# Hesitant Fuzzy Generalised Bonferroni Mean Operators Based on Archimedean Copula for Multiple-Attribute Decision-Making

Authors: Ju Wu; Fang Liu; Yuan Rong; Yi Liu; Chengxi Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Information fusion is an important part of multiple-attribute decision-making, and aggregation operator is an important tool of decision information fusion. Integration operators in a variety of fuzzy information environments have a slight lack of consideration for the correlation between variables. Archimedean copula provides information fusion patterns that rely on the intrinsic relevance of information. This paper extends the Archimedean copula to the aggregation of hesitant fuzzy information. Firstly, the Archimedean copula is used to generate the operation rules of the hesitant fuzzy elements. Secondly, the hesitant fuzzy copula Bonferroni mean operator and hesitant fuzzy weighted copula Bonferroni mean operator are propounded, and several properties are proved in detail. Furthermore, a decision-making method based on the operators is proposed, and the specific decision steps are given. Finally, an example is presented to illustrate the practical advantages of the method, and the sensitivity analysis of the decision results with the change of parameters is carried out.


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## Body

## 1. Introduction

In the practical problem of multiple-attribute decision-making (MADM), the decision maker (DM) vacillates between several values on the evaluation of the alternative. So traditional fuzzy sets show limitations in describing this uncertain information. As an important generalisation of fuzzy set [1], hesitant fuzzy set (HFS) is proposed by Torra [2, 3]. It allows DM to give multiple possible values flexibility, which is better to solve the indecision of DM and difficulty of reaching consensus. As a new information description tool for uncertain decision-making, the theory and its application in decision-making have attracted the attention of scholars at home and abroad. Xu and Xia [4, 5] proposed the mathematical form of the HFS and studied the hesitant fuzzy (HF) integration operator, similarity measure, distance of the hesitant fuzzy elements (HFEs), and so on. The following researchers further studied the theory of HF and applied it to the MADM problem [6–11].The aggregation operator is the basis of many decision methods in the MADM problems. Therefore, the study of aggregation operator is particularly important under the HF environment. Common aggregation operators include weighted average (WA) operator, weighted geometric (WG) operator, power average (PA) operator, power geometric (PG) operator, Choquet integral operator, Bonferroni mean (BM) operator, and so on. The researchers improved these operators and applied them to the aggregation of HF information. Xia and Xu [4] presented eight HF aggregation operators, such as hesitant fuzzy weighted averaging (HFWA) operator and hesitant fuzzy weighted geometric (HFWG) operator. Zhang [12] propounded ten operators like hesitant fuzzy power average (HFPA) operator and hesitant fuzzy power geometric (HFPG) operator. Wei [13] defined hesitant fuzzy Choquet ordered averaging (GHFCOA) operator and hesitant fuzzy Choquet ordered geometric (HFCOG) operator to solve the MADM problem with HF information. Zhu [14] developed hesitant fuzzy Bonferroni mean (HFBM) operator and weighted hesitant fuzzy Bonferroni mean (WHFBM) operator, which can deal with the MADM problem well. He [15] combined the power average operator with the Bonferroni mean in hesitant fuzzy environments and developed HFPBM and HFPGBM for hesitant fuzzy multiple-attribute group decision-making.Each operator has its own characteristics, which can solve the corresponding problems well. These operators are basically derived from the t-norms and t-conorms. Most of the above operators only consider the influence of the location of data, but the correlation between attributes is slightly deficient. Because many decision problems do not satisfy the independence principle, there is some connection between attributes. For example, a factory needs to purchase a kind of equipment according to the four attributes of product price, technology, after-sales service level, and supplier reputation. Price may depend upon technology and supplier reputation. The after-sales service level may influence the supplier reputation. On the basis of t-norms and s-norms, the correlation between variables has gradually become an important aspect of consideration in MADM.The copulas and co-copulas [16–18] are the generalisations of various t-norms and t-conorms. At first, copulas and co-copulas were widely used in the fields of finance and insurance, etc. In recent years, the application of copulas and co-copulas in decision problems has widely been concerned by scholars. This is mainly based on the following points: (1) Copulas and co-copulas can reveal the dependence among attributes; (2) Copulas and co-copulas can prevent information losing in the midst of aggregation; and (3) Copulas and co-copulas are flexible because DMs can select different types of copulas and co-copulas to define the operations under fuzzy environment, and the results obtained from these operations are closed. Archimedean copulas (ACs) and co-copulas [19] have the advantages of symmetry, associability, and simple operation. On the basis of AC, Tao [20] studied a new computational model for unbalanced linguistic variables. Chen [21] defined new aggregation operators in linguistic neutrosophic set based on copulas and applied them to settle MCDM problems.So we propose the more general and flexible aggregation operators on the basis of BM operator and AC to solve MADM problem. Hesitant fuzzy copula Bonferroni mean (HFCBM) operator and hesitant fuzzy weight copula Bonferroni mean (HFWCBM) operator can not only reflect the interrelation among attributes but also have more flexible and diversified forms. They can solve the MADM problem well and provide a preference choice for DM. For the above goals, the structure of this work is arranged as follows: some notions on HFS, BM, and AC are reviewed firstly in Section2. The HFCBM and HFWCBM are given in Section 3, and different forms of aggregation operators are investigated based on different AC functions. In Section 4, the MADM approach based on the propounded operators is constructed, case analysis will be carried out, some comparisons with existing approaches under the HF environment, and merits of the proposed MADM approach based on HFCBM operators are analysed. The conclusion is obtained in Section 5.


## 2. Preliminaries

In this section, several basis knowledge such as HFS, BM operator, HFBM operator, and AC is succinctly retrospected.


### 2.1. Hesitant Fuzzy Sets

Definition 1.
(see [2]). Let S be a finite reference set. A hesitant fuzzy set G on S is in terms of a function when applied to S returning a subset of 0,1 denoted by(1)G=s,gh|∀s∈S,wheregh is a collection of numbers hi from 0,1, indicating the possible membership degrees of ∀s∈S to G. We call gh a hesitant fuzzy element (HFE) and G the set of all HFEs.To compare the HFEs, the comparison laws are defined as follows.Definition 2.
(see [4]). For a HFE gh=∪i=1èghi, πg=1/èg∑i=1èghi is called the score function of gh, where èg is the number of possible elements in gh.
For two HFEsg1h and g2h,(i)
ifπg1>πg2, then g1≻g2(ii)
ifπg1=πg2, then g1=g2Definition 3.
(see [4]). Let g1h=∪m1=1èg1h1m1, g2h=∪m2=1èg2h2m2, and gh=∪i=1èghi be three HFEs, ρ≥0, and the novel operational rules of HFEs are given as follows:(2)g1⊕g2=∪h1m1∈g1h2m2∈g2h1m1+h2m2−h1m1h2m2,g1⊗g2=∪h1m1∈g1h2m2∈g2h1m1h2m2,ρg=∪hi∈g1−1−hiρ|i=1,2,…,èg,gρ=∪hi∈ghiρ|i=1,2,…,èg,wherem1=1,2,…,èg1andm2=1,2,…,èg2.


### 2.2. BM Operator and HFBM Operator

Definition 4.
(see [22]). Assume akk=1,2,…,n is a family of positive real number and μ,ν≥0, then the aggregated mapping(3)BMμ,νa1,a2,…,an=1nn−1∑k,j=1nk≠jakμajν1/μ+ν.is named BM operator.Definition 5.
(see [14]). Let gih=∪mi=1ègihimi|i=1,2,…,n and μ,ν≥0. Then the hesitant fuzzy Bonferroni mean (HFBM) operator is expressed as(4)HFBMμ,νg1,g2,…,gn=1nn−1∑k,j=1nk≠jgkμ⊗gjν1/μ+ν=∪himi∈gihjmj∈gj1−∏i,j=1i≠jn1−himiμhjmjν1/nn−11/μ+ν.Definition 6.
(see [14]). Let gih=∪mi=1ègihimi|i=1,2,…,n and μ,ν≥0. Then the hesitant fuzzy weight Bonferroni mean (HFWBM) operator is expressed as(5)HFWBMμ,νg1,g2,…,gn=1nn−1∑k,j=1k≠jnnωkgkμ⊗nωjgjν1/μ+ν=∪himi∈gihjmj∈gj1−∏i,j=1i≠jn1−1−1−himinωiμ1−1−hjmjnωjν1/nn−11/μ+ν.


### 2.3. Archimedean Copula

Definition 7.
(see [16]). A two-dimensional mapping Ϝ:0,1×0,1⟶0,1 is called a copula, if it contents the following properties:(1)
ôs,1=ô1,s=s;(2)
ôs,0=ô0,u=0;(3)
ôs1,t1−ôs1,t2−ôs2,t1+ôs2,t2≥0.Definition 8.
(see [17]). A copula ô:0,1×0,1⟶0,1 is said to be the Archimedean copula if there exists a continuous and strictly decreasing function ς from 0,1 to 0,∞ with ς1=0, and function τ from 0,∞ to 0,1 is defined by(6)τr=ς−1r,r∈0,ς0;0,r∈ς0,+∞.such that for anyt1,t2∈0,1×0,1,(7)Ϝt1,t2=τςt1+ςt2.The AC is propounded based on the following statements: (1) Ifô is strictly increasing on 0,1×0,1, then we can obtain ς0=+∞; (2) If τ accords ς−1 on 0,+∞. Then equation (7) is transformed as(8)Ϝt1,t2=ς−1ςt1+ςt2,where the functions ς and ô are named strict generator and Archimedean copulas, respectively.Definition 9.
(see [18]). Suppose Ϝ: 0,1×0,1⟶0,1 be a copula, then co-copula is expressed as(9)Ϝ∗t1,t2=1−Ϝ1−t1,1−t2.


## 2.1. Hesitant Fuzzy Sets

Definition 1.
(see [2]). Let S be a finite reference set. A hesitant fuzzy set G on S is in terms of a function when applied to S returning a subset of 0,1 denoted by(1)G=s,gh|∀s∈S,wheregh is a collection of numbers hi from 0,1, indicating the possible membership degrees of ∀s∈S to G. We call gh a hesitant fuzzy element (HFE) and G the set of all HFEs.To compare the HFEs, the comparison laws are defined as follows.Definition 2.
(see [4]). For a HFE gh=∪i=1èghi, πg=1/èg∑i=1èghi is called the score function of gh, where èg is the number of possible elements in gh.
For two HFEsg1h and g2h,(i)
ifπg1>πg2, then g1≻g2(ii)
ifπg1=πg2, then g1=g2Definition 3.
(see [4]). Let g1h=∪m1=1èg1h1m1, g2h=∪m2=1èg2h2m2, and gh=∪i=1èghi be three HFEs, ρ≥0, and the novel operational rules of HFEs are given as follows:(2)g1⊕g2=∪h1m1∈g1h2m2∈g2h1m1+h2m2−h1m1h2m2,g1⊗g2=∪h1m1∈g1h2m2∈g2h1m1h2m2,ρg=∪hi∈g1−1−hiρ|i=1,2,…,èg,gρ=∪hi∈ghiρ|i=1,2,…,èg,wherem1=1,2,…,èg1andm2=1,2,…,èg2.


## 2.2. BM Operator and HFBM Operator

Definition 4.
(see [22]). Assume akk=1,2,…,n is a family of positive real number and μ,ν≥0, then the aggregated mapping(3)BMμ,νa1,a2,…,an=1nn−1∑k,j=1nk≠jakμajν1/μ+ν.is named BM operator.Definition 5.
(see [14]). Let gih=∪mi=1ègihimi|i=1,2,…,n and μ,ν≥0. Then the hesitant fuzzy Bonferroni mean (HFBM) operator is expressed as(4)HFBMμ,νg1,g2,…,gn=1nn−1∑k,j=1nk≠jgkμ⊗gjν1/μ+ν=∪himi∈gihjmj∈gj1−∏i,j=1i≠jn1−himiμhjmjν1/nn−11/μ+ν.Definition 6.
(see [14]). Let gih=∪mi=1ègihimi|i=1,2,…,n and μ,ν≥0. Then the hesitant fuzzy weight Bonferroni mean (HFWBM) operator is expressed as(5)HFWBMμ,νg1,g2,…,gn=1nn−1∑k,j=1k≠jnnωkgkμ⊗nωjgjν1/μ+ν=∪himi∈gihjmj∈gj1−∏i,j=1i≠jn1−1−1−himinωiμ1−1−hjmjnωjν1/nn−11/μ+ν.


## 2.3. Archimedean Copula

Definition 7.
(see [16]). A two-dimensional mapping Ϝ:0,1×0,1⟶0,1 is called a copula, if it contents the following properties:(1)
ôs,1=ô1,s=s;(2)
ôs,0=ô0,u=0;(3)
ôs1,t1−ôs1,t2−ôs2,t1+ôs2,t2≥0.Definition 8.
(see [17]). A copula ô:0,1×0,1⟶0,1 is said to be the Archimedean copula if there exists a continuous and strictly decreasing function ς from 0,1 to 0,∞ with ς1=0, and function τ from 0,∞ to 0,1 is defined by(6)τr=ς−1r,r∈0,ς0;0,r∈ς0,+∞.such that for anyt1,t2∈0,1×0,1,(7)Ϝt1,t2=τςt1+ςt2.The AC is propounded based on the following statements: (1) Ifô is strictly increasing on 0,1×0,1, then we can obtain ς0=+∞; (2) If τ accords ς−1 on 0,+∞. Then equation (7) is transformed as(8)Ϝt1,t2=ς−1ςt1+ςt2,where the functions ς and ô are named strict generator and Archimedean copulas, respectively.Definition 9.
(see [18]). Suppose Ϝ: 0,1×0,1⟶0,1 be a copula, then co-copula is expressed as(9)Ϝ∗t1,t2=1−Ϝ1−t1,1−t2.


## 3. New Operators Based on AC and BM

### 3.1. Generalised Operations for HFEs

To deal with HF information, we propose a generalised version of operational rules based on AC.Definition 10.
(see [23]). Let g1h=∪m1=1èg1h1m1, g2h=∪m2=1èg2h2m2, and gh=∪i=1èghi be three HFEs, ρ≥0, and the novel operational rules of HFEs are given as follows:(10)g1⊕g2=∪h1m1∈g1h2m2∈g21−ς−1ς1−h1m1+ς1−h2m2,g1⊗g2=∪h1m1∈g1h2m2∈g2ς−1ςh1m1+ςh2m2,ρg=∪hi∈g1−ς−1ρς1−hi|i=1,2,…,èg,gρ=∪hi∈gς−1ρςhi|i=1,2,…,èg,wherem1=1,2,…,èg1andm2=1,2,…,èg2.


### 3.2. HFCBM Operator

Based on Definitions5 and 10, the HFCBM operator can be proposed.Theorem 1.
Letgih=∪mi=1ègihimi|i=1,2,…,n, and μ,ν≥0. Then the hesitant fuzzy copula Bonferroni mean (HFCBM) operator is expressed as(11)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj.Proof.
Since(12)giμ=∪himi∈giς−1μςhimi|i=1,2,…,ègi,gjν=∪hjmj∈gjς−1νςhjmj|j=1,2,…,ègj.Then(13)giμ⊗gjν=∪himi∈gihjmj∈gjς−1ςς−1μςhimi+ςς−1νςhjmj|mi=1,2,…,ègi,mj=1,2,…,ègj,∑i,j=1i≠jngiμ⊗gjν=∪himi∈gihjmj∈gj1−ς−1∑i,j=1i≠jnς1−ς−1μςhimi+νςhjmj.Furthermore(14)1nn−1∑i,j=1i≠jngiμ⊗gjν=∪himi∈gihjmj∈gj1−ς−11nn−1ςς−1∑i,j=1i≠jnς1−ς−1μςhimi+νςhjmj.Accordingly(15)1nn−1∑i,j=1i≠jngiμ⊗gjν1/μ+ν=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑i,j=1i≠jnς1−ς−1μςhimi+νςhjmj.Accordingly, we can attain the establishment of Theorem1.In what follows, several desired characteristics of HFCBM operator are proved.Theorem 2.
Letgih=∪mi=1ègihimi|i=1,2,…,n, and μ,ν≥0, then(1)
(Idempotency) Ifg1=g2=⋯gn=h, then HFCBMμ,νg1,g2,…,gn=h(2)
(Monotonicity) Letgi∗h=∪mi=1ègi∗himi∗|i=1,2,…,n, if himi≤himi∗, so(16)HFCBMμ,νg1,g2,…,gn≤HFCBMμ,νg1∗,g2∗,…,gn∗.(3)
(Boudedness) Ifh−=mini=1,2,…,nhimi ,h+=maxi=1,2,…,nhimi,(17)h−≤HFCBMμ,νg1,g2,…,gn≤h+.Proof.
(1)
Ifg1=g2=⋯gn=h,(18)ς−11μ+νς1−ς−11nn−1∑i,j=1i≠jnς1−ς−1μ+νςh=ς−11μ+νς1−ς−11nn−1nn−1ς1−ς−1μ+νςh=ς−11μ+νς1−ς−1ς1−ς−1μ+νςh=ς−11μ+νμ+νςα=ς−1ςh=h.Then(19)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑k,j=1nk≠jς1−ς−1μ+νςh=h.Accordingly, the idempotency of the HFCBM operator is proved.(2)
Sincehimi≤himi∗, hjmj≤hijmj∗, ςt and ς−1t are decreasing functions and 1−ςt and 1−ς−1t are increasing functions. Then, we have(20)μςhimi+νςhjmj≥μςhimi∗+νςhjmj∗,1−ς−1μςhimi+νςhjmj≥1−ς−1μςhimi∗+νςhjmj∗.Then,(21)∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≤∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗,1nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≤1nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗.Furthermore,(22)1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≤1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗,ς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≥ς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗.Moreover,(23)1μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≥1μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗.Hence(24)ς−11μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≤ς−11μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗.Accordingly, the monotonicity of the HFCBM operator is proved.(3)
Letg−=h− and g+=h+; according to the monotonicity, we can acquire(25)HFCBMμ,νg−,g−,…,g−≤HFCBMμ,νg1,g2,…,gn≤HFCBMμ,νg+,g+,…,g+.According to idempotency, we can acquire(26)HFCBMμ,νg−,g−,…,g−=h−;HFCBMμ,νg+,g+,…,g+=h+.Accordingly, we can acquireh−≤HFCBMμ,νg1,g2,…,gn≤h+.


### 3.3. HFWCBM Operator

To overcome the disadvantage of the HFCBM operator not considering attribute weight, the HFWCBM operator is given as follows.Theorem 3.
Letgih=∪mi=1ègihimi|i=1,2,…,n, and μ,ν≥0. Then the hesitant fuzzy weight copula Bonferroni mean (HFWCBM) operator is expressed as(27)HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1ζ,where(28)ζ=μς1−ς−1nωkς1−hkmk+νς1−ς−1nωjς1−hjmj.Proof.
The proof of Theorem3 is similar to Theorem 1, we can utilize the proof of nωkgkμ⊗nωjgjν to replace ωkgkμ⊗ωjgjν in equation (1).
Then(29)nωkgk=1−ς−1nωkς1−hkmk,nωjgj=1−ς−1nωjς1−hjmj,nωkgkμ=ς−1μς1−ς−1nωkς1−hkmk,nωjgjν=ς−1νς1−ς−1nωjς1−hjmj.
Then(30)nωkgkμ⊗nωjgjν=ς−1μς1−ς−1nωkς1−hkmk+νς1−ς−1nωjς1−hjmj.
Hence(31)1nn−1∑k,j=1nk≠jnωkgkμ⊗nωjgjν1/μ+ν=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑k,j=1nk≠jς1−ς−1ζ,where(32)ζ=μς1−ς−1nωkς1−hkmk+νς1−ς−1nωjς1−hjmj.
So we can attain the establishment of Theorem3. □
In what follows, several desired characteristics of HFWCBM operator are proved.Theorem 4.
Letgih=∪mi=1ègihimi|i=1,2,…,n, and μ,ν≥0, then(1)
(Idempotency) Ifg1=g2=⋯gn=h, HFWCBMμ,νg1,g2,…,gn=h(2)
(Monotonicity) Letgi∗h=∪mi=1ègi∗himi∗|i=1,2,…,n; if himi≤himi∗,(33)HFWCBMμ,νg1,g2,…,gn≤HFWCBMμ,νg1∗,g2∗,…,gn∗;(3)
(Boudedness) Ifh−=mini=1,2,…,nhimi ,h+=maxi=1,2,…,nhimi,(34)h−≤HFWCBMμ,νg1,g2,…,gn≤h+.
Theorem4 is similar to Theorem 2, so the proofs are omitted here.


### 3.4. Different Forms of HFCBM and HFWCBM

Next, we will explore sever particular cases through taking diverse AC functionςt.Case 1.
Whenςt=−lntκκ≥1, then(35)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gje−1/μ+ν−ln1−e−ζ11/κκ1/κ,where(36)ζ1=1nn−1∑i,j=1i≠jn−ln1−e−ζ1′1/κκ,ζ1′=μ−lnhimiκ+ν−lnhjmjκ,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gje−1/μ+ν−ln1−e−φ11/κκ1/κ,where(37)φ1=1nn−1∑i,j=1i≠jn−ln1−e−φ1′1/κκ,φ1′=μ−ln1−e−nωk−ln1−himiκ1/κκ+ν−ln1−e−nωj−ln1−hjmjκ1/κκ.Specifically, whenκ=1,ςt=−lnt, then they are the operators proposed by Zhu [14].Case 2.
Whenςt=t−κ−1 with κ≥−1,κ≠0, then(38)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1μ+ν1−ζ2+1−1/κ−κ−1/κ,where(39)ζ2=1nn−1∑i,j=1i≠jn1−ζ2′+1−1/κ−κ−1,ζ2′=μhimi−κ+νhjmj−κ−2,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1μ+ν1−φ2+1−1/κ−κ−1/κ,where(40)φ2=1nn−1∑i,j=1i≠jn1−φ2′+1−1/κ−κ−1,φ2′=μ1−nωi1−himi−κ−1/κ−κ+ν1−nωj1−hjmj−κ−1/κ−κ−2.Case 3.
Whenςt=lne−κt−1/e−κ−1κ≠0, then(41)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj−1κlnζ3e−κ−1+1,where(42)ζ3=e−κ1+1/κlnζ3′e−κ−1+1−1e−κ−11/μ+ν,ζ3′=∏i,j=1i≠jne−κ1+1/κlnζ3″e−κ−1+1−1e−κ−11/nn−1,ζ3″=e−κhimi−1e−κ−1μe−κhjmj−1e−κ−1ν,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj−1κlnφ3e−κ−1+1,where(43)φ3=e−κ1+1/κlnφ3′e−κ−1+1−1e−κ−11/μ+ν,φ3′=∏i,j=1i≠jne−κ1+1/κlnφ3″e−κ−1+1−1e−κ−11/nn−1,φ3″=e−κφ3″i−1e−κ−1μe−κφ3″j−1e−κ−1ν,φ3″i=1+1κlne−κ1−himi−1e−κ−1nωke−κ−1+1,φ3″j=1+1κlne−κ1−hjmj−1e−κ−1nωje−κ−1+1.Case 4.
Whenςt=ln1−κ1−t/t with κ∈−1,1, we have(44)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1−κζ4−κ,where(45)ζ4=1−κ1−κ/ζ4′−κ1−1−κ/ζ4′−κ1/μ+ν,ζ4′=∏i,j=1i≠jn1−κ1−κ/ζ4″−κ1−1−κ/ζ4″−κ1/nn−1,ζ4″=1−κ1−himihimiμ1−κ1−hjmjhjmjν,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1−κφ4−κ,where(46)φ4=1−κ1−κ/φ4′−κ1−1−κ/φ4′−κ1/μ+ν,φ4′=∏i,j=1i≠jn1−κ1−κ/φ4″−κ1−1−κ/φ4″−κ1/nn−1,φ4″=1−ς1−ς/φ4″i−ς1−1−ς/φ4″i−ςμ1−ς1−ς/φ4″j−ς1−1−ς/φ4″j−ςν,φ4″i=1−κhimi1−himinωi,φ4j=1−κhjmj1−hjmjnωj.Case 5.
Whenςt=−ln1−1−tκ,κ≥1, then(47)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1−1−1−ζ5κ1/μ+ν1/κ,where(48)ζ5=∏i,j=1i≠jn1−1−1−ζ5′1/κκ1/nn−1,ζ5′=1−1−himiκμ1−1−hjmjκν,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1−1−1−φ5κ1/μ+ν1/κ,where(49)φ5=∏i,j=1i≠jn1−1−1−φ5′1/κκ1/nn−1,φ5′=1−1−1−1−himiκnωk1/κκμ1−1−1−1−hjmjκnωj1/κκν.


## 3.1. Generalised Operations for HFEs

To deal with HF information, we propose a generalised version of operational rules based on AC.Definition 10.
(see [23]). Let g1h=∪m1=1èg1h1m1, g2h=∪m2=1èg2h2m2, and gh=∪i=1èghi be three HFEs, ρ≥0, and the novel operational rules of HFEs are given as follows:(10)g1⊕g2=∪h1m1∈g1h2m2∈g21−ς−1ς1−h1m1+ς1−h2m2,g1⊗g2=∪h1m1∈g1h2m2∈g2ς−1ςh1m1+ςh2m2,ρg=∪hi∈g1−ς−1ρς1−hi|i=1,2,…,èg,gρ=∪hi∈gς−1ρςhi|i=1,2,…,èg,wherem1=1,2,…,èg1andm2=1,2,…,èg2.


## 3.2. HFCBM Operator

Based on Definitions5 and 10, the HFCBM operator can be proposed.Theorem 1.
Letgih=∪mi=1ègihimi|i=1,2,…,n, and μ,ν≥0. Then the hesitant fuzzy copula Bonferroni mean (HFCBM) operator is expressed as(11)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj.Proof.
Since(12)giμ=∪himi∈giς−1μςhimi|i=1,2,…,ègi,gjν=∪hjmj∈gjς−1νςhjmj|j=1,2,…,ègj.Then(13)giμ⊗gjν=∪himi∈gihjmj∈gjς−1ςς−1μςhimi+ςς−1νςhjmj|mi=1,2,…,ègi,mj=1,2,…,ègj,∑i,j=1i≠jngiμ⊗gjν=∪himi∈gihjmj∈gj1−ς−1∑i,j=1i≠jnς1−ς−1μςhimi+νςhjmj.Furthermore(14)1nn−1∑i,j=1i≠jngiμ⊗gjν=∪himi∈gihjmj∈gj1−ς−11nn−1ςς−1∑i,j=1i≠jnς1−ς−1μςhimi+νςhjmj.Accordingly(15)1nn−1∑i,j=1i≠jngiμ⊗gjν1/μ+ν=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑i,j=1i≠jnς1−ς−1μςhimi+νςhjmj.Accordingly, we can attain the establishment of Theorem1.In what follows, several desired characteristics of HFCBM operator are proved.Theorem 2.
Letgih=∪mi=1ègihimi|i=1,2,…,n, and μ,ν≥0, then(1)
(Idempotency) Ifg1=g2=⋯gn=h, then HFCBMμ,νg1,g2,…,gn=h(2)
(Monotonicity) Letgi∗h=∪mi=1ègi∗himi∗|i=1,2,…,n, if himi≤himi∗, so(16)HFCBMμ,νg1,g2,…,gn≤HFCBMμ,νg1∗,g2∗,…,gn∗.(3)
(Boudedness) Ifh−=mini=1,2,…,nhimi ,h+=maxi=1,2,…,nhimi,(17)h−≤HFCBMμ,νg1,g2,…,gn≤h+.Proof.
(1)
Ifg1=g2=⋯gn=h,(18)ς−11μ+νς1−ς−11nn−1∑i,j=1i≠jnς1−ς−1μ+νςh=ς−11μ+νς1−ς−11nn−1nn−1ς1−ς−1μ+νςh=ς−11μ+νς1−ς−1ς1−ς−1μ+νςh=ς−11μ+νμ+νςα=ς−1ςh=h.Then(19)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑k,j=1nk≠jς1−ς−1μ+νςh=h.Accordingly, the idempotency of the HFCBM operator is proved.(2)
Sincehimi≤himi∗, hjmj≤hijmj∗, ςt and ς−1t are decreasing functions and 1−ςt and 1−ς−1t are increasing functions. Then, we have(20)μςhimi+νςhjmj≥μςhimi∗+νςhjmj∗,1−ς−1μςhimi+νςhjmj≥1−ς−1μςhimi∗+νςhjmj∗.Then,(21)∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≤∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗,1nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≤1nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗.Furthermore,(22)1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≤1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗,ς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≥ς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗.Moreover,(23)1μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≥1μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗.Hence(24)ς−11μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi+νςhjmj≤ς−11μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1μςhimi∗+νςhjmj∗.Accordingly, the monotonicity of the HFCBM operator is proved.(3)
Letg−=h− and g+=h+; according to the monotonicity, we can acquire(25)HFCBMμ,νg−,g−,…,g−≤HFCBMμ,νg1,g2,…,gn≤HFCBMμ,νg+,g+,…,g+.According to idempotency, we can acquire(26)HFCBMμ,νg−,g−,…,g−=h−;HFCBMμ,νg+,g+,…,g+=h+.Accordingly, we can acquireh−≤HFCBMμ,νg1,g2,…,gn≤h+.


## 3.3. HFWCBM Operator

To overcome the disadvantage of the HFCBM operator not considering attribute weight, the HFWCBM operator is given as follows.Theorem 3.
Letgih=∪mi=1ègihimi|i=1,2,…,n, and μ,ν≥0. Then the hesitant fuzzy weight copula Bonferroni mean (HFWCBM) operator is expressed as(27)HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑i,j=1ni≠jς1−ς−1ζ,where(28)ζ=μς1−ς−1nωkς1−hkmk+νς1−ς−1nωjς1−hjmj.Proof.
The proof of Theorem3 is similar to Theorem 1, we can utilize the proof of nωkgkμ⊗nωjgjν to replace ωkgkμ⊗ωjgjν in equation (1).
Then(29)nωkgk=1−ς−1nωkς1−hkmk,nωjgj=1−ς−1nωjς1−hjmj,nωkgkμ=ς−1μς1−ς−1nωkς1−hkmk,nωjgjν=ς−1νς1−ς−1nωjς1−hjmj.
Then(30)nωkgkμ⊗nωjgjν=ς−1μς1−ς−1nωkς1−hkmk+νς1−ς−1nωjς1−hjmj.
Hence(31)1nn−1∑k,j=1nk≠jnωkgkμ⊗nωjgjν1/μ+ν=∪himi∈gihjmj∈gjς−11μ+νς1−ς−11nn−1∑k,j=1nk≠jς1−ς−1ζ,where(32)ζ=μς1−ς−1nωkς1−hkmk+νς1−ς−1nωjς1−hjmj.
So we can attain the establishment of Theorem3. □
In what follows, several desired characteristics of HFWCBM operator are proved.Theorem 4.
Letgih=∪mi=1ègihimi|i=1,2,…,n, and μ,ν≥0, then(1)
(Idempotency) Ifg1=g2=⋯gn=h, HFWCBMμ,νg1,g2,…,gn=h(2)
(Monotonicity) Letgi∗h=∪mi=1ègi∗himi∗|i=1,2,…,n; if himi≤himi∗,(33)HFWCBMμ,νg1,g2,…,gn≤HFWCBMμ,νg1∗,g2∗,…,gn∗;(3)
(Boudedness) Ifh−=mini=1,2,…,nhimi ,h+=maxi=1,2,…,nhimi,(34)h−≤HFWCBMμ,νg1,g2,…,gn≤h+.
Theorem4 is similar to Theorem 2, so the proofs are omitted here.


## 3.4. Different Forms of HFCBM and HFWCBM

Next, we will explore sever particular cases through taking diverse AC functionςt.Case 1.
Whenςt=−lntκκ≥1, then(35)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gje−1/μ+ν−ln1−e−ζ11/κκ1/κ,where(36)ζ1=1nn−1∑i,j=1i≠jn−ln1−e−ζ1′1/κκ,ζ1′=μ−lnhimiκ+ν−lnhjmjκ,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gje−1/μ+ν−ln1−e−φ11/κκ1/κ,where(37)φ1=1nn−1∑i,j=1i≠jn−ln1−e−φ1′1/κκ,φ1′=μ−ln1−e−nωk−ln1−himiκ1/κκ+ν−ln1−e−nωj−ln1−hjmjκ1/κκ.Specifically, whenκ=1,ςt=−lnt, then they are the operators proposed by Zhu [14].Case 2.
Whenςt=t−κ−1 with κ≥−1,κ≠0, then(38)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1μ+ν1−ζ2+1−1/κ−κ−1/κ,where(39)ζ2=1nn−1∑i,j=1i≠jn1−ζ2′+1−1/κ−κ−1,ζ2′=μhimi−κ+νhjmj−κ−2,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1μ+ν1−φ2+1−1/κ−κ−1/κ,where(40)φ2=1nn−1∑i,j=1i≠jn1−φ2′+1−1/κ−κ−1,φ2′=μ1−nωi1−himi−κ−1/κ−κ+ν1−nωj1−hjmj−κ−1/κ−κ−2.Case 3.
Whenςt=lne−κt−1/e−κ−1κ≠0, then(41)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj−1κlnζ3e−κ−1+1,where(42)ζ3=e−κ1+1/κlnζ3′e−κ−1+1−1e−κ−11/μ+ν,ζ3′=∏i,j=1i≠jne−κ1+1/κlnζ3″e−κ−1+1−1e−κ−11/nn−1,ζ3″=e−κhimi−1e−κ−1μe−κhjmj−1e−κ−1ν,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj−1κlnφ3e−κ−1+1,where(43)φ3=e−κ1+1/κlnφ3′e−κ−1+1−1e−κ−11/μ+ν,φ3′=∏i,j=1i≠jne−κ1+1/κlnφ3″e−κ−1+1−1e−κ−11/nn−1,φ3″=e−κφ3″i−1e−κ−1μe−κφ3″j−1e−κ−1ν,φ3″i=1+1κlne−κ1−himi−1e−κ−1nωke−κ−1+1,φ3″j=1+1κlne−κ1−hjmj−1e−κ−1nωje−κ−1+1.Case 4.
Whenςt=ln1−κ1−t/t with κ∈−1,1, we have(44)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1−κζ4−κ,where(45)ζ4=1−κ1−κ/ζ4′−κ1−1−κ/ζ4′−κ1/μ+ν,ζ4′=∏i,j=1i≠jn1−κ1−κ/ζ4″−κ1−1−κ/ζ4″−κ1/nn−1,ζ4″=1−κ1−himihimiμ1−κ1−hjmjhjmjν,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1−κφ4−κ,where(46)φ4=1−κ1−κ/φ4′−κ1−1−κ/φ4′−κ1/μ+ν,φ4′=∏i,j=1i≠jn1−κ1−κ/φ4″−κ1−1−κ/φ4″−κ1/nn−1,φ4″=1−ς1−ς/φ4″i−ς1−1−ς/φ4″i−ςμ1−ς1−ς/φ4″j−ς1−1−ς/φ4″j−ςν,φ4″i=1−κhimi1−himinωi,φ4j=1−κhjmj1−hjmjnωj.Case 5.
Whenςt=−ln1−1−tκ,κ≥1, then(47)HFCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1−1−1−ζ5κ1/μ+ν1/κ,where(48)ζ5=∏i,j=1i≠jn1−1−1−ζ5′1/κκ1/nn−1,ζ5′=1−1−himiκμ1−1−hjmjκν,HFWCBMμ,νg1,g2,…,gn=∪himi∈gihjmj∈gj1−1−1−φ5κ1/μ+ν1/κ,where(49)φ5=∏i,j=1i≠jn1−1−1−φ5′1/κκ1/nn−1,φ5′=1−1−1−1−himiκnωk1/κκμ1−1−1−1−hjmjκnωj1/κκν.


## 4. Novel MADM Approach Based on the Propounded Operators

In this section, we will propound a novel approach based on the presented operators for handling MADM issues.Suppose that there arem alternatives ϒii=1,2,…,m, n attributes Gjj=1,2,…,n. If DMs provide several values for the alternative ϒi under the attribute Gjj=1,2,…,n with anonymity, these values can be considered as a HFE hij. In the case, if two DMs provide the same value, then the value emerges only once in hij. In what follows, the specific algorithm for MADM problems under the HF environment will be designed.Step 1: DMs provide their evaluations about the alternativeϒi under the attribute Gj , denoted by the hesitant fuzzy elements hij,i=1,2,…,m;j=1,2,…,nStep 2: use the HFCBM or HFWCBM to fuse the HFEyii=1,2,…,m for Υii=1,2,…,mStep 3: the score valuesπyii=1,2,…,m of yi are calculated using Definition 2 and comparedStep 4: the optimal alternativesϒii=1,2,…,m are made by ranking πyii=1,2,…,m


### 4.1. Empirical Example

Example 1.
Selection of Equipment Purchase. Assume that there are four short-term stocksϒ1,ϒ2,ϒ3,andϒ4. The following four attributes: G1,G2,G3,andG4, should be considered. G1: product price; G2: technology; G3: after-sales service level; G4: supplier reputation.
Next, we use the developed method to find the ranking of the alternatives and the optimal choice.Step 1: the decision matrix is given by expert and shown in Table1Table 1
Hesitant fuzzy decision matrix [4].
AlternativesG1G2G3G4ϒ10.2,0.4,0.70.2,0.6,0.80.2,0.3,0.6,0.7,0.90.3,0.4,0.5,0.7,0.8ϒ20.2,0.4,0.7,0.90.1,0.2,0.4,0.50.3,0.4,0.6,0.90.5,0.6,0.8,0.9ϒ30.3,0.5,0.6,0.70.2,0.4,0.5,0.60.3,0.5,0.7,0.80.2,0.5,0.6,0.7ϒ40.3,0.5,0.60.2,0.40.5,0.6,0.70.8,0.9Step 2: use equation (35), let ςt=−lnt1.2 and μ=1, ν=1 to aggregate the HFE yii=1,…,4 for ϒii=1,…,4.Step 3: compute the score valuesπyii=1,…,4 of yii=1,…,4 by Definition 2. We have πy1=0.5794, πy2=0.5744, πy3=0.5487, and πy4=0.5738.Step 4: rank the alternativesϒii=1,…,4, and select the desirable one in term of comparison rules. Since πy1>πy2>πy4>πy3, we can obtain the rank of alternatives ϒ1≻ϒ2≻ϒ4≻ϒ3, and ϒ1 is the best alternative.


### 4.2. Sensitivity Analysis

Next, we first analyse the sensitivity of sorting results obtained by diverse operations from AC. In addition, the effect of parametersμ,ν, and κ on the final sorting results of alternatives are also analysed in detail.Firstly, we analyse the sorting results on the basis of different generators produced diverse copulas. The sorting results of alternatives using diverse generators are listed in Table2. From Table 2, although the score values of alternatives obtained by diverse operations from AC, the final sorting relation are basically same, which can further illustrate that the diverse copulas are relatively stable for the ultimate decision outcomes.Table 2
Decision results obtained by the HFCBM operator with diverse copulas.
Generatorςtπϒ1πϒ2πϒ3πϒ4RankingCase 1−lntκκ=10.58100.57780.54950.5791ϒ1≻ϒ4≻ϒ2≻ϒ3Case 2t−κ−1κ=10.82750.82360.75460.7976ϒ1≻ϒ2≻ϒ4≻ϒ3Case 3lne−κt−1/e−κ−1κ=10.57820.57470.54760.5746ϒ1≻ϒ2≻ϒ4≻ϒ3Case 4ln1−κ1−t/tκ=−10.58680.58330.55260.5862ϒ1≻ϒ4≻ϒ2≻ϒ3Case 5−ln1−1−tκκ=10.58100.57780.54950.5791ϒ1≻ϒ4≻ϒ2≻ϒ3Secondly, we research the effect of parameterκ in the HFCBM operator on the ultimate orders of alternatives. Table 3 shows the values of score functions with the different κ when μ=1 and ν=1. Figure 1 shows how the score functions obtained by Case 1 change with the parameter κ under different values of μ and ν. As can be seen from the results of the chart, the score values increase with the increase of κ for the same alternatives.Table 3
Decision results obtained by Case 1 with diverseκ.
κπϒ1πϒ2πϒ3πϒ4Ranking20.58320.56890.54900.5606ϒ1≻ϒ2≻ϒ4≻ϒ350.61940.58490.56290.5611ϒ1≻ϒ2≻ϒ3≻ϒ470.63340.59500.57030.5689ϒ1≻ϒ2≻ϒ3≻ϒ4100.64600.60540.57800.5782ϒ1≻ϒ2≻ϒ4≻ϒ3Figure 1
The score functions obtained by Case 1 with respect to parameterκ.
(a)(b)(c)(d)Thirdly, we explore the effect of parametersμ and ν in the HFCBM operator on the ranking results of alternatives. The ranking results of alternatives based on diverse values of μ and ν can be generated in Table 4 (we select Case 1 as an example and let κ=2). Figure 2 shows the variation of the score function with μ and ν. The results indicate that the scores of alternatives by the HFCBM operator increase as the parameter μ ranges from 0 to 10 and the parameter ν ranges from 0 to 10, but the ranking results of factors vary considerably, which provide DMs with flexible options. The DMs can adjust the value of μ and ν to get the desired ranking result.Table 4
Decision results obtained by Case 1 with diverseμ and ν.
ParameterμParameterνπϒ1πϒ2πϒ3πϒ4Ranking010.68320.67340.63860.6873ϒ4≻ϒ1≻ϒ2≻ϒ330.70410.69220.60460.7104ϒ4≻ϒ1≻ϒ2≻ϒ360.71860.70610.61390.7278ϒ4≻ϒ1≻ϒ2≻ϒ3100.72940.71690.62150.7415ϒ4≻ϒ1≻ϒ2≻ϒ3110.58320.56890.54900.5606ϒ1≻ϒ2≻ϒ4≻ϒ330.61240.59440.56490.5903ϒ1≻ϒ2≻ϒ4≻ϒ360.63590.62650.58310.6300ϒ1≻ϒ4≻ϒ2≻ϒ3100.67120.65350.59780.6640ϒ1≻ϒ2≻ϒ4≻ϒ3210.59750.58070.55660.5738ϒ1≻ϒ2≻ϒ4≻ϒ330.63780.58410.56000.5753ϒ1≻ϒ2≻ϒ4≻ϒ360.62940.60720.57390.6036ϒ1≻ϒ2≻ϒ4≻ϒ3100.65370.63220.58790.6352ϒ1≻ϒ4≻ϒ2≻ϒ3510.63590.61730.57800.6185ϒ1≻ϒ4≻ϒ2≻ϒ330.61800.59390.56700.5854ϒ1≻ϒ2≻ϒ4≻ϒ360.62330.59640.56980.5867ϒ1≻ϒ2≻ϒ4≻ϒ3100.63780.61030.57820.6038ϒ1≻ϒ2≻ϒ4≻ϒ3Figure 2
The score functions obtained by Case 1 with respect toμ and ν.
(a)(b)(c)(d)


### 4.3. Comparative Analysis

Next, we conduct a contrastive analysis between the propounded MADM method with the previous approaches to confirm the validity and to reveal the significant advantages of the propounded approach.


#### 4.3.1. Testification of Validity

To demonstrate the validity of the propounded approach, we utilize three previously existing methodologies to address the aforementioned example and perform a comparison study. The previous methods include HFWA and HFWG operators proposed by Xia and Xu [4], HFBM operator proposed by Zhu and Xu [14], and HFFWA and HFFWG operators proposed by Qin et al. [24]. Using the data in Reference [24] and different operators, Table 5 is obtained.Table 5
The scores obtained by different operators.
OperatorParameterNeed weight or notRankingHFWA [4]NoneYesA6≻A2≻A3≻A5≻A1≻A4HFWG [4]NoneYesA6≻A2≻A3≻A5≻A1≻A4HFBM [14]p=2,q=1NoA6≻A2≻A3≻A1≻A5≻A4HFFWA [24]λ=2YesA6≻A2≻A3≻A5≻A1≻A4HFFWG [24]λ=2YesA6≻A2≻A3≻A5≻A1≻A4HFCBM (Case1)κ=2,μ=2,ν=1NoA6≻A2≻A3≻A1≻A5≻A4HFWCBM (Case1)κ=2,μ=2,ν=1YesA6≻A2≻A3≻A1≻A5≻A4From Table5, we can observe that the orders are marginally diverse, but the optimal selections are all ϒ6. So the propounded approach in this paper is workable and efficient.


#### 4.3.2. Further Comparison and Analysis

(i)
Compared with the method based upon the HFWA (or HFWG) operator propounded by Xia and Xu [4], although the HFWA operator can fuse HF information and the computation is simple, it has the following weakness: (1) it assumes that all attributes are independent in the course of information integration; (2) it ignores the correction of dissimilar attribute assessment information; and (3) it lacks flexibility and robustness because its operational rules are constructed by the Algebraic operations. However, the proposed approach can effectively overcome these disadvantages through the following aspects: (a) the BM operator can take into consideration the interconnection of diverse attributes in this paper, and the operational laws are established based upon different copula with a flexible parameter, which can overcome the weakness (1) and (2); (b) the AC can make decision course more general and flexible because five copulas with their flexible parameters can be selected according to DMs’ attitude, which can conquer the weakness (3).(ii)
Compared with the approach based upon the HFFWA operator propounded by Qin et al. [24], although it can aggregate vagueness information validly and make the aggregation process flexible by the adjustable parameter, it ignores the interrelationship of different attributes during the production of fusion. In comparison, the propounded method in this paper based on the HFCBM operator not only takes into account the correction of diverse attributes but also provides a more universal and flexible aggregation operator, and the HFCBM can regard five aggregation functions through assigning diverse copulas to it.(iii)Compared with the method based upon the HFBM operator propounded by Zhu and Xu [14], it can consider the relevance of dissimilar attributes through the BM operator. But the operational laws of the HFBM operator are defined by the algebraic operation, which lacks flexibility and robustness in the course of information aggregation. In contrast, the HFBM operator supplies universal operations based on the Archimedean copula. Specially, when κ=1 in the Gumbel copula and Joe copula, they shall degenerate into algebraic operations, i.e., the HFBM operator is a particular instance of HFCBM operator. Hence, the proposed method is more universal and flexible. Hence, the developed approach can utilize an ocean of practical decision issues and especially needs to consider the relevance of diverse attributes.


#### 4.3.3. The Distinct Merits of the Propounded Approach in This Paper

A detailed contrastive analysis for aforementioned approaches is displayed in Table5, and the propounded method based on the HFCBM operator has the following distinct advantages during the course of information fusion: (1) It provides a universal method through selection different copulas to attain diverse aggregation operator; (2) The flexibility and robustness of the presented methodology is shown through the parameter in each copula, and it supplies more selection to choose parameters that accord with their preferences attitude; and (3) The interconnection between diverse attributes is taken into consideration during the information aggregation proceeding.


## 4.1. Empirical Example

Example 1.
Selection of Equipment Purchase. Assume that there are four short-term stocksϒ1,ϒ2,ϒ3,andϒ4. The following four attributes: G1,G2,G3,andG4, should be considered. G1: product price; G2: technology; G3: after-sales service level; G4: supplier reputation.
Next, we use the developed method to find the ranking of the alternatives and the optimal choice.Step 1: the decision matrix is given by expert and shown in Table1Table 1
Hesitant fuzzy decision matrix [4].
AlternativesG1G2G3G4ϒ10.2,0.4,0.70.2,0.6,0.80.2,0.3,0.6,0.7,0.90.3,0.4,0.5,0.7,0.8ϒ20.2,0.4,0.7,0.90.1,0.2,0.4,0.50.3,0.4,0.6,0.90.5,0.6,0.8,0.9ϒ30.3,0.5,0.6,0.70.2,0.4,0.5,0.60.3,0.5,0.7,0.80.2,0.5,0.6,0.7ϒ40.3,0.5,0.60.2,0.40.5,0.6,0.70.8,0.9Step 2: use equation (35), let ςt=−lnt1.2 and μ=1, ν=1 to aggregate the HFE yii=1,…,4 for ϒii=1,…,4.Step 3: compute the score valuesπyii=1,…,4 of yii=1,…,4 by Definition 2. We have πy1=0.5794, πy2=0.5744, πy3=0.5487, and πy4=0.5738.Step 4: rank the alternativesϒii=1,…,4, and select the desirable one in term of comparison rules. Since πy1>πy2>πy4>πy3, we can obtain the rank of alternatives ϒ1≻ϒ2≻ϒ4≻ϒ3, and ϒ1 is the best alternative.


## 4.2. Sensitivity Analysis

Next, we first analyse the sensitivity of sorting results obtained by diverse operations from AC. In addition, the effect of parametersμ,ν, and κ on the final sorting results of alternatives are also analysed in detail.Firstly, we analyse the sorting results on the basis of different generators produced diverse copulas. The sorting results of alternatives using diverse generators are listed in Table2. From Table 2, although the score values of alternatives obtained by diverse operations from AC, the final sorting relation are basically same, which can further illustrate that the diverse copulas are relatively stable for the ultimate decision outcomes.Table 2
Decision results obtained by the HFCBM operator with diverse copulas.
Generatorςtπϒ1πϒ2πϒ3πϒ4RankingCase 1−lntκκ=10.58100.57780.54950.5791ϒ1≻ϒ4≻ϒ2≻ϒ3Case 2t−κ−1κ=10.82750.82360.75460.7976ϒ1≻ϒ2≻ϒ4≻ϒ3Case 3lne−κt−1/e−κ−1κ=10.57820.57470.54760.5746ϒ1≻ϒ2≻ϒ4≻ϒ3Case 4ln1−κ1−t/tκ=−10.58680.58330.55260.5862ϒ1≻ϒ4≻ϒ2≻ϒ3Case 5−ln1−1−tκκ=10.58100.57780.54950.5791ϒ1≻ϒ4≻ϒ2≻ϒ3Secondly, we research the effect of parameterκ in the HFCBM operator on the ultimate orders of alternatives. Table 3 shows the values of score functions with the different κ when μ=1 and ν=1. Figure 1 shows how the score functions obtained by Case 1 change with the parameter κ under different values of μ and ν. As can be seen from the results of the chart, the score values increase with the increase of κ for the same alternatives.Table 3
Decision results obtained by Case 1 with diverseκ.
κπϒ1πϒ2πϒ3πϒ4Ranking20.58320.56890.54900.5606ϒ1≻ϒ2≻ϒ4≻ϒ350.61940.58490.56290.5611ϒ1≻ϒ2≻ϒ3≻ϒ470.63340.59500.57030.5689ϒ1≻ϒ2≻ϒ3≻ϒ4100.64600.60540.57800.5782ϒ1≻ϒ2≻ϒ4≻ϒ3Figure 1
The score functions obtained by Case 1 with respect to parameterκ.
(a)(b)(c)(d)Thirdly, we explore the effect of parametersμ and ν in the HFCBM operator on the ranking results of alternatives. The ranking results of alternatives based on diverse values of μ and ν can be generated in Table 4 (we select Case 1 as an example and let κ=2). Figure 2 shows the variation of the score function with μ and ν. The results indicate that the scores of alternatives by the HFCBM operator increase as the parameter μ ranges from 0 to 10 and the parameter ν ranges from 0 to 10, but the ranking results of factors vary considerably, which provide DMs with flexible options. The DMs can adjust the value of μ and ν to get the desired ranking result.Table 4
Decision results obtained by Case 1 with diverseμ and ν.
ParameterμParameterνπϒ1πϒ2πϒ3πϒ4Ranking010.68320.67340.63860.6873ϒ4≻ϒ1≻ϒ2≻ϒ330.70410.69220.60460.7104ϒ4≻ϒ1≻ϒ2≻ϒ360.71860.70610.61390.7278ϒ4≻ϒ1≻ϒ2≻ϒ3100.72940.71690.62150.7415ϒ4≻ϒ1≻ϒ2≻ϒ3110.58320.56890.54900.5606ϒ1≻ϒ2≻ϒ4≻ϒ330.61240.59440.56490.5903ϒ1≻ϒ2≻ϒ4≻ϒ360.63590.62650.58310.6300ϒ1≻ϒ4≻ϒ2≻ϒ3100.67120.65350.59780.6640ϒ1≻ϒ2≻ϒ4≻ϒ3210.59750.58070.55660.5738ϒ1≻ϒ2≻ϒ4≻ϒ330.63780.58410.56000.5753ϒ1≻ϒ2≻ϒ4≻ϒ360.62940.60720.57390.6036ϒ1≻ϒ2≻ϒ4≻ϒ3100.65370.63220.58790.6352ϒ1≻ϒ4≻ϒ2≻ϒ3510.63590.61730.57800.6185ϒ1≻ϒ4≻ϒ2≻ϒ330.61800.59390.56700.5854ϒ1≻ϒ2≻ϒ4≻ϒ360.62330.59640.56980.5867ϒ1≻ϒ2≻ϒ4≻ϒ3100.63780.61030.57820.6038ϒ1≻ϒ2≻ϒ4≻ϒ3Figure 2
The score functions obtained by Case 1 with respect toμ and ν.
(a)(b)(c)(d)


## 4.3. Comparative Analysis

Next, we conduct a contrastive analysis between the propounded MADM method with the previous approaches to confirm the validity and to reveal the significant advantages of the propounded approach.


### 4.3.1. Testification of Validity

To demonstrate the validity of the propounded approach, we utilize three previously existing methodologies to address the aforementioned example and perform a comparison study. The previous methods include HFWA and HFWG operators proposed by Xia and Xu [4], HFBM operator proposed by Zhu and Xu [14], and HFFWA and HFFWG operators proposed by Qin et al. [24]. Using the data in Reference [24] and different operators, Table 5 is obtained.Table 5
The scores obtained by different operators.
OperatorParameterNeed weight or notRankingHFWA [4]NoneYesA6≻A2≻A3≻A5≻A1≻A4HFWG [4]NoneYesA6≻A2≻A3≻A5≻A1≻A4HFBM [14]p=2,q=1NoA6≻A2≻A3≻A1≻A5≻A4HFFWA [24]λ=2YesA6≻A2≻A3≻A5≻A1≻A4HFFWG [24]λ=2YesA6≻A2≻A3≻A5≻A1≻A4HFCBM (Case1)κ=2,μ=2,ν=1NoA6≻A2≻A3≻A1≻A5≻A4HFWCBM (Case1)κ=2,μ=2,ν=1YesA6≻A2≻A3≻A1≻A5≻A4From Table5, we can observe that the orders are marginally diverse, but the optimal selections are all ϒ6. So the propounded approach in this paper is workable and efficient.


### 4.3.2. Further Comparison and Analysis

(i)
Compared with the method based upon the HFWA (or HFWG) operator propounded by Xia and Xu [4], although the HFWA operator can fuse HF information and the computation is simple, it has the following weakness: (1) it assumes that all attributes are independent in the course of information integration; (2) it ignores the correction of dissimilar attribute assessment information; and (3) it lacks flexibility and robustness because its operational rules are constructed by the Algebraic operations. However, the proposed approach can effectively overcome these disadvantages through the following aspects: (a) the BM operator can take into consideration the interconnection of diverse attributes in this paper, and the operational laws are established based upon different copula with a flexible parameter, which can overcome the weakness (1) and (2); (b) the AC can make decision course more general and flexible because five copulas with their flexible parameters can be selected according to DMs’ attitude, which can conquer the weakness (3).(ii)
Compared with the approach based upon the HFFWA operator propounded by Qin et al. [24], although it can aggregate vagueness information validly and make the aggregation process flexible by the adjustable parameter, it ignores the interrelationship of different attributes during the production of fusion. In comparison, the propounded method in this paper based on the HFCBM operator not only takes into account the correction of diverse attributes but also provides a more universal and flexible aggregation operator, and the HFCBM can regard five aggregation functions through assigning diverse copulas to it.(iii)Compared with the method based upon the HFBM operator propounded by Zhu and Xu [14], it can consider the relevance of dissimilar attributes through the BM operator. But the operational laws of the HFBM operator are defined by the algebraic operation, which lacks flexibility and robustness in the course of information aggregation. In contrast, the HFBM operator supplies universal operations based on the Archimedean copula. Specially, when κ=1 in the Gumbel copula and Joe copula, they shall degenerate into algebraic operations, i.e., the HFBM operator is a particular instance of HFCBM operator. Hence, the proposed method is more universal and flexible. Hence, the developed approach can utilize an ocean of practical decision issues and especially needs to consider the relevance of diverse attributes.


### 4.3.3. The Distinct Merits of the Propounded Approach in This Paper

A detailed contrastive analysis for aforementioned approaches is displayed in Table5, and the propounded method based on the HFCBM operator has the following distinct advantages during the course of information fusion: (1) It provides a universal method through selection different copulas to attain diverse aggregation operator; (2) The flexibility and robustness of the presented methodology is shown through the parameter in each copula, and it supplies more selection to choose parameters that accord with their preferences attitude; and (3) The interconnection between diverse attributes is taken into consideration during the information aggregation proceeding.


## 4.3.1. Testification of Validity

To demonstrate the validity of the propounded approach, we utilize three previously existing methodologies to address the aforementioned example and perform a comparison study. The previous methods include HFWA and HFWG operators proposed by Xia and Xu [4], HFBM operator proposed by Zhu and Xu [14], and HFFWA and HFFWG operators proposed by Qin et al. [24]. Using the data in Reference [24] and different operators, Table 5 is obtained.Table 5
The scores obtained by different operators.
OperatorParameterNeed weight or notRankingHFWA [4]NoneYesA6≻A2≻A3≻A5≻A1≻A4HFWG [4]NoneYesA6≻A2≻A3≻A5≻A1≻A4HFBM [14]p=2,q=1NoA6≻A2≻A3≻A1≻A5≻A4HFFWA [24]λ=2YesA6≻A2≻A3≻A5≻A1≻A4HFFWG [24]λ=2YesA6≻A2≻A3≻A5≻A1≻A4HFCBM (Case1)κ=2,μ=2,ν=1NoA6≻A2≻A3≻A1≻A5≻A4HFWCBM (Case1)κ=2,μ=2,ν=1YesA6≻A2≻A3≻A1≻A5≻A4From Table5, we can observe that the orders are marginally diverse, but the optimal selections are all ϒ6. So the propounded approach in this paper is workable and efficient.


## 4.3.2. Further Comparison and Analysis

(i)
Compared with the method based upon the HFWA (or HFWG) operator propounded by Xia and Xu [4], although the HFWA operator can fuse HF information and the computation is simple, it has the following weakness: (1) it assumes that all attributes are independent in the course of information integration; (2) it ignores the correction of dissimilar attribute assessment information; and (3) it lacks flexibility and robustness because its operational rules are constructed by the Algebraic operations. However, the proposed approach can effectively overcome these disadvantages through the following aspects: (a) the BM operator can take into consideration the interconnection of diverse attributes in this paper, and the operational laws are established based upon different copula with a flexible parameter, which can overcome the weakness (1) and (2); (b) the AC can make decision course more general and flexible because five copulas with their flexible parameters can be selected according to DMs’ attitude, which can conquer the weakness (3).(ii)
Compared with the approach based upon the HFFWA operator propounded by Qin et al. [24], although it can aggregate vagueness information validly and make the aggregation process flexible by the adjustable parameter, it ignores the interrelationship of different attributes during the production of fusion. In comparison, the propounded method in this paper based on the HFCBM operator not only takes into account the correction of diverse attributes but also provides a more universal and flexible aggregation operator, and the HFCBM can regard five aggregation functions through assigning diverse copulas to it.(iii)Compared with the method based upon the HFBM operator propounded by Zhu and Xu [14], it can consider the relevance of dissimilar attributes through the BM operator. But the operational laws of the HFBM operator are defined by the algebraic operation, which lacks flexibility and robustness in the course of information aggregation. In contrast, the HFBM operator supplies universal operations based on the Archimedean copula. Specially, when κ=1 in the Gumbel copula and Joe copula, they shall degenerate into algebraic operations, i.e., the HFBM operator is a particular instance of HFCBM operator. Hence, the proposed method is more universal and flexible. Hence, the developed approach can utilize an ocean of practical decision issues and especially needs to consider the relevance of diverse attributes.


## 4.3.3. The Distinct Merits of the Propounded Approach in This Paper

A detailed contrastive analysis for aforementioned approaches is displayed in Table5, and the propounded method based on the HFCBM operator has the following distinct advantages during the course of information fusion: (1) It provides a universal method through selection different copulas to attain diverse aggregation operator; (2) The flexibility and robustness of the presented methodology is shown through the parameter in each copula, and it supplies more selection to choose parameters that accord with their preferences attitude; and (3) The interconnection between diverse attributes is taken into consideration during the information aggregation proceeding.


## 5. Conclusions

In order to better accomplish the procedure of information aggregation and decrease information loss under the HF environment, this paper propounds HFCBM operator and HFWCBM operator to aggregate HF information and exert it to tackle MADM issues. The primarily contribution of our research was that the proposed operators can capture the connection between attributes and provide additional choice for DMs by using the generalised Archimedean copula functions. The comparative analysis showed that this method was more flexible, practical, general, and better than the existing methods. However, this article also has some shortcomings and limitations. First, the proposed method only solved a numerical example making the study incomplete. Second, we overlooked the scope of each Archimedean copula function making the study less rigorous. In the future, the propounded method will be applied to other actual application such as supplier selection, investment analysis, and pattern recognition. We will also continue to study the MADM problems under the HF environment [25–27]. Besides, we will explore other theories of copulas in different fuzzy settings [28–31], as well as the large-scale decision-making algorithm based upon linguistic assessment theory and methodology will become our focal point research target.



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