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; liqingguoli@aliyun.com

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).

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