# Application of Dual Hesitant Fuzzy Geometric Bonferroni Mean Operators in Deciding an Energy Policy for the Society

Authors: Raja Noshad Jamil; Tabasam Rashid
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Dual hesitant fuzzy geometric Bonferroni mean is defined for dual hesitant fuzzy sets. Different properties of dual hesitant fuzzy geometric Bonferroni mean are discussed. Some special cases are studied in detail for dual hesitant fuzzy geometric Bonferroni mean. In addition, dual hesitant fuzzy weighted geometric Bonferroni mean and dual hesitant fuzzy Choquet geometric Bonferroni mean are proposed. A multicriteria decision-making method is discussed to find the best alternative among different alternatives by using proposed aggregated operators and an illustrated example is also given to understand our proposal.


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## Body

## 1. Introduction

Fuzzy sets (FS) initiated by Zadeh [1] are great invention in the field of sciences. The idea of fuzzy sets received great intension for handling uncertainty and vagueness situations. Often the information taken from human preferences has vagueness. Bellman and Zadeh succeeded to develop a useful technique in decision making for these situations by the help of fuzzy sets theory [2]. Due to use of fuzzy sets as the effective tools to collect and represent arguments in different fields like [3–11], several well-known extensions of fuzzy sets have been developed including interval-valued fuzzy sets (IVFS) [12], intuitionistic fuzzy sets (IFS) [13], interval-valued intuitionistic fuzzy sets (IVIFS) [4], hesitant fuzzy set [14], and dual hesitant fuzzy set (DFHS) [15].InFS, membership degree of an element a is taken as m∈[0,1], a single value, but in real life, the degree of nonmembership of a is not equal to 1-m(say m′) because there may exist some hesitation degree. Due to that, intuitionistic fuzzy set (IFS) which is a generalization of fuzzy sets theory was suggested by Atanassov [13, 16]. IFS integrates the membership degree of hesitation defined as m′′=1-(m+m′) where 0≤m′′≤1. IFS as the generalization of FS motivates the researchers to apply it in different real life applications. Atanassov [16, 17] showed that the information and semantic illustration of intuitionistic fuzzy set become more significant, practical, and applicable due to its degree of belongingness, degree of nonbelongingness, and the hesitation boundary. IFS is a useful representation of uncertainty and ambiguity of an entity and hence is able to be used as a great instrument to express data information under a range of different fuzzy situations. Torra [14] extended the concept of fuzzy sets to hesitant fuzzy sets (HFS). In HFS, a decision maker tries to manage those situations where hesitation involves in decision to provide membership value for particular element [18–23]. Zhu et al. initiated to merge both IFS and HFS and introduced the idea of dual hesitant fuzzy sets (DHFS) [15].Multicriteria decision making is the general observable routine in daily life, which is to select the most suitable selection that possibly takes place from several likely options or to obtain their position by aggregating the performances of each choice under several criteria. Aggregation is a procedure in which different available options are aggregated with different method and return a single value [24, 25]. Dual hesitant fuzzy sets (DHFS) have ability to deal with the circumstances when the assessment of an alternative under each condition is corresponded to several possible values for membership and nonmembership for the same element [15]. In existing literature there is very fewer work existing on aggregation operator for DHFS; particularly no work exists when arguments interrelated with each other. This inadequacy motivated us to develop some aggregative operators for DHFS based on geometric Bonferroni mean and Choquet integral. Geometric Bonferroni mean has the properties to capture the interrelationships among arguments [26]. The Choquet integral also is an important tool to consider the correlations among arguments [27]. Both of these operators have ability to aggregate all arguments when they interrelated with each other.In this research article, we purposed some aggregation operators forDHFS as dual hesitant fuzzy geometric Bonferroni mean (DHFGBM) and dual hesitant fuzzy weighted geometric Bonferroni mean (DHFWGBM). Further in this research, Choquet integral with geometric Bonferroni mean is used to introduce aggregation operators for DHFS called dual hesitant fuzzy Choquet geometric Bonferroni mean (DHFCBGM).The rest of the paper is organized as follows: Section2 briefly reviewed some basic concepts including definition of fuzzy set, its extension, and aggregated operators defined on these generalized fuzzy sets. Section 3 defined a ranking method for dual hesitant fuzzy sets and also defined some operational laws for manipulation of DHFS. Section 4 defined DHFGBM and discussed different properties of DHFGBM. Section 5 is devoted to discussion on Choquet integral (CI) based geometric Bonferroni mean in dual hesitant fuzzy environment. In Section 6, a multicriteria decision-making approach is discussed based on DHFCGBM and an illustrated example is given to understand our method. The research article finished in Section 7 with some concluding remarks.


## 2. Preliminaries

This section is devoted to some significant basic theories which are important to understand the article.Definition 1 (see [28]).
Letm1,m2>0 and ai≥0 for all i=1,2,…,n. Then Bonferroni geometric mean (BGM) is defined as(1)BGMm1,m2a1,a2,…,an=1m1+m2∏i,j=1,i≠jnm1ai+m2aj1/nn-1Definition 2 (see [1]).
LetX be a set of crisp elements, afuzzy setA on X is a function hA:X→[0,1] for any x∈X, and the value hA(x) is said to be the degree of membership of x in A.Definition 3 (see [13]).
LetX be a nonempty set; an intuitionistic fuzzy set A on X is combination of functions hA:X→[0,1] and hA′:X→[0,1], where hA(x) and hA′(x) represent the degrees of membership and nonmembership for every element x∈X and must satisfy the condition (2)0≤hAx+hA′x≤1Let=(h,h′), e1=(h1,h1′), and e2=(h2,h2′) be elements of intuitionistic fuzzy set A, then following basic operations introduced by Xu and Yager [29] hold:(1)
αe=1-(1-h)α,h′α, α>0;(2)
eα=hα,1-(1-h′)α, α>0;(3)
e1⊕e2=h1+h2-h1h2,h1′h2′;(4)
e1⊗e2=h1h2,h1′+h2′-h1′h2′.Based on these operation laws intuitionistic fuzzy Bonferroni mean and intuitionistic fuzzy geometric Bonferroni mean are defined as follows.Definition 4 (see [11]).
Letei=(εi,εi′) for all i=1,2,…,n be the elements of intuitionistic fuzzy set I for any m1, m2>0,(3)IFBm1,m2e1,e2,…,en=1nn-1⨁i,j=1i≠jneim1⊗ejm21/m1+m2,thenIFBm1,m2 is said to be intuitionistic fuzzy Bonferroni mean (IFBM).Definition 5 (see [30]).
Letei=(εi,εi′) for all i=1,2,…,n be the elements of intuitionistic fuzzy set I for any m1, m2>0,(4)IFGBm1,m2e1,e2,…,en=1m1+m2⨂i,j=1,i≠jnm1ei⊕m2ej⊗m1ej⊕m2ei2/nn-1,thenIFGBm1,m2 is said to be intuitionistic fuzzy Bonferroni Geometric mean (IFGBM).Torra [14] proposed HFS, which is a more general fuzzy set and permits the membership to include a set of possible values.Definition 6 (see [14]).
LetX be a fixed set; a hesitant fuzzy set A on X is defined in terms of a function hA(x) that when applied to X it returns to a finite subset of [0,1].To be easily understood, Xia and Xu [31] express the HFS by a mathematical symbol: (5)E=x,hEx∣x∈X,where hE(x) is a set of some values in [0,1], denoting the possible membership degrees of the element x∈X to the set E, and called a hesitant fuzzy element (HFE).Let=h, e1=h1, and e2=h2 be elements of hesitant fuzzy set A, then following basic operations introduced by Xia et al. [28] hold:(1)
eα=∪h∈ehα, α>0;(2)
αe=∪h∈e1-1-hα,α>0;(3)
h1⊕h2=∪h1∈e1,h2∈e2h1+h2-h1h2;(4)
h1⊗h2=∪h1∈e1,h2∈e2h1h2.Based on these operational laws hesitant fuzzy geometric Bonferroni mean is defined as follows.Definition 7 (see [32]).
Letei≥0 for all i=1,2,…,n be a collection of HFEs. For any m1,m2>0, then hesitant fuzzy geometric Bonferroni mean (HFGBM) is defined as(6)HFGBMm1,m2e1,e2,…,en=1m1+m2⨂i,j=1,i≠jnm1ei⊕m2ej⊗m1ej⊕m2ei2/nn-1In 2012, Zhu, et al. introduced dual hesitant fuzzy set which has both membership and nonmembership degree of an element in hesitant environment [15]. In 2014, Beg and Rashid [5] further discussed dual hesitant fuzzy set with new name intuitionistic fuzzy hesitant set and defined TOPSIS technique.Definition 8 (see [15]).
LetX be a nonempty set. A dual hesitant fuzzy set A on X is defined as pair of functions hA:X→[0,1] and hA′:X→[0,1], which can return to the subsets of [0,1] and is represented as A={x,hAx,hA′x∣x∈X. For every element x∈X, the condition 0≤max(hA(x))+minhA′x≤1 and 0≤min(hA(x))+max(hA′(x))≤1 must be satisfied.   e=(hA(x),hA′(x)) is called an dual hesitant fuzzy element (DHFE), where hA(x) is the possible membership values x while hA′(x) is the possible nonmembership values of x.


## 3. Manipulation with Dual Hesitant Fuzzy Elements

Ranking is a basic necessity of decision making in any fuzzy environment; therefore, dual hesitant fuzzy numbers must be ranked before any action is taken by a decision maker. To find order between twoDHFE, we defined accuracy and similarity function as follows.Definition 9.
Lete=(ε,ε′) be an DHFE where ε={ε1,ε2,…,εn} and ε′={ε1′,ε2′,…,εm′}, then an accuracy function for DHFE is defined as(7)φe=12∑i=1nεin+∑j=1mεj′mDefinition 10.
Ife=(ε,ε′) is DHFE where ε={ε1,ε2,…,εn} and ε′={ε1′,ε2′,…,εm′}, then a similarity function for DHFE is defined as(8)ψe=∑j=1mεj′m2-φeOrder between twoDHFE  e1=(ε1,ε1′) and e2=(ε2,ε2′) based on (7) and (8) is defined as follows.Definition 11.
Lete1=(ε1,ε1′) and e2=(ε2,ε2′) be two DHFE where ε=ε1,ε2,…,εn and ε′={ε1′,ε2′,…,εm′}, then order between e1 and e1 is as follows:(1)
ifψ(e1)>ψ(e2) then e1≻e2;(2)
ifψ(e1)=ψ(e2) and φ(e1)>φ(e2) then e1≻e2;(3)
ifψ(e1)=ψ(e2) and φ(e1)=φ(e2) then e1≈e2.Definition 12.
Letλ>0 and ek=(ηk,ηk′)(k=1,2,…,p) be collection of DHFE, where ηk=ε1,ε2,…,εm, ηk′={ε1′,ε2′,…,εn′} and εi,εj′∈[0,1] for i=1,2,…,m and j=1,2,…,n with condition (9)0≤max⁡εi+min⁡εj′≤1,and(10)0≤min⁡εi+max⁡εj′≤1,
then the following operations are hold:
(1)(11)λe1=⋃η1,η1′∈e11-1-εiλ,εj′λ∀i,j,alsoεi∈η1andεj′∈η1′
(2)(12)e1λ=⋃η1,η1′∈e1εiλ,1-1-εj′λ∀i,jalsoεi∈η1andεj′∈η1′
(3)(13)⨁k=1pek=⋃ηk,ηk′∈ek1-∏i=1m1-εk,i,∏j=1nεk,j′∀i,jalsoεk,i∈ηkandεk,j′∈ηk′
(4)(14)⨂k=1pek=⋃ηk,ηk′∈ek∏i=1mεk,i,1-∏j=1n1-εk,j′∀i,jalsoεk,i∈ηkandεk,j′∈ηk′Theorem 13.
Lete1=(ε1,ε1′) and e2=(ε2,ε2′) be DHFEs and λ>0, then(i)
λ(e1⊕e2)=λe1⊕λe2;(ii)
(e1⊗e2)λ=e1λ⊗e2λ.Theorem 14.
Leteαi and eβi be two collections of DHFE, if for any m1,m2>0, one has (15)ζαi,j,i<j=m1eαi⊕m2eαj⊗m1eαj⊕m2eαiwhereηαi,ηαi′∈eαiforεαi∈ηαiandεαi′∈ηαi′and(16)ζβi,j,i<j=m1eβi⊕m2eβj⊗m1eβj⊕m2eβiwhereηβi,ηβi′∈eβiforεβi∈ηβiandεβi′∈ηβi′and if for all εαi≤εβi and for all εαi′≥εβi′ then one has (17)ζαi,j,i<j≤ζβi,j,i<jProof.
Asεαi≤εβi,εαj≤εβj and εαi′≥εβi′,εαj′≥εβj′   ∀i,j=1,2,…,n,i≠j, we have(18)1-1-εαim11-εαjm2≤1-1-εβim11-εβjm2(19)1-1-εαjm11-εαim2≤1-1-εβjm11-εβim2(20)εαi′m1εαj′m2≥εβi′m1εβj′m2(21)εαj′m1εαi′m2≥εβj′m1εβi′m2;
take(22)ζαi,j,i<j=m1eαi⊕m2eαj⊗m1eαj⊕m2eαi=⋃εαi,εαi′∈eαi,εαj,εαj′∈eαj1-1-εαim1,εαi′m1⊕1-1-εαjm2,εαj′m2⊗⋃εαi,εαi′∈eαi,εαj,εαj′∈eαj1-1-εαjm1,εαj′m1⊕1-1-εαim2,εαi′m2ζαi,j,i<j=⋃εαi,εαi′∈eαi,εαj,εαj′∈eαj1-1-εαim11-εαjm2,εαi′m1εαj′m2⊗⋃εαi,εαi′ϵeαi,εαj,εαj′ϵeαj1-1-εαjm11-εαim2,εαj′m1εαi′m2;
similarly,(23)ζβi,j,i<j=⋃εβi,εβi′∈eβi,εβj,εβj′∈eβj1-1-εβim11-εβjm2,εβi′m1εβj′m2⊗⋃εβi,εβi′∈eβi,εβj,εβj′∈eβj1-1-εβjm11-εβim2,εβj′m1εβi′m2
Letταi,j,i<j∈ζαi,j,i<j and τβi,j,i<j∈ζβi,j,i<j∀i,j=1,2,3,…,n;i<j then by Definition 12 and (18)−(23), we have ταi,j,i<j≤τβi,j,i<j∀i,j=1,2,3,…,n with i<j and hence(24)ζαi,j,i<j≤ζβi,j,i<jis the required result.Theorem 15.
Letei be a collection of DHFE and for any m1,m2>0, one has (25)ζi,j,i<j=m1ei⊕m2ej⊗m1ej⊕m2ei,ei-=⋃εi,εi′ϵeimin⁡εi,max⁡εi′
and(26)ei+=⋃εi,εi′∈eimax⁡εi,min⁡εi′;if (ε-,ε′+)∈ei- and (ε+,ε′-)∈ei+ for i,j=1,2,…,n;i<j with ε-≤ε+ and ε′+≥ε′- then(27)⋃ε-,ε′+∈ei-1-1-ε-m1+m22,1-1-ε′+m1+m22≤ζαi,j,i<j≤⋃ε+,ε′-∈ei+1-1-ε+m1+m22,1-1-ε′-m1+m22Proof.
Sinceε-≤εi≤ε+ and ε′-≤εi′≤ε′+(i=1,2,…,n) then(28)1-1-ε-m1+m2≤1-1-εim11-εjm2≤1-1-ε+m1+m2,(29)1-1-ε-m1+m2≤1-1-εjm11-εim2≤1-1-ε+m1+m2,(30)ε′-m1+m2≤εi′m1εj′m2≤ε′+m1+m2,and(31)ε′-m1+m2≤εj′m1εi′m2≤ε′+m1+m2as(32)ζi,j,i<j=m1ei⊕m2ej⊗m1ej⊕m2ei=⋃εi,εi′∈ei,εj,εj′∈ej1-1-εim11-εjm2,εi′m1εj′m2⊗⋃εi,εi′∈ei,εj,εj′∈ej1-1-εjm11-εim2,εj′m1εi′m2;
according to Definition12 and (28)−(32) we have (33)⋃ε-,ε′+∈ei-1-1-ε-m1+m22,1-1-ε′+m1+m22≤ζαi,j,i<j≤⋃ε+,ε′-∈ei+1-1-ε+m1+m22,1-1-ε′-m1+m22
which is required.Remark 16.
Letei be a collection of DHFE then for any m1,m2>0 we have (34)m1ei⊕m2ej⊗m1ej⊕m2ei=m2ei⊕m1ej⊗m2ej⊕m1ei


## 4. Dual Hesitant Fuzzy Geometric Bonferroni Operators

In multicriteria decision making, the qualifications of an option under a certain condition may be characterized by many potential options. To aggregate all the potential options of an alternative under some criteria, we presented a generalization of theGBM in dual hesitant fuzzy environments, which is defined as follows.Definition 17.
Letm1,m2>0, and ei(i=1,2,…,n) be the collection of DHFE, then dual hesitant fuzzy geometric Bonferroni mean (DHFGBM) is defined as(35)DHFGBMm1,m2e1,e2,…,en=1m1+m2⨂i,j=1i≠jnm1ei⊕m2ej⊗m1ej⊕m2ei2/nn-1;
with the help of operational laws (11)-(14), defined on DHFE, we further derive the following results.Theorem 18.
Letm1,m2>0 and ei(i=1,2,…,n) be the collection of DHFE. Then DHFGBM is an DHFE and (36)DHFGBMm1,m2e1,e2,…,en=⋃εi,j,εi,j′∈ζi,j1-1-∏i,j=1,i<jnεi,j2/nn-11/m1+m2,1-∏i,j=1,i<jn1-εi,j′2/nn-11/m1+m2where(37)ζi,j,i<j=m1ei⊕m2ej⊗m1ej⊕m2eiProof.
From (11)−(14) and Theorem 13 we see that ζi,j is also an DHFE. By (35) and (37) we have(38)DHFGBMm1,m2e1,e2,…,en=1m1+m2⨂i,j=1nζi,j,i<j2/nn-1Theorem 13 applies on (38); we obtain (39)DHFGBMm1,m2e1,e2,…,en=1m1+m2⨂i,j=1nζi,j,i<j2/nn-1;by (14) (40)=1m1+m2⋃εi,j,εi,j′∈ζi,j;i<j∏i,j=1,i<jnεi,j,1-∏i,j=1,i<jn1-εi,j′2/nn-1; from (12) (41)=1m1+m2⋃εi,j,εi,j′ϵζi,j;i<j∏i,j=1,i<jnεi,j2/nn-1,1-∏i,j=1,i<jn1-εi,j′2/nn-1 using (11), we have(42)DHFGBMm1,m2e1,e2,…,en=⋃εi,j,εi,j′ϵζi,j;i<j1-1-∏i,j=1,i<jnεi,j2/nn-11/m1+m2,1-∏i,j=1,i<jn1-εi,j′2/nn-11/m1+m2;the proof is completed.Example 19.
Let us consider threeDHFE, e1=0.1,0.3, e2=0.2,0.4,0.1, and e3=0.3,0.1,0.2. If m1=m2=1, then(43)DHGBM1,1e1,e2,e3=0.0660,0.0726,0.0784,0.0799,0.0863,0.0932,0.0950,0.1027,0.1132,0.2150,0.2223,0.2294,0.2369,0.2436,0.2500,0.2569,0.2629,0.2688Theorem 20.
Letei=εi,εi′,(i=1,2,…,n) be the collection of DHFEs, then by theorem one has the following results:(1)
Ifei=εi,εi′=({0},{0})∀i=1,2,…,n then DHFGBM=({0},{0})(2)
Ifei=εi,εi′=({1},{0})∀i=1,2,…,n then DHFGBM=({1},{0})(3)
Ifei=εi,εi′=({0},{1})∀i=1,2,…,n then DHFGBM=({0},{1})Next we discussed some special cases ofDHFGBM by replacing the parameters m1 and m2 as follows.Case 1.
Ifm2→0 then by (35) we have (44)DHFGBMm1,m2e1,e2,…,en=1m1+m2⨂i,j=1,i<jnm1ei⊕m2ej⊗m1ej⊕m2ei2/nn-1=1m1⨂i,j=1,i<jnm1ei⊗m1ej2/nn-1=1m1⋃εi,εi′∈ei,εj,εj′∈ej∏i,j=1,i<jn1-1-εim11-1-εjm12/nn-1,1-∏i,j=1,i<jn1-1-1-εi′m11-εj′m12/nn-1=⋃εi,εi′∈ei,εj,εj′∈ej1-1-∏i,j=1,i<jn1-1-εim11-1-εjm12/nn-11/m1,1-∏i,j=1,i<jn1-1-1-εi′m11-εj′m12/nn-11/m1=DHFGBMm1,0e1,e2,…,enwhich is called generalized dual hesitant fuzzy geometric Bonferroni mean (GDHFGBM).Case 2.
Ifm1=1 and m2→0 then (35) becomes (45)DHFGBMm1,m2e1,e2,…,en=⨂i,j=1nei⊗ej2/nn-1=⋃εi,εi′∈ei,εj,εj′∈ej1-1-∏i,j=1,i<jn1-1-εi22/nn-1,1-∏i,j=1,i<jn1-1-1-εi′22/nn-1=DHFGBM1,0eiwhich is called dual hesitant fuzzy unit geometric Bonferroni mean.Case 3.
Ifm1=2 and m2→0, (46)DHFGBMm1,m2e1,e2,…,en=12⨂i,j=1i<jn2ei⊗2ej2/nn-1=⋃εi,εi′∈ei,εj,εj′∈ej1-1-∏i,j=1,i<jn1-1-εi21-1-εj22/nn-11/2,1-∏i,j=1,i<jn1-1-1-εi′21-εj′22/nn-11/2=DHFGBM2,0e1,e2,…,enDHFGBM2,0(e1,e2,…,en) is called dual hesitant fuzzy square geometric Bonferroni mean.Case 4.
Ifm1=1 and m2=1, let (47)ζi,j,i<j1,1=ei⊕ej⊗ej⊕ei=⋃εi,j1,1,εi,j′1,1∈ζi,j1,1i<jεi,j1,1,εi,j′1,1, then (35) reduces to the following: (48)DHFGBMm1,m2e1,e2,…,en=12⨂i,j=1i<jnei⊕ej⊗ej⊕ei2/nn-1=12⋃εi,j1,1,εi,j′1,1∈ζi,j1,1i<j∏i,j=1,i<jnεi,j1,1,1-∏i,j=1,i<jn1-εi,j′1,12/nn-1=12⋃εi,j1,1,εi,j′1,1∈ζi,j1,1i<j∏i,j=1,i<jnεi,j1,12/nn-1,1-1-1-∏i,j=1,i<jn1-εi,j′1,12/nn-1=⋃εi,j1,1,εi,j′1,1∈ζi,j1,1i<j1-1-∏i,j=1,i<jnεi,j1,12/nn-11/2,1-1-1-∏i,j=1,i<jn1-εi,j′1,12/nn-11/2 which is called an dual hesitant fuzzy interrelated square geometric Bonferroni mean.Corollary 21.
Ifei=(εi,εi′)=(ε,ε′)=e, such that ε,ε′⊂0,1 for all i=1,2,…,n, then for any m1,m2>0 is (49)DHFGBMm1,m2e1,e2,…,en=1m1+m2⨂i,j=1,i<jnm1ei⊕m2ej⊗m1ej⊕m2ei2/nn-1=⋃η,η′∈ζ1-1-η1/m1+m2,η′1/m1+m2whereη,η′∈ζ and (50)ζ=m1e⊕m2e⊗m1e⊕m2eProof.
From (11)-(14) and Theorem 13 we see that ζi,j is also an DHFE(51)ζi,j,i<j=m1ei⊕m2ej⊗m1ej⊕m2ei=m1e⊕m2e⊗m1e⊕m2e=ζLet n be total number of DHFEs and Υ be the collection of all ζi,j,i<j, then the total number of elements in Υ is(52)OΥ=nn-12;
take(53)DHFGBMm1,m2e1,e2,…,en=1m1+m2⨂i,j=1i<jnζi,j2/nn-1=1m1+m2⨂i,j=1i<jnζ2/nn-1and  ifη,η′∈ζ,then(54)=1m1+m2⋃η,η′∈ζ∏i,j=1nη,1-∏i,j=1n1-η′2/nn-1=1m1+m2⋃η,η′∈ζ∏i,j=1nη2/nn-1,1-∏i,j=1n1-η′2/nn-1=⋃η,η′∈ζ1-1-∏i,j=1nη2/nn-11/m1+m2,1-∏i,j=1n1-η′2/nn-11/m1+m2=⋃η,η′∈ζ1-1-∏i,j=1nη2/nn-11/m1+m2,1-∏i,j=1n1-η′2/nn-11/m1+m2by equation (52),(55)∏i,j=1nη=ηnn-1/2,∏i,j=1n1-η=1-ηnn-1/2therefore,DHFGBMm1,m2e1,e2,…,en=⋃η,η′∈ζ1-1-η1/m1+m2,η′1/m1+m2Corollary 22 (monotonicity).
Let(ηαi,ηαi′)∈eαi and (ηβi,ηβi′)∈eβi be two collections of DHFE; also εαi∈ηαi, εβi∈ηβi,εαi′∈ηαi′, and εβi′∈ηβi′ with the conditions εαi≤εβi and εαi′≥εβi′∀i,j=1,2,…,n,i≠j, then DHFGBMm1,m2(eαi)≤DHFGBMm1,m2(eβi).Proof.
By (24) and (42) we have(56)1-1-∏i,j=1;i<jnεαi,j2/nn-11/m1+m2≤1-1-∏i,j=1,i<jnεβi,j2/nn-11/m1+m2(57)1-∏i,j=1,i<jn1-εαi,j′2/nn-11/m1+m2≥1-∏i,j=1,i<jn1-εβi,j′2/nn-11/m1+m2;
by Definition12 and (56) and (57) we have(58)1m1+m2⨂i,j=1nζαi,j2/nn-1=⋃εi,j,εi,j′∈ζi,j;i<j1-1-∏i,j=1,i<jnεαi,j2/nn-11/m1+m2,1-∏i,j=1,i<jn1-εαi,j′2/nn-11/m1+m2≤⋃εi,j,εi,j′∈ζi,j;i<j1-1-∏i,j=1,i<jnεβi,j2/nn-11/m1+m2,1-∏i,j=1,i<jn1-εβi,j′2/nn-11/m1+m2≤1m1+m2⨂i,j=1nζβi,j2/nn-1; hence (59)1m1+m2⨂i,j=1nζαi,j2/nn-1≤1m1+m2⨂i,j=1nζβi,j2/nn-1; the proof is completed.Corollary 23 (bounded).
Let(εαi,εαi′)ϵeαi and (εβi,εβi′)ϵeβi be two collections of DHFE, ei-=∪(εi,εi′)∈eimin{εi},max{εi′} and ei+=∪(εi,εi′)∈eimax{εi},min{εi′}; if (ε-,ε′+)∈ei- and (ε+,ε′-)∈ei+,i,j=1,2,…,n;i≠j with the condition ε-≤ε+ and ε′+≥ε′-, then for any m1,m2>0, one has(60)⋃ε-,ε′+∈ei-1-1-1-1-ε-m1+m221/m1+m2,1-1-1-ε′+m1+m221/m1+m2≤DHFGBm1,m2≤⋃ε+,ε′-∈ei+1-1-1-1-ε+m1+m221/m1+m2,1-1-1-ε′-m1+m221/m1+m2Proof.
By (27) and (42) we have(61)1-1-1-1-ε-m1+m221/m1+m2,1-1-1-ε′+m1+m221/m1+m2≤1-1-∏i,j=1,i<jnεβi,j2/nn-11/m1+m2,1-∏i,j=1,i<jn1-εβi,j′2/nn-11/m1+m2≤1-1-1-1-ε+m1+m221/m1+m2,1-1-1-ε′-m1+m221/m1+m2 By (27) and (61) we have the required result.Corollary 24.
Letei=(ηi,ηi′)(i=1,2,…,n) be collection of DHFEs. Let εi=(ε1,ε2,…,εp)∈ηi, εi′=(ε1′,ε2′,…,εq′)∈ηi′, m1,m2∈N, εi.=(ε1.,ε2.,…,εp.)∈ηi., and εi.′=(ε1.′,ε2.′,…,εq.′)∈ηi.′, where “.” represents any permutations of εi and εi′, respectively, then (62)DHFGBMm1,m2ei=DHFGBMm1,m2ei. where ζi,j;i<j=(m1ei⊕m2ej)⊗(m1ej⊕m2ei) and ζi,j;i<j.=(m1ei.⊕m2ej.)⊗(m1ej.⊕m2ei.)i,j=1,2,…,n,i≠j.Proof.
Given thatei=(ηi,ηi′) and εi∈ηi , εi′∈ηi′, m1,m2∈N, εi.∈ηi., and εi.′∈ηi.′, then (63)DHFGBMm1,m2ηi,ηi′=1m1+m2⨂i,j=1nζi,j;i<j2/nn-1=1m1+m2⨂i,j=1nζi,j;i<j.2/nn-1=DHFGBMm1,m2ηi.,ηi.′Corollary 25.
Letei=(ηi,ηi′)(i=1,2,…,n) be collection of DHFEs, where εi∈ηi, εi′∈ηi′ with εi=(ε1,ε2,…,εp) and εi′=(ε1′,ε2′,…,εq′), m1,m2∈N, then (64)DHFGBMm1,m2e1,e2,…,en=DHFGBMm2,m1e1,e2,…,enProof.
One has(65)DHFGBMm1,m2e1,e2,…,en=1m1+m2⨂i,j=1;i<jnm1ei⊕m2ej⊗m1ej⊕m2ei2/nn-1=1m2+m1⨂i,j=1;i<jnm2ei⊕m1ej⊗m2ej⊕m1ei2/nn-1=DHFGBMm2,m1e1,e2,…,en which is required proof.In practical applications often, we have to deal with complicated situations, considering not only the importance of individual arguments but also the relations among them. In order to consider the connections among criteria in multicriteria decision-making problems, we developed a dual hesitant fuzzy weighted geometric Bonferroni mean(DHFWGBM).Definition 26.
Letei(i=1,2,…,n) be a collection of DHFE and w=(wi) be the weight vector of ei, where wi indicates the importance degree of ei, satisfying wi>0(i=1,2,…,n) and ∑i=1nwi=1. Then dual hesitant fuzzy weighted geometric Bonferroni mean (DHFWGBM) for m1,m2>0 is(66)DHFWGBMm1,m2e1,e2,…,en=1m1+m2⨂i,j=1:i<jnm1eiwi⊕m2ejwj⊗m1ejwj⊕m2eiwi2/nn-1


## 5. Choquet Integral and Geometric Bonferroni Mean

As Bonferroni mean has the subjective characteristics to capture interrelationships among arguments, Choquet integral has ability to represent objective characteristics of arguments. Sometimes in multicriteria decision-making problems, it should be required to take the choice of the decision makers; the Choquet integral can deal with these situations; i.e., it is possible to take not only the importance of each criteria but also the correlations of criteria. Therefore, both techniques are often used in practical situations that why in this research paper we merged theDHFGBM with the Choquet integral as follows.Let(67)Aw~i,j:i<j=ζσi,j1w,ζσi,j2w,…,ζσi,jn-1wsuchthatζi,j:i<jw∉A~i,j:i<jandζσi,jl-1w≥ζσi,jlw,∀l=2,3,…,n-1and(68)Bσi,jkw=∅whenk=1andBσi,jkw=ζσi,jl:l≤kwwhenk≥2IfE=e1,e2,…,en is the set of criteria and E~i,j=E-{ei,ej} is the set of all criteria except ei and ej, then a fuzzy measure ωi,j:i<j on E~i,j is(69)ωi,j:i<jE~i,j⟶0,1which satisfied the following conditions:(1)
ωi,j:i<j(∅)=0(2)
ωi,j:i<j(E~i,j)=1(3)
ωi,j:i<j(G)≤ωi,j:i<j(H)Ifωi,j:i<j is a fuzzy measure on E~i,j defined in (69) and if we take(70)aw~i,j:i<j=m1eiwi⊕m2ejwj⊗m1ejwj⊕m2eiwi then(71)μi,j:i<jw=⨁i,j=1,i≠jnaw~i,j:i<jωi,j:i<jBσi,j:i<jkw-ωi,j:i<jBσi,j:i<jk-1wwhereaw~i,j:i<j∈A~i,j:i<jandk=2,3,…,n.Then the Choquet integral ofaw~i,j:i<j with respect to fuzzy measure ωi,j:i<j is defined as follows:(72)Cωi,j:i<jwaw~i,j:i<j=∑k2=1n2εi,j:i<j′wn22-φaw~i,j:i<jfori,j=1,2,…,nwhere,φaw~i,j:i<j=12∑k1=1n1εi,j:i<jwn1+∑k2=1n2εi,j:i<j′wn2whereεi,j:i<jw,εi,j:i<j′w∈μi,j:i<jwDefinition 27.
Letei(i=1,2,…,n) be a collection of DHFE and w=(wi) be the weight vector of ei, where wi indicates the importance degree of ei, satisfying wi>0(i=1,2,…,n) and ∑i=1nwi=1. Then dual hesitant fuzzy Choquet geometric Bonferroni mean (DHFCGBM) is as follows for any m1,m2>0:(73)DHFCGBMwm1,m2e1,e2,…,en=1m1+m2⨂i,j=1:i<jnm1eiwi⊕m2ejwj⊗m1ejwj⊕m2eiwiCωi,j:i<jwaw~i,j:i<j2/nn-1where Cωi,j:i<jw(aw~i,j:i<j) be the Choquet integral defined in (72)Theorem 28.
Letm1,m2>0, ei=(εi,εi′)  for i=1,2,…,n be the collection of DHFE, then aggregated value of DHFCGBM is also an DHFE and equal to(74)DHFCGBMwm1,m2e1,e2,…,en=1m1+m2⋃εi,jwi,j,εi,j′wi,j∈ζi,j:i<jwi,j⨂i,j=1,i<jnεi,jwi,jCωi,j:i<jwaw~i,j:i<j2/nn-1,1-1-εi,j′wi,jCωi,j:i<jwaw~i,j:i<j2/nn-1Proof.
It is obvious from (11)-(14) and Theorem 13 that (75)ζi,j:i<jwi,j=m1eiwi⊕m2ejwj⊗m1ejwj⊕m2eiwi,an DHFE. Therefore from (73) and (75) we have (76)DHFCGBMwm1,m2e1,e2,…,en=1m1+m2⨂i,j=1:i<jnm1eiwi⊕m2ejwj⊗m1ejwj⊕m2eiwiCωi,j:i<jwaw~i,j:i<j2/nn-1=1m1+m2⨂i,j=1,i<jnζi,j:i<jwi,jCωi,j:i<jwaw~i,j:i<j2/nn-1=1m1+m2⋃εi,jwi,j,εi,j′wi,j∈ζi,j:i<jwi,j⨂i,j=1,i<jnεi,jwi,jCωi,j:i<jwaw~i,j:i<j,1-1-εi,j′wi,jCωi,j:i<jwaw~i,j:i<j2/nn-1=1m1+m2⋃εi,jwi,j,εi,j′wi,j∈ζi,j:i<jwi,j⨂i,j=1,i<jnεi,jwi,jCωi,j:i<jwaw~i,j:i<j2/nn-1,1-1-εi,j′wi,jCωi,j:i<jwaw~i,j:i<j2/nn-1=1m1+m2⋃εi,jwi,j,εi,j′wi,j∈ζi,j:i<jwi,j∏i,j=1,i<jnεi,jwi,jCωi,j:i<jwaw~i,j:i<j2/nn-1,1-∏i,j=1,i<jn1-εi,j′wi,jCωi,j:i<jwaw~i,j:i<j2/nn-1=⋃εi,jwi,j,εi,j′wi,j∈ζi,j:i<jwi,j1-1-∏i,j=1,i<jnεi,jwi,jCωi,j:i<jwaw~i,j:i<j2/nn-11/m1+m2,1-∏i,j=1,i<jn1-εi,j′wi,jCωi,j:i<jwaw~i,j,i<j2/nn-11/m1+m2


## 6. An Approach to Multicriteria Decision Making with CI Based GBM

In this section, we developed a method by applyingDHFWGBM and DHFCGBM operators for multicriteria decision making. There are m decision makers (DMs) who want to select the best alternative from the collection of Pi(i=1,2,…,n) options; W={w1,w2,…,wk} are the weights for Ck(k=1,2,…,p) criteria to be considered by DMs. DMs used DHFS for their opinion for each alternative; i.e., DMs predilections are in the form of optimistic felling with hesitant situations (membership value) and denial felling also with hesitant situations (nonmembership value). We now introduced the following methodology to deal with the above discussed situation. The following steps are required to find the best choice among the participated alternatives.Step 1.
Firstly represent eachDMs opinion in matrix form, called dual hesitant fuzzy matrix (DHFM)  Dj=({εr1},{εr2})n×pr1,r2∈N, n and p represented the total numbers of alternatives and criteria, respectively. Let wm be the weight vector for DMs and wc be weight vector for criteria.Step 2.
Find a matrixDn×p by aggregating each corresponding element of Dj(j=1,2,…,m) matrices by aggregation operator DHFWGBM and also taking aggregation weights for resultant element by calculating simple arithmetic mean on the weights wc suggested by each DMs for every criteria.Step 3.
Aggregate all values of each alternative in the matrixDn×p by DHFCGBM and then find ψi(Pi)∀i=1,2,…,n.Step 4.
The best alternative ismax(ψi).


### 6.1. Illustrative Example

The following example illustrated the suggested method.Example 29.
Energy is an essential part for the growth of social-economic development of any society. Thus it is very important to adopt proper energy policy for economic development and environment. Let us assume that there are five alternatives (energy projects)Pi(i=1,2,3,4,5) to be invested and three criteria to be considered: C1- technological; C2- environmental; C3- economic. Two DMs are requested to evaluate the performances of the five alternatives. To deal with this energy policy problem, consider weights according to preference of DMs for each alternative in group decision making.Case 1.
⁡
Step  1. The DMs provide their preferences over all the alternatives Pi(i=1,2,3,4,5)  w.r.t. the criteria Ck(k=1,2,3) based on DHFS; let weight vectors for the criteria be wc1=0.4,0.3,0.3 and wc2=0.3,0.4,0.3, with respect to DMs, respectively. The following Dj(j=1,2) represented the suggestion of DMs, in DHFE: (77)D1=0.3,0.6,0.30.6,0.30.7,0.20.6,0.30.3,0.5,0.40.7,0.1,0.20.4,0.3,0.50.6,0.2,0.30.5,0.50.6,0.7,0.1,0.20.4,0.50.6,0.30.6,0.30.5,0.6,0.30.8,0.1D2=0.4,0.40.6,0.7,0.1,0.20.4,0.50.6,0.7,0.1,0.20.6,0.30.3,0.4,0.60.5,0.6,0.30.4,0.5,0.30.5,0.7,0.2,0.30.4,0.50.4,0.20.3,0.4,0.50.6,0.7,0.1,0.20.4,0.4,0.50.3,0.4,0.4
Step  2. Apply aggregation operator DHFWGBM as taking wp1=0.5=wp2 on Dj(j=1,2). The following D5×3 matrix represented aggregation values of each alternative Pi(i=1,2,…,5): (78)D5×3=0.6621,0.7075,0.7633,0.15710.8151,0.8284,0.8428,0.0748,0.0926,0.10740.7989,0.14400.8152,0.8284,0.8484,0.0748,0.0925,0.10740.7370,0.7614,0.7890,0.15720.7767,0.7874,0.7989,0.1123,0.1391,0.16140.7296,0.7458,0.7633,0.1337,0.1582,0.17930.7633,0.7757,0.7890,0.1074,0.1212,0.13370.7591,0.7874,0.8201,0.1439,0.1626,0.17930.7633,0.7803,0.7989,0.1002,0.1241,0.14400.6963,0.10740.7367,0.7498,0.7633,0.17930.8152,0.8284,0.8428,0.0748,0.0925,0.10740.7296,0.7458,0.7633,0.1571,0.1686,0.17930.8209,0.8295,0.8386,0.1572
Step  3. Aggregate all values of each alternative, by applying DHFCGBM on D5×3 and find ψ(P1)=0.002201, ψ(P2)=0.001646, ψ(P3)=0.002166, ψ(P4)=0.002195, and ψ(P5)=0.002020.
Step  4. ψ(P1)>ψ(P4)>ψ(P3)>ψ(P5)>ψ(P2)⇒P1≻P4≻P3≻P5≻P2.
Hence projectP1 is the best choice and P2 is the worst choice in the given scenario.Case 2.
⁡
Step  1. Repeat Step 1 as in Case 1.
Step  2. Now apply aggregation operator DHFWGBM as taking wp1=0.3 and wp2=0.7 on Dj(j=1,2). The following D5×3 matrix represented aggregation values of each alternative Pi(i=1,2,…,5):(79)D5×3=0.6867,0.7314,0.7870,0.14300.8306,0.8432,0.8570,0.0677,0.0841,0.097780.8203,0.12900.8306,0.8472,0.8570,0.0677,0.0841,0.09780.7515,0.7765,0.8050,0.14500.8087,0.8113,0.8203,0.0992,0.1233,0.14330.7498,0.7640,0.7792,0.1225,0.1464,0.16690.7870,0.7973,0.8083,0.0978,0.1108,0.12250.7792,0.8042,0.8381,0.1348,0.1517,0.16690.7870,0.8029,0.8203,0.0893,0.1111,0.12910.7218,0.099030.7661,0.7763,0.7870,0.16180.8306,0.8423,0.8549,0.0694,0.0855,0.09900.7547,0.7702,0.7870,0.1430,0.1527,0.16180.8427,0.8495,0.8567,0.1430
Step  3. Aggregate all values of each alternative, by applying DHFCGBM on updated D5×3 and calculated new values ψ(P1)=0.00159, ψ(P2)=0.00139, ψ(P3)=0.00180, ψ(P4)=0.00177, and ψ(P5)=0.00167.
Step  4. ψ(P3)>ψ(P4)>ψ(P5)>ψ(P1)>ψ(P2)⇒P3≻P4≻P5≻P1≻P2.
Hence projectP3 is the best choice and P2 is the worst choice, when D2 has some priority with D1.


## 6.1. Illustrative Example

The following example illustrated the suggested method.Example 29.
Energy is an essential part for the growth of social-economic development of any society. Thus it is very important to adopt proper energy policy for economic development and environment. Let us assume that there are five alternatives (energy projects)Pi(i=1,2,3,4,5) to be invested and three criteria to be considered: C1- technological; C2- environmental; C3- economic. Two DMs are requested to evaluate the performances of the five alternatives. To deal with this energy policy problem, consider weights according to preference of DMs for each alternative in group decision making.Case 1.
⁡
Step  1. The DMs provide their preferences over all the alternatives Pi(i=1,2,3,4,5)  w.r.t. the criteria Ck(k=1,2,3) based on DHFS; let weight vectors for the criteria be wc1=0.4,0.3,0.3 and wc2=0.3,0.4,0.3, with respect to DMs, respectively. The following Dj(j=1,2) represented the suggestion of DMs, in DHFE: (77)D1=0.3,0.6,0.30.6,0.30.7,0.20.6,0.30.3,0.5,0.40.7,0.1,0.20.4,0.3,0.50.6,0.2,0.30.5,0.50.6,0.7,0.1,0.20.4,0.50.6,0.30.6,0.30.5,0.6,0.30.8,0.1D2=0.4,0.40.6,0.7,0.1,0.20.4,0.50.6,0.7,0.1,0.20.6,0.30.3,0.4,0.60.5,0.6,0.30.4,0.5,0.30.5,0.7,0.2,0.30.4,0.50.4,0.20.3,0.4,0.50.6,0.7,0.1,0.20.4,0.4,0.50.3,0.4,0.4
Step  2. Apply aggregation operator DHFWGBM as taking wp1=0.5=wp2 on Dj(j=1,2). The following D5×3 matrix represented aggregation values of each alternative Pi(i=1,2,…,5): (78)D5×3=0.6621,0.7075,0.7633,0.15710.8151,0.8284,0.8428,0.0748,0.0926,0.10740.7989,0.14400.8152,0.8284,0.8484,0.0748,0.0925,0.10740.7370,0.7614,0.7890,0.15720.7767,0.7874,0.7989,0.1123,0.1391,0.16140.7296,0.7458,0.7633,0.1337,0.1582,0.17930.7633,0.7757,0.7890,0.1074,0.1212,0.13370.7591,0.7874,0.8201,0.1439,0.1626,0.17930.7633,0.7803,0.7989,0.1002,0.1241,0.14400.6963,0.10740.7367,0.7498,0.7633,0.17930.8152,0.8284,0.8428,0.0748,0.0925,0.10740.7296,0.7458,0.7633,0.1571,0.1686,0.17930.8209,0.8295,0.8386,0.1572
Step  3. Aggregate all values of each alternative, by applying DHFCGBM on D5×3 and find ψ(P1)=0.002201, ψ(P2)=0.001646, ψ(P3)=0.002166, ψ(P4)=0.002195, and ψ(P5)=0.002020.
Step  4. ψ(P1)>ψ(P4)>ψ(P3)>ψ(P5)>ψ(P2)⇒P1≻P4≻P3≻P5≻P2.
Hence projectP1 is the best choice and P2 is the worst choice in the given scenario.Case 2.
⁡
Step  1. Repeat Step 1 as in Case 1.
Step  2. Now apply aggregation operator DHFWGBM as taking wp1=0.3 and wp2=0.7 on Dj(j=1,2). The following D5×3 matrix represented aggregation values of each alternative Pi(i=1,2,…,5):(79)D5×3=0.6867,0.7314,0.7870,0.14300.8306,0.8432,0.8570,0.0677,0.0841,0.097780.8203,0.12900.8306,0.8472,0.8570,0.0677,0.0841,0.09780.7515,0.7765,0.8050,0.14500.8087,0.8113,0.8203,0.0992,0.1233,0.14330.7498,0.7640,0.7792,0.1225,0.1464,0.16690.7870,0.7973,0.8083,0.0978,0.1108,0.12250.7792,0.8042,0.8381,0.1348,0.1517,0.16690.7870,0.8029,0.8203,0.0893,0.1111,0.12910.7218,0.099030.7661,0.7763,0.7870,0.16180.8306,0.8423,0.8549,0.0694,0.0855,0.09900.7547,0.7702,0.7870,0.1430,0.1527,0.16180.8427,0.8495,0.8567,0.1430
Step  3. Aggregate all values of each alternative, by applying DHFCGBM on updated D5×3 and calculated new values ψ(P1)=0.00159, ψ(P2)=0.00139, ψ(P3)=0.00180, ψ(P4)=0.00177, and ψ(P5)=0.00167.
Step  4. ψ(P3)>ψ(P4)>ψ(P5)>ψ(P1)>ψ(P2)⇒P3≻P4≻P5≻P1≻P2.
Hence projectP3 is the best choice and P2 is the worst choice, when D2 has some priority with D1.


## 7. Conclusion

In this paper, we defined a ranking method forDHFE. DHFE is a generalization of both intuitionistic and hesitant fuzzy elements so it inherited properties of both intuitionistic and hesitant fuzzy sets. In the article dual hesitant fuzzy geometric Bonferroni mean developed and proved the fact that DHFGBM is also DHFE. Several particular cases and properties of DHFGBM have been examined, defined weighted dual hesitant fuzzy geometric Bonferroni mean. In the article we discussed a fuzzy measure for dual hesitant fuzzy environment and then defined Choquet integral for DHFE, Choquet integral combined with DHFGBM and defined dual hesitant fuzzy Choquet geometric Bonferroni mean. A multicriteria decision-making technique developed based on the aggregation operators is defined in the article. An illustrated example is also discussed to understand the developed multicriteria decision-making method. We also observed and saw in Figures 1 and 2 that when a decision maker dominates others opinion, then overall results change accordingly.Figure 1
Graphical representation of case 1.Figure 2
Graphical representation of case 2.



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*Source: 4541982-2018-06-26.xml*