)) = 0.5646 and 𝑠(HFEWG
𝜀(ℎ1, ℎ2,
ℎ3)) = 0.5568. It follows that HFEWG
𝜀(ℎ1, ℎ2, ℎ3) ≻
HFEWG𝜀(ℎ1, ℎ2, ℎ3). Clearly, ℎ
𝑗≺ ℎ𝑗for 𝑗 = 1, 2, 3, but
HFEWG𝜀(ℎ1, ℎ2, ℎ3) ≻ HFEWG
𝜀(ℎ
1, ℎ
2, ℎ
3).
4.2. Hesitant Fuzzy Einstein Ordered Weighted AveragingOperator.
Similar to the HFOWG operator introduced byXia and Xu [36] (i.e.,
(15)), in what follows, we developan (HFEOWG
𝜀) operator, which is an extension of OWA
operator proposed by Yager [50].
Definition 24. For a collection of the HFEs ℎ𝑗(𝑗 =
1, 2, . . . , 𝑛), a hesitant fuzzy Einstein ordered weighted
aver-aging (HFEOWG
𝜀) operator is a mapping HFEWG
𝜀: 𝐻𝑛
→
𝐻 such that
HFEOWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛)
=
𝑛
⨂𝜀
𝑗=1
ℎ∧𝜀𝑤𝑗
𝜎(𝑗)
= ⋃𝛾𝜎(1)∈ℎ𝜎(1),𝛾𝜎(2)∈ℎ𝜎(2),...,𝛾𝜎(𝑛)∈ℎ𝜎(𝑛)
{
{
{
(2
𝑛
∏𝑗=1
𝛾𝑤𝑗
𝜎(𝑗))
× (
𝑛
∏𝑗=1
(2 − 𝛾𝜎(𝑗)
)𝑤𝑗
+
𝑛
∏𝑗=1
𝛾𝑤𝑗
𝜎(𝑗))
−1
}
}
}
,
(37)
where (𝜎(1), 𝜎(2), . . . , 𝜎(𝑛)) is a permutation of (1, 2, . .
. , 𝑛),such that ℎ
𝜎(𝑗−1)≻ ℎ
𝜎(𝑗)for all 𝑗 = 2, . . . , 𝑛 and
𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 is aggregation-associated vector with
𝑤𝑗
∈ [0, 1] and ∑𝑛𝑗=1
𝑤𝑗
= 1. In particular, if 𝑤 =(1/𝑛, 1/𝑛, . . . , 1/𝑛)
𝑇, then the HFEOWG𝜀operator is reduced
to the HFEA𝜀operator of dimension 𝑛 (i.e., (17)).
-
10 Journal of Applied Mathematics
Note that the HFEOWG𝜀weights can be obtained similar
to the OWA weights. Several methods have been introducedto
determine the OWA weights in [20, 21, 50–53].
Similar to the HFEWG𝜀operator, the HFEOWG
𝜀opera-
tor has the following properties.
Theorem 25. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of
HFEs
and 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 the weight vector of ℎ
𝑗(𝑗 =
1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛
𝑗=1𝑤𝑗= 1. Then
HFEOWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) ⪰ HFOWG (ℎ
1, ℎ2, . . . , ℎ
𝑛) ,
(38)
where the equality holds if only if all ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛)
are
equal and the number of values in ℎ𝑗is only one.
From Theorem 25, we can conclude that the valuesobtained by the
HFEOWG
𝜀operator are not less than the
ones obtained by the HFOWA operator proposed by Xiaand Xu [36].
To illustrate that, let us consider the followingexample.
Example 26. Let ℎ1
= (0.1, 0.4, 0.7), ℎ2
= (0.3, 0.5), andℎ3
= (0.2, 0.6) be three HFEs and suppose that 𝑤 =(0.2, 0.45,
0.35)
𝑇 is the associated vector of the aggregationoperator.
By Definitions 3 and 4, we calculate the score values andthe
accuracy values of ℎ
1, ℎ2, and ℎ
3as follows, respectively:
𝑠(ℎ1) = 𝑠(ℎ
2) = 𝑠(ℎ
3) = 0.5, 𝑘(ℎ
1) = 0.7551, 𝑘(ℎ
2) = 0.9,
𝑘(ℎ3) = 0.8.According to Definition 5, we have ℎ
2≺ ℎ3≺ ℎ1. Then
ℎ𝜎(1)
= ℎ2, ℎ𝜎(2)
= ℎ3, ℎ𝜎(3)
= ℎ1.
By the definition of HFEOWG𝜀, we have
HFEOWG𝜀(ℎ1, ℎ2, ℎ3)
=
3
⨂𝜀
𝑗=1
ℎ∧𝜀𝑤𝑗
𝜎(𝑗)
= ⋃𝛾𝜎(1)∈ℎ𝜎(1),𝛾𝜎(2)∈ℎ𝜎(2),𝛾𝜎(3)∈ℎ𝜎(3)
{{
{{
{
2∏3
𝑗=1𝛾𝑤𝑗
𝜎(𝑗)
∏3
𝑗=1(2 − 𝛾
𝜎(𝑗))𝑤𝑗
+∏3
𝑗=1𝛾𝑤𝑗
𝜎(𝑗)
}}
}}
}
= {0.1716, 0.2787, 0.3495, 0.2939, 0.4582, 0.5598, 0.1926,
0.3106, 0.3877, 0.3272, 0.5047, 0.6125} .
(39)
If we use the HFOWA operator, which was given by Xia andXu [36]
(i.e., (15)), to aggregate the HFEs ℎ
𝑗(𝑖 = 1, 2, 3), then
we have
HFOWG (ℎ1, ℎ2, . . . , ℎ
𝑛)
=
3
⨂𝑗=1
ℎ𝑤𝑗
𝜎(𝑗)= ⋃𝛾𝜎(1)∈ℎ𝜎(1),𝛾𝜎(2)∈ℎ𝜎(2),𝛾𝜎(3)∈ℎ𝜎(3)
{
{
{
3
∏𝑗=1
𝛾𝑤𝑗
𝜎(𝑗)
}
}
}
= {0.1702, 0.2764, 0.3363, 0.2790, 0.4532, 0.5513,
0.1885, 0.3062, 0.3724, 0.3090, 0.5020, 0.6106} .
(40)
Clearly, 𝑠(HFEOWG𝜀(ℎ1, ℎ2, ℎ3)) = 0.3706 > 0.3629 =
𝑠(HFOWG(ℎ1, ℎ2, . . . , ℎ
𝑛)). By Definition 3, we have
HFEOWG𝜀(ℎ1, ℎ2, ℎ3) ≻ HFOWG(ℎ
1, ℎ2, ℎ3).
Theorem 27. Let 𝛼 > 0, ℎ be an HFE, let ℎ𝑗and ℎ
𝑗
(𝑗 = 1, 2, . . . , 𝑛) be two collection of HFEs, and let 𝑤 =(𝑤1,
𝑤2, . . . , 𝑤
𝑛)𝑇 be an aggregation-associated vector with
𝑤𝑗∈ [0, 1] and ∑𝑛
𝑗=1𝑤𝑗= 1. Then
(1) HFEWG𝜀(ℎ∧𝜀𝛼
1, ℎ∧𝜀𝛼
2, . . . , ℎ∧𝜀𝛼
𝑛) =
(HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛))∧𝜀𝛼,
(2) HFEWG𝜀(ℎ1⊗𝜀ℎ, ℎ2⊗𝜀ℎ, . . . , ℎ
𝑛⊗𝜀ℎ) = HFEWG
𝜀(ℎ1,
ℎ2, . . . , ℎ
𝑛)⊗𝜀ℎ,
(3) HFEWG𝜀(ℎ∧𝜀𝛼
1⊗𝜀ℎ, ℎ∧𝜀𝛼
1⊗𝜀ℎ, . . . , ℎ∧𝜀𝛼
𝑛⊗𝜀ℎ) =
(HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛)⊗𝜀ℎ)∧𝜀𝛼,
(4) HFEWG𝜀(ℎ1⊗𝜀ℎ1, ℎ2⊗𝜀ℎ2, . . . , ℎ
𝑛⊗𝜀ℎ𝑛) =
HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛)⊗𝜀HFEWG
𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛).
Theorem 28. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of
HFEs
and let 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 be an aggregation-associated
vector with 𝑤𝑗∈ [0, 1] and ∑𝑛
𝑗=1𝑤𝑗= 1. Then
ℎ−
min ⪯ HFEOWG𝜀 (ℎ1, ℎ2, . . . , ℎ𝑛) ⪯ ℎ+
max, (41)
where ℎ−min = min𝑗{ℎ−
𝑗| ℎ−𝑗= min{𝛾
𝑗∈ ℎ𝑗}} and ℎ+max =
max𝑗{ℎ+𝑗| ℎ+𝑗= max{𝛾
𝑗∈ ℎ𝑗}}.
Besides the above properties, we can get the followingdesirable
results on the HFOWG
𝜀operator.
Theorem 29. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of
HFEs,
and let 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 be an aggregation-associated
vector with 𝑤𝑗∈ [0, 1] and ∑𝑛
𝑗=1𝑤𝑗= 1. Then
HFEOWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) = HFEOWG
𝜀(ℎ
1, ℎ
2, . . . , ℎ
𝑛) ,
(42)
where (ℎ1, ℎ2, . . . , ℎ
𝑛) is any permutation of (ℎ
1, ℎ2, . . . , ℎ
𝑛).
Proof. Let HFEOWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) = ⊗
𝜀
𝑛
𝑗=1ℎ∧𝜀𝑤𝑗
𝜎(𝑗)
and HFEOWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) = ⊗
𝜀
𝑛
𝑗=1ℎ𝜎(𝑗)
∧𝜀𝑤𝑗 . Since
(ℎ1, ℎ2, . . . , ℎ
𝑛) is any permutation of (ℎ
1, ℎ2, . . . , ℎ
𝑛),
then we have ℎ𝜎(𝑗)
= ℎ𝜎(𝑗)
(𝑗 = 1, 2, . . . , 𝑛). ThusHFEOWG
𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛) = HFEOWG
𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛).
Theorem 30. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of
HFEs,
and let 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 be an aggregation-associated
vector with 𝑤𝑗∈ [0, 1] and ∑𝑛
𝑗=1𝑤𝑗= 1. Then
(1) if 𝑤 = (0, 0, . . . , 1), then HFOWG𝜀(ℎ1, ℎ2, . . . , ℎ
𝑛)=
min{ℎ1, ℎ2, . . . , ℎ
𝑛};
-
Journal of Applied Mathematics 11
(2) if 𝑤 = (1, 0, . . . , 0), then HFOWG𝜀(ℎ1, ℎ2, . . .,
ℎ𝑛)=max{ℎ
1, ℎ2, . . . , ℎ
𝑛};
(3) if 𝑤𝑗= 1 and 𝑤
𝑖= 0 (𝑖 ̸= 𝑗), then HFOWG
𝜀(ℎ1, ℎ2, . . .,
ℎ𝑛) = ℎ
𝜎(𝑗), where ℎ
𝜎(𝑗)is the 𝑗th largest of ℎ
𝑖(𝑖 =
1, 2, . . . , 𝑛).
5. An Application in HesitantFuzzy Decision Making
In this section, we apply the HFEWG𝜀and HFEOWG
𝜀
operators to multiple attribute decision making with
hesitantfuzzy information.
For hesitant fuzzy multiple attribute decision makingproblems,
let 𝑌 = {𝑌
1, 𝑌2, . . . , 𝑌
𝑚} be a discrete set of
alternatives, let 𝐴 = {𝐴1, 𝐴2, . . . , 𝐴
𝑛} be a collection of
attributes, and let 𝜔 = (𝜔1, 𝜔2, . . . , 𝜔
𝑛)𝑇 be the weight vector
of 𝐴𝑗(𝑗 = 1, 2, . . . , 𝑛) with 𝜔
𝑗≥ 0, 𝑗 = 1, 2, . . . , 𝑛, and
∑𝑛
𝑗=1𝜔𝑗= 1. If the decision makers provide several values for
the alternative 𝑌𝑖(𝑖 = 1, 2, . . . , 𝑚) under the attribute
𝐴
𝑗(𝑗 =
1, 2, . . . , 𝑛)with anonymity, these values can be considered
asan HFE ℎ
𝑖𝑗. In the case where two decision makers provide
the same value, the value emerges only once in ℎ𝑖𝑗. Suppose
that the decision matrix 𝐻 = (ℎ𝑖𝑗)𝑚×𝑛
is the hesitant fuzzydecision matrix, where ℎ
𝑖𝑗(𝑖 = 1, 2, . . . , 𝑚, 𝑗 = 1, 2, . . . , 𝑛) are
in the form of HFEs.To get the best alternative, we can utilize
the HFEWG
𝜀
operator or the HFEOWG𝜀operator; that is,
ℎ𝑖= HFEWG
𝜀(ℎ𝑖1, ℎ𝑖2, . . . , ℎ
𝑖𝑛)
= ⋃𝛾𝑖1∈ℎ𝑖1,𝛾𝑖2∈ℎ𝑖2,...,𝛾𝑖𝑛∈ℎ𝑖𝑛
{
{
{
2∏𝑛
𝑗=1𝛾𝜔𝑗
𝑖𝑗
∏𝑛
𝑗=1(2 − 𝛾
𝑖𝑗)𝜔𝑗
+∏𝑛
𝑗=1𝛾𝜔𝑗
𝑖𝑗
}
}
}
(43)
or
ℎ𝑖= HFEOWG
𝜀(ℎ𝑖1, ℎ𝑖2, . . . , ℎ
𝑖𝑛)
= ⋃𝛾𝑖𝜎(𝑗)∈ℎ𝑖𝜎(𝑗),𝑗=1,2,...,𝑛
{{
{{
{
2∏𝑛
𝑗=1𝛾𝑤𝑗
𝑖𝜎(𝑗)
∏𝑛
𝑗=1(2 − 𝛾
𝑖𝜎(𝑗))𝑤𝑗
+∏𝑛
𝑗=1𝛾𝑤𝑗
𝑖𝜎(𝑗)
}}
}}
}(44)
to derive the overall value ℎ𝑖of the alternatives 𝑌
𝑖(𝑖 =
1, 2, . . . , 𝑚), where𝑤 = (𝑤1, 𝑤2, . . . , 𝑤
𝑛)𝑇 is the weight vector
related to the HFEOWA𝜀operator, such that 𝑤
𝑗≥ 0, 𝑗 =
1, 2, . . . , 𝑛, and ∑𝑛𝑗=1
𝑤𝑗= 1, which can be obtained by the
normal distribution based method [20].Then by Definition 3, we
compute the scores 𝑠(ℎ
𝑖) (𝑖 =
1, 2, . . . , 𝑚) of the overall values ℎ𝑖(𝑖 = 1, 2, . . . , 𝑚)
and use
the scores 𝑠(ℎ𝑖) (𝑖 = 1, 2, . . . , 𝑚) to rank the
alternatives
𝑌 = {𝑌1, 𝑌2, . . . , 𝑌
𝑚} and then select the best one (note that
if there is no difference between the two scores ℎ𝑖and ℎ
𝑗,
thenwe need to compute the accuracy degrees 𝑘(ℎ𝑖) and 𝑘(ℎ
𝑗)
of the overall values ℎ𝑖and ℎ
𝑗by Definition 4, respectively,
and then rank the alternatives 𝑌𝑖and 𝑌
𝑗in accordance with
Definition 5).In the following, an example on multiple attribute
deci-
sion making problem involving a customer buying a car,
which is adopted from Herrera and Martinez [54], is givento
illustrate the proposed method using the HFEOWG
𝜀
operator.
Example 31. Consider that a customer wants to buy a car,which
will be chosen from five types 𝑌
𝑖(𝑖 = 1, 2, . . . , 5).
In the process of choosing one of the cars, four factors
areconsidered:𝐴
1is the consumption petrol,𝐴
2is the price,𝐴
3
is the degree of comfort, and 𝐴4is the safety factor.
Suppose
that the characteristic information of the alternatives 𝑌𝑖(𝑖
=
1, 2, . . . , 5) can be represented by HFEs ℎ𝑖𝑗(𝑖 = 1, 2, . . .
, 5; 𝑗 =
1, 2, . . . , 4), and the hesitant fuzzy decision matrix is
given inTable 1.
To use HFEOWG𝜀operator, we first reorder the ℎ
𝑖𝑗(𝑗 =
1, 2, . . . , 4) for each alternative 𝑌𝑖(𝑖 = 1, 2, . . . , 5).
According
to Definitions 3 and 4, we compute the score values andaccuracy
degrees of 𝑠(ℎ
𝑖𝑗) (𝑖 = 1, 2, . . . , 5; 𝑗 = 1, 2, . . . , 4) as
follows:
𝑠 (ℎ11) = 0.45, 𝑠 (ℎ
12) = 0.75, 𝑠 (ℎ
13) = 0.3,
𝑠 (ℎ14) = 0.3, 𝑘 (ℎ
13) = 0.9184, 𝑘 (ℎ
14) = 0.9;
𝑠 (ℎ21) = 0.5, 𝑠 (ℎ
22) = 0.7, 𝑠 (ℎ
23) = 0.7,
𝑠 (ℎ24) = 0.5, 𝑘 (ℎ
21) = 0.7551, 𝑘 (ℎ
24) = 0.8129,
𝑘 (ℎ22) = 0.8367, 𝑘 (ℎ
23) = 0.9;
𝑠 (ℎ31) = 0.85, 𝑠 (ℎ
32) = 0.4, 𝑠 (ℎ
33) = 0.35,
𝑠 (ℎ34) = 0.4, 𝑘 (ℎ
32) = 0.8367, 𝑘 (ℎ
34) = 0.7764;
𝑠 (ℎ41) = 0.6, 𝑠 (ℎ
42) = 0.6, 𝑠 (ℎ
43) = 0.3,
𝑠 (ℎ44) = 0.4, 𝑘 (ℎ
41) = 0.772, 𝑘 (ℎ
42) = 0.8367;
𝑠 (ℎ51) = 0.5, 𝑠 (ℎ
52) = 0.3, 𝑠 (ℎ
53) = 0.5,
𝑠 (ℎ54) = 0.35, 𝑘 (ℎ
51) = 0.8367, 𝑘 (ℎ
53) = 0.8129.
(45)
Then by Definition 5, we have
ℎ1𝜎(1)
= ℎ12, ℎ1𝜎(2)
= ℎ11, ℎ1𝜎(3)
= ℎ13, ℎ1𝜎(4)
= ℎ14;
ℎ2𝜎(1)
= ℎ23, ℎ2𝜎(2)
= ℎ22, ℎ2𝜎(3)
= ℎ24, ℎ2𝜎(4)
= ℎ21;
ℎ3𝜎(1)
= ℎ31, ℎ3𝜎(2)
= ℎ32, ℎ3𝜎(3)
= ℎ34, ℎ3𝜎(4)
= ℎ33;
ℎ4𝜎(1)
= ℎ42, ℎ4𝜎(2)
= ℎ41, ℎ4𝜎(3)
= ℎ44, ℎ4𝜎(4)
= ℎ43;
ℎ5𝜎(1)
= ℎ51, ℎ5𝜎(2)
= ℎ53, ℎ5𝜎(3)
= ℎ54, ℎ5𝜎(4)
= ℎ52.
(46)
Suppose that 𝑤 = (0.1835, 0.3165, 0.3165, 0.1835)𝑇 is
theweighted vector related to the HFEOWA
𝜀operator and it
is derived by the normal distribution based method [20].Thenwe
utilize theHFEOWA
𝜀operator to obtain the hesitant
-
12 Journal of Applied Mathematics
Table 1: Hesitant fuzzy decision making matrix.
𝐴1
𝐴2
𝐴3
𝐴4
𝑌1
{0.4, 0.5} {0.7, 0.8} {0.2, 0.3, 0.4} {0.2, 0.4}
𝑌2
{0.2, 0.5, 0.8} {0.5, 0.7, 0.9} {0.6, 0.8} {0.2, 0.5, 0.6,
0.7}
𝑌3
{0.8, 0.9} {0.2, 0.4, 0.6} {0.2, 0.3, 0.4, 0.5} {0.1, 0.3, 0.5,
0.7}
𝑌4
{0.3, 0.4, 0.6, 0.8, 0.9} {0.4, 0.6, 0.8} {0.1, 0.2, 0.4, 0.5}
{0.2, 0.3, 0.5, 0.6}
𝑌5
{0.3, 0.5, 0.7} {0.2, 0.3, 0.4} {0.2, 0.5, 0.6, 0.7} {0.1, 0.3,
0.4, 0.6}
fuzzy elements ℎ𝑖(𝑖 = 1, 2, 3, 4, 5) for the alternatives 𝑋
𝑖
(𝑖 = 1, 2, 3, 4, 5). Take alternative𝑋1for an example; we
have
ℎ1= HFEOWG
𝜀(ℎ11, ℎ12, . . . , ℎ
14)
= ⋃𝛾1𝜎(𝑗)∈ℎ1𝜎(𝑗),𝑗=1,2,3,4
{{
{{
{
2∏4
𝑗=1𝛾𝑤𝑗
1𝜎(𝑗)
∏4
𝑗=1(2 − 𝛾
1𝜎(𝑗))𝑤𝑗
+∏4
𝑗=1𝛾𝑤𝑗
1𝜎(𝑗)
}}
}}
}
= {0.3220, 0.3642, 0.3635, 0.4099, 0.3974, 0.4470,
0.3473, 0.3921, 0.3914, 0.4403, 0.4272, 0.4794,
0.3327, 0.3760, 0.3753, 0.4228, 0.4101, 0.4607,
0.3587, 0.4046, 0.4039, 0.4539, 0.4405, 0.4938} .
(47)
The results can be obtained similarly for the other
alterna-tives; here we will not list them for vast amounts of data.
ByDefinition 3, the score values 𝑠(ℎ
𝑖) of ℎ𝑖(𝑖 = 1, 2, 3, 4, 5) can
be computed as follows:
𝑠 (ℎ1) = 0.4048, 𝑠 (ℎ
2) = 0.5758, 𝑠 (ℎ
3) = 0.4311,
𝑠 (ℎ4) = 0.4479, 𝑠 (ℎ
5) = 0.3620.
(48)
According to the scores 𝑠(ℎ𝑖) of the overall hesitant fuzzy
values ℎ𝑖(𝑖 = 1, 2, 3, 4, 5), we can rank all the alternatives
𝑋
𝑖:
𝑋2≻ 𝑋4≻ 𝑋3≻ 𝑋1≻ 𝑋5. Thus the optimal alternative is
𝑋2.If we use the HFWG operator introduced by Xia and Xu
[36] to aggregate the hesitant fuzzy values, then
𝑠 (ℎ1) = 0.3960, 𝑠 (ℎ
2) = 0.5630, 𝑠 (ℎ
3) = 0.4164,
𝑠 (ℎ4) = 0.4344, 𝑠 (ℎ
5) = 0.3548.
(49)
By Definition 5, we have𝑋2≻ 𝑋4≻ 𝑋3≻ 𝑋1≻ 𝑋5.
Note that the rankings are the same in such two cases, butthe
overall values of alternatives by the HFEOWG
𝜀operator
are not smaller than the ones by the HFOWG operator.It shows
that the attitude of the decision maker using theproposed
HFEOWG
𝜀operator is more optimistic than the
one using the HFOWG operator introduced by Xia andXu [36] in
aggregation process. Therefore, according to thedecision makers’
optimistic (or pessimistic) attitudes, thedifferent hesitant fuzzy
aggregation operators can be used toaggregate the hesitant fuzzy
information in decision makingprocess.
6. Conclusions
The purpose of multicriteria decision making is to select
theoptimal alternative from several alternatives or to get
theirranking by aggregating the performances of each
alternativeunder some attributes, which is the pervasive
phenomenonin modern life. Hesitancy is the most common problemin
decision making, for which hesitant fuzzy set can beconsidered as a
suitable means allowing several possibledegrees for an element to a
set. Therefore, the hesitant fuzzymultiple attribute decision
making problems have receivedmore and more attention. In this
paper, an accuracy functionof HFEs has been defined for
distinguishing between thetwo HFEs having the same score values,
and a new orderrelation between two HFEs has been provided. Some
Ein-stein operations on HFEs and their basic properties havebeen
presented. With the help of the proposed operations,several new
hesitant fuzzy aggregation operators includingthe HFEWG
𝜀operator and HFEOWG
𝜀operator have been
developed, which are extensions of the weighted
geometricoperator and the OWGoperator with hesitant fuzzy
informa-tion, respectively. Moreover, some desirable properties of
theproposed operators have been discussed and the
relationshipsbetween the proposed operators and the existing
hesitantfuzzy aggregation operators introduced by Xia and Xu
[36]have been established. Finally, based on the HFEOWG
𝜀
operator, an approach of hesitant fuzzy decision making hasbeen
given and a practical example has been presented todemonstrate its
practicality and effectiveness.
Conflict of Interests
The authors declared that there is no conflict of
interestsregarding the publication of this paper.
Acknowledgments
The authors are very grateful to the editors and the anony-mous
reviewers for their insightful and constructive com-ments and
suggestions that have led to an improved version ofthis paper. This
work was supported by the National NaturalScience Foundation of
China (nos. 11071061 and 11101135)and the National Basic Research
Program of China (no.2011CB311808).
References
[1] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets
andSystems, vol. 20, no. 1, pp. 87–96, 1986.
-
Journal of Applied Mathematics 13
[2] K. T. Atanassov, “More on intuitionistic fuzzy sets,” Fuzzy
Setsand Systems, vol. 33, no. 1, pp. 37–45, 1989.
[3] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8,
no. 3,pp. 338–353, 1965.
[4] S. Chen and C. Hwang, Fuzzy Multiple Attribute
DecisionMaking, Springer, Berlin, Germany, 1992.
[5] Z. Xu, “Intuitionistic fuzzy aggregation operators,” IEEE
Trans-actions on Fuzzy Systems, vol. 15, no. 6, pp. 1179–1187,
2007.
[6] Z. Xu and R. R. Yager, “Some geometric aggregation
operatorsbased on intuitionistic fuzzy sets,” International Journal
ofGeneral Systems, vol. 35, no. 4, pp. 417–433, 2006.
[7] D.-F. Li, “Multiattribute decision making method based
ongeneralized OWA operators with intuitionistic fuzzy sets,”Expert
Systems with Applications, vol. 37, no. 12, pp. 8673–8678,2010.
[8] D.-F. Li, “TheGOWAoperator based approach
tomultiattributedecision making using intuitionistic fuzzy sets,”
Mathematicaland Computer Modelling, vol. 53, no. 5-6, pp.
1182–1196, 2011.
[9] J. M. Merigó and M. Casanovas, “Fuzzy generalized
hybridaggregation operators and its application in fuzzy
decisionmaking,” International Journal of Fuzzy Systems, vol. 12,
no. 1,pp. 15–24, 2010.
[10] W. Wang and X. Liu, “Intuitionistic fuzzy information
aggre-gation using einstein operations,” IEEE Transactions on
FuzzySystems, vol. 20, no. 5, pp. 923–938, 2012.
[11] W.Wang andX. Liu, “Intuitionistic fuzzy geometric
aggregationoperators based on einstein operations,” International
Journal ofIntelligent Systems, vol. 26, no. 11, pp. 1049–1075,
2011.
[12] G. Wei, “Some induced geometric aggregation operators
withintuitionistic fuzzy information and their application to
groupdecisionmaking,”Applied Soft Computing Journal, vol. 10, no.
2,pp. 423–431, 2010.
[13] G. W. Wei, “Some geometric aggregation functions and
theirapplication to dynamic multiple attribute decision makingin
the intuitionistic fuzzy setting,” International Journal
ofUncertainty, Fuzziness and Knowlege-Based Systems, vol. 17, no.2,
pp. 179–196, 2009.
[14] G. Wei, “Some induced geometric aggregation operators
withintuitionistic fuzzy information and their application to
groupdecisionmaking,”Applied Soft Computing Journal, vol. 10, no.
2,pp. 423–431, 2010.
[15] G. Wei and X. Zhao, “Some induced correlated
aggregatingoperators with intuitionistic fuzzy information and
their appli-cation to multiple attribute group decision making,”
ExpertSystems with Applications, vol. 39, no. 2, pp. 2026–2034,
2012.
[16] M. Xia and Z. Xu, “Entropy/cross entropy-based group
decisionmaking under intuitionistic fuzzy environment,”
InformationFusion, vol. 13, no. 1, pp. 31–47, 2012.
[17] Z. Xu, “Some similarity measures of intuitionistic fuzzy
sets andtheir applications to multiple attribute decision making,”
FuzzyOptimization and Decision Making, vol. 6, no. 2, pp.
109–121,2007.
[18] Z. Xu, “On consistency of the weighted geometric
meancomplex judgement matrix in AHP,” European Journal of
Oper-ational Research, vol. 126, no. 3, pp. 683–687, 2000.
[19] Z. Xu, “Approaches to multiple attribute group
decisionmakingbased on intuitionistic fuzzy power aggregation
operators,”Knowledge-Based Systems, vol. 24, no. 6, pp. 749–760,
2011.
[20] Z. Xu, “An overview ofmethods for
determiningOWAweights,”International Journal of Intelligent
Systems, vol. 20, no. 8, pp.843–865, 2005.
[21] Z. S. Xu and Q. L. Da, “The uncertain OWA operator,”
Interna-tional Journal of Intelligent Systems, vol. 17, no. 6, pp.
569–575,2002.
[22] Z. S. Xu, “Models for multiple attribute decision
makingwith intuitionistic fuzzy information,” International Journal
ofUncertainty, Fuzziness and Knowlege-Based Systems, vol. 15, no.3,
pp. 285–297, 2007.
[23] Z. Xu, “Multi-person multi-attribute decision making
modelsunder intuitionistic fuzzy environment,” Fuzzy
Optimizationand Decision Making, vol. 6, no. 3, pp. 221–236,
2007.
[24] Z. Xu and R. R. Yager, “Intuitionistic fuzzy bonferroni
means,”IEEE Transactions on Systems, Man, and Cybernetics B, vol.
41,no. 2, pp. 568–578, 2011.
[25] Z. Xu and X. Cai, “Recent advances in intuitionistic
fuzzyinformation aggregation,” Fuzzy Optimization and
DecisionMaking, vol. 9, no. 4, pp. 359–381, 2010.
[26] R. R. Yager, “Some aspects of intuitionistic fuzzy sets,”
FuzzyOptimization andDecisionMaking, vol. 8, no. 1, pp. 67–90,
2009.
[27] J. Ye, “Fuzzy decision-making method based on the
weightedcorrelation coefficient under intuitionistic fuzzy
environment,”European Journal of Operational Research, vol. 205,
no. 1, pp.202–204, 2010.
[28] J. Ye, “Multicriteria fuzzy decision-making method
usingentropy weights-based correlation coefficients of
interval-valued intuitionistic fuzzy sets,” Applied Mathematical
Mod-elling, vol. 34, no. 12, pp. 3864–3870, 2010.
[29] J. Ye, “Cosine similarity measures for intuitionistic fuzzy
setsand their applications,”Mathematical and Computer
Modelling,vol. 53, no. 1-2, pp. 91–97, 2011.
[30] G. Wei and X. Zhao, “Some induced correlated
aggregatingoperators with intuitionistic fuzzy information and
their appli-cation to multiple attribute group decision making,”
ExpertSystems with Applications, vol. 39, no. 2, pp. 2026–2034,
2012.
[31] V. Torra andY.Narukawa, “On hesitant fuzzy sets and
decision,”in Proceedings of the IEEE International Conference on
FuzzySystems, pp. 1378–1382, Jeju Island, Korea, August 2009.
[32] V. Torra, “Hesitant fuzzy sets,” International Journal of
IntelligentSystems, vol. 25, no. 6, pp. 529–539, 2010.
[33] X. Gu, Y. Wang, and B. Yang, “A method for hesitant
fuzzymultiple attribute decision making and its application to
riskinvestment,” Journal of Convergence Information Technology,vol.
6, no. 6, pp. 282–287, 2011.
[34] R. M. Rodriguez, L. Martinez, and F. Herrera, “Hesitant
fuzzylinguistic term sets for decision making,” IEEE Transactions
onFuzzy Systems, vol. 20, no. 1, pp. 109–119, 2012.
[35] G. Wei, “Hesitant fuzzy prioritized operators and their
appli-cation to multiple attribute decision making,”
Knowledge-BasedSystems, vol. 31, pp. 176–182, 2012.
[36] M. Xia and Z. Xu, “Hesitant fuzzy information aggregationin
decision making,” International Journal of ApproximateReasoning,
vol. 52, no. 3, pp. 395–407, 2011.
[37] M. Xia, Z. Xu, and N. Chen, “Some hesitant fuzzy
aggregationoperators with their application ingroup decision
making,”GroupDecision andNegotiation, vol. 22, no. 2, pp. 259–279,
2013.
[38] M. Xia, Z. Xu, and N. Chen, “Induced aggregation under
confi-dence levels,” International Journal of Uncertainty,
Fuzziness andKnowlege-Based Systems, vol. 19, no. 2, pp. 201–227,
2011.
[39] Z. Xu andM.Xia, “Distance and similaritymeasures for
hesitantfuzzy sets,” Information Sciences, vol. 181, no. 11, pp.
2128–2138,2011.
-
14 Journal of Applied Mathematics
[40] Z. Xu and M. Xia, “On distance and correlation measures
ofhesitant fuzzy information,” International Journal of
IntelligentSystems, vol. 26, no. 5, pp. 410–425, 2011.
[41] Z. Xu and M. Xia, “Hesitant fuzzy entropy and
cross-entropyand their use in multiattributedecision-making,”
InternationalJournal of Intelligent System, vol. 27, no. 9, pp.
799–822, 2012.
[42] D. Yu, Y. Wu, and W. Zhou, “Multi-criteria decision
makingbased on Choquet integral under hesitant fuzzy
environment,”Journal of Computational Information Systems, vol. 7,
no. 12, pp.4506–4513, 2011.
[43] B. Zhu, Z. Xu, andM.Xia, “Hesitant fuzzy geometric
Bonferronimeans,” Information Sciences, vol. 205, pp. 72–85,
2012.
[44] W.Wang andX. Liu, “Interval-valued intuitionistic fuzzy
hybridweighted averaging operatorbased on Einstein operation and
itsapplication to decisionmaking,” Journal of Intelligent and
FuzzySystems, vol. 25, no. 2, pp. 279–290, 2013.
[45] E. P. Klement, R. Mesiar, and E. Pap, “Triangular
norms.Position paper I: basic analytical and algebraic
properties,”Fuzzy Sets and Systems, vol. 143, no. 1, pp. 5–26,
2004.
[46] D. H. Hong and C.-H. Choi, “Multicriteria fuzzy
decision-making problems based on vague set theory,” Fuzzy Sets
andSystems, vol. 114, no. 1, pp. 103–113, 2000.
[47] C. Benjamin, L. Ehie, and Y. Omurtag, “Planning facilities
at theuniversity of missourirolla,” Interface, vol. 22, no. 4,
1992.
[48] Z. S. Xu and Q. L. Da, “The ordered weighted
geometricaveraging operators,” International Journal of Intelligent
Systems,vol. 17, no. 7, pp. 709–716, 2002.
[49] V. Torra and Y. Narukawa, Modeling Decisions:
InformationFusion and Aggregation Operators, Springer, 2007.
[50] R. R. Yager, “On ordered weighted averaging
aggregationoperators in multicriteria decisionmaking,” IEEE
Transactionson Systems,Man andCybernetics, vol. 18, no. 1, pp.
183–190, 1988.
[51] R. R. Yager and Z. Xu, “The continuous ordered
weightedgeometric operator and its application to decision
making,”Fuzzy Sets and Systems, vol. 157, no. 10, pp. 1393–1402,
2006.
[52] R. R. Yager andD. P. Filev, “Induced orderedweighted
averagingoperators,” IEEE Transactions on Systems, Man, and
CyberneticsB, vol. 29, no. 2, pp. 141–150, 1999.
[53] D. Filev and R. R. Yager, “On the issue of obtaining
OWAoperator weights,” Fuzzy Sets and Systems, vol. 94, no. 2, pp.
157–169, 1998.
[54] F. Herrera and L. Martinez, “An approach for
combininglinguistic and numerical information based on the
2-tuplefuzzy linguistic representation model in
decision-making,”International Journal of Uncertainty, Fuzziness
and Knowlege-Based Systems, vol. 8, no. 5, pp. 539–562, 2000.
-
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Probability and StatisticsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
CombinatoricsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical
Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
The Scientific World JournalHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014
Stochastic AnalysisInternational Journal of