# Multiple Attribute Decision Making Based on Hesitant Fuzzy Einstein Geometric Aggregation Operators

Authors: Xiaoqiang Zhou; Qingguo Li
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

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. We also develop several new hesitant fuzzy aggregation operators, including the hesitant fuzzy Einstein weighted geometric (HFEWGε) 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 discuss various properties of the proposed operators. Finally, we apply the HFEWGε operator to solve the hesitant fuzzy decision making problems.


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## Body

## 1. Introduction

Atanassov [1, 2] introduced the concept of intuitionistic fuzzy set (IFS) characterized by a membership function and a nonmembership function. It is more suitable to deal with fuzziness and uncertainty than the ordinary fuzzy set proposed by Zadeh [3] characterized by one membership function. Information aggregation is an important research topic in many applications such as fuzzy logic systems and multiattribute decision making as discussed by Chen and Hwang [4]. Research on aggregation operators with intuitionistic fuzzy information has received increasing attention as shown in the literature. Xu [5] developed some basic arithmetic aggregation operators based on intuitionistic fuzzy values (IFVs), such as the intuitionistic fuzzy weighted averaging operator and intuitionistic fuzzy ordered weighted averaging operator, while Xu and Yager [6] presented some basic geometric aggregation operators for aggregating IFVs, including the intuitionistic fuzzy weighted geometric operator and intuitionistic fuzzy ordered weighted geometric operator. Based on these basic aggregation operators proposed in [6] and [5], many generalized intuitionistic fuzzy aggregation operators have 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 characteristic of HFS is that it allows membership degree to have a set of possible values. Therefore, HFS is a very useful tool in the situations where there are some difficulties in determining the membership of an element to a set. Lately, research on aggregation methods and multiple attribute decision making theories under hesitant fuzzy environment is very active, and a lot of results have been obtained for hesitant fuzzy information [33–43]. For example, Xia et al. [38] developed some confidence induced aggregation operators for hesitant fuzzy information. Xia et al. [37] gave several series of hesitant fuzzy aggregation operators with the help of quasiarithmetic means. Wei [35] explored several hesitant fuzzy prioritized aggregation operators and applied them to hesitant fuzzy decision making problems. Zhu et al. [43] investigated the geometric Bonferroni mean combining the Bonferroni mean and the geometric mean under hesitant fuzzy environment. Xia and Xu [36] presented some hesitant fuzzy operational laws based on the relationship between the HFEs and the IFVs. They also proposed a series of aggregation operators, such as hesitant fuzzy weighted geometric (HFWG) operator and hesitant fuzzy ordered weighted geometric (HFOWG) operator. Furthermore, they applied the proposed aggregation operators to solve the multiple attribute decision making problems.Note that all aggregation operators introduced previously are based on the algebraic product and algebraic sum ofIFVs (or HFEs) to carry out the combination process. However, the algebraic operations include algebraic product and algebraic sum, which are not the unique operations that can be used to perform the intersection and union. There are many instances of various t-norms and t-conorms families which can be chosen to model the corresponding intersections and unions, among which Einstein product and Einstein sum are good alternatives for they typically give the same smooth approximation 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 intuitionistic fuzzy aggregation operators with the help of Einstein operations. For hesitant fuzzy information, however, it seems that in the literature there is little investigation on aggregation techniques using the Einstein operations to aggregate hesitant fuzzy information. Therefore, it is necessary to develop some hesitant fuzzy information aggregation operators based on Einstein operations.The remainder of this paper is structured as follows. In Section2, we briefly review some basic concepts and operations related to IFS and HFS. we also define an accuracy function of HFEs to distinguish the two HFEs having the same score values, based on which we give the new comparison laws on HFEs. In Section 3, we present some new operations for HFEs and discuss some basic properties of the proposed operations. In Section 4, we develop some novel hesitant fuzzy geometric aggregation operators with the help of Einstein operations, such as the HFEWGɛ operator and the HFEOWGɛ operator, and we further study various properties of these operators. Section 5 gives an approach to solve the multiple attribute hesitant fuzzy decision making problems based on the HFEOWGɛ operator. Finally, Section 6 concludes the paper.


## 2. Preliminaries

In this section, we briefly introduce Einstein operations and some notions ofIFS and HFS. Meantime, we define an accuracy function of HFEs and redefine the comparison laws between two HFEs.


### 2.1. Einstein Operations

Since the appearance of fuzzy set theory, the set theoretical operators have played an important role and received more and more attention. It is well known that thet-norms and t-conorms are the general concepts including all types of the specific operators, and they satisfy the requirements of the conjunction and disjunction operators, respectively. There are various t-norms and t-conorms families that can be used to perform the corresponding intersections and unions. Einstein sum ⊕ɛ and Einstein product ⊗ɛ are examples of t-conorms and t-norms, respectively. They are called Einstein operations and defined as [45]
(1)x⊕ɛy=x·y1+(1-x)·(1-y),x⊕ɛy=x+y1+x·y,222222222222222222222222222∀x,y∈[0,1].


### 2.2. Intuitionistic Fuzzy Set

Atanassov [1, 2] generalized the concept of fuzzy set [3] and defined the concept of intuitionistic fuzzy set (IFS) as follows.Definition 1.
LetU be fixed an IFSA on U is given by;
(2)A={〈x,μA(x),νA(x)〉∣x∈U},
where μA:U→[0,1] and νA:U→[0,1], with the condition 0≤μA(x)+νA(x)≤1 for all x∈U. Xu [5] called a~=(μa~,νa~) an IFV.ForIFVs, Wang and Liu [11] introduced some operations as follows.Letλ>0, a~1=(μa~1,νa~1) and a~2=(μa~2,νa~2) be two IFVs; then
(3)(1)a~1⊕ɛa~2=(μa~1+μa~21+μa~1μa~2,νa~1νa~21+(1-νa~1)(1-νa~2))(2)a~1⊕ɛa~2=(μa~1μa~21+(1-μa~1)(1-μa~2),νa~1+νa~21+νa~1νa~2)(3)a~1∧ɛλ=(2νa~1λ(2-νa~1)λ+νa~1λ,(1+μa~1)λ-(1-μa~1)λ(1+μa~1)λ+(1-μa~1)λ).


### 2.3. Hesitant Fuzzy Set

As another generalization of fuzzy set,HFS was first introduced by Torra and Narukawa [31, 32].Definition 2.
LetX be a reference set; an HFS on X is in terms of a function that when applied to X returns a subset of [0,1].To be easily understood, Xia and Xu use the following mathematical symbol to express theHFS:
(4)H={hH(x)x∣x∈X},
where hH(x) is a set of some values in [0,1], denoting the possible membership degrees of the element x∈X to the set H. For convenience, Xu and Xia [40] called hH(x) a hesitant fuzzy element (HFE).Leth be an HFE, h-=min⁡⁡{γ∣γ∈h}, and h+=max⁡{γ∣γ∈h}. Torra and Narukawa [31, 32] define the IFVAenv(h) as the envelope of h, where Aenv(h)=(h-,1-h+).Letα>0, h1 and h2 be two HFEs. Xia and Xu [36] defined some operations as follows:
(5)(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}.In [36], Xia and Xu defined the score function of an HFEh to compare the HFEs and gave the comparison laws.Definition 3.
Leth be an HFE; s(h)=(1/n(h))∑γ∈h‍γ is called the score function of h, where n(h) is the number of values of h. For two HFEsh1 and h2, if s(h1)>s(h2), then h1>h2; if s(h1)=s(h2), then h1=h2.From Definition3, it can be seen that all HFEs are regarded as the same if their score values are equal. In hesitant fuzzy decision making process, however, we usually need to compare two HFEs for reordering or ranking. In the case where two HFEs have the same score values, they can not be distinguished by Definition 3. Therefore, it is necessary to develop a new method to overcome the difficulty.For anIFV, Hong and Choi [46] showed that the relation between the score function and the accuracy function is similar to the relation between mean and variance in statistics. From Definition 3, we know that the score value of HFEh is just the mean of the values in h. Motivated by the idea of Hong and Choi [46], we can define the accuracy function of HFEh by using the variance of the values in h.Definition 4.
Leth be an HFE; k(h)=1-(1/n(h))∑γ∈h‍(γ-s(h))2 is called the accuracy function of h, where n(h) is the number of values in h and s(h) is the score function of h.It is well known that an efficient estimator is a measure of the variance of an estimate’s sampling distribution in statistics: the smaller the variance, the better the performance of the estimator. Motivated by this idea, it is meaningful and appropriate to stipulate that the higher the accuracy degree ofHFE, the better the HFE. Therefore, in the following, we develop a new method to compare two HFEs, which is based on the score function and the accuracy function, defined as follows.Definition 5.
Leth1 and h2 be two HFEs and let s(·) and k(·) be the score function and accuracy function of HFEs, respectively. Then(1)
ifs(h1)<s(h2), then h1 is smaller than h2, denoted by h1≺h2;(2)
ifs(h1)=s(h2), then
(i)
ifk(h1)<k(h2), then h1 is smaller than h2, denoted by h1≺h2;(ii)
ifk(h1)=k(h2), then h1 and h2 represent the same information, denoted by h1≐h2. In particular, if γ1=γ2 for any γ1∈h1 and γ2∈h2, then h1 is equal to h2, denoted by h1=h2.Example 6.
Leth1={0.5}, h2={0.1,0.9}, h3={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(h2)=s(h3)=s(h4)=s(h5)=s(h6)=0.5, k(h1)=1, k(h2)=0.6, k(h3)=0.8, k(h4)=0.6838, k(h5)=0.7764, and k(h6)=0.7418. By Definition 5, we have h1≻h3≻h5≻h6≻h4≻h2.


## 2.1. Einstein Operations

Since the appearance of fuzzy set theory, the set theoretical operators have played an important role and received more and more attention. It is well known that thet-norms and t-conorms are the general concepts including all types of the specific operators, and they satisfy the requirements of the conjunction and disjunction operators, respectively. There are various t-norms and t-conorms families that can be used to perform the corresponding intersections and unions. Einstein sum ⊕ɛ and Einstein product ⊗ɛ are examples of t-conorms and t-norms, respectively. They are called Einstein operations and defined as [45]
(1)x⊕ɛy=x·y1+(1-x)·(1-y),x⊕ɛy=x+y1+x·y,222222222222222222222222222∀x,y∈[0,1].


## 2.2. Intuitionistic Fuzzy Set

Atanassov [1, 2] generalized the concept of fuzzy set [3] and defined the concept of intuitionistic fuzzy set (IFS) as follows.Definition 1.
LetU be fixed an IFSA on U is given by;
(2)A={〈x,μA(x),νA(x)〉∣x∈U},
where μA:U→[0,1] and νA:U→[0,1], with the condition 0≤μA(x)+νA(x)≤1 for all x∈U. Xu [5] called a~=(μa~,νa~) an IFV.ForIFVs, Wang and Liu [11] introduced some operations as follows.Letλ>0, a~1=(μa~1,νa~1) and a~2=(μa~2,νa~2) be two IFVs; then
(3)(1)a~1⊕ɛa~2=(μa~1+μa~21+μa~1μa~2,νa~1νa~21+(1-νa~1)(1-νa~2))(2)a~1⊕ɛa~2=(μa~1μa~21+(1-μa~1)(1-μa~2),νa~1+νa~21+νa~1νa~2)(3)a~1∧ɛλ=(2νa~1λ(2-νa~1)λ+νa~1λ,(1+μa~1)λ-(1-μa~1)λ(1+μa~1)λ+(1-μa~1)λ).


## 2.3. Hesitant Fuzzy Set

As another generalization of fuzzy set,HFS was first introduced by Torra and Narukawa [31, 32].Definition 2.
LetX be a reference set; an HFS on X is in terms of a function that when applied to X returns a subset of [0,1].To be easily understood, Xia and Xu use the following mathematical symbol to express theHFS:
(4)H={hH(x)x∣x∈X},
where hH(x) is a set of some values in [0,1], denoting the possible membership degrees of the element x∈X to the set H. For convenience, Xu and Xia [40] called hH(x) a hesitant fuzzy element (HFE).Leth be an HFE, h-=min⁡⁡{γ∣γ∈h}, and h+=max⁡{γ∣γ∈h}. Torra and Narukawa [31, 32] define the IFVAenv(h) as the envelope of h, where Aenv(h)=(h-,1-h+).Letα>0, h1 and h2 be two HFEs. Xia and Xu [36] defined some operations as follows:
(5)(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}.In [36], Xia and Xu defined the score function of an HFEh to compare the HFEs and gave the comparison laws.Definition 3.
Leth be an HFE; s(h)=(1/n(h))∑γ∈h‍γ is called the score function of h, where n(h) is the number of values of h. For two HFEsh1 and h2, if s(h1)>s(h2), then h1>h2; if s(h1)=s(h2), then h1=h2.From Definition3, it can be seen that all HFEs are regarded as the same if their score values are equal. In hesitant fuzzy decision making process, however, we usually need to compare two HFEs for reordering or ranking. In the case where two HFEs have the same score values, they can not be distinguished by Definition 3. Therefore, it is necessary to develop a new method to overcome the difficulty.For anIFV, Hong and Choi [46] showed that the relation between the score function and the accuracy function is similar to the relation between mean and variance in statistics. From Definition 3, we know that the score value of HFEh is just the mean of the values in h. Motivated by the idea of Hong and Choi [46], we can define the accuracy function of HFEh by using the variance of the values in h.Definition 4.
Leth be an HFE; k(h)=1-(1/n(h))∑γ∈h‍(γ-s(h))2 is called the accuracy function of h, where n(h) is the number of values in h and s(h) is the score function of h.It is well known that an efficient estimator is a measure of the variance of an estimate’s sampling distribution in statistics: the smaller the variance, the better the performance of the estimator. Motivated by this idea, it is meaningful and appropriate to stipulate that the higher the accuracy degree ofHFE, the better the HFE. Therefore, in the following, we develop a new method to compare two HFEs, which is based on the score function and the accuracy function, defined as follows.Definition 5.
Leth1 and h2 be two HFEs and let s(·) and k(·) be the score function and accuracy function of HFEs, respectively. Then(1)
ifs(h1)<s(h2), then h1 is smaller than h2, denoted by h1≺h2;(2)
ifs(h1)=s(h2), then
(i)
ifk(h1)<k(h2), then h1 is smaller than h2, denoted by h1≺h2;(ii)
ifk(h1)=k(h2), then h1 and h2 represent the same information, denoted by h1≐h2. In particular, if γ1=γ2 for any γ1∈h1 and γ2∈h2, then h1 is equal to h2, denoted by h1=h2.Example 6.
Leth1={0.5}, h2={0.1,0.9}, h3={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(h2)=s(h3)=s(h4)=s(h5)=s(h6)=0.5, k(h1)=1, k(h2)=0.6, k(h3)=0.8, k(h4)=0.6838, k(h5)=0.7764, and k(h6)=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 operations onHFEs and analyze some desirable properties of these operations. Motivated by the operational laws (1)–(3) on IFVs and based on the interconnection between HFEs and IFVs, we give some new operations of HFEs as follows.Letα>0, h, h1, and h2 be three HFEs; then
(6)(8)h1⊕ɛh2=⋃γ1∈h1,γ2∈h2{γ1+γ21+γ1γ2},(9)h1⊕ɛh2=⋃γ1∈h1,γ2∈h2{γ1γ21+(1-γ1)(1-γ2)},(10)h∧ɛα=⋃γ∈h{2γα(2-γ)α+γα}.Proposition 7.
Letα>0, α1>0, α2>0, h, h1 and h2 be three HFEs; then(1)
h1⊗ɛh2=h2⊗ɛh1,(2)
(h1⊗ɛh2)⊗ɛh3=h1⊗ɛ(h2⊗ɛh3),(3)
(h1⊗ɛh2)∧ɛα=h1∧ɛα⊗ɛh2∧ɛα,(4)
(h∧ɛα1)∧ɛα2=h∧ɛ(α1α2);(5)
Aenv(h∧ɛα)=(Aenv(h))∧ɛα,(6)
Aenv(h1⊗ɛh2)=Aenv(h1)⊗ɛAenv(h2).Proof.
(
1
) It is trivial.
(
2
) By the operational law (9), we have
(7)(h1⊕ɛh2)⊕ɛh3=⋃γ1∈h1,γ2∈h2,γ3∈h3{((c))-1((γ1γ2/(1+(1-γ1)(1-γ2)))γ3)=∪γ1∈h1,γ2∈h2,γ3∈h3×(1+(1-(γ1γ2/(1+(1-γ1)(1-γ2))))22222222222222222×(1-γ3))-1}=⋃γ1∈h1,γ2∈h2,γ3∈h3{+(1-γ2)(1-γ3))-1(γ1γ2γ3)×(1+(1-γ1)(1-γ2)2222222222222222222222222+(1-γ1)(1-γ3)+(1-γ2)2222222222222222222222222×(1-γ3))-1}=⋃γ1∈h1,γ2∈h2,γ3∈h3{(1-(γ2γ3/(1+(1-γ2)(1-γ3)))))-1(γ1(γ2γ3/(1+(1-γ2)(1-γ3))))=∪γ1∈h1,γ2∈h2,γ3∈h3×(((1-γ3))1+(1-γ1)=∪γ1∈h1,γ2∈h2,γ3∈h3×(1-(γ2γ3/(1+(1-γ2)(1-γ3)))))-1}=h1⊕ɛ(h2⊕ɛh3).
(
3
) Let h=h1⊗ɛh2; then h=h1⊗ɛh2=⋃γ1∈h1,γ2∈h2{γ1γ2/(1+(1-γ1)(1-γ2))}(8)(h1⊕ɛh2)∧ɛα=h∧ɛα=⋃γ∈h{2γα(2-γ)α+γα}=⋃γ1∈h1,γ2∈h2{(γ1γ2/(1+(1-γ1)(1-γ2)))α)-1(2(γ1γ2/(1+(1-γ1)(1-γ2)))α)=∪γ1∈h1,γ2∈h2×((2-(γ1γ2/(1+(1-γ1)(1-γ2))))α=∪γ1∈h1,γ2∈h2222+(γ1γ2/(1+(1-γ1)(1-γ2)))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}.
Since h1∧ɛα=⋃γ1∈h{2γ1α/((2-γ1)α+γ1α)} and h2∧ɛα=⋃γ2∈h{2γ2α/((2-γ2)α+γ2α)}, then
(9)h1∧ɛα⊕ɛh2∧ɛα=⋃γ1∈h1,γ2∈h2{(1-(2γ2α/((2-γ2)α+γ2α))))-1}((2γ1α/((2-γ1)α+γ1α))=∪γ1∈h1,γ2∈h2·(2γ2α/((2-γ2)α+γ2α)))=∪γ1∈h1,γ2∈h2×((())-11+(1-(2γ1α/((2-γ1)α+γ1α)))=∪γ1∈h1,γ2∈h2××(1-(2γ2α/((2-γ2)α+γ2α))))-1}=⋃γ1∈h1,γ2∈h2{2(γ1γ2)α(2-γ1)α(2-γ2)α+(γ1γ2)α}.
Thus (h1⊗ɛh2)∧ɛα=h1∧ɛα⊗ɛh2∧ɛα.
(
4
) Since h∧ɛα1=⋃γ∈h{2γα1/((2-γ)α1+γα1)}, then
(10)(h∧ɛα1)∧ɛα2=⋃γ∈h{(2γα1/((2-γ)α1+γα1))α2)-1(2(2γα1/((2-γ)α1+γα1))α2)=∪γ∈h×((2-(2γα1/((2-γ)α1+γα1)))α2=∪γ1∈h1,γ2∈h2+(2γα1/((2-γ)α1+γα1))α2)-1(1-(γ2γ3/(1+(1-γ2)(1-γ3)))))-1}=⋃γ∈h{2γ(α1α2)(2-γ)(α1α2)+γ(α1α2)}=h∧ɛ(α1α2).
(
5
) By the definition of the envelope of an HFE and the operation laws (3) and (10), we have
(11)(Aenv(h))∧ɛα=(h-,1-h+)∧ɛα=(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∧ɛα)=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).
Thus, Aenv(h∧ɛα)=(Aenv(h))∧ɛα.
(
6
) By the definition of the envelope of an HFE and the operation laws (2) and (9), we have
(12)Aenv(h1)⊕ɛAenv(h2)=(h1-,1-h1+)⊕ɛ(h2-,1-h2+)=(h1-h2-1+(1-h1-)(1-h2-),(1-h1+)+(1-h2+)1+(1-h1+)(1-h2+))Aenv(h1⊕ɛh2)=Aenv(⋃γ1∈h1,γ2∈h2{γ1γ21+(1-γ1)(1-γ2)})=(h1-h2-1+(1-h1-)(1-h2-),1-h1+h2+1+(1-h1+)(1-h2+))=(h1-h2-1+(1-h1-)(1-h2-),(1-h1+)+(1-h2+)1+(1-h1+)(1-h2+)).
Thus, Aenv(h1⊗ɛh2)=Aenv(h1)⊗ɛAenv(h2).Remark 8.
Letα1>0, α2>0, and h be an HFE. It is worth noting that h∧ɛα1⊗ɛh∧ɛα2≐h∧ɛ(α1+α2) does not hold necessarily in general. To illustrate that, an example is given as follows.Example 9.
Leth=(0.3,0.5), α1=α2=1; then h∧ɛα1⊗ɛh∧ɛα2=h⊗ɛh=⋃γi∈h,γj∈h,(i,j=1,2){γiγj/(1+(1-γi)(1-γj))}=(0.0604,0.1111,0.2), and h∧ɛ(α1+α2)=h∧ɛ2=⋃γ∈h{2γ2/((2-γ)2+γ2)}=(0.0604,0.2). Clearly, s(h∧ɛα1⊗ɛh∧ɛα2)=0.1238<0.1302=s(h∧ɛ(α1+α2)). Thus h∧ɛα1⊗ɛh∧ɛα1≺h∧ɛ(α1+α2).However, if the number of the values inh is only one, that is, HFEh is reduced to a fuzzy value, then the above result holds.Proposition 10.
Letα1>0, α2>0, and h be an HFE, in which the number of the values is only one, that is, h={γ}; then h∧ɛα1⊗ɛh∧ɛα2=h∧ɛ(α1+α2).Proof.
Sinceh∧ɛα1=⋃γ∈h{2γα1/((2-γ)α1+γα1)} and h∧ɛα2=⋃γ∈h{2γα2/((2-γ)α2+γα2)}, then
(13)h∧ɛα1⊕ɛh∧ɛα1=⋃γ∈h{(1-(2γα2/((2-γ)α2+γα2))))-1((2γα1/((2-γ)α1+γα1))=∪γ1∈h1,γ2∈h2·(2γα2/((2-γ)α2+γα2)))⋃×((())-11+(1-(2γα1/((2-γ)α1+γα1)))⋃××(1-(2γα2/((2-γ)α2+γα2))))-1}=⋃γ∈h{(1-(2γα2/((2-γ)α2+γα2))))-1(2γα1·2γα2)×([(2-γ)α1+γα1]·[(2-γ)α2+γα2]=∪γ∈h(2γα1·2γα2)××+[(2-γ)α1-γα1]=∪γ∈h(2γα1·2γα2)××·[(2-γ)α2-γα2])-1}=⋃γ∈h{2γα1+α2(2-γ)α1+α2+γα1+α2}=h∧ɛ(α1+α2).Proposition10 shows that it is consistent with the result (iii) in Theorem 2 in the literature [11].


## 4. Hesitant Fuzzy Einstein Geometric Aggregation Operators

The weighted geometric operator [47] and the ordered weighted geometric operator [48] are two of the most common 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 using Einstein operations.


### 4.1. Hesitant Fuzzy Einstein Geometric Weighted Aggregation Operator

Based on the operational laws(5) and (7) on HFEs, Xia and Xu [36] developed some hesitant fuzzy aggregation operators as listed below.Lethj(j=1,2,…,n) be a collection of HFEs; then.(
1
) the hesitant fuzzy weighted geometric (HFWG) operator
(14)HFWG(h1,h2,…,hn)=⨂j=1nhjωj=⋃γ1∈h1,γ2∈h2,…,γn∈hn{∏j=1n‍γjωj},
where ω=(ω1,ω2,…,ωn)T is the weight vector of hj(j=1,2,…,n) with ωj∈[0,1] and ∑j=1nωj=1.(
2
) the hesitant fuzzy ordered weighted geometric (HFOWG) operator
(15)HFOWG(h1,h2,…,hn)=⨂j=1n⁡wjhσ(j)=⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),…,γσ(n)∈hσ(n){∏j=1n‍γσ(j)wj},
where σ(1),σ(2),…,σ(n) is a permutation of 1,2,…,n, such that hσ(j-1)>hσ(j) for all j=2,…,n and w=(w1,w2,…,wn)T is aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1.For convenience, letH be the set of all HFEs. Based on the proposed Einstein operations on HFEs, we develop some new aggregation operators for HFEs and discuss their desirable properties.Definition 11.
Lethj(j=1,2,…,n) be a collection of HFEs. A hesitant fuzzy Einstein weighted geometric (HFEWGɛ) operator of dimension n is a mapping HFEWGɛ:Hn→H defined as follows:
(16)HFEWGɛ(h1,h2,…,hn)=⨂εj=1nhj∧ɛωj=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγjωj∏j=1n(2-γj)ωj+∏j=1nγjωj},
where w=(w1,w2,…,wn)T is the weight vector of hj(j=1,2,…,n) and wj>0, ∑j=1nwj=1. In particular, when wj=1/n, j=1,2,…,n, the HFEWGɛ operator is reduced to the hesitant fuzzy Einstein geometric (HFEGɛ) operator:
(17)HFEGɛ(h1,h2,…,hn)=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγj1/n∏j=1n(2-γj)1/n+∏j=1nγj1/n}.From Proposition10, we easily get the following result.Corollary 12.
If allhj(j=1,2,…,n) are equal and the number of values in hj is only one, that is, hj=h={γ} for all j=1,2,…,n, then
(18)HFEWGɛ(h1,h2,…,hn)=h.Note that theHFEWGɛ operator is not idempotent in general; we give the following example to illustrate this case.Example 13.
Leth1=h2=h3=h=(0.3,0.7), w=(0.4,0.25,0.35)T; then HFEWGɛ(h1,h2,h3)={0.3,0.4137,0.3782,0.5126,0.4323,0.579,0.5342,0.7}. By Definition 3, we have s(HFEWGɛ(h1,h2,h3))=0.4812<0.5=s(h). Hence HFEWGɛ(h1,h2,h3)≺h.Lemma 14 (see [18, 49]).
Letγj>0,wj>0, j=1,2,…,n, and ∑j=1nwj=1. Then
(19)∏j=1n‍γjwj≤∑j=1n‍wjγj
with equality if and only if γ1=γ2=⋯=γn.Theorem 15.
Lethj(j=1,2,…,n) be a collection of HFEs and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑j=1nwj=1. Then
(20)HFEWGɛ(h1,h2,…,hn)≽HFWG(h1,h2,…,hn),
where the equality holds if only if all hj(j=1,2,…,n) are equal and the number of values in hj is only one.Proof.
For anyγj∈hj(j=1,2,…,n), by Lemma 14, we have
(21)∏j=1n(2-γj)wj+∏j=1nγjwj≤∑j=1n‍wj(2-γj)+∑j=1n‍wjγj=2.
Then
(22)2∏j=1nγjωj∏j=1n(2-γj)ωj+∏j=1nγjωj≥∏j=1n‍γjωj.
It follows that s(⊗εj=1nhj∧ɛωj)≥s(⊗εj=1nhjωj), which completes the proof of Theorem 15.Theorem15 tells us the result that the HFEWGɛ operator shows the decision maker’s more optimistic attitude than the HFWA operator proposed by Xia and Xu [36] (i.e., (15)) in aggregation process. To illustrate that, we give an example adopted from Example  1 in [36] as follows.Example 16.
Leth1=(0.2,0.3,0.5), h2=(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
(23)HFEWGɛ(h1,h2)=⋃γ1∈h1,γ2∈h2{2∏j=12γjωj∏j=12(2-γj)ωj+∏j=12γjωj}={0.2482,0.2856,0.3276,0.3744,2220.4683,0.5288}.
However, Xia and Xu [36] used the HFWG operator to aggregate the hj(j=1,2) and got
(24)HFEG(h1,h2)=⋃γ1∈h1,γ2∈h2{∏j=12‍γjwj}={0.2462,0.2781,0.3270,0.3693,0.4676,0.5281}.
It is clear that s(HFEWGɛ(h1,h2))=0.3722>0.3694=s(HFEG(h1,h2)). Thus HFEWGɛ(h1,h2)≻HFEG(h1,h2).Based on Definition11 and the proposed operational laws, we can obtain the following properties on HFEWGɛ operator.Theorem 17.
Letα>0, hj(j=1,2,…,n), be a collection of HFEs and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(25)HFEWGɛ(h1∧ɛα,h2∧ɛα,…,hn∧ɛα)=(HFEWGɛ(h1,h2,…,hn))∧ɛα.Proof.
Sincehj∧ɛα=⋃γ∈hj{2γjα/((2-γj)α+γjα)} for all j=1,2,…,n, by the definition of HFEWGɛ, we have
(26)HFEWGɛ(h1∧ɛα,h2∧ɛα,…,hn∧ɛα)=⋃γ1∈h1,γ2∈h2,…,γn∈hn{(∏j=1n‍)(2∏j=1n‍(2γjα/((2-γj)α+γjα))ωj)=∪γ1∈h1,γ2∈h2,…,γn∈hn×(∏j=1n‍(2-(2γjα/((2-γj)α+γjα)))ωj=∪γ1∈h1,γ2∈h2,…,γn∈hn×+∏j=1n‍(2γjα/((2-γj)α+γjα))ωj)-1}=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγjαωj∏j=1n(2-γj)αωj+∏j=1nγjαωj}.
Since HFEWGɛ(h1,h2,…,hn)=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj)}, then
(27)(HFEWGɛ(h1,h2,…,hn))∧ɛα=⋃γ1∈h1,γ2∈h2,…,γn∈hn{(∏j=1n‍())(2(2∏j=1n‍γjωj/(∏j=1n‍(2-γj)ωj=∪γ1∈h1,γ2∈h2,…,γn∈hn2∏j=1n‍γjωj+∏j=1n‍γjωj))α)=∪γ1∈h1,γ2∈h2,…,γn∈hn×((2-(2∏j=1n‍γjωj/(∏j=1n‍(2-γj)ωj=∪γ1∈h1,γ2∈h2,…,γn∈hn2∏j=1n‍γjωj+∏j=1n‍γjωj)))α=∪γ1∈h1,γ2∈h2,…,γn∈hn++(2∏j=1n‍γjωj/(∏j=1n‍(2-γj)ωj=∪γ1∈h1,γ2∈h2,…,γn∈hn2∏j=1n‍γjωj+∏j=1n‍γjωj))α)-1}=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2(∏j=1nγjωj)α(∏j=1n(2-γj)ωj)α+(∏j=1nγjαωj)α}=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγjαωj∏j=1n(2-γj)αωj+∏j=1nγjαωj}.Theorem 18.
Leth be an HFE, hj(j=1,2,…,n) a collection of HFEs, and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(28)HFEWGɛ(h1⊗ɛh,h2⊗ɛh,…,hn⊗ɛh)=HFEWGɛ(h1,h2,…,hn)⊗ɛh.Proof.
By the definition ofHFEWGɛ and Einstein product operator of HFEs, we have(29)HFEWGɛ(h1,h2,…,hn)⊕ɛh=⋃γ∈h,γj∈hj,j=1,…,n{(2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj))·γ1+(1-(2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj)))(1-γ)}=⋃γ∈h,γj∈hj,j=1,…,n{2γ∏j=1nγjωj(2-γ)∏j=1n(2-γj)ωj+γ∏j=1nγjωj}. Since hj⊗ɛh=⋃γj∈hj,γ∈h{γjγ/(1+(1-γj)(1-γ))} for all j=1,2,…,n, by the definition of HFEWGɛ, we have
(30)HFEWGɛ(h1⊗ɛh,h2⊗ɛh,…,hn⊕ɛh)=⋃γ∈h,γj∈hj,j=1,…,n{(2∏j=1n‍(γjγ/(1+(1-γj)(1-γ)))ωj)=∪γ∈h,γj∈hj,j=1,…,n×(∏j=1n‍(2-(γjγ/(1+(1-γj)=∪γ∈h,γj∈hj,j=1,…,n+222222222×(1-γ))))ωj=∪γ∈h,γj∈hj,j=1,…,n+∏j=1n(γjγ/(1+(1-γj)‍=∪γ∈h,γj∈hj,j=1,…,n+222222222×(1-γ)))wj∏j=1n‍)-12∏j=1n‍)}=⋃γ∈h,γj∈hj,j=1,…,n{2∏j=1n(γjγ)ωj∏j=1n((2-γj)(2-γ))ωj+∏j=1n(γjγ)ωj}=⋃γ∈h,γj∈hj,j=1,…,n{(∏j=1n‍)(2∏j=1n‍γωj·∏j=1n‍γjωj)=∪γ∈h,γj∈hj,j=1,…,n×(∏j=1n‍(2-γ)ωj·∏j=1n‍(2-γj)ωj+∏j=1n‍γωj=∪γ∈h,γj∈hj,j=1,…,n×·∏j=1n‍γjωj)-1}=⋃γ∈h,γj∈hj,j=1,…,n{(2γ∑j=1nωj·∏j=1n‍γjωj)=∪γ∈h,γj∈hj,j=1,…,n×((2-γ)∑j=1nωj·∏j=1n‍(2-γj)ωj=∪γ∈h,γj∈hj,j=1,…,n×+γ∑j=1nωj·∏j=1n‍γjωj)-1}=⋃γ∈h,γj∈hj,j=1,…,n{2γ∏j=1nγjωj(2-γ)∏j=1n(2-γj)ωj+γ∏j=1nγjωj}.Based on Theorems17 and 18, the following property can be obtained easily.Theorem 19.
Letα>0, h be an HFE, let hj(j=1,2,…,n) be a collection of HFEs, and let w=(w1,w2,…,wn)T be the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(31)HFEWGɛ(h1∧ɛα⊗ɛh,h1∧ɛα⊗ɛh,…,hn∧ɛα⊗ɛh)=(HFEWGɛ(h1,h2,…,hn)⊗ɛh)∧ɛα.Theorem 20.
Lethj and hj′(j=1,2,…,n) be two collections of HFEs and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(32)HFEWGɛ(h1⊗ɛh1′,h2⊗ɛh2′,…,hn⊗ɛhn′)=HFEWGɛ(h1,h2,…,hn)⊗ɛHFEWGɛ(h1′,h2′,…,hn′).Proof.
By the definition ofHFEWGɛ and Einstein product operator of HFEs, we have(33)HFEWGε(h1,h2,…,hn)⊗εHFEWGε(h1′,h2′,…,hn′)=⋃γj∈hj,γj′∈hj′,j=1,…,n{(2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj))·(2∏j=1nγj′ωj/(∏j=1n(2-γj′)ωj+∏j=1nγj′ωj))1+(1-(2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj)))(1-(2∏j=1nγj′ωj/(∏j=1n(2-γj′)ωj+∏j=1nγj′ωj)))}=⋃γj∈hj,γj′∈hj′,j=1,…,n{2∏j=1nγjωj·∏j=1nγj′ωj∏j=1n(2-γj)ωj·∏j=1n(2-γj′)ωj+∏j=1nγjωj·∏j=1nγj′ωj}.Since hj⊗ɛhj′=⋃γj∈hj,γj′∈hj′{γjγj′/(1+(1-γj)(1-γj′))} for all j=1,2,…,n, by the definition of HFEWGɛ, we have
(34)HFEWGɛ(h1⊕ɛh1′,h2⊕ɛh2′,…,hn⊕ɛhn′)=⋃γj∈hj,γj′∈hj′,j=1,…,n{(2∏j=1n‍(γjγj′/(1+(1-γj)(1-γj′)))ωj)=∪γj∈hj,γj′∈hj′,j=1,…,n×(∏j=1n‍(2-(γjγj′/(1+(1-γj)=∪γj∈hj,γj′∈hj′,j=1,…,n×22222222222222×(1-γj′))))ωj=∪γj∈hj,γj′∈hj′,j=1,…,n×+∏j=1n‍(γjγj′/(1+(1-γj)=∪γj∈hj,γj′∈hj′,j=1,…,n×∏j=1n‍111111×(1-γj′)))ωj∏j=1n‍)-1∏j=1n‍)}=⋃γj∈hj,γj′∈hj′,j=1,…,n{(2∏j=1n(γjγj′)ωj)⋃γj∈hj,γj′∈hj′,j=1,…,n×(∏j=1n[(2-γj)(2-γj′)]ωj⋃γj∈hj,γj′∈hj′,j=1,…,n×+∏j=1n(γjγj′)ωj)-1}=⋃γj∈hj,γj′∈hj′,j=1,…,n{(2∏j=1n‍γjωj·∏j=1n‍γj′ωj)=∪γj∈hj,γj′∈hj′,j=1,…,n×(∏j=1n‍(2-γj)ωj·∏j=1n‍(2-γj′)ωj=∪γj∈hj,γj′∈hj′,j=1,…,n×+∏j=1n‍γjωj·∏j=1n‍γj′ωj)-1}.Theorem 21.
Lethj(j=1,2,…,n) be a collection of HFEs, hmin⁡-=min⁡j⁡{hj-∣hj-=min⁡{γj∈hj}}, and hmax⁡+=max⁡j⁡{hj+∣hj+=max⁡{γj∈hj}}, and let w=(w1,w2,…,wn)T be the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(35)hmin⁡-⪯HFEWGɛ(h1,h2,…,hn)⪯hmax⁡+,
where the equality holds if only if all hj(j=1,2,…,n) are equal and the number of values in hj is only one.Proof.
Letf(t)=(2-t)/t, t∈[0,1]. Then f′(t)=-2/t2<0. Hence f(t) is a decreasing function. Since hmin⁡-≤hj-≤γj≤hj+≤hmax⁡+ for any γj∈hj(j=1,2,…,n), then f(hmax⁡+)≤f(γj)≤f(hmin⁡-); that is, (2-hmax⁡+)/hmax⁡+≤(2-γj)/γj≤(2-hmin⁡-)/hmin⁡-. Then for any γj∈hj(j=1,2,…,n), we have
(36)∏j=1n‍(2-hmax⁡+hmax⁡+)wj≤∏j=1n‍(2-γjγj)wj≤∏j=1n‍(1-hmin⁡-1+hmin⁡-)wj⇔(2-hmax⁡+hmax⁡+)∑j=1nwj≤∏j=1n‍(2-γjγj)wj≤(1-hmin⁡-1+hmin⁡-)∑j=1nwj⇔(2-hmax⁡+hmax⁡+)≤∏j=1n‍(2-γjγj)wj≤(1-hmin⁡-1+hmin⁡-)⇔2hmax⁡+≤∏j=1n‍(2-γjγj)wj+1≤2hmin⁡-⇔hmin⁡-2≤1∏j=1n((2-γj)/γj)wj+1≤(hmax⁡+2)⇔hmin⁡-≤2∏j=1n((2-γj)/γj)wj+1≤hmax⁡+⇔hmin⁡-≤2∏j=1nγjωj∏j=1n(2-γj)ωj+∏j=1nγjωj≤hmax⁡+.
It follows that hmin⁡-≤s(HFEWGɛ(h1,h2,…,hn))≤hmax⁡+. Thus we have hmin⁡-⪯HFEWGɛ(h1,h2,…,hn)⪯hmax⁡+.Remark 22.
Lethj and hj′(j=1,2,…,n) be two collections of HFEs, and hj≺hj′ for all j; then HFEWGɛ(h1,h2,…,hn)≺HFEWGɛ(h1′,h2′,…,hn′) does not hold necessarily in general. To illustrate that, an example is given as follows.Example 23.
Leth1=(0.45,0.6), h2=(0.6,0.7), h3=(0.5,0.6), h1′=(0.2,0.9), h2′=(0.45,0.95), h3′=(0.35,0.8), and w=(0.5,0.3,0.2)T; then HFEWGɛ(h1,h2,h3)={0.5024,0.5215,0.5286,0.5483,0.5791,0.6,0.6077,0.6291} and HFEWGɛ(h1′,h2′,h3′)={0.2778,0.3372,0.3835,0.4595,0.6088,0.7099,0.7833,0.8947}. By Definition 3, we have s(HFEWGɛ(h1,h2,h3))=0.5646 and s(HFEWGɛ(h1′,h2′,h3′))=0.5568. It follows that HFEWGɛ(h1,h2,h3)≻HFEWGɛ(h1′,h2′,h3′). Clearly, hj≺hj′ for j=1,2,3, but HFEWGɛ(h1,h2,h3)≻HFEWGɛ(h1′,h2′,h3′).


### 4.2.  Hesitant Fuzzy Einstein Ordered Weighted Averaging Operator

Similar to theHFOWG operator introduced by Xia and Xu [36] (i.e., (15)), in what follows, we develop an (HFEOWGɛ) operator, which is an extension of OWA operator proposed by Yager [50].Definition 24.
For a collection of theHFEshj(j=1,2,…,n), a hesitant fuzzy Einstein ordered weighted averaging (HFEOWGɛ) operator is a mapping HFEWGɛ:Hn→H such that
(37)HFEOWGɛ(h1,h2,…,hn)=⨂εj=1nhσ(j)∧ɛwj=⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),…,γσ(n)∈hσ(n){∏j=1nγσ(j)wj(2∏j=1nγσ(j)wj)⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),…,γσ(n)∈hσ(n)×(∏j=1n(2-γσ(j))wj⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),…,γσ(n)∈hσ(n)×+∏j=1nγσ(j)wj)-1},
where (σ(1),σ(2),…,σ(n)) is a permutation of (1,2,…,n), such that hσ(j-1)≻hσ(j) for all j=2,…,n and w=(w1,w2,…,wn)T is aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. In particular, if w=(1/n,1/n,…,1/n)T, then the HFEOWGɛ operator is reduced to the HFEAɛ operator of dimension n (i.e., (17)).Note that theHFEOWGɛ weights can be obtained similar to the OWA weights. Several methods have been introduced to determine the OWA weights in [20, 21, 50–53].Similar to theHFEWGɛ operator, the HFEOWGɛ operator has the following properties.Theorem 25.
Lethj(j=1,2,…,n) be a collection of HFEs and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑j=1nwj=1. Then
(38)HFEOWGɛ(h1,h2,…,hn)≽HFOWG(h1,h2,…,hn),
where the equality holds if only if all hj(j=1,2,…,n) are equal and the number of values in hj is only one.From Theorem25, we can conclude that the values obtained by the HFEOWGɛ operator are not less than the ones obtained by the HFOWA operator proposed by Xia and Xu [36]. To illustrate that, let us consider the following example.Example 26.
Leth1=(0.1,0.4,0.7), h2=(0.3,0.5), and h3=(0.2,0.6) be three HFEs and suppose that w=(0.2,0.45,0.35)T is the associated vector of the aggregation operator.
By Definitions3 and 4, we calculate the score values and the accuracy values of h1, h2, and h3 as follows, respectively:
s
(
h
1
)
=
s
(
h
2
)
=
s
(
h
3
)
=
0.5, k(h1)=0.7551, k(h2)=0.9, k(h3)=0.8.
According to Definition5, we have h2≺h3≺h1. Then hσ(1)=h2, hσ(2)=h3, hσ(3)=h1.
By the definition ofHFEOWGɛ, we have
(39)HFEOWGɛ(h1,h2,h3)=⨂εj=13hσ(j)∧ɛwj=⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),γσ(3)∈hσ(3){2∏j=13γσ(j)wj∏j=13(2-γσ(j))wj+∏j=13γσ(j)wj}={0.1716,0.2787,0.3495,0.2939,0.4582,0.5598,0.1926,222220.3106,0.3877,0.3272,0.5047,0.6125}.
If we use the HFOWA operator, which was given by Xia and Xu [36] (i.e., (15)), to aggregate the HFEshj  (i=1,2,3), then we have
(40)HFOWG(h1,h2,…,hn)=⨂3j=1hσ(j)wj=⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),γσ(3)∈hσ(3){∏j=13‍γσ(j)wj}={0.1702,0.2764,0.3363,0.2790,0.4532,0.5513,2220.1885,0.3062,0.3724,0.3090,0.5020,0.6106}.
Clearly, s(HFEOWGɛ(h1,h2,h3))=0.3706>0.3629=s(HFOWG(h1,h2,…,hn)). By Definition 3, we have HFEOWGɛ(h1,h2,h3)≻HFOWG(h1,h2,h3).Theorem 27.
Letα>0, h be an HFE, let hj and hj′(j=1,2,…,n) be two collection of HFEs, and let w=(w1,w2,…,wn)T be an aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. Then (1)
HFEWGɛ(h1∧ɛα,h2∧ɛα,…,hn∧ɛα)=(HFEWGɛ(h1,h2,…,hn))∧ɛα,(2)
HFEWGɛ(h1⊗ɛh,h2⊗ɛh,…,hn⊗ɛh)=HFEWGɛ(h1,h2,…,hn)⊗ɛh,(3)
HFEWGɛ(h1∧ɛα⊗ɛh,h1∧ɛα⊗ɛh,…,hn∧ɛα⊗ɛh)=(HFEWGɛ(h1,h2,…,hn)⊗ɛh)∧ɛα,(4)
HFEWGɛ(h1⊗ɛh1′,h2⊗ɛh2′,…,hn⊗ɛhn′)=HFEWGɛ(h1,h2,…,hn)⊗ɛHFEWGɛ(h1′,h2′,…,hn′).Theorem 28.
Lethj(j=1,2,…,n) be a collection of HFEs and let w=(w1,w2,…,wn)T be an aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. Then
(41)hmin⁡-⪯HFEOWGɛ(h1,h2,…,hn)⪯hmax⁡+,
where hmin⁡-=min⁡j⁡{hj-∣hj-=min⁡{γj∈hj}} and hmax⁡+=max⁡j⁡{hj+∣hj+=max⁡{γj∈hj}}.Besides the above properties, we can get the following desirable results on theHFOWGɛ operator.Theorem 29.
Lethj(j=1,2,…,n) be a collection of HFEs, and let w=(w1,w2,…,wn)T be an aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. Then
(42)HFEOWGɛ(h1,h2,…,hn)=HFEOWGɛ(h1′,h2′,…,hn′),
where (h1′,h2′,…,hn′) is any permutation of (h1,h2,…,hn).Proof.
LetHFEOWGɛ(h1,h2,…,hn)=⊗εj=1nhσ(j)∧ɛwj and HFEOWGɛ(h1′,h2′,…,hn′)=⊗εj=1nhσ(j)′∧ɛwj. Since (h1′,h2′,…,hn′) is any permutation of (h1,h2,…,hn), then we have hσ(j)=hσ(j)′(j=1,2,…,n). Thus HFEOWGɛ(h1,h2,…,hn)=HFEOWGɛ(h1′,h2′,…,hn′).Theorem 30.
Lethj(j=1,2,…,n) be a collection of HFEs, and let w=(w1,w2,…,wn)T be an aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. Then(1)
ifw=(0,0,…,1), then HFOWGɛ(h1,h2,…,hn)=min⁡{h1,h2,…,hn};(2)
ifw=(1,0,…,0), then HFOWGɛ(h1,h2,…,hn)=max⁡{h1,h2,…,hn};(3)
ifwj=1 and wi=0(i≠j), then HFOWGɛ(h1,h2,…,hn)=hσ(j), where hσ(j) is the jth largest of hi(i=1,2,…,n).


## 4.1. Hesitant Fuzzy Einstein Geometric Weighted Aggregation Operator

Based on the operational laws(5) and (7) on HFEs, Xia and Xu [36] developed some hesitant fuzzy aggregation operators as listed below.Lethj(j=1,2,…,n) be a collection of HFEs; then.(
1
) the hesitant fuzzy weighted geometric (HFWG) operator
(14)HFWG(h1,h2,…,hn)=⨂j=1nhjωj=⋃γ1∈h1,γ2∈h2,…,γn∈hn{∏j=1n‍γjωj},
where ω=(ω1,ω2,…,ωn)T is the weight vector of hj(j=1,2,…,n) with ωj∈[0,1] and ∑j=1nωj=1.(
2
) the hesitant fuzzy ordered weighted geometric (HFOWG) operator
(15)HFOWG(h1,h2,…,hn)=⨂j=1n⁡wjhσ(j)=⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),…,γσ(n)∈hσ(n){∏j=1n‍γσ(j)wj},
where σ(1),σ(2),…,σ(n) is a permutation of 1,2,…,n, such that hσ(j-1)>hσ(j) for all j=2,…,n and w=(w1,w2,…,wn)T is aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1.For convenience, letH be the set of all HFEs. Based on the proposed Einstein operations on HFEs, we develop some new aggregation operators for HFEs and discuss their desirable properties.Definition 11.
Lethj(j=1,2,…,n) be a collection of HFEs. A hesitant fuzzy Einstein weighted geometric (HFEWGɛ) operator of dimension n is a mapping HFEWGɛ:Hn→H defined as follows:
(16)HFEWGɛ(h1,h2,…,hn)=⨂εj=1nhj∧ɛωj=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγjωj∏j=1n(2-γj)ωj+∏j=1nγjωj},
where w=(w1,w2,…,wn)T is the weight vector of hj(j=1,2,…,n) and wj>0, ∑j=1nwj=1. In particular, when wj=1/n, j=1,2,…,n, the HFEWGɛ operator is reduced to the hesitant fuzzy Einstein geometric (HFEGɛ) operator:
(17)HFEGɛ(h1,h2,…,hn)=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγj1/n∏j=1n(2-γj)1/n+∏j=1nγj1/n}.From Proposition10, we easily get the following result.Corollary 12.
If allhj(j=1,2,…,n) are equal and the number of values in hj is only one, that is, hj=h={γ} for all j=1,2,…,n, then
(18)HFEWGɛ(h1,h2,…,hn)=h.Note that theHFEWGɛ operator is not idempotent in general; we give the following example to illustrate this case.Example 13.
Leth1=h2=h3=h=(0.3,0.7), w=(0.4,0.25,0.35)T; then HFEWGɛ(h1,h2,h3)={0.3,0.4137,0.3782,0.5126,0.4323,0.579,0.5342,0.7}. By Definition 3, we have s(HFEWGɛ(h1,h2,h3))=0.4812<0.5=s(h). Hence HFEWGɛ(h1,h2,h3)≺h.Lemma 14 (see [18, 49]).
Letγj>0,wj>0, j=1,2,…,n, and ∑j=1nwj=1. Then
(19)∏j=1n‍γjwj≤∑j=1n‍wjγj
with equality if and only if γ1=γ2=⋯=γn.Theorem 15.
Lethj(j=1,2,…,n) be a collection of HFEs and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑j=1nwj=1. Then
(20)HFEWGɛ(h1,h2,…,hn)≽HFWG(h1,h2,…,hn),
where the equality holds if only if all hj(j=1,2,…,n) are equal and the number of values in hj is only one.Proof.
For anyγj∈hj(j=1,2,…,n), by Lemma 14, we have
(21)∏j=1n(2-γj)wj+∏j=1nγjwj≤∑j=1n‍wj(2-γj)+∑j=1n‍wjγj=2.
Then
(22)2∏j=1nγjωj∏j=1n(2-γj)ωj+∏j=1nγjωj≥∏j=1n‍γjωj.
It follows that s(⊗εj=1nhj∧ɛωj)≥s(⊗εj=1nhjωj), which completes the proof of Theorem 15.Theorem15 tells us the result that the HFEWGɛ operator shows the decision maker’s more optimistic attitude than the HFWA operator proposed by Xia and Xu [36] (i.e., (15)) in aggregation process. To illustrate that, we give an example adopted from Example  1 in [36] as follows.Example 16.
Leth1=(0.2,0.3,0.5), h2=(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
(23)HFEWGɛ(h1,h2)=⋃γ1∈h1,γ2∈h2{2∏j=12γjωj∏j=12(2-γj)ωj+∏j=12γjωj}={0.2482,0.2856,0.3276,0.3744,2220.4683,0.5288}.
However, Xia and Xu [36] used the HFWG operator to aggregate the hj(j=1,2) and got
(24)HFEG(h1,h2)=⋃γ1∈h1,γ2∈h2{∏j=12‍γjwj}={0.2462,0.2781,0.3270,0.3693,0.4676,0.5281}.
It is clear that s(HFEWGɛ(h1,h2))=0.3722>0.3694=s(HFEG(h1,h2)). Thus HFEWGɛ(h1,h2)≻HFEG(h1,h2).Based on Definition11 and the proposed operational laws, we can obtain the following properties on HFEWGɛ operator.Theorem 17.
Letα>0, hj(j=1,2,…,n), be a collection of HFEs and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(25)HFEWGɛ(h1∧ɛα,h2∧ɛα,…,hn∧ɛα)=(HFEWGɛ(h1,h2,…,hn))∧ɛα.Proof.
Sincehj∧ɛα=⋃γ∈hj{2γjα/((2-γj)α+γjα)} for all j=1,2,…,n, by the definition of HFEWGɛ, we have
(26)HFEWGɛ(h1∧ɛα,h2∧ɛα,…,hn∧ɛα)=⋃γ1∈h1,γ2∈h2,…,γn∈hn{(∏j=1n‍)(2∏j=1n‍(2γjα/((2-γj)α+γjα))ωj)=∪γ1∈h1,γ2∈h2,…,γn∈hn×(∏j=1n‍(2-(2γjα/((2-γj)α+γjα)))ωj=∪γ1∈h1,γ2∈h2,…,γn∈hn×+∏j=1n‍(2γjα/((2-γj)α+γjα))ωj)-1}=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγjαωj∏j=1n(2-γj)αωj+∏j=1nγjαωj}.
Since HFEWGɛ(h1,h2,…,hn)=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj)}, then
(27)(HFEWGɛ(h1,h2,…,hn))∧ɛα=⋃γ1∈h1,γ2∈h2,…,γn∈hn{(∏j=1n‍())(2(2∏j=1n‍γjωj/(∏j=1n‍(2-γj)ωj=∪γ1∈h1,γ2∈h2,…,γn∈hn2∏j=1n‍γjωj+∏j=1n‍γjωj))α)=∪γ1∈h1,γ2∈h2,…,γn∈hn×((2-(2∏j=1n‍γjωj/(∏j=1n‍(2-γj)ωj=∪γ1∈h1,γ2∈h2,…,γn∈hn2∏j=1n‍γjωj+∏j=1n‍γjωj)))α=∪γ1∈h1,γ2∈h2,…,γn∈hn++(2∏j=1n‍γjωj/(∏j=1n‍(2-γj)ωj=∪γ1∈h1,γ2∈h2,…,γn∈hn2∏j=1n‍γjωj+∏j=1n‍γjωj))α)-1}=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2(∏j=1nγjωj)α(∏j=1n(2-γj)ωj)α+(∏j=1nγjαωj)α}=⋃γ1∈h1,γ2∈h2,…,γn∈hn{2∏j=1nγjαωj∏j=1n(2-γj)αωj+∏j=1nγjαωj}.Theorem 18.
Leth be an HFE, hj(j=1,2,…,n) a collection of HFEs, and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(28)HFEWGɛ(h1⊗ɛh,h2⊗ɛh,…,hn⊗ɛh)=HFEWGɛ(h1,h2,…,hn)⊗ɛh.Proof.
By the definition ofHFEWGɛ and Einstein product operator of HFEs, we have(29)HFEWGɛ(h1,h2,…,hn)⊕ɛh=⋃γ∈h,γj∈hj,j=1,…,n{(2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj))·γ1+(1-(2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj)))(1-γ)}=⋃γ∈h,γj∈hj,j=1,…,n{2γ∏j=1nγjωj(2-γ)∏j=1n(2-γj)ωj+γ∏j=1nγjωj}. Since hj⊗ɛh=⋃γj∈hj,γ∈h{γjγ/(1+(1-γj)(1-γ))} for all j=1,2,…,n, by the definition of HFEWGɛ, we have
(30)HFEWGɛ(h1⊗ɛh,h2⊗ɛh,…,hn⊕ɛh)=⋃γ∈h,γj∈hj,j=1,…,n{(2∏j=1n‍(γjγ/(1+(1-γj)(1-γ)))ωj)=∪γ∈h,γj∈hj,j=1,…,n×(∏j=1n‍(2-(γjγ/(1+(1-γj)=∪γ∈h,γj∈hj,j=1,…,n+222222222×(1-γ))))ωj=∪γ∈h,γj∈hj,j=1,…,n+∏j=1n(γjγ/(1+(1-γj)‍=∪γ∈h,γj∈hj,j=1,…,n+222222222×(1-γ)))wj∏j=1n‍)-12∏j=1n‍)}=⋃γ∈h,γj∈hj,j=1,…,n{2∏j=1n(γjγ)ωj∏j=1n((2-γj)(2-γ))ωj+∏j=1n(γjγ)ωj}=⋃γ∈h,γj∈hj,j=1,…,n{(∏j=1n‍)(2∏j=1n‍γωj·∏j=1n‍γjωj)=∪γ∈h,γj∈hj,j=1,…,n×(∏j=1n‍(2-γ)ωj·∏j=1n‍(2-γj)ωj+∏j=1n‍γωj=∪γ∈h,γj∈hj,j=1,…,n×·∏j=1n‍γjωj)-1}=⋃γ∈h,γj∈hj,j=1,…,n{(2γ∑j=1nωj·∏j=1n‍γjωj)=∪γ∈h,γj∈hj,j=1,…,n×((2-γ)∑j=1nωj·∏j=1n‍(2-γj)ωj=∪γ∈h,γj∈hj,j=1,…,n×+γ∑j=1nωj·∏j=1n‍γjωj)-1}=⋃γ∈h,γj∈hj,j=1,…,n{2γ∏j=1nγjωj(2-γ)∏j=1n(2-γj)ωj+γ∏j=1nγjωj}.Based on Theorems17 and 18, the following property can be obtained easily.Theorem 19.
Letα>0, h be an HFE, let hj(j=1,2,…,n) be a collection of HFEs, and let w=(w1,w2,…,wn)T be the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(31)HFEWGɛ(h1∧ɛα⊗ɛh,h1∧ɛα⊗ɛh,…,hn∧ɛα⊗ɛh)=(HFEWGɛ(h1,h2,…,hn)⊗ɛh)∧ɛα.Theorem 20.
Lethj and hj′(j=1,2,…,n) be two collections of HFEs and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(32)HFEWGɛ(h1⊗ɛh1′,h2⊗ɛh2′,…,hn⊗ɛhn′)=HFEWGɛ(h1,h2,…,hn)⊗ɛHFEWGɛ(h1′,h2′,…,hn′).Proof.
By the definition ofHFEWGɛ and Einstein product operator of HFEs, we have(33)HFEWGε(h1,h2,…,hn)⊗εHFEWGε(h1′,h2′,…,hn′)=⋃γj∈hj,γj′∈hj′,j=1,…,n{(2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj))·(2∏j=1nγj′ωj/(∏j=1n(2-γj′)ωj+∏j=1nγj′ωj))1+(1-(2∏j=1nγjωj/(∏j=1n(2-γj)ωj+∏j=1nγjωj)))(1-(2∏j=1nγj′ωj/(∏j=1n(2-γj′)ωj+∏j=1nγj′ωj)))}=⋃γj∈hj,γj′∈hj′,j=1,…,n{2∏j=1nγjωj·∏j=1nγj′ωj∏j=1n(2-γj)ωj·∏j=1n(2-γj′)ωj+∏j=1nγjωj·∏j=1nγj′ωj}.Since hj⊗ɛhj′=⋃γj∈hj,γj′∈hj′{γjγj′/(1+(1-γj)(1-γj′))} for all j=1,2,…,n, by the definition of HFEWGɛ, we have
(34)HFEWGɛ(h1⊕ɛh1′,h2⊕ɛh2′,…,hn⊕ɛhn′)=⋃γj∈hj,γj′∈hj′,j=1,…,n{(2∏j=1n‍(γjγj′/(1+(1-γj)(1-γj′)))ωj)=∪γj∈hj,γj′∈hj′,j=1,…,n×(∏j=1n‍(2-(γjγj′/(1+(1-γj)=∪γj∈hj,γj′∈hj′,j=1,…,n×22222222222222×(1-γj′))))ωj=∪γj∈hj,γj′∈hj′,j=1,…,n×+∏j=1n‍(γjγj′/(1+(1-γj)=∪γj∈hj,γj′∈hj′,j=1,…,n×∏j=1n‍111111×(1-γj′)))ωj∏j=1n‍)-1∏j=1n‍)}=⋃γj∈hj,γj′∈hj′,j=1,…,n{(2∏j=1n(γjγj′)ωj)⋃γj∈hj,γj′∈hj′,j=1,…,n×(∏j=1n[(2-γj)(2-γj′)]ωj⋃γj∈hj,γj′∈hj′,j=1,…,n×+∏j=1n(γjγj′)ωj)-1}=⋃γj∈hj,γj′∈hj′,j=1,…,n{(2∏j=1n‍γjωj·∏j=1n‍γj′ωj)=∪γj∈hj,γj′∈hj′,j=1,…,n×(∏j=1n‍(2-γj)ωj·∏j=1n‍(2-γj′)ωj=∪γj∈hj,γj′∈hj′,j=1,…,n×+∏j=1n‍γjωj·∏j=1n‍γj′ωj)-1}.Theorem 21.
Lethj(j=1,2,…,n) be a collection of HFEs, hmin⁡-=min⁡j⁡{hj-∣hj-=min⁡{γj∈hj}}, and hmax⁡+=max⁡j⁡{hj+∣hj+=max⁡{γj∈hj}}, and let w=(w1,w2,…,wn)T be the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑i=1nwj=1. Then
(35)hmin⁡-⪯HFEWGɛ(h1,h2,…,hn)⪯hmax⁡+,
where the equality holds if only if all hj(j=1,2,…,n) are equal and the number of values in hj is only one.Proof.
Letf(t)=(2-t)/t, t∈[0,1]. Then f′(t)=-2/t2<0. Hence f(t) is a decreasing function. Since hmin⁡-≤hj-≤γj≤hj+≤hmax⁡+ for any γj∈hj(j=1,2,…,n), then f(hmax⁡+)≤f(γj)≤f(hmin⁡-); that is, (2-hmax⁡+)/hmax⁡+≤(2-γj)/γj≤(2-hmin⁡-)/hmin⁡-. Then for any γj∈hj(j=1,2,…,n), we have
(36)∏j=1n‍(2-hmax⁡+hmax⁡+)wj≤∏j=1n‍(2-γjγj)wj≤∏j=1n‍(1-hmin⁡-1+hmin⁡-)wj⇔(2-hmax⁡+hmax⁡+)∑j=1nwj≤∏j=1n‍(2-γjγj)wj≤(1-hmin⁡-1+hmin⁡-)∑j=1nwj⇔(2-hmax⁡+hmax⁡+)≤∏j=1n‍(2-γjγj)wj≤(1-hmin⁡-1+hmin⁡-)⇔2hmax⁡+≤∏j=1n‍(2-γjγj)wj+1≤2hmin⁡-⇔hmin⁡-2≤1∏j=1n((2-γj)/γj)wj+1≤(hmax⁡+2)⇔hmin⁡-≤2∏j=1n((2-γj)/γj)wj+1≤hmax⁡+⇔hmin⁡-≤2∏j=1nγjωj∏j=1n(2-γj)ωj+∏j=1nγjωj≤hmax⁡+.
It follows that hmin⁡-≤s(HFEWGɛ(h1,h2,…,hn))≤hmax⁡+. Thus we have hmin⁡-⪯HFEWGɛ(h1,h2,…,hn)⪯hmax⁡+.Remark 22.
Lethj and hj′(j=1,2,…,n) be two collections of HFEs, and hj≺hj′ for all j; then HFEWGɛ(h1,h2,…,hn)≺HFEWGɛ(h1′,h2′,…,hn′) does not hold necessarily in general. To illustrate that, an example is given as follows.Example 23.
Leth1=(0.45,0.6), h2=(0.6,0.7), h3=(0.5,0.6), h1′=(0.2,0.9), h2′=(0.45,0.95), h3′=(0.35,0.8), and w=(0.5,0.3,0.2)T; then HFEWGɛ(h1,h2,h3)={0.5024,0.5215,0.5286,0.5483,0.5791,0.6,0.6077,0.6291} and HFEWGɛ(h1′,h2′,h3′)={0.2778,0.3372,0.3835,0.4595,0.6088,0.7099,0.7833,0.8947}. By Definition 3, we have s(HFEWGɛ(h1,h2,h3))=0.5646 and s(HFEWGɛ(h1′,h2′,h3′))=0.5568. It follows that HFEWGɛ(h1,h2,h3)≻HFEWGɛ(h1′,h2′,h3′). Clearly, hj≺hj′ for j=1,2,3, but HFEWGɛ(h1,h2,h3)≻HFEWGɛ(h1′,h2′,h3′).


## 4.2.  Hesitant Fuzzy Einstein Ordered Weighted Averaging Operator

Similar to theHFOWG operator introduced by Xia and Xu [36] (i.e., (15)), in what follows, we develop an (HFEOWGɛ) operator, which is an extension of OWA operator proposed by Yager [50].Definition 24.
For a collection of theHFEshj(j=1,2,…,n), a hesitant fuzzy Einstein ordered weighted averaging (HFEOWGɛ) operator is a mapping HFEWGɛ:Hn→H such that
(37)HFEOWGɛ(h1,h2,…,hn)=⨂εj=1nhσ(j)∧ɛwj=⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),…,γσ(n)∈hσ(n){∏j=1nγσ(j)wj(2∏j=1nγσ(j)wj)⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),…,γσ(n)∈hσ(n)×(∏j=1n(2-γσ(j))wj⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),…,γσ(n)∈hσ(n)×+∏j=1nγσ(j)wj)-1},
where (σ(1),σ(2),…,σ(n)) is a permutation of (1,2,…,n), such that hσ(j-1)≻hσ(j) for all j=2,…,n and w=(w1,w2,…,wn)T is aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. In particular, if w=(1/n,1/n,…,1/n)T, then the HFEOWGɛ operator is reduced to the HFEAɛ operator of dimension n (i.e., (17)).Note that theHFEOWGɛ weights can be obtained similar to the OWA weights. Several methods have been introduced to determine the OWA weights in [20, 21, 50–53].Similar to theHFEWGɛ operator, the HFEOWGɛ operator has the following properties.Theorem 25.
Lethj(j=1,2,…,n) be a collection of HFEs and w=(w1,w2,…,wn)T the weight vector of hj(j=1,2,…,n) with wj∈[0,1] and ∑j=1nwj=1. Then
(38)HFEOWGɛ(h1,h2,…,hn)≽HFOWG(h1,h2,…,hn),
where the equality holds if only if all hj(j=1,2,…,n) are equal and the number of values in hj is only one.From Theorem25, we can conclude that the values obtained by the HFEOWGɛ operator are not less than the ones obtained by the HFOWA operator proposed by Xia and Xu [36]. To illustrate that, let us consider the following example.Example 26.
Leth1=(0.1,0.4,0.7), h2=(0.3,0.5), and h3=(0.2,0.6) be three HFEs and suppose that w=(0.2,0.45,0.35)T is the associated vector of the aggregation operator.
By Definitions3 and 4, we calculate the score values and the accuracy values of h1, h2, and h3 as follows, respectively:
s
(
h
1
)
=
s
(
h
2
)
=
s
(
h
3
)
=
0.5, k(h1)=0.7551, k(h2)=0.9, k(h3)=0.8.
According to Definition5, we have h2≺h3≺h1. Then hσ(1)=h2, hσ(2)=h3, hσ(3)=h1.
By the definition ofHFEOWGɛ, we have
(39)HFEOWGɛ(h1,h2,h3)=⨂εj=13hσ(j)∧ɛwj=⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),γσ(3)∈hσ(3){2∏j=13γσ(j)wj∏j=13(2-γσ(j))wj+∏j=13γσ(j)wj}={0.1716,0.2787,0.3495,0.2939,0.4582,0.5598,0.1926,222220.3106,0.3877,0.3272,0.5047,0.6125}.
If we use the HFOWA operator, which was given by Xia and Xu [36] (i.e., (15)), to aggregate the HFEshj  (i=1,2,3), then we have
(40)HFOWG(h1,h2,…,hn)=⨂3j=1hσ(j)wj=⋃γσ(1)∈hσ(1),γσ(2)∈hσ(2),γσ(3)∈hσ(3){∏j=13‍γσ(j)wj}={0.1702,0.2764,0.3363,0.2790,0.4532,0.5513,2220.1885,0.3062,0.3724,0.3090,0.5020,0.6106}.
Clearly, s(HFEOWGɛ(h1,h2,h3))=0.3706>0.3629=s(HFOWG(h1,h2,…,hn)). By Definition 3, we have HFEOWGɛ(h1,h2,h3)≻HFOWG(h1,h2,h3).Theorem 27.
Letα>0, h be an HFE, let hj and hj′(j=1,2,…,n) be two collection of HFEs, and let w=(w1,w2,…,wn)T be an aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. Then (1)
HFEWGɛ(h1∧ɛα,h2∧ɛα,…,hn∧ɛα)=(HFEWGɛ(h1,h2,…,hn))∧ɛα,(2)
HFEWGɛ(h1⊗ɛh,h2⊗ɛh,…,hn⊗ɛh)=HFEWGɛ(h1,h2,…,hn)⊗ɛh,(3)
HFEWGɛ(h1∧ɛα⊗ɛh,h1∧ɛα⊗ɛh,…,hn∧ɛα⊗ɛh)=(HFEWGɛ(h1,h2,…,hn)⊗ɛh)∧ɛα,(4)
HFEWGɛ(h1⊗ɛh1′,h2⊗ɛh2′,…,hn⊗ɛhn′)=HFEWGɛ(h1,h2,…,hn)⊗ɛHFEWGɛ(h1′,h2′,…,hn′).Theorem 28.
Lethj(j=1,2,…,n) be a collection of HFEs and let w=(w1,w2,…,wn)T be an aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. Then
(41)hmin⁡-⪯HFEOWGɛ(h1,h2,…,hn)⪯hmax⁡+,
where hmin⁡-=min⁡j⁡{hj-∣hj-=min⁡{γj∈hj}} and hmax⁡+=max⁡j⁡{hj+∣hj+=max⁡{γj∈hj}}.Besides the above properties, we can get the following desirable results on theHFOWGɛ operator.Theorem 29.
Lethj(j=1,2,…,n) be a collection of HFEs, and let w=(w1,w2,…,wn)T be an aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. Then
(42)HFEOWGɛ(h1,h2,…,hn)=HFEOWGɛ(h1′,h2′,…,hn′),
where (h1′,h2′,…,hn′) is any permutation of (h1,h2,…,hn).Proof.
LetHFEOWGɛ(h1,h2,…,hn)=⊗εj=1nhσ(j)∧ɛwj and HFEOWGɛ(h1′,h2′,…,hn′)=⊗εj=1nhσ(j)′∧ɛwj. Since (h1′,h2′,…,hn′) is any permutation of (h1,h2,…,hn), then we have hσ(j)=hσ(j)′(j=1,2,…,n). Thus HFEOWGɛ(h1,h2,…,hn)=HFEOWGɛ(h1′,h2′,…,hn′).Theorem 30.
Lethj(j=1,2,…,n) be a collection of HFEs, and let w=(w1,w2,…,wn)T be an aggregation-associated vector with wj∈[0,1] and ∑j=1nwj=1. Then(1)
ifw=(0,0,…,1), then HFOWGɛ(h1,h2,…,hn)=min⁡{h1,h2,…,hn};(2)
ifw=(1,0,…,0), then HFOWGɛ(h1,h2,…,hn)=max⁡{h1,h2,…,hn};(3)
ifwj=1 and wi=0(i≠j), then HFOWGɛ(h1,h2,…,hn)=hσ(j), where hσ(j) is the jth largest of hi(i=1,2,…,n).


## 5. An Application in Hesitant Fuzzy Decision Making

In this section, we apply theHFEWGɛ and HFEOWGɛ operators to multiple attribute decision making with hesitant fuzzy information.For hesitant fuzzy multiple attribute decision making problems, letY={Y1,Y2,…,Ym} be a discrete set of alternatives, let A={A1,A2,…,An} be a collection of attributes, and let ω=(ω1,ω2,…,ωn)T be the weight vector of Aj(j=1,2,…,n) with ωj≥0, j=1,2,…,n, and ∑j=1nωj=1. If the decision makers provide several values for the alternative Yi  (i=1,2,…,m) under the attribute Aj(j=1,2,…,n) with anonymity, these values can be considered as an HFEhij. 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 fuzzy decision matrix, where hij(i=1,2,…,m,j=1,2,…,n) are in the form of HFEs.To get the best alternative, we can utilize theHFEWGɛ operator or the HFEOWGɛ operator; that is,
(43)hi=HFEWGɛ(hi1,hi2,…,hin)=⋃γi1∈hi1,γi2∈hi2,…,γin∈hin{2∏j=1nγijωj∏j=1n(2-γij)ωj+∏j=1nγijωj}
or
(44)hi=HFEOWGɛ(hi1,hi2,…,hin)=⋃γiσ(j)∈hiσ(j),j=1,2,…,n‍{2∏j=1nγiσ(j)wj∏j=1n(2-γiσ(j))wj+∏j=1nγiσ(j)wj}
to derive the overall value hi of the alternatives Yi(i=1,2,…,m), where w=(w1,w2,…,wn)T is the weight vector related to the HFEOWAɛ operator, such that wj≥0, j=1,2,…,n, and ∑j=1nwj=1, which can be obtained by the normal distribution based method [20].Then by Definition3, we compute the scores s(hi)(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,…,Ym} and then select the best one (note that if there is no difference between the two scores hi and hj, then we need to compute the accuracy degrees k(hi) and k(hj) of the overall values hi and hj by Definition 4, respectively, and then rank the alternatives Yi and Yj in accordance with Definition 5).In the following, an example on multiple attribute decision making problem involving a customer buying a car, which is adopted from Herrera and Martinez [54], is given to illustrate the proposed method using the HFEOWGɛ operator.Example 31.
Consider that a customer wants to buy a car, which will be chosen from five typesYi(i=1,2,…,5). In the process of choosing one of the cars, four factors are considered: A1 is the consumption petrol, A2 is the price, A3 is the degree of comfort, and A4 is the safety factor. Suppose that the characteristic information of the alternatives Yi(i=1,2,…,5) can be represented by HFEshij(i=1,2,…,5;j=1,2,…,4), and the hesitant fuzzy decision matrix is given in Table 1.Table 1
Hesitant fuzzy decision making matrix.
A
1
A
2
A
3
A
4
Y
1
{
0.4,0.5
}
{
0.7,0.8
}
{
0.2,0.3,0.4
}
{
0.2,0.4
}
Y
2
{
0.2,0.5,0.8
}
{
0.5,0.7,0.9
}
{
0.6,0.8
}
{
0.2,0.5,0.6,0.7
}
Y
3
{
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
}
Y
4
{
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
}
Y
5
{
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
}To useHFEOWGɛ operator, we first reorder the hij(j=1,2,…,4) for each alternative Yi(i=1,2,…,5). According to Definitions 3 and 4, we compute the score values and accuracy degrees of s(hij)(i=1,2,…,5;j=1,2,…,4) as follows:
(45)s(h11)=0.45,s(h12)=0.75,s(h13)=0.3,s(h14)=0.3,k(h13)=0.9184,k(h14)=0.9;s(h21)=0.5,s(h22)=0.7,s(h23)=0.7,s(h24)=0.5,k(h21)=0.7551,k(h24)=0.8129,k(h22)=0.8367,k(h23)=0.9;s(h31)=0.85,s(h32)=0.4,s(h33)=0.35,s(h34)=0.4,k(h32)=0.8367,k(h34)=0.7764;s(h41)=0.6,s(h42)=0.6,s(h43)=0.3,s(h44)=0.4,k(h41)=0.772,k(h42)=0.8367;s(h51)=0.5,s(h52)=0.3,s(h53)=0.5,s(h54)=0.35,k(h51)=0.8367,k(h53)=0.8129.Then by Definition5, we have
(46)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.Suppose thatw=(0.1835,0.3165,0.3165,0.1835)T is the weighted vector related to the HFEOWAɛ operator and it is derived by the normal distribution based method [20]. Then we utilize the HFEOWAɛ operator to obtain the hesitant fuzzy elements hi(i=1,2,3,4,5) for the alternatives Xi(i=1,2,3,4,5). Take alternative X1 for an example; we have
(47)h1=HFEOWGɛ(h11,h12,…,h14)=⋃γ1σ(j)∈h1σ(j),j=1,2,3,4‍{2∏j=14γ1σ(j)wj∏j=14(2-γ1σ(j))wj+∏j=14γ1σ(j)wj}={0.3220,0.3642,0.3635,0.4099,0.3974,0.4470,2220.3473,0.3921,0.3914,0.4403,0.4272,0.4794,2220.3327,0.3760,0.3753,0.4228,0.4101,0.4607,2220.3587,0.4046,0.4039,0.4539,0.4405,0.4938}.
The results can be obtained similarly for the other alternatives; here we will not list them for vast amounts of data. By Definition 3, the score values s(hi) of hi(i=1,2,3,4,5) can be computed as follows:
(48)s(h1)=0.4048,s(h2)=0.5758,s(h3)=0.4311,s(h4)=0.4479,s(h5)=0.3620.According to the scoress(hi) of the overall hesitant fuzzy values hi(i=1,2,3,4,5), we can rank all the alternatives Xi: X2≻X4≻X3≻X1≻X5. Thus the optimal alternative is X2.If we use theHFWG operator introduced by Xia and Xu [36] to aggregate the hesitant fuzzy values, then
(49)s(h1)=0.3960,s(h2)=0.5630,s(h3)=0.4164,s(h4)=0.4344,s(h5)=0.3548.By Definition5, we have X2≻X4≻X3≻X1≻X5.Note that the rankings are the same in such two cases, but the overall values of alternatives by theHFEOWGɛ operator are not smaller than the ones by the HFOWG operator. It shows that the attitude of the decision maker using the proposed HFEOWGɛ operator is more optimistic than the one using the HFOWG operator introduced by Xia and Xu [36] in aggregation process. Therefore, according to the decision makers’ optimistic (or pessimistic) attitudes, the different hesitant fuzzy aggregation operators can be used to aggregate the hesitant fuzzy information in decision making process.


## 6. Conclusions

The purpose of multicriteria decision making is to select the optimal alternative from several alternatives or to get their ranking by aggregating the performances of each alternative under some attributes, which is the pervasive phenomenon in modern life. Hesitancy is the most common problem in decision making, for which hesitant fuzzy set can be considered as a suitable means allowing several possible degrees for an element to a set. Therefore, the hesitant fuzzy multiple attribute decision making problems have received more and more attention. In this paper, an accuracy function ofHFEs has been defined for distinguishing between the two HFEs having the same score values, and a new order relation between two HFEs has been provided. Some Einstein operations on HFEs and their basic properties have been presented. With the help of the proposed operations, several new hesitant fuzzy aggregation operators including the HFEWGɛ operator and HFEOWGɛ operator have been developed, which are extensions of the weighted geometric operator and the OWG operator with hesitant fuzzy information, respectively. Moreover, some desirable properties of the proposed operators have been discussed and the relationships between the proposed operators and the existing hesitant fuzzy aggregation operators introduced by Xia and Xu [36] have been established. Finally, based on the HFEOWGɛ operator, an approach of hesitant fuzzy decision making has been given and a practical example has been presented to demonstrate its practicality and effectiveness.



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*Source: 745617-2014-01-16.xml*