# Archimedean Copula-Based Hesitant Fuzzy Information Aggregation Operators for Multiple Attribute Decision Making

Authors: Ju Wu; Lianming Mou; Fang Liu; Haobin Liu; Yi Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In view of the good properties of copulas and their effective use in various fuzzy environments, the goal of the current study is to develop a series of aggregation operators for hesitant fuzzy information based on Archimedean copula and cocopula, which are applied to the MADM problems. Firstly, operational laws of hesitant fuzzy elements on the basis of copulas and cocopulas are defined which can show the relevance between hesitant fuzzy values. Secondly, four aggregation operators (AC-HFWA, AC-GHFWA, AC-HFWG, and AC-GHFWG) under hesitant fuzzy environment are developed according to the proposed operational laws. The properties of these operators are also studied in detail, including idempotence, monotonicity, boundedness, etc. Subsequently, five special cases of copula are also given and the special forms of aggregation operator are obtained. In the end, an example is used to illustrate the application of the proposed approach in MADM problems. The influences of different generated functions and parameters are shown, and the feasibility of the proposed method is validated through comparative analyses.


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## Body

## 1. Introduction

Multiple attribute decision making (MADM), also known as limited scheme multiobjective decision, is to select the optimal alternatives or ranking decision making problems in the case of considering multiple attributes. It is a vital part of modern decision science; its theories and methods have been widely utilized in engineering, technology, economy, management, military, and many other fields. One of the most important tasks of MADM is to fuse the attribute values given to each alternative by the decision maker and then summarize the decision maker’s opinion on each alternative. In this process, a primary issue is to describe the values of criteria. For this issue, many experts proposed to adopt fuzzy sets. MADM problems with different kinds of fuzzy information are handled by utilizing fuzzy set (FS) [1] which is proposed by Zadeh and their various extensions, including the intuitionistic fuzzy set (IFS) [2], interval-valued intuitionistic fuzzy set (IVIFS) [3], hesitant fuzzy set (HFS) [4, 5], Pythagorean fuzzy set (PFS) [6], neutrosophic set (NS) [7], and so on.In the numerous extensions of the FS, IFS as one of the most important, was introduced by Atanassov [2]. Because it provides a membership degree (MD), a nonmembership degree (NMD), and a hesitancy degree (HD) to each element, IFS is better at handling uncertainty and vagueness than FS. Since its emergence, IFS has attracted more and more researchers’ attention. However, when giving the membership degree of an element, the difficulty of establishing the membership degree is not because we have a margin of error or some possibility distribution on the possibility values but because we have several possible values. For such cases, Torra and Narukawa [4] proposed hesitant fuzzy set (HFS) and indicated that the envelope of a hesitant fuzzy element (HFE) is an intuitionistic fuzzy value (IFV). So, all the operations on IFS can be suitable for HFS, and many research studies of IFS can be extended to HFS.The aggregation operator, which fuses multiple information sources, plays a key role in the realization of collective opinions in MADM. In order to deal with information in different fuzzy environments, various aggregation operators are proposed. Weighted average (WA) operator and weighted geometry (WG) operator are the most commonly used integration operators in classical decision science theory. In the process of MADM, they have been deeply studied by scholars [8–12], which have been extended to the integration of different types of decision information, such as ordered weighted averaging operator (OWA) and ordered weighted geometry operator (OWG). Based on the defined operations for IFS, Xia and Xu [13] presented eight hesitant fuzzy aggregation operators, such as hesitant fuzzy weighted averaging (HFWA) operator, hesitant fuzzy weighted geometric (HFWG) operator, and so on. According to the operators mentioned above, many scholars investigated many operators to solve MCDM problems under hesitant fuzzy environment [14–21]. Qin et al. [22] developed some hesitant fuzzy aggregation operators based on Frank operations, such as HFFWA operator, HFFOWA operator, and so on. Yu et al. [23] studied a set of hesitant fuzzy Einstein aggregation operators, such as HFECOA operator, HFECOG operator, HFEPWA operator, and HFEPWG operator. Using the technique of obtaining values in the interval, Du et al. [24] proposed the generalized hesitant fuzzy harmonic mean operators including GHFWHM operator, GHFOWHM operator, and GHFHHM operator. Li and Chen [25] presented two new aggregation operators: belief structure hesitant fuzzy induced ordered weighted averaging operator and belief structure hesitant fuzzy induced ordered weighted geometric operator. Although the research and application of the integration operator have been well developed, the decision problem based on the integration operator has certain complexity, so it is necessary to conduct in-depth research on it and explore new information integration methods.In the aforementioned aggregation operators under hesitant fuzzy environment, the operational laws of any two HFEs are built on the t-norms (TCs) and t-conorms (TCs). Commonly, TNs are applied to integrate MD of fuzzy sets, while copulas are tools to deal with probability distributions. Besides, there exist also TNs which are copulas and vice versa. Thus, the application of copulas in fuzzy sets has important practical significance. Copulas [26] can not only reveal the dependence among attributes but also prevent information loss in the midst of aggregation. There are two distinguishing features of copula: (1) copulas and cocopulas are flexible because decision makers can select different types of copulas and cocopulas to define the operations under fuzzy environment, and the results obtained from these operations are closed; (2) copula functions are flexible to capture the correlations among attributes in MADMs. Based on the two obvious characteristic, copulas have been applied to some MADMs. In the light of Archimedean copula, Tao et al. [27] studied a new computational model for unbalanced linguistic variables. Chen et al. [28] defined new aggregation operators in linguistic neutrosophic set based on copula and applied them to settle MCDM problems.In this paper, based on the current research, the copulas are generalized to the HFS, and two kinds of hesitating fuzzy information integration operators based on copulas are proposed, which are applied to the MADM problems. For the goals, the structure of this work is arranged as follows. Some notions on hesitant fuzzy set and copulas are reviewed firstly in Section2. The hesitant fuzzy weighted averaging operator-based Archimedean copulas (AC-HFWA) are defined in Section 3; before AC-HFWA is given, the operations of hesitant fuzzy elements based on Archimedean copula are also defined. After AC-HFWA is given, the generalized hesitant fuzzy weighted averaging operator-based Archimedean copulas (AC-GHFWA) are introduced, and the properties of AC-HFWA and AC-HFWG are investigated along with the different cases. The hesitant fuzzy weighted geometry operator-based Archimedean copulas (AC-HFWG) are defined in Section 4; before AC-HFWG is given, the operations of hesitant fuzzy elements based on Archimedean copula are also defined. After AC-HFWG is given, the generalized hesitant fuzzy weighted geometry operator-based Archimedean copulas (AC-GHFWG) are introduced, and the properties of AC-HFWA and AC-HFWG are investigated along with the different cases. In Section 5, the algorithm of MADM with hesitant fuzzy information based on AC-HFWA/AC-HFWG is constructed firstly; next, case analysis will be carried out and some comparisons with existing approaches in the hesitant fuzzy environment and merits of the proposed MADM approach based on AC-HFWA/AC-HFWG operators are analysed, and the conclusion will be obtained in Section 6.


## 2. Preliminaries

In this section, we will retrospect the related concepts of HFS and copula and cocopula; these notions are the basis of this work.


### 2.1. Hesitant Fuzzy Sets

Definition 1 (see [5]).
LetS be a finite reference set. A hesitant fuzzy set G on S in terms of a function when applied to S returns a subset of 0,1 denoted by(1)G=s,gh∀s∈S,where gh 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 [5].Definition 2 (see [5]).
For a HFEgh=∪i=1♮ghi, μg=1/♮g∑i=1♮ghi is called the score function of gh, where ♮g is the number of possible elements in gh.
For two HFEsg1h and g2h,Ifμg1>μg2, then g1≻g2;Ifμg1=μg2, then g1=g2.


### 2.2. Copulas and Cocopulas

Definition 3 (see [26]).
A two-dimensional functionΩ:0,12⟶0,1 is called a copula, if the following conditions are met:(1)
Ωm,1=Ω1,m=m, Ωm,0=Ω0,m=0(2)
Ωm1,n1−Ωm2,n1−Ωm1,n2+Ωm2,n2≥0where m,m1,m2,n1,n2∈0,1 and m1≤m2,n1≤n2.Definition 4 (see [29]).
A copulaΩ is named as an Archimedean copula, if there is a strictly decreasing and continuous function ςδ:0,1⟶0,∞ with ς1=0, and σ from 0,∞ to 0,1 is defined as follows:(2)σδ=ς−1δ,δ∈0,ς0;0,δ∈ς0,+∞.
For allδ,ε∈0,12, we have(3)σδ,ε=σςδ+ςε.
IfΩ is strictly increasing on 0,12,ς0=+∞, and σ coincides with ς−1 on 0,+∞, then Ω is written as [30](4)Ωδ,ε=ς−1ςδ+ςε,and the function ς is called a strict generator and Ω is called a strict Archimedean copula.Definition 5 (see [31]).
LetΩ be a copula, and the cocopula is introduced as follows:(5)Ω∗δ,ε=1−Ω1−δ,1−ε.
IfΩ is a strict Archimedean copula, Ω∗ is also changed to be(6)Ω∗δ,ε=1−Ω1−δ,1−ε=1−ς−1ς1−δ+ς1−ε.
In order to introduce some new operations based on copulas and cocopulas mentioned above, the following conclusion is given firstly.Theorem 1.
For∀δ,ε∈0,1, then 0≤Ωδ,ε≤1,0≤Ω∗δ,ε≤1.Proof.
If0≤δ≤ε≤1, then 0≤1−ε≤1−δ≤1. As ς is strictly decreasing and ς1=0,ς0=+∞,(7)0≤ςε≤ςδ≤+∞,0≤ς1−δ≤ς1−ε≤+∞.
So,(8)ςδ≤ςδ+ςε≤2ςδ≤+∞,ς1−ε≤ς1−δ+ς1−ε≤2ς1−ε≤+∞.
We have(9)0≤ς−1ςδ+ςε≤δ≤ε≤1−ς−1ς1−δ+ς1−ε≤1.
Thus, Theorem1 holds.Definition 6.
Letδ,ε∈0,1; the algebra operations based on copula and cocopula are defined as follows:(10)1δ⊕ε=Ω∗δ,ε=1−ς−1ς1−δ+ς1−ε,2δ⊗ε=Ωδ,ε=ς−1ςδ+ςε.
It is easy to verify that⊕ and ⊗ satisfy associative law, that is, for ∀δ,ε,ν∈0,1,(11)δ⊕ε⊕ν=δ⊕ε⊕ν,δ⊗ε⊗ν=δ⊗ε⊗ν.Theorem 2.
For∀δ∈0,1, ρ≥0, we have ρδ=1−ς−1ρς1−δ, δρ=ς−1ρςδ.


## 2.1. Hesitant Fuzzy Sets

Definition 1 (see [5]).
LetS be a finite reference set. A hesitant fuzzy set G on S in terms of a function when applied to S returns a subset of 0,1 denoted by(1)G=s,gh∀s∈S,where gh 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 [5].Definition 2 (see [5]).
For a HFEgh=∪i=1♮ghi, μg=1/♮g∑i=1♮ghi is called the score function of gh, where ♮g is the number of possible elements in gh.
For two HFEsg1h and g2h,Ifμg1>μg2, then g1≻g2;Ifμg1=μg2, then g1=g2.


## 2.2. Copulas and Cocopulas

Definition 3 (see [26]).
A two-dimensional functionΩ:0,12⟶0,1 is called a copula, if the following conditions are met:(1)
Ωm,1=Ω1,m=m, Ωm,0=Ω0,m=0(2)
Ωm1,n1−Ωm2,n1−Ωm1,n2+Ωm2,n2≥0where m,m1,m2,n1,n2∈0,1 and m1≤m2,n1≤n2.Definition 4 (see [29]).
A copulaΩ is named as an Archimedean copula, if there is a strictly decreasing and continuous function ςδ:0,1⟶0,∞ with ς1=0, and σ from 0,∞ to 0,1 is defined as follows:(2)σδ=ς−1δ,δ∈0,ς0;0,δ∈ς0,+∞.
For allδ,ε∈0,12, we have(3)σδ,ε=σςδ+ςε.
IfΩ is strictly increasing on 0,12,ς0=+∞, and σ coincides with ς−1 on 0,+∞, then Ω is written as [30](4)Ωδ,ε=ς−1ςδ+ςε,and the function ς is called a strict generator and Ω is called a strict Archimedean copula.Definition 5 (see [31]).
LetΩ be a copula, and the cocopula is introduced as follows:(5)Ω∗δ,ε=1−Ω1−δ,1−ε.
IfΩ is a strict Archimedean copula, Ω∗ is also changed to be(6)Ω∗δ,ε=1−Ω1−δ,1−ε=1−ς−1ς1−δ+ς1−ε.
In order to introduce some new operations based on copulas and cocopulas mentioned above, the following conclusion is given firstly.Theorem 1.
For∀δ,ε∈0,1, then 0≤Ωδ,ε≤1,0≤Ω∗δ,ε≤1.Proof.
If0≤δ≤ε≤1, then 0≤1−ε≤1−δ≤1. As ς is strictly decreasing and ς1=0,ς0=+∞,(7)0≤ςε≤ςδ≤+∞,0≤ς1−δ≤ς1−ε≤+∞.
So,(8)ςδ≤ςδ+ςε≤2ςδ≤+∞,ς1−ε≤ς1−δ+ς1−ε≤2ς1−ε≤+∞.
We have(9)0≤ς−1ςδ+ςε≤δ≤ε≤1−ς−1ς1−δ+ς1−ε≤1.
Thus, Theorem1 holds.Definition 6.
Letδ,ε∈0,1; the algebra operations based on copula and cocopula are defined as follows:(10)1δ⊕ε=Ω∗δ,ε=1−ς−1ς1−δ+ς1−ε,2δ⊗ε=Ωδ,ε=ς−1ςδ+ςε.
It is easy to verify that⊕ and ⊗ satisfy associative law, that is, for ∀δ,ε,ν∈0,1,(11)δ⊕ε⊕ν=δ⊕ε⊕ν,δ⊗ε⊗ν=δ⊗ε⊗ν.Theorem 2.
For∀δ∈0,1, ρ≥0, we have ρδ=1−ς−1ρς1−δ, δρ=ς−1ρςδ.


## 3. Archimedean Copula-Based Hesitant Fuzzy Weighted Averaging Operator (AC-HFWA)

In this part, we will put forward the Archimedean copula-based HF weighted averaging operator (AC-HFWA). Before AC-HFWA is introduced, the new operations of HFE based on copula will be defined, and then some properties of AC-HFWA are also investigated.


### 3.1. New Operations for HFEs Based on Copulas

We will give a new version of operational rules based on copulas and cocopulas.Definition 7.
Letg1h=∪m1=1♮g1h1m1, g2h=∪m2=1♮g2h2m2, and gh=∪i=1♮ghi be three HFEs and ρ≥0; the novel operational rules of HFEs are given as follows:(12)g1⊕g2=∪h1m1∈g1h2m2∈g21−ς−1ς1−h1m1+ς1−h2m2m1=1,2,…,♮g1,m2=1,2,…,♮g2,g1⊗g2=∪h1m1∈g1h2m2∈g2ς−1ςh1m1+ςh2m2m1=1,2,…,♮g1,m2=1,2,…,♮g2,ρg=∪hi∈g1−ς−1ρς1−hii=1,2,…,♮g,gρ=∪hi∈gς−1ρςhii=1,2,…,♮g.
From the above definition, the following conclusions can be easily drawn.Theorem 3.
Letg1, g2, and g3 be three HFEs and a,b,c∈R+; then, we have(13)1g1⊕g2=g2⊕g1,2g1⊕g2⊕g3=g1⊕g2⊕g3,3ag1⊕bg1=a+bg1,4abg1⊕cg2=abg1⊕acg2,5abg1=abg1,6g1⊗g2=g2⊗g1,7g1⊗g2⊗g3=g1⊗g2⊗g3.The algorithms can be used to fuse the HF information and investigate their ideal properties which is the focus of the following sections.


### 3.2. AC-HFWA

In this section, the AC-HFWA will be introduced and the proposed operations of HFEs based on copula as well as the properties of AC-HFWA are investigated.Definition 8.
LetG=g1,g2,…,gn be a set of n HFEs and Φ be a function on G, Φ:0,1n⟶0,1; then, ΦG=∪Φg1,g2,…,gn.Definition 9.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1. Archimedean copula-based hesitant fuzzy weighted averaging operator (AC-HFWA) is defined as follows:(14)AC−HFWAg1,g2,…,gn=ω1g1⊕ω2g2⊕⋯⊕ωngn.Theorem 4.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(15)AC−HFWAg1,g2,…,gn=⊕i=1nωigi=∪himi∈gi1−τ−1∑i=1nωiτ1−himimi=1,2,…,♮gi.Proof.
Forn=2, we have(16)AC−HFWAg1,g2=ω1g1⊕ω2g2=∪h1m1∈g1h2m2∈g21−ς−1ω1ς1−h1m1+ω2ς1−h2m2m1=1,2,…,♮g1,m2=1,2,…,♮g2.
Suppose that equation (15) holds for n=k, that is,(17)AC−HFWAg1,g2,…,gk=ω1g1⊕ω2g2⊕⋯⊕ωkgk=∪himi∈gi1−τ−1∑i=1kωiτ1−himimi=1,2,…,♮gi.
Then,(18)AC−HFWAg1,g2,…,gk,gk+1=⊕i=1kωigi⊕ωk+1gk+1=∪himi∈gi1−τ−1∑i=1kωiτ1−himimi=1,2,…,♮gi⊕∪hk+1,mk+1∈gk+11−τ−1ωk+1τ1−hk+1,mk+1mk+1=1,2,…,♮gk+1=∪himi∈gi1−τ−1∑i=1kωiτ1−himi+ωk+1τ1−hk+1,mk+1mi=1,2,…,♮gi=∪himi∈gi1−τ−1∑i=1k+1ωiτ1−himimi=1,2,…,♮gi.
Equation (15) holds for n=k+1. Thus, equation (15) holds for all n.Theorem 5.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(1)
(Idempotency) Ifg1=g2=⋯gn=h, AC−HFWAg1,g2,…,gn=h.(2)
(Monotonicity) Letgi∗h=∪mi=1♮gi∗himi∗i=1,2,…,n; if himi≤himi∗,(19)AC−HFWAg1,g2,…,gn≤AC−HFWAg1∗,g2∗,…,gn∗.(3)
(Boundedness) Ifh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi,(20)h−≤AC−HFWAg1,g2,…,gn≤h+.Proof.
(1)
AC−HFWAg1,g2,…,gn=∪1−ς−1∑i=1nωiς1−hh∈gi,i=1,2,…,n=h.(2)
Ifhimi≤himi∗, ς1−himi≤ς1−himi∗ and ∑i=1nωiς1−himi≤∑i=1nωiς1−himi∗.Then,ς−1∑i=1nωiς1−himi≥ς−1∑i=1nωiς1−himi∗ and 1−ς−1∑i=1nωiς1−himi≤1−ς−1∑i=1nωiς1−himi∗.So,AC−HFWAg1,g2,…,gn≤AC−HFWAg1∗,g2∗,…,gn∗.(3)
Supposeh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi.
Therefore,1−h+≤1−himi, 1−h−≥1−himi, for all i and mi.
Sinceς is strictly decreasing, ς−1 is also strictly decreasing.
Then,ς1−h−≤ς1−himi≤ς1−h+,∀i=1,2,…,n, and so(21)∑i=1nωiς1−h−≤∑i=1nωiς1−himi≤∑i=1nωiς1−h+.
That is,ς1−h−≤∑i=1nωiς1−himi≤ς1−h+.
Therefore,ς−1ς1−h+≤ς−1∑i=1nωiς1−himi≤ς−1ς1−h−, h0≤1−ς−1∑i=1nωiς1−himi≤h+.Definition 10.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1. The Archimedean copula-based generalized hesitant fuzzy averaging operator (AC-GHFWA) is given by(22)AC−GHFWAθg1,g2,…,gn=ω1g1θ⊕ω2g2θ⊕⋯⊕ωngnθ1/θ=⊕i=1nωigiθ1/θ.
Especially, whenθ=1, the AC-GHFWA operator becomes the AC-HFWA operator.
The following theorems are easily obtained from Theorem4 and the operations of HFEs.Theorem 6.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(23)AC−GHFWAθg1,g2,…,gn=∪himi∈gi−ς−11θς1−ς−1∑i=1nωiς1−ς−1θςhimimi=1,2,…,♮gi.
Similar to Theorem5, the properties of AC-GHFWA can be obtained easily.Theorem 7.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(1)
(Idempotency) Ifg1=g2=⋯gn=h, AC−GHFWAθg1,g2,…,gn=h.(2)
(Monotonicity) Letgi∗h=∪mi=1♮gi∗himi∗i=1,2,…,n; if himi≤himi∗,(24)AC−GHFWAθg1,g2,…,gn≤AC−GHFWAθg1∗,g2∗,…,gn∗.(3)
(Boundedness) Ifh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi,(25)h−≤AC−GHFWAθg1,g2,…,gn≤h+.


### 3.3. Different Forms of AC-HFWA

We can see from Theorem4 that some specific AC-HFWAs can be obtained when ς is assigned different generators.Case 1. If ςt=−lntκ,κ≥1, then ς−1t=e−t1/κ. So, δ⊕ε=1−e−−ln1−δκ+−ln1−εκ1/κ, δ⊗ε=1−e−−lnδκ+−lnεκ1/κ.(26)AC−HFWAg1,g2,…,gn=∪himi∈gi1−e−∑i=1nωi−ln1−himiλ1/λmi=1,2,…,♮gi.Specifically, whenκ=1,ςt=−lnt, then δ⊕ε=1−1−δ1−ε, δ⊗ε=δε, and the AC-HFWA becomes the following:(27)HFWAg1,g2,…,gn=∪himi∈gi1−∏i=1n1−himiωimi=1,2,…,♮gi,GHFWAθg1,g2,…,gn=∪himi∈gi1−∏i=1n1−himiωi1/θmi=1,2,…,♮gi.They are the HF operators defined by Xia and Xu [13].Case 2. If ςt=t−κ−1,κ>0, then ς−1t=t+1−1/κ. So, δ⊕ε=1−1−δ−κ+1−ε−κ−1−1/κ, δ⊗ε=δ−κ+ε−κ−1−1/κ.(28)AC−HFWAg1,g2,…,gn=∪himi∈gi1−∑i=1nωi1−himi−κ−1/κmi=1,2,…,♮gi.Case 3. If ςt=−lne−κt−1/e−κ−1,κ≠0, then ς−1t=−1/κlne−te−κ−1+1. So, δ⊕ε=1+1/κlne−κ1−δ−1e−κ1−ε−1/e−κ−1+1, δ⊗ε=−1/κlne−κδ−1e−κε−1/e−κ−1+1.(29)AC−HFWAg1,g2,…,gn=∪himi∈gi1+1κln∏i=1ne−κ1−himi−1ωi+1mi=1,2,…,♮gi.Case 4. If ςt=ln1−κ1−t/t,−1≤κ<1, then ς−1t=1−κ/et−κ. So, δ⊕ε=1−1−δ1−ε/1−κδε, δ⊗ε=δε/1−κ1−δ1−ε.(30)AC−HFWAg1,g2,…,gn=∪himi∈gi∏i=1n1−κhimiωi−∏i=1n1−himiωi∏i=1n1−κhimiωi−κ∏i=1n1−himiωimi=1,2,…,♮gi.Case 5. If ςt=−ln1−1−tκ,κ≥1, then ς−1t=1−1−e−t1/κ. So δ⊕ε=δκ+εκ−δκεκ1/κ, δ⊗ε=1−1−δκ+1−εκ−1−δκ1−εκ1/κ.(31)AC−HFWAg1,g2,…,gn=∪himi∈gi1−∏i=1n1−himiκωi1/κmi=1,2,…,♮gi.


## 3.1. New Operations for HFEs Based on Copulas

We will give a new version of operational rules based on copulas and cocopulas.Definition 7.
Letg1h=∪m1=1♮g1h1m1, g2h=∪m2=1♮g2h2m2, and gh=∪i=1♮ghi be three HFEs and ρ≥0; the novel operational rules of HFEs are given as follows:(12)g1⊕g2=∪h1m1∈g1h2m2∈g21−ς−1ς1−h1m1+ς1−h2m2m1=1,2,…,♮g1,m2=1,2,…,♮g2,g1⊗g2=∪h1m1∈g1h2m2∈g2ς−1ςh1m1+ςh2m2m1=1,2,…,♮g1,m2=1,2,…,♮g2,ρg=∪hi∈g1−ς−1ρς1−hii=1,2,…,♮g,gρ=∪hi∈gς−1ρςhii=1,2,…,♮g.
From the above definition, the following conclusions can be easily drawn.Theorem 3.
Letg1, g2, and g3 be three HFEs and a,b,c∈R+; then, we have(13)1g1⊕g2=g2⊕g1,2g1⊕g2⊕g3=g1⊕g2⊕g3,3ag1⊕bg1=a+bg1,4abg1⊕cg2=abg1⊕acg2,5abg1=abg1,6g1⊗g2=g2⊗g1,7g1⊗g2⊗g3=g1⊗g2⊗g3.The algorithms can be used to fuse the HF information and investigate their ideal properties which is the focus of the following sections.


## 3.2. AC-HFWA

In this section, the AC-HFWA will be introduced and the proposed operations of HFEs based on copula as well as the properties of AC-HFWA are investigated.Definition 8.
LetG=g1,g2,…,gn be a set of n HFEs and Φ be a function on G, Φ:0,1n⟶0,1; then, ΦG=∪Φg1,g2,…,gn.Definition 9.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1. Archimedean copula-based hesitant fuzzy weighted averaging operator (AC-HFWA) is defined as follows:(14)AC−HFWAg1,g2,…,gn=ω1g1⊕ω2g2⊕⋯⊕ωngn.Theorem 4.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(15)AC−HFWAg1,g2,…,gn=⊕i=1nωigi=∪himi∈gi1−τ−1∑i=1nωiτ1−himimi=1,2,…,♮gi.Proof.
Forn=2, we have(16)AC−HFWAg1,g2=ω1g1⊕ω2g2=∪h1m1∈g1h2m2∈g21−ς−1ω1ς1−h1m1+ω2ς1−h2m2m1=1,2,…,♮g1,m2=1,2,…,♮g2.
Suppose that equation (15) holds for n=k, that is,(17)AC−HFWAg1,g2,…,gk=ω1g1⊕ω2g2⊕⋯⊕ωkgk=∪himi∈gi1−τ−1∑i=1kωiτ1−himimi=1,2,…,♮gi.
Then,(18)AC−HFWAg1,g2,…,gk,gk+1=⊕i=1kωigi⊕ωk+1gk+1=∪himi∈gi1−τ−1∑i=1kωiτ1−himimi=1,2,…,♮gi⊕∪hk+1,mk+1∈gk+11−τ−1ωk+1τ1−hk+1,mk+1mk+1=1,2,…,♮gk+1=∪himi∈gi1−τ−1∑i=1kωiτ1−himi+ωk+1τ1−hk+1,mk+1mi=1,2,…,♮gi=∪himi∈gi1−τ−1∑i=1k+1ωiτ1−himimi=1,2,…,♮gi.
Equation (15) holds for n=k+1. Thus, equation (15) holds for all n.Theorem 5.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(1)
(Idempotency) Ifg1=g2=⋯gn=h, AC−HFWAg1,g2,…,gn=h.(2)
(Monotonicity) Letgi∗h=∪mi=1♮gi∗himi∗i=1,2,…,n; if himi≤himi∗,(19)AC−HFWAg1,g2,…,gn≤AC−HFWAg1∗,g2∗,…,gn∗.(3)
(Boundedness) Ifh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi,(20)h−≤AC−HFWAg1,g2,…,gn≤h+.Proof.
(1)
AC−HFWAg1,g2,…,gn=∪1−ς−1∑i=1nωiς1−hh∈gi,i=1,2,…,n=h.(2)
Ifhimi≤himi∗, ς1−himi≤ς1−himi∗ and ∑i=1nωiς1−himi≤∑i=1nωiς1−himi∗.Then,ς−1∑i=1nωiς1−himi≥ς−1∑i=1nωiς1−himi∗ and 1−ς−1∑i=1nωiς1−himi≤1−ς−1∑i=1nωiς1−himi∗.So,AC−HFWAg1,g2,…,gn≤AC−HFWAg1∗,g2∗,…,gn∗.(3)
Supposeh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi.
Therefore,1−h+≤1−himi, 1−h−≥1−himi, for all i and mi.
Sinceς is strictly decreasing, ς−1 is also strictly decreasing.
Then,ς1−h−≤ς1−himi≤ς1−h+,∀i=1,2,…,n, and so(21)∑i=1nωiς1−h−≤∑i=1nωiς1−himi≤∑i=1nωiς1−h+.
That is,ς1−h−≤∑i=1nωiς1−himi≤ς1−h+.
Therefore,ς−1ς1−h+≤ς−1∑i=1nωiς1−himi≤ς−1ς1−h−, h0≤1−ς−1∑i=1nωiς1−himi≤h+.Definition 10.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1. The Archimedean copula-based generalized hesitant fuzzy averaging operator (AC-GHFWA) is given by(22)AC−GHFWAθg1,g2,…,gn=ω1g1θ⊕ω2g2θ⊕⋯⊕ωngnθ1/θ=⊕i=1nωigiθ1/θ.
Especially, whenθ=1, the AC-GHFWA operator becomes the AC-HFWA operator.
The following theorems are easily obtained from Theorem4 and the operations of HFEs.Theorem 6.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(23)AC−GHFWAθg1,g2,…,gn=∪himi∈gi−ς−11θς1−ς−1∑i=1nωiς1−ς−1θςhimimi=1,2,…,♮gi.
Similar to Theorem5, the properties of AC-GHFWA can be obtained easily.Theorem 7.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(1)
(Idempotency) Ifg1=g2=⋯gn=h, AC−GHFWAθg1,g2,…,gn=h.(2)
(Monotonicity) Letgi∗h=∪mi=1♮gi∗himi∗i=1,2,…,n; if himi≤himi∗,(24)AC−GHFWAθg1,g2,…,gn≤AC−GHFWAθg1∗,g2∗,…,gn∗.(3)
(Boundedness) Ifh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi,(25)h−≤AC−GHFWAθg1,g2,…,gn≤h+.


## 3.3. Different Forms of AC-HFWA

We can see from Theorem4 that some specific AC-HFWAs can be obtained when ς is assigned different generators.Case 1. If ςt=−lntκ,κ≥1, then ς−1t=e−t1/κ. So, δ⊕ε=1−e−−ln1−δκ+−ln1−εκ1/κ, δ⊗ε=1−e−−lnδκ+−lnεκ1/κ.(26)AC−HFWAg1,g2,…,gn=∪himi∈gi1−e−∑i=1nωi−ln1−himiλ1/λmi=1,2,…,♮gi.Specifically, whenκ=1,ςt=−lnt, then δ⊕ε=1−1−δ1−ε, δ⊗ε=δε, and the AC-HFWA becomes the following:(27)HFWAg1,g2,…,gn=∪himi∈gi1−∏i=1n1−himiωimi=1,2,…,♮gi,GHFWAθg1,g2,…,gn=∪himi∈gi1−∏i=1n1−himiωi1/θmi=1,2,…,♮gi.They are the HF operators defined by Xia and Xu [13].Case 2. If ςt=t−κ−1,κ>0, then ς−1t=t+1−1/κ. So, δ⊕ε=1−1−δ−κ+1−ε−κ−1−1/κ, δ⊗ε=δ−κ+ε−κ−1−1/κ.(28)AC−HFWAg1,g2,…,gn=∪himi∈gi1−∑i=1nωi1−himi−κ−1/κmi=1,2,…,♮gi.Case 3. If ςt=−lne−κt−1/e−κ−1,κ≠0, then ς−1t=−1/κlne−te−κ−1+1. So, δ⊕ε=1+1/κlne−κ1−δ−1e−κ1−ε−1/e−κ−1+1, δ⊗ε=−1/κlne−κδ−1e−κε−1/e−κ−1+1.(29)AC−HFWAg1,g2,…,gn=∪himi∈gi1+1κln∏i=1ne−κ1−himi−1ωi+1mi=1,2,…,♮gi.Case 4. If ςt=ln1−κ1−t/t,−1≤κ<1, then ς−1t=1−κ/et−κ. So, δ⊕ε=1−1−δ1−ε/1−κδε, δ⊗ε=δε/1−κ1−δ1−ε.(30)AC−HFWAg1,g2,…,gn=∪himi∈gi∏i=1n1−κhimiωi−∏i=1n1−himiωi∏i=1n1−κhimiωi−κ∏i=1n1−himiωimi=1,2,…,♮gi.Case 5. If ςt=−ln1−1−tκ,κ≥1, then ς−1t=1−1−e−t1/κ. So δ⊕ε=δκ+εκ−δκεκ1/κ, δ⊗ε=1−1−δκ+1−εκ−1−δκ1−εκ1/κ.(31)AC−HFWAg1,g2,…,gn=∪himi∈gi1−∏i=1n1−himiκωi1/κmi=1,2,…,♮gi.


## 4. Archimedean Copula-Based Hesitant Fuzzy Weighted Geometric Operator (AC-HFWG)

In this section, the Archimedean copula-based hesitant fuzzy weighted geometric operator (AC-HFWG) will be introduced, and some special forms of AC-HFWG operators will be discussed when the generatorς takes different functions.


### 4.1. AC-HFWG

Definition 11.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1. The Archimedean copula-based hesitant fuzzy weighted geometric operator (AC-HFWG) is defined as follows:(32)AC−HFWGg1,g2,…,gn=g1ω1⊗g2ω2⊗⋯⊗gnωn.Theorem 8.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(33)AC−HFWGg1,g2,…,gn=∪himi∈giς−1∑i=1nωiςhimimi=1,2,…,♮gi.Proof.
Forn=2, we have(34)AC−HFWGg1,g2=g1ω1⊗g2ω2=∪h1m1∈g1h2m2∈g2ς−1ω1ςh1m1+ω2ςh2m2m1=1,2,…,♮g1,m2=1,2,…,♮g2.
Suppose that equation (33) holds for n=k, that is,(35)AC−HFWGg1,g2,…,gk=∪himi∈giς−1∑i=1kωiςhimimi=1,2,…,♮gi.
Then,(36)AC−HFWGg1,g2,…,gk,gk+1=⊗i=1kgiωi⊕gk+1ωk+1=∪himi∈giς−1∑i=1k+1ωiςhimimi=1,2,…,♮gi.
Equation (33) holds for n=k+1. Thus, equation (33) holds for all n.Theorem 9.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(1)
Ifg1=g2=⋯gn=h, AC−HFWGg1,g2,…,gn=h.(2)
Letgi∗h=∪mi=1♮gi∗himi∗i=1,2,…,n; if himi≤himi∗,(37)AC−HFWGg1,g2,…,gn≤AC−HFWGg1∗,g2∗,…,gn∗.(3)
Ifh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi,(38)h−≤AC−HFWGg1,g2,…,gn≤h+.Proof.
Supposeh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi.
Sinceς is strictly decreasing, ς−1 is also strictly decreasing.
Then,ςh+≤ςhimi≤ςh−,∀i=1,2,…,n.
∑i=1nωiςh+≤∑i=1nωiςhimi≤∑i=1nωiςh−, ςh+≤∑i=1nωiςhimi≤ςh−, so ς−1ςh−≤ς−1∑i=1nωiςhimi≤ς−1ςh+h−≤ς−1∑i=1nωiςhimi≤h+.Definition 12.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then, the generalized hesitant fuzzy weighted geometric operator based on Archimedean copulas (AC-CHFWG) is defined as follows:(39)AC−GHFWGθg1,g2,…,gn=1θθg1ω1⊗θg2ω2⊗⋯⊗θgnωn=1θ⊗i=1nθgiωi.
Especially, whenθ=1, the AC-GHFWG operator becomes the AC-HFWG operator.Theorem 10.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(40)AC−GHFWGθg1,g2,…,gn=∪himi∈gi1−τ−11θτ1−τ−1∑i=1nωiτ1−τ−1θτ1−himimi=1,2,…,♮gi.Theorem 11.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(1)
Ifg1=g2=⋯gn=h, AC−GHFWGθg1,g2,…,gn=h.(2)
Letgi∗h=∪mi=1♮gi∗himi∗i=1,2,…,n; if himi≤himi∗,(41)AC−GHFWGθg1,g2,…,gn≤AC−GHFWGθg1∗,g2∗,…,gn∗.(3)
Ifh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi,(42)h−≤AC−GHFWGθg1,g2,…,gn≤h+.


### 4.2. Different Forms of AC-HFWG Operators

We can see from Theorem8 that some specific AC-HFWGs can be obtained when ς is assigned different generators.Case 1. If ςt=−lntκ,κ≥1, then ς−1t=e−t1/κ. So, δ⊕ε=1−e−−ln1−δκ+−ln1−εκ1/κ, δ⊗ε=e−−lnδκ+−lnεκ1/κ.(43)AC−HFWGg1,g2,…,gn=∪himi∈gie−∑i=1nωi−lnhimiκ1/κmi=1,2,…,♮gi.Specifically, whenκ=1 and ςt=−lnt, then δ⊕ε=1−1−δ1−ε and δ⊗ε=δε, and the AC-HFWG operator reduces to HFWG and GHFWG [13]:(44)HFWGg1,g2,…,gn=∪himi∈gi∏i=1nhimiωimi=1,2,…,♮gi,GHFWGθg1,g2,…,gn=∪himi∈gi1−1−∏i=1n1−1−himiθωi1/θmi=1,2,…,♮gi.Case 2. If ςt=t−κ−1,κ>0, then ς−1t=t+1−1/κ. So, δ⊕ε=1−1−δ−κ+1−ε−κ−1−1/κ, δ⊗ε=δ−κ+ε−κ−1−1/κ.(45)AC−HFWGg1,g2,…,gn=∪himi∈gi∑i=1nωihimi−κ−1/κmi=1,2,…,♮gi.Case 3. If ςt=−lne−κt−1/e−κ−1,κ≠0, then ς−1t=−1/κlne−te−κ−1+1. So, δ⊕ε=1+1/κlne−κ1−δ−1e−κ1−ε−1/e−κ−1+1, δ⊗ε=−1/klne−κδ−1e−κε−1/e−κ−1+1.(46)AC−HFWGg1,g2,…,gn=∪himi∈gi−1κln∏i=1ne−κhimi−1ωi+1mi=1,2,…,♮gi.Case 4. If ςt=ln1−κ1−t/t,−1≤κ<1, then ς−1t=1−κ/et−κ. So, δ⊕ε=1−1−δ1−ε/1−κδε, δ⊗ε=δε/1−κ1−δ1−ε.(47)AC−HFWGg1,g2,…,gn=∪himi∈gi1−κ∏i=1nhimiωi∏i=1n1−κ1−himiωi−κ∏i=1nhimiωimi=1,2,…,♮gi.Case 5. If ςt=−ln1−1−tκ,κ≥1, then ς−1t=1−1−e−t1/κ. So, δ⊕ε=δκ+εκ−δκεκ1/κ, δ⊗ε=1−1−δκ+1−εκ−1−δκ1−εκ1/κ.(48)AC−HFWGg1,g2,…,gn=∪himi∈gi1−1−∏i=1n1−1−himiκωi1/κmi=1,2,…,♮gi.


### 4.3. The Properties of AC-HFWA and AC-HFWG

It is seen from above discussion that AC-HFWA and AC-HFWG are functions with respect to the parameter which is from the generatorς. In this section, we will introduce the properties of the AC-HFWA and AC-HFWG operator regarding to the parameter κ.Theorem 12.
Letςt be the generator function of copula, and it takes five cases proposed in Section 4.2; then μAC−HFWAg1,g2,…,gn is an increasing function of κ, μAC−HFWGg1,g2,…,gn is an decreasing function of κ, and μAC−HFWAg1,g2,…,gn≥μAC−HFWGg1,g2,…,gn.Proof.
(Case 1) Whenςt=−lntκ with κ≥1. By Definition 6, we have(49)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−e−∑i=1nωi−ln1−himiκ1/κ,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gne−∑i=1nωi−ln1−himiκ1/κ.
Suppose1≤κ1<κ2; according to reference [10], ∑i=1nωiaiκ1/κ is an increasing function of κ. So,(50)∑i=1nωi−ln1−himiκ11/κ1≤∑i=1nωi−ln1−himiκ21/κ2,∑i=1nωi−lnhimiκ11/κ1≤∑i=1nωi−lnhimiκ21/κ2.
Furthermore,(51)1−e−∑i=1nωi−ln1−himiκ11/κ1≤1−e−∑i=1nωi−ln1−himiκ21/κ2,e−∑i=1nωi−ln1−himiκ11/κ1≥e−∑i=1nωi−ln1−himiκ21/κ2.
Therefore,μAC−HFWAκ1≤μAC−HFWAκ2, μAC−HFWGκ1≥μAC−HFWGκ2. Because κ≥1,(52)e−∑i=1nωi−ln1−himiκ1/κ+e−∑i=1nωi−lnhimiκ1/κ≤e−∑i=1nωi−ln1−himi+e−∑i=1nωi−lnhimi=∏i=1n1−himiωi+∏i=1nhimiωi≤∑i=1nωi1−himi+∑ωihimi=∑i=1nωi=1.
So,1−e−∑i=1nωi−ln1−himiκ1/κ≥e−∑i=1nωi−lnhimiκ1/κ.
That is,μAC−HFWAg1,g2,…,gn≥μAC−HFWGg1,g2,…,gn.
So Theorem12 holds under Case 1.
Case 2. When ςt=t−κ−1,κ>0. Firstly,(53)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−ς−1∑i=1nωiς1−himi,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gnς−1∑i=1kωiςhimi.
Secondly, becauseςκ,t=t−κ−1,κ>0,0<t<1, ∂ς/∂κ=t−κlnt, t−κ>0, lnt<0, and then ∂ς/∂κ<0, ςκ,t is decreasing with respect to κ.
ς−1κ,t=t+1−1/κ,κ>0,0<t<1, so ∂ς−1/∂κ=t+1−1/κlnt+11/κ2, t+1−1/κ>0,lnt+1>0,1/κ2>0, and then ∂ς−1/∂κ>0, ς−1κ,t is decreasing with respect to κ.
Thus,1−ς−1∑i=1nωiς1−himi is decreasing with respect to κ, and ς−1∑i=1kωiςhimi is decreasing with respect to κ. Suppose 1≤κ1<κ2; we have
μAC−HFWAκ1≤μAC−HFWAκ2 and μAC−HFWGκ1≥μAC−HFWGκ2.
Lastly,(54)∑i=1nωi1−himi−κ−1/κ+∑i=1nωihimi−κ−1/κ<limκ⟶0∑i=1nωi1−himi−κ−1/κ+limκ⟶0∑i=1nωihimi−κ−1/κ=e∑i=1nωiln1−himi+e∑i=1nωilnhimi=∏i=1n1−himiωi+∏i=1nhimiωi≤∑i=1nωi1−himi+∑ωihimi=∑i=1nωi=1.
That is, Theorem12 holds under Case 2.
Case 3. When ςt=−lne−κt−1/e−κ−1,κ≠0. Firstly,(55)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−ς−1∑i=1nωiς1−himi,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gnς−1∑i=1kωiςhimi.
Secondly,ςκ,t=−lne−κt−1/e−κ−1,κ≠0,0<t<1.
Ifς=−lnς1, ς1=ς2t−1/ς2−1, ς2=e−κ,κ≠0,0<t<1.
Becauseς is decreasing with respect to ς1, ς1 is decreasing with respect to ς2, and ς2 is decreasing with respect to κ, ςκ,t is decreasing with respect to κ.(56)ς−1κ,t=−1κlne−te−κ−1+1,κ≠0,0<t<1.
Supposeς−1=−lnψ1, ψ1=ψ21/κ, ψ2=e−tψ3+1, ς3=e−κ−1,κ≠0,0<t<1.
Becauseς−1 is decreasing with respect to ψ1, ψ1 is decreasing with respect to ψ2, and ψ2 is decreasing with respect to ψ3, ψ3 is decreasing with respect to κ. Then, ς−1κ,t is decreasing with respect to κ.
Thus,1−ς−1∑i=1nωiς1−himi is decreasing with respect to κ and ς−1∑i=1kωiςhimi is decreasing with respect to κ.
Suppose1≤κ1<κ2.
Therefore,μAC−HFWAκ1≤μAC−HFWAκ2 and μAC−HFWGκ1≥μAC−HFWGκ2.
Lastly, when−∝<κ<+∝,(57)−1κln∏i=1ne−κhimi−1ωi+1≤limκ⟶−∝ln∏i=1ne−κhimi−1ωi+1−κ=limκ⟶−∝∏i=1ne−κhimi−1ωi/∏i=1ne−κhimi−1ωi+1∑i=1nωi−himie−κhimi/e−κhimi−1−1=limκ⟶−∝∏i=1ne−κhimi−1ωi∏i=1ne−κhimi−1ωi+1∑i=1nωihimie−κhimie−κhimi−1=∑i=1nωihimi,−1κln∏i=1ne−κ1−himi−1ωi+1≤∑i=1nωi1−himi.
So,−1/κln∏i=1ne−κ1−himi−1ωi+1−1/kln∏i=1ne−κhimi−1ωi+1≤∑i=1nωi1−himi+∑i=1nωihimi=1.
That is, Theorem12 holds under Case 3.
Case 4. When ςt=ln1−κ1−t/t,−1≤κ<1. Firstly,(58)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−ς−1∑i=1nωiς1−himi,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gnς−1∑i=1kωiςhimi.
Secondly, becauseςκ,t=ln1−κ1−t/t,−1≤κ<1,0<t<1, so ∂ς/∂κ=tt−1/1−κ1−t, tt−1<0.
Supposeς1κ,t=1−κ1−t,−1≤κ<1,0<t<1, ∂ς1/∂κ=t−1<0, ς1κ,t is decreasing with respect to κ.
So,ς1κ,t>ς11,t=t>0; then, ∂ς/∂κ<0, and ςκ,t is decreasing with respect to κ.
Asς−1κ,t=1−κ/et−κ,−1≤κ<1,0<t<1, ∂ς−1/∂κ=et+1/et−κ2, et+1>0, et−κ2>0; then ∂ς−1/∂κ>0, ς−1κ,t is decreasing with respect to κ. Thus, 1−ς−1∑i=1nωiς1−himi is decreasing with respect to κ and ς−1∑i=1kωiςhimi is decreasing with respect to κ.
Suppose1≤κ1<κ2. We have μAC−HFWAκ1≤μAC−HFWAκ2 and μAC−HFWGκ1≥μAC−HFWGκ2.
Lastly, becausey=xωi,ωi∈0,1 is decreasing with respect to κ, 1−hi2ωi≥1, 1−1−hi2ωi≥1; then,(59)∏i=1n1+himiωi+∏i=1n1−himiωi≥2∏i=1n1+himiωi∏i=1n1−himiωi1/2=2∏i=1n1−himi2ωi1/2≥2∏i=1n2−himiωi+∏i=1nhimiωi≥2∏i=1n2−himiωi∏i=1nhimiωi1/2=2∏i=1n1−1−himi2ωi1/2≥2.
When−1≤κ<1, we have(60)∏i=1n1−himiωi−κ∏i=1n1−himiωi∏i=1n1−κhimiωi−κ∏i=1n1−himiωi+1−κ∏i=1nhimiωi∏i=1n1−κ1−himiωi−κ∏i=1nhimiωi≤∏i=1n1−himiωi+∏i=1n1−himiωi∏i=1n1+himiωi+∏i=1n1−himiωi+2∏i=1nhimiωi∏i=1n1+1−himiωi+∏i=1nhimiωi=2∏i=1n1−himiωi∏i=1n1+himiωi+∏i=1n1−himiωi+2∏i=1nhimiωi∏i=1n2−himiωi+∏i=1nhimiωi≤∏i=1n1−himiωi+∏i=1nhimiωi≤∑i=1nωi1−himi+∑ωihimi=∑i=1nωi=1.
So,(61)∏i=1n1−κhimiωi−∏i=1n1−himiωi∏i=1n1−κhimiωi−κ∏i=1n1−himiωi≥1−κ∏i=1nhimiωi∏i=1n1−κ1−himiωi−κ∏i=1nhimiωi.
Therefore,μAC−HFWAg1,g2,…,gn≥μAC−HFWGg1,g2,…,gn. That is, Theorem 12 holds under Case 4.
Case 5. When ςt=−ln1−1−tκ,κ≥1.
Firstly,(62)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−ς−1∑i=1nωiς1−himi,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gnς−1∑i=1kωiςhimi.
Secondly, becauseςκ,t=−ln1−1−tκ,κ≥1,0<t<1, ∂ς/∂κ=1−tκln1−t/1−1−tκ, ln1−t<0.
Supposeς1κ,t=1−tκ,κ≥1,0<t<1, ∂ς1/∂κ=1−tκln1−t<0, ς1κ,t is decreasing with respect to κ, so 0<ς1κ,t≤ς11,t=1−t<1, 1−1−tκ>0.
Then,∂ς/∂κ<0, ςκ,t is decreasing with respect to κ.
ς−1κ,t=1−1−e−t1/κ,κ≥1,0<t<1, so ∂ς−1/∂κ=1−e−t1/κln1−e−t−1/κ2, 1−e−t1/κ>0, ln1−e−t<0, −1/κ2<0; then, ∂ς−1/∂κ>0, ς−1κ,t is decreasing with respect to κ.
Thus,1−ς−1∑i=1nωiς1−himi is decreasing with respect to κ and ς−1∑i=1nωiςhimi is decreasing with respect to κ.
Lastly, suppose1≤κ1<κ2, Therefore, μAC−HFWAκ1≤μAC−HFWAκ2, μAC−HFWGκ1≥μAC−HFWGκ2.
Becauseκ≥1,(63)∏i=1n1−himiκωi1/κ+1−1−∏i=1n1−1−himiκωi1/κ≤∏i=1n1−himi1ωi1+1−1−∏i=1n1−1−himi1ωi1=∏i=1n1−himiωi+∏i=1nhimiωi≤∑i=1nωi1−himi+∑ωihimi=∑i=1nωi=1.
So,1−e−∑i=1nωi−ln1−himiκ1/κ≥e−∑i=1nωi−lnhimiκ1/κ, μAC−HFWAg1,g2,…,gn≥μAC−HFWGg1,g2,…,gn.
That is, Theorem12 holds under Case 5.


## 4.1. AC-HFWG

Definition 11.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1. The Archimedean copula-based hesitant fuzzy weighted geometric operator (AC-HFWG) is defined as follows:(32)AC−HFWGg1,g2,…,gn=g1ω1⊗g2ω2⊗⋯⊗gnωn.Theorem 8.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(33)AC−HFWGg1,g2,…,gn=∪himi∈giς−1∑i=1nωiςhimimi=1,2,…,♮gi.Proof.
Forn=2, we have(34)AC−HFWGg1,g2=g1ω1⊗g2ω2=∪h1m1∈g1h2m2∈g2ς−1ω1ςh1m1+ω2ςh2m2m1=1,2,…,♮g1,m2=1,2,…,♮g2.
Suppose that equation (33) holds for n=k, that is,(35)AC−HFWGg1,g2,…,gk=∪himi∈giς−1∑i=1kωiςhimimi=1,2,…,♮gi.
Then,(36)AC−HFWGg1,g2,…,gk,gk+1=⊗i=1kgiωi⊕gk+1ωk+1=∪himi∈giς−1∑i=1k+1ωiςhimimi=1,2,…,♮gi.
Equation (33) holds for n=k+1. Thus, equation (33) holds for all n.Theorem 9.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(1)
Ifg1=g2=⋯gn=h, AC−HFWGg1,g2,…,gn=h.(2)
Letgi∗h=∪mi=1♮gi∗himi∗i=1,2,…,n; if himi≤himi∗,(37)AC−HFWGg1,g2,…,gn≤AC−HFWGg1∗,g2∗,…,gn∗.(3)
Ifh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi,(38)h−≤AC−HFWGg1,g2,…,gn≤h+.Proof.
Supposeh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi.
Sinceς is strictly decreasing, ς−1 is also strictly decreasing.
Then,ςh+≤ςhimi≤ςh−,∀i=1,2,…,n.
∑i=1nωiςh+≤∑i=1nωiςhimi≤∑i=1nωiςh−, ςh+≤∑i=1nωiςhimi≤ςh−, so ς−1ςh−≤ς−1∑i=1nωiςhimi≤ς−1ςh+h−≤ς−1∑i=1nωiςhimi≤h+.Definition 12.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then, the generalized hesitant fuzzy weighted geometric operator based on Archimedean copulas (AC-CHFWG) is defined as follows:(39)AC−GHFWGθg1,g2,…,gn=1θθg1ω1⊗θg2ω2⊗⋯⊗θgnωn=1θ⊗i=1nθgiωi.
Especially, whenθ=1, the AC-GHFWG operator becomes the AC-HFWG operator.Theorem 10.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(40)AC−GHFWGθg1,g2,…,gn=∪himi∈gi1−τ−11θτ1−τ−1∑i=1nωiτ1−τ−1θτ1−himimi=1,2,…,♮gi.Theorem 11.
Letgih=∪mi=1♮gihimii=1,2,…,n, ωi be the weight vector of gi with ωi∈0,1 and ∑i=1nωi=1; then,(1)
Ifg1=g2=⋯gn=h, AC−GHFWGθg1,g2,…,gn=h.(2)
Letgi∗h=∪mi=1♮gi∗himi∗i=1,2,…,n; if himi≤himi∗,(41)AC−GHFWGθg1,g2,…,gn≤AC−GHFWGθg1∗,g2∗,…,gn∗.(3)
Ifh−=mini=1,2,…,nhimi and h+=maxi=1,2,…,nhimi,(42)h−≤AC−GHFWGθg1,g2,…,gn≤h+.


## 4.2. Different Forms of AC-HFWG Operators

We can see from Theorem8 that some specific AC-HFWGs can be obtained when ς is assigned different generators.Case 1. If ςt=−lntκ,κ≥1, then ς−1t=e−t1/κ. So, δ⊕ε=1−e−−ln1−δκ+−ln1−εκ1/κ, δ⊗ε=e−−lnδκ+−lnεκ1/κ.(43)AC−HFWGg1,g2,…,gn=∪himi∈gie−∑i=1nωi−lnhimiκ1/κmi=1,2,…,♮gi.Specifically, whenκ=1 and ςt=−lnt, then δ⊕ε=1−1−δ1−ε and δ⊗ε=δε, and the AC-HFWG operator reduces to HFWG and GHFWG [13]:(44)HFWGg1,g2,…,gn=∪himi∈gi∏i=1nhimiωimi=1,2,…,♮gi,GHFWGθg1,g2,…,gn=∪himi∈gi1−1−∏i=1n1−1−himiθωi1/θmi=1,2,…,♮gi.Case 2. If ςt=t−κ−1,κ>0, then ς−1t=t+1−1/κ. So, δ⊕ε=1−1−δ−κ+1−ε−κ−1−1/κ, δ⊗ε=δ−κ+ε−κ−1−1/κ.(45)AC−HFWGg1,g2,…,gn=∪himi∈gi∑i=1nωihimi−κ−1/κmi=1,2,…,♮gi.Case 3. If ςt=−lne−κt−1/e−κ−1,κ≠0, then ς−1t=−1/κlne−te−κ−1+1. So, δ⊕ε=1+1/κlne−κ1−δ−1e−κ1−ε−1/e−κ−1+1, δ⊗ε=−1/klne−κδ−1e−κε−1/e−κ−1+1.(46)AC−HFWGg1,g2,…,gn=∪himi∈gi−1κln∏i=1ne−κhimi−1ωi+1mi=1,2,…,♮gi.Case 4. If ςt=ln1−κ1−t/t,−1≤κ<1, then ς−1t=1−κ/et−κ. So, δ⊕ε=1−1−δ1−ε/1−κδε, δ⊗ε=δε/1−κ1−δ1−ε.(47)AC−HFWGg1,g2,…,gn=∪himi∈gi1−κ∏i=1nhimiωi∏i=1n1−κ1−himiωi−κ∏i=1nhimiωimi=1,2,…,♮gi.Case 5. If ςt=−ln1−1−tκ,κ≥1, then ς−1t=1−1−e−t1/κ. So, δ⊕ε=δκ+εκ−δκεκ1/κ, δ⊗ε=1−1−δκ+1−εκ−1−δκ1−εκ1/κ.(48)AC−HFWGg1,g2,…,gn=∪himi∈gi1−1−∏i=1n1−1−himiκωi1/κmi=1,2,…,♮gi.


## 4.3. The Properties of AC-HFWA and AC-HFWG

It is seen from above discussion that AC-HFWA and AC-HFWG are functions with respect to the parameter which is from the generatorς. In this section, we will introduce the properties of the AC-HFWA and AC-HFWG operator regarding to the parameter κ.Theorem 12.
Letςt be the generator function of copula, and it takes five cases proposed in Section 4.2; then μAC−HFWAg1,g2,…,gn is an increasing function of κ, μAC−HFWGg1,g2,…,gn is an decreasing function of κ, and μAC−HFWAg1,g2,…,gn≥μAC−HFWGg1,g2,…,gn.Proof.
(Case 1) Whenςt=−lntκ with κ≥1. By Definition 6, we have(49)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−e−∑i=1nωi−ln1−himiκ1/κ,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gne−∑i=1nωi−ln1−himiκ1/κ.
Suppose1≤κ1<κ2; according to reference [10], ∑i=1nωiaiκ1/κ is an increasing function of κ. So,(50)∑i=1nωi−ln1−himiκ11/κ1≤∑i=1nωi−ln1−himiκ21/κ2,∑i=1nωi−lnhimiκ11/κ1≤∑i=1nωi−lnhimiκ21/κ2.
Furthermore,(51)1−e−∑i=1nωi−ln1−himiκ11/κ1≤1−e−∑i=1nωi−ln1−himiκ21/κ2,e−∑i=1nωi−ln1−himiκ11/κ1≥e−∑i=1nωi−ln1−himiκ21/κ2.
Therefore,μAC−HFWAκ1≤μAC−HFWAκ2, μAC−HFWGκ1≥μAC−HFWGκ2. Because κ≥1,(52)e−∑i=1nωi−ln1−himiκ1/κ+e−∑i=1nωi−lnhimiκ1/κ≤e−∑i=1nωi−ln1−himi+e−∑i=1nωi−lnhimi=∏i=1n1−himiωi+∏i=1nhimiωi≤∑i=1nωi1−himi+∑ωihimi=∑i=1nωi=1.
So,1−e−∑i=1nωi−ln1−himiκ1/κ≥e−∑i=1nωi−lnhimiκ1/κ.
That is,μAC−HFWAg1,g2,…,gn≥μAC−HFWGg1,g2,…,gn.
So Theorem12 holds under Case 1.
Case 2. When ςt=t−κ−1,κ>0. Firstly,(53)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−ς−1∑i=1nωiς1−himi,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gnς−1∑i=1kωiςhimi.
Secondly, becauseςκ,t=t−κ−1,κ>0,0<t<1, ∂ς/∂κ=t−κlnt, t−κ>0, lnt<0, and then ∂ς/∂κ<0, ςκ,t is decreasing with respect to κ.
ς−1κ,t=t+1−1/κ,κ>0,0<t<1, so ∂ς−1/∂κ=t+1−1/κlnt+11/κ2, t+1−1/κ>0,lnt+1>0,1/κ2>0, and then ∂ς−1/∂κ>0, ς−1κ,t is decreasing with respect to κ.
Thus,1−ς−1∑i=1nωiς1−himi is decreasing with respect to κ, and ς−1∑i=1kωiςhimi is decreasing with respect to κ. Suppose 1≤κ1<κ2; we have
μAC−HFWAκ1≤μAC−HFWAκ2 and μAC−HFWGκ1≥μAC−HFWGκ2.
Lastly,(54)∑i=1nωi1−himi−κ−1/κ+∑i=1nωihimi−κ−1/κ<limκ⟶0∑i=1nωi1−himi−κ−1/κ+limκ⟶0∑i=1nωihimi−κ−1/κ=e∑i=1nωiln1−himi+e∑i=1nωilnhimi=∏i=1n1−himiωi+∏i=1nhimiωi≤∑i=1nωi1−himi+∑ωihimi=∑i=1nωi=1.
That is, Theorem12 holds under Case 2.
Case 3. When ςt=−lne−κt−1/e−κ−1,κ≠0. Firstly,(55)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−ς−1∑i=1nωiς1−himi,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gnς−1∑i=1kωiςhimi.
Secondly,ςκ,t=−lne−κt−1/e−κ−1,κ≠0,0<t<1.
Ifς=−lnς1, ς1=ς2t−1/ς2−1, ς2=e−κ,κ≠0,0<t<1.
Becauseς is decreasing with respect to ς1, ς1 is decreasing with respect to ς2, and ς2 is decreasing with respect to κ, ςκ,t is decreasing with respect to κ.(56)ς−1κ,t=−1κlne−te−κ−1+1,κ≠0,0<t<1.
Supposeς−1=−lnψ1, ψ1=ψ21/κ, ψ2=e−tψ3+1, ς3=e−κ−1,κ≠0,0<t<1.
Becauseς−1 is decreasing with respect to ψ1, ψ1 is decreasing with respect to ψ2, and ψ2 is decreasing with respect to ψ3, ψ3 is decreasing with respect to κ. Then, ς−1κ,t is decreasing with respect to κ.
Thus,1−ς−1∑i=1nωiς1−himi is decreasing with respect to κ and ς−1∑i=1kωiςhimi is decreasing with respect to κ.
Suppose1≤κ1<κ2.
Therefore,μAC−HFWAκ1≤μAC−HFWAκ2 and μAC−HFWGκ1≥μAC−HFWGκ2.
Lastly, when−∝<κ<+∝,(57)−1κln∏i=1ne−κhimi−1ωi+1≤limκ⟶−∝ln∏i=1ne−κhimi−1ωi+1−κ=limκ⟶−∝∏i=1ne−κhimi−1ωi/∏i=1ne−κhimi−1ωi+1∑i=1nωi−himie−κhimi/e−κhimi−1−1=limκ⟶−∝∏i=1ne−κhimi−1ωi∏i=1ne−κhimi−1ωi+1∑i=1nωihimie−κhimie−κhimi−1=∑i=1nωihimi,−1κln∏i=1ne−κ1−himi−1ωi+1≤∑i=1nωi1−himi.
So,−1/κln∏i=1ne−κ1−himi−1ωi+1−1/kln∏i=1ne−κhimi−1ωi+1≤∑i=1nωi1−himi+∑i=1nωihimi=1.
That is, Theorem12 holds under Case 3.
Case 4. When ςt=ln1−κ1−t/t,−1≤κ<1. Firstly,(58)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−ς−1∑i=1nωiς1−himi,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gnς−1∑i=1kωiςhimi.
Secondly, becauseςκ,t=ln1−κ1−t/t,−1≤κ<1,0<t<1, so ∂ς/∂κ=tt−1/1−κ1−t, tt−1<0.
Supposeς1κ,t=1−κ1−t,−1≤κ<1,0<t<1, ∂ς1/∂κ=t−1<0, ς1κ,t is decreasing with respect to κ.
So,ς1κ,t>ς11,t=t>0; then, ∂ς/∂κ<0, and ςκ,t is decreasing with respect to κ.
Asς−1κ,t=1−κ/et−κ,−1≤κ<1,0<t<1, ∂ς−1/∂κ=et+1/et−κ2, et+1>0, et−κ2>0; then ∂ς−1/∂κ>0, ς−1κ,t is decreasing with respect to κ. Thus, 1−ς−1∑i=1nωiς1−himi is decreasing with respect to κ and ς−1∑i=1kωiςhimi is decreasing with respect to κ.
Suppose1≤κ1<κ2. We have μAC−HFWAκ1≤μAC−HFWAκ2 and μAC−HFWGκ1≥μAC−HFWGκ2.
Lastly, becausey=xωi,ωi∈0,1 is decreasing with respect to κ, 1−hi2ωi≥1, 1−1−hi2ωi≥1; then,(59)∏i=1n1+himiωi+∏i=1n1−himiωi≥2∏i=1n1+himiωi∏i=1n1−himiωi1/2=2∏i=1n1−himi2ωi1/2≥2∏i=1n2−himiωi+∏i=1nhimiωi≥2∏i=1n2−himiωi∏i=1nhimiωi1/2=2∏i=1n1−1−himi2ωi1/2≥2.
When−1≤κ<1, we have(60)∏i=1n1−himiωi−κ∏i=1n1−himiωi∏i=1n1−κhimiωi−κ∏i=1n1−himiωi+1−κ∏i=1nhimiωi∏i=1n1−κ1−himiωi−κ∏i=1nhimiωi≤∏i=1n1−himiωi+∏i=1n1−himiωi∏i=1n1+himiωi+∏i=1n1−himiωi+2∏i=1nhimiωi∏i=1n1+1−himiωi+∏i=1nhimiωi=2∏i=1n1−himiωi∏i=1n1+himiωi+∏i=1n1−himiωi+2∏i=1nhimiωi∏i=1n2−himiωi+∏i=1nhimiωi≤∏i=1n1−himiωi+∏i=1nhimiωi≤∑i=1nωi1−himi+∑ωihimi=∑i=1nωi=1.
So,(61)∏i=1n1−κhimiωi−∏i=1n1−himiωi∏i=1n1−κhimiωi−κ∏i=1n1−himiωi≥1−κ∏i=1nhimiωi∏i=1n1−κ1−himiωi−κ∏i=1nhimiωi.
Therefore,μAC−HFWAg1,g2,…,gn≥μAC−HFWGg1,g2,…,gn. That is, Theorem 12 holds under Case 4.
Case 5. When ςt=−ln1−1−tκ,κ≥1.
Firstly,(62)μAC−HFWAκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gn1−ς−1∑i=1nωiς1−himi,μAC−HFWGκ=1♮g1♮g2⋯♮gn∑m1=1♮g1∑m2=1♮g2⋯∑mn=1♮gnς−1∑i=1kωiςhimi.
Secondly, becauseςκ,t=−ln1−1−tκ,κ≥1,0<t<1, ∂ς/∂κ=1−tκln1−t/1−1−tκ, ln1−t<0.
Supposeς1κ,t=1−tκ,κ≥1,0<t<1, ∂ς1/∂κ=1−tκln1−t<0, ς1κ,t is decreasing with respect to κ, so 0<ς1κ,t≤ς11,t=1−t<1, 1−1−tκ>0.
Then,∂ς/∂κ<0, ςκ,t is decreasing with respect to κ.
ς−1κ,t=1−1−e−t1/κ,κ≥1,0<t<1, so ∂ς−1/∂κ=1−e−t1/κln1−e−t−1/κ2, 1−e−t1/κ>0, ln1−e−t<0, −1/κ2<0; then, ∂ς−1/∂κ>0, ς−1κ,t is decreasing with respect to κ.
Thus,1−ς−1∑i=1nωiς1−himi is decreasing with respect to κ and ς−1∑i=1nωiςhimi is decreasing with respect to κ.
Lastly, suppose1≤κ1<κ2, Therefore, μAC−HFWAκ1≤μAC−HFWAκ2, μAC−HFWGκ1≥μAC−HFWGκ2.
Becauseκ≥1,(63)∏i=1n1−himiκωi1/κ+1−1−∏i=1n1−1−himiκωi1/κ≤∏i=1n1−himi1ωi1+1−1−∏i=1n1−1−himi1ωi1=∏i=1n1−himiωi+∏i=1nhimiωi≤∑i=1nωi1−himi+∑ωihimi=∑i=1nωi=1.
So,1−e−∑i=1nωi−ln1−himiκ1/κ≥e−∑i=1nωi−lnhimiκ1/κ, μAC−HFWAg1,g2,…,gn≥μAC−HFWGg1,g2,…,gn.
That is, Theorem12 holds under Case 5.


## 5. MADM Approach Based on AC-HFWA and AC-HFWG

From the above analysis, a novel decision making way will be given to address MADM problems under HF environment based on the proposed operators.LetY=Ψii=1,2,…,m be a collection of alternatives and C=Gjj=1,2,…,n be the set of attributes, whose weight vector is W=ω1,ω2,…,ωnT, satisfying ωj∈0,1 and ∑j=1nωj=1. 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 that case, if two DMs give the same value, then the value emerges only once in hij. In what follows, the specific algorithm for MADM problems under HF environment will be designed.Step 1. The DMs provide their evaluations about the alternative ϒi under the attribute Gj, denoted by the HFEs hij,i=1,2,…,m;j=1,2,…,n.Step 2. Use the proposed AC-HFWA (AC-HFWG) to fuse the HFE yii=1,2,…,m for ϒii=1,2,…,m.Step 3. The score values μyii=1,2,…,m of yi are calculated using Definition 6 and compared.Step 4. The order of the attributes ϒii=1,2,…,m is given by the size of μyii=1,2,…,m.


### 5.1. Illustrative Example

An example of stock investment will be given to illustrate the application of the proposed method. Assume that there are four short-term stocksϒ1,ϒ2,ϒ3,ϒ4. The following four attributes G1,G2,G3,G4 should be considered: G1—net value of each share; G2—earnings per share; G3—capital reserve per share; and G4—accounts receivable. Assume the weight vector of each attributes is W=0.2,0.3,0.15,0.35T.Next, we will use the developed method to find the ranking of the alternatives and the optimal choice.Step 1. The decision matrix is given by DM and is shown in Table 1.Step 2. Use the AC-HFWA operator to fuse the HEE yii=1,…,4 for ϒii=1,…,4.Takeϒ4 as an example, and let ςt=−lnt1.2 in Case 1; we have(64)y4=AC−HFWAg1,g2,g3,g4=⊕j=14ωjgj=∪hjmj∈gj1−e−∑j=14ωi1−hjmj1.25/6mj=1,2,…,♮gj=0.5870,0.6139,0.6021,0.6278,0.6229,0.6470,0.7190,0.7361,0.7286,0.6117,0.6367,0.6257,0.6496,0.6450,0.6675,0.7347,0.7508,0.7437,0.6303,0.6538,0.6435,0.6660,0.6617,0.6829,0.7466,0.7618,0.7551,0.7451,0.7419,0.7574,0.7591,0.7561,0.7707,0.7697,0.7669,0.7808.Step 3. Compute the score values μyii=1,…,4 of yii=1,…,4 by Definition 6. μy4=0.6944. Similarly, μy1=0.5761,μy2=0.6322,μy3=0.5149.Step 4. Rank the alternatives ϒii=1,…,4 and select the desirable one in terms of comparison rules. Since μy4>μy2>μy1>μy3, we can obtain the rank of alternatives as ϒ4≻ϒ2≻ϒ1≻ϒ3, and Υ4 is the best alternative.Table 1
HF decision matrix [13].
AlternativesG1G2G3G4ϒ10.2,0.4,0.70.2,0.6,0.80.2,0.3,0.6,0.7,0.90.3,0.4,0.5,0.7,0.8ϒ20.2,0.4,0.7,0.90.1,0.2,0.4,0.50.3,0.4,0.6,0.90.5,0.6,0.8,0.9ϒ30.3,0.5,0.6,0.70.2,0.4,0.5,0.60.3,0.5,0.7,0.80.2,0.5,0.6,0.7ϒ40.3,0.5,0.60.2,0.40.5,0.6,0.70.8,0.9


### 5.2. Sensitivity Analysis

#### 5.2.1. The Effect of Parameters on the Results

We take Case 1 as an example to analyse the effect of parameter changes on the result. The results are shown in Table2 and Figure 1.Table 2
The ordering results obtained by AC-HFWA in Case 1.
Parameterκμϒ1μϒ2μϒ3μϒ4Ranking order10.56120.60090.51780.6524ϒ4≻ϒ2≻ϒ1≻ϒ31.20.57200.61600.52490.6678ϒ4≻ϒ2≻ϒ1≻ϒ320.60840.66210.54980.7156ϒ4≻ϒ2≻ϒ1≻ϒ32.50.62580.68230.56260.7365ϒ4≻ϒ2≻ϒ1≻ϒ330.63990.69810.57360.7525ϒ4≻ϒ2≻ϒ1≻ϒ350.67570.73560.60450.7893ϒ4≻ϒ2≻ϒ1≻ϒ3Figure 1
The score values obtained by AC-HFWA in Case 1.It is easy to see from Table2 that the scores of alternatives by the AC-HFWA operator increase as the value of the parameter κ ranges from 1 to 5. This is consistent with Theorem 12. But the order has not changed, and the best alternative is consistent.


#### 5.2.2. The Effect of Different Types of Generators on the Results in AC-HFWA

The ordering results of alternatives using other generators proposed in present work are listed in Table3. Figure 2 shows more intuitively how the score function varies with parameters.Table 3
The ordering results obtained by AC-HFWA in Cases 2–5.
FunctionsParameterλμϒ1μϒ2μϒ3μϒ4Ranking orderCase 210.60590.66810.54250.7256ϒ4≻ϒ2≻ϒ1≻ϒ320.64110.71100.56560.7694ϒ4≻ϒ2≻ϒ1≻ϒ330.66500.73510.58510.7932ϒ4≻ϒ2≻ϒ1≻ϒ3Case 310.57100.61430.52410.6665ϒ4≻ϒ2≻ϒ1≻ϒ320.58150.62780.53110.6806ϒ4≻ϒ2≻ϒ1≻ϒ330.59220.64100.53860.6943ϒ4≻ϒ2≻ϒ1≻ϒ3Case 40.250.56660.60840.52120.6604ϒ4≻ϒ2≻ϒ1≻ϒ30.50.57390.61880.52560.6717ϒ4≻ϒ2≻ϒ1≻ϒ30.750.58490.63490.53200.6894ϒ4≻ϒ2≻ϒ1≻ϒ3Case 510.56120.60090.51780.6524ϒ4≻ϒ2≻ϒ1≻ϒ320.58300.62790.53370.6781ϒ4≻ϒ2≻ϒ1≻ϒ330.60220.64940.54840.6996ϒ4≻ϒ2≻ϒ1≻ϒ3Figure 2
The score functions obtained by AC-HFWA.
(a)(b)(c)(d)(e)The results show that the order of alternatives obtained by applying different generators and parameters is different, the ranking order of alternatives is almost the same, and the desirable one is alwaysϒ4, which indicates that AC-HFWA operator is highly stable.


#### 5.2.3. The Effect of Different Types of Generators on the Results in AC-HFWG

In step 2, if we utilize the AC-HFWG operator instead of the AC-HFWA operator to calculate the values of the alternatives, the results are shown in Table4.Table 4
The ordering results obtained by AC-HFWG.
FunctionsParameterλμϒ1μϒ2μϒ3μϒ4Ranking orderCase 11.10.46870.45410.46270.5042ϒ4≻ϒ1≻ϒ3≻ϒ220.42470.39700.43580.4437ϒ4≻ϒ3≻ϒ1≻ϒ280.32490.29640.36050.3365ϒ3≻ϒ4≻ϒ1≻ϒ2Case 210.43190.40100.43890.4573ϒ4≻ϒ3≻ϒ1≻ϒ220.39620.36010.41590.4147ϒ3≻ϒ4≻ϒ1≻ϒ230.36990.33460.39820.3858ϒ3≻ϒ4≻ϒ1≻ϒ2Case 310.46420.44820.45990.5257ϒ4≻ϒ1≻ϒ3≻ϒ230.44170.41900.44620.5212ϒ4≻ϒ3≻ϒ1≻ϒ280.39040.36280.41240.4036ϒ3≻ϒ4≻ϒ1≻ϒ2Case 4−0.90.48660.48030.47320.5317ϒ4≻ϒ1≻ϒ2≻ϒ30.250.46890.45470.46270.5051ϒ4≻ϒ1≻ϒ3≻ϒ20.750.45070.42900.45110.4804ϒ4≻ϒ3≻ϒ1≻ϒ2Case 510.47440.46250.46610.5210ϒ4≻ϒ1≻ϒ3≻ϒ220.45030.42950.45260.5157ϒ4≻ϒ3≻ϒ1≻ϒ280.36450.33990.39220.3750ϒ3≻ϒ4≻ϒ1≻ϒ2Figure3 shows the variation of the values with parameter κ. We can find that the ranking of the alternatives may change when κ changes in the AC-HFWG operator. With the increase in κ, the ranking results change from ϒ4≻ϒ1≻ϒ3≻ϒ2 to ϒ4≻ϒ3≻ϒ1≻ϒ2 and finally settle at ϒ3≻ϒ4≻ϒ1≻ϒ2, which indicates that AC-HFWG has certain stability. In the decision process, DMs can confirm the value of κ in accordance with their preferences.Figure 3
The score functions obtained by AC-HFWG.
(a)(b)(c)(d)(e)Through the analysis of Tables3 and 4, we can find that the score values calculated by the AC-HFWG operator decrease with the increase of parameter κ. For the same generator and parameter κ, the values obtained by the AC-HFWA operator are always greater than those obtained by the AC-HFWG operator, which is consistent with Theorem 12.


#### 5.2.4. The Effect of Parameterθ on the Results in AC-GHFWA and AC-GHFWG

We employ the AC-GHFWA operator and AC-GHFWG operator to aggregate the values of the alternatives, taking Case 1 as an example. The results are listed in Tables5 and 6. Figures 4 and 5 show the variation of the score function with κ and θ.Table 5
The ordering results obtained by AC-GHFWA in Case 1.
ParameterκParameterθμϒ1μϒ2μϒ3μϒ4Ranking order110.56120.60090.51780.6524ϒ4≻ϒ2≻ϒ1≻ϒ350.63170.68060.57220.7313ϒ4≻ϒ2≻ϒ1≻ϒ3100.67010.72360.60790.7745ϒ4≻ϒ2≻ϒ1≻ϒ3510.67570.73560.60450.7893ϒ4≻ϒ2≻ϒ1≻ϒ350.68520.74420.61420.7972ϒ4≻ϒ2≻ϒ1≻ϒ3100.68920.74790.61860.8005ϒ4≻ϒ2≻ϒ1≻ϒ31010.70810.76880.63860.8199ϒ4≻ϒ2≻ϒ1≻ϒ350.71120.77120.64200.8219ϒ4≻ϒ2≻ϒ1≻ϒ3100.71260.77230.64350.8227ϒ4≻ϒ2≻ϒ1≻ϒ3Table 6
The ordering results obtained by AC-GHFWG in Case 1.
ParameterκParameterθμϒ1μϒ2μϒ3μϒ4Ranking order110.47440.46250.46610.5131ϒ4≻ϒ1≻ϒ3≻ϒ250.39620.37060.41710.4083ϒ3≻ϒ4≻ϒ1≻ϒ2100.35620.32620.38080.3607ϒ3≻ϒ4≻ϒ1≻ϒ2510.35230.32220.38360.3636ϒ3≻ϒ4≻ϒ1≻ϒ250.34170.31240.37460.3527ϒ3≻ϒ4≻ϒ1≻ϒ2100.33720.30830.37060.3481ϒ3≻ϒ4≻ϒ1≻ϒ21010.32000.28700.35170.3266ϒ3≻ϒ4≻ϒ1≻ϒ250.31650.28410.34840.3234ϒ3≻ϒ4≻ϒ1≻ϒ2100.31500.28290.34700.3220ϒ3≻ϒ4≻ϒ1≻ϒ2Figure 4
The score functions obtained by AC-GHFWA in Case 1.
(a)(b)(c)(d)Figure 5
The score functions obtained by AC-GHFWG in Case 1.
(a)(b)(c)(d)The results indicate that the scores of alternatives by the AC-GHFWA operator increase as the parameterκ goes from 0 to 10 and the parameter θ ranges from 0 to 10, and the ranking of alternatives has not changed. But the score values of alternatives reduce, and the order of alternatives obtained by the AC-GHFWG operator changes significantly.


### 5.3. Comparisons with Existing Approach

The above results indicate the effectiveness of our method, which can solve MADM problems. To further prove the validity of our method, this section compares the existing methods with our proposed method. The previous methods include hesitant fuzzy weighted averaging (HFWA) operator (or HFWG operator) by Xia and Xu [13], hesitant fuzzy Bonferroni means (HFBM) operator by Zhu and Xu [32], and hesitant fuzzy Frank weighted average (HFFWA) operator (or HFFWG operator) by Qin et al. [22]. Using the data in reference [22] and different operators, Table 7 is obtained. The order of these operators is A6≻A2≻A3≻A5≻A1≻A4, except for the HFBM operator which is A6≻A2≻A3≻A1≻A5≻A4.(1)
From the previous argument, it is obvious to find that Xia’s operators [13] and Qin’s operators [22] are special cases of our operators. When κ=1 in Case 1 of our proposed operators, AC-HFWA reduces to HFWA and AC-HFWG reduces to HFWG. When κ=1 in Case 3 of our proposed operators, AC-HFWA becomes HFFWA and AC-HFWG becomes HFFWG. The operational rules of Xia’s method are algebraic t-norm and algebraic t-conorm, and the operational rules of Qin’s method are Frank t-norm and Frank t-conorm, which are all special forms of copula and cocopula. Therefore, our proposed operators are more general. Furthermore, the proposed approach will supply more choice for DMs in real MADM problems.(2)
The results of Zhu’s operator [32] are basically the same as ours, but Zhu’s calculations were more complicated. Although we all introduce parameters, which can resize the aggregate value on account of actual decisions, the regularity of our parameter changes is stronger than Zhu’s method. The AC-HFWA operator has an ideal property of monotonically increasing with respect to the parameter, and the AC-HFWG operator has an ideal property of monotonically decreasing with respect to the parameter, which provides a basis for DMs to select appropriate values according to their risk appetite. If the DM is risk preference, we can choose the largest parameter possible, and if the DM is risk aversion, we can choose the smallest parameter possible.Table 7
The scores obtained by different operators.
OperatorParameterμA1μA2μA3μA4μA5μA6HFWA [13]None0.47220.60980.59880.43700.54790.6969HFWG [13]None0.40330.50640.48960.33540.46110.6472HFBM [32]p=2,q=10.53720.57580.55760.39730.52750.7072HFFWA [22]λ=20.36910.53220.47820.37590.47080.6031HFFWG [22]λ=20.54190.63350.62320.47120.60290.7475AC-HFWA (Case 2)λ=10.50690.66470.59850.49160.58880.7274AC-HFWG (Case 2)λ=10.37060.46160.45740.28730.41580.6308


## 5.1. Illustrative Example

An example of stock investment will be given to illustrate the application of the proposed method. Assume that there are four short-term stocksϒ1,ϒ2,ϒ3,ϒ4. The following four attributes G1,G2,G3,G4 should be considered: G1—net value of each share; G2—earnings per share; G3—capital reserve per share; and G4—accounts receivable. Assume the weight vector of each attributes is W=0.2,0.3,0.15,0.35T.Next, we will use the developed method to find the ranking of the alternatives and the optimal choice.Step 1. The decision matrix is given by DM and is shown in Table 1.Step 2. Use the AC-HFWA operator to fuse the HEE yii=1,…,4 for ϒii=1,…,4.Takeϒ4 as an example, and let ςt=−lnt1.2 in Case 1; we have(64)y4=AC−HFWAg1,g2,g3,g4=⊕j=14ωjgj=∪hjmj∈gj1−e−∑j=14ωi1−hjmj1.25/6mj=1,2,…,♮gj=0.5870,0.6139,0.6021,0.6278,0.6229,0.6470,0.7190,0.7361,0.7286,0.6117,0.6367,0.6257,0.6496,0.6450,0.6675,0.7347,0.7508,0.7437,0.6303,0.6538,0.6435,0.6660,0.6617,0.6829,0.7466,0.7618,0.7551,0.7451,0.7419,0.7574,0.7591,0.7561,0.7707,0.7697,0.7669,0.7808.Step 3. Compute the score values μyii=1,…,4 of yii=1,…,4 by Definition 6. μy4=0.6944. Similarly, μy1=0.5761,μy2=0.6322,μy3=0.5149.Step 4. Rank the alternatives ϒii=1,…,4 and select the desirable one in terms of comparison rules. Since μy4>μy2>μy1>μy3, we can obtain the rank of alternatives as ϒ4≻ϒ2≻ϒ1≻ϒ3, and Υ4 is the best alternative.Table 1
HF decision matrix [13].
AlternativesG1G2G3G4ϒ10.2,0.4,0.70.2,0.6,0.80.2,0.3,0.6,0.7,0.90.3,0.4,0.5,0.7,0.8ϒ20.2,0.4,0.7,0.90.1,0.2,0.4,0.50.3,0.4,0.6,0.90.5,0.6,0.8,0.9ϒ30.3,0.5,0.6,0.70.2,0.4,0.5,0.60.3,0.5,0.7,0.80.2,0.5,0.6,0.7ϒ40.3,0.5,0.60.2,0.40.5,0.6,0.70.8,0.9


## 5.2. Sensitivity Analysis

### 5.2.1. The Effect of Parameters on the Results

We take Case 1 as an example to analyse the effect of parameter changes on the result. The results are shown in Table2 and Figure 1.Table 2
The ordering results obtained by AC-HFWA in Case 1.
Parameterκμϒ1μϒ2μϒ3μϒ4Ranking order10.56120.60090.51780.6524ϒ4≻ϒ2≻ϒ1≻ϒ31.20.57200.61600.52490.6678ϒ4≻ϒ2≻ϒ1≻ϒ320.60840.66210.54980.7156ϒ4≻ϒ2≻ϒ1≻ϒ32.50.62580.68230.56260.7365ϒ4≻ϒ2≻ϒ1≻ϒ330.63990.69810.57360.7525ϒ4≻ϒ2≻ϒ1≻ϒ350.67570.73560.60450.7893ϒ4≻ϒ2≻ϒ1≻ϒ3Figure 1
The score values obtained by AC-HFWA in Case 1.It is easy to see from Table2 that the scores of alternatives by the AC-HFWA operator increase as the value of the parameter κ ranges from 1 to 5. This is consistent with Theorem 12. But the order has not changed, and the best alternative is consistent.


### 5.2.2. The Effect of Different Types of Generators on the Results in AC-HFWA

The ordering results of alternatives using other generators proposed in present work are listed in Table3. Figure 2 shows more intuitively how the score function varies with parameters.Table 3
The ordering results obtained by AC-HFWA in Cases 2–5.
FunctionsParameterλμϒ1μϒ2μϒ3μϒ4Ranking orderCase 210.60590.66810.54250.7256ϒ4≻ϒ2≻ϒ1≻ϒ320.64110.71100.56560.7694ϒ4≻ϒ2≻ϒ1≻ϒ330.66500.73510.58510.7932ϒ4≻ϒ2≻ϒ1≻ϒ3Case 310.57100.61430.52410.6665ϒ4≻ϒ2≻ϒ1≻ϒ320.58150.62780.53110.6806ϒ4≻ϒ2≻ϒ1≻ϒ330.59220.64100.53860.6943ϒ4≻ϒ2≻ϒ1≻ϒ3Case 40.250.56660.60840.52120.6604ϒ4≻ϒ2≻ϒ1≻ϒ30.50.57390.61880.52560.6717ϒ4≻ϒ2≻ϒ1≻ϒ30.750.58490.63490.53200.6894ϒ4≻ϒ2≻ϒ1≻ϒ3Case 510.56120.60090.51780.6524ϒ4≻ϒ2≻ϒ1≻ϒ320.58300.62790.53370.6781ϒ4≻ϒ2≻ϒ1≻ϒ330.60220.64940.54840.6996ϒ4≻ϒ2≻ϒ1≻ϒ3Figure 2
The score functions obtained by AC-HFWA.
(a)(b)(c)(d)(e)The results show that the order of alternatives obtained by applying different generators and parameters is different, the ranking order of alternatives is almost the same, and the desirable one is alwaysϒ4, which indicates that AC-HFWA operator is highly stable.


### 5.2.3. The Effect of Different Types of Generators on the Results in AC-HFWG

In step 2, if we utilize the AC-HFWG operator instead of the AC-HFWA operator to calculate the values of the alternatives, the results are shown in Table4.Table 4
The ordering results obtained by AC-HFWG.
FunctionsParameterλμϒ1μϒ2μϒ3μϒ4Ranking orderCase 11.10.46870.45410.46270.5042ϒ4≻ϒ1≻ϒ3≻ϒ220.42470.39700.43580.4437ϒ4≻ϒ3≻ϒ1≻ϒ280.32490.29640.36050.3365ϒ3≻ϒ4≻ϒ1≻ϒ2Case 210.43190.40100.43890.4573ϒ4≻ϒ3≻ϒ1≻ϒ220.39620.36010.41590.4147ϒ3≻ϒ4≻ϒ1≻ϒ230.36990.33460.39820.3858ϒ3≻ϒ4≻ϒ1≻ϒ2Case 310.46420.44820.45990.5257ϒ4≻ϒ1≻ϒ3≻ϒ230.44170.41900.44620.5212ϒ4≻ϒ3≻ϒ1≻ϒ280.39040.36280.41240.4036ϒ3≻ϒ4≻ϒ1≻ϒ2Case 4−0.90.48660.48030.47320.5317ϒ4≻ϒ1≻ϒ2≻ϒ30.250.46890.45470.46270.5051ϒ4≻ϒ1≻ϒ3≻ϒ20.750.45070.42900.45110.4804ϒ4≻ϒ3≻ϒ1≻ϒ2Case 510.47440.46250.46610.5210ϒ4≻ϒ1≻ϒ3≻ϒ220.45030.42950.45260.5157ϒ4≻ϒ3≻ϒ1≻ϒ280.36450.33990.39220.3750ϒ3≻ϒ4≻ϒ1≻ϒ2Figure3 shows the variation of the values with parameter κ. We can find that the ranking of the alternatives may change when κ changes in the AC-HFWG operator. With the increase in κ, the ranking results change from ϒ4≻ϒ1≻ϒ3≻ϒ2 to ϒ4≻ϒ3≻ϒ1≻ϒ2 and finally settle at ϒ3≻ϒ4≻ϒ1≻ϒ2, which indicates that AC-HFWG has certain stability. In the decision process, DMs can confirm the value of κ in accordance with their preferences.Figure 3
The score functions obtained by AC-HFWG.
(a)(b)(c)(d)(e)Through the analysis of Tables3 and 4, we can find that the score values calculated by the AC-HFWG operator decrease with the increase of parameter κ. For the same generator and parameter κ, the values obtained by the AC-HFWA operator are always greater than those obtained by the AC-HFWG operator, which is consistent with Theorem 12.


### 5.2.4. The Effect of Parameterθ on the Results in AC-GHFWA and AC-GHFWG

We employ the AC-GHFWA operator and AC-GHFWG operator to aggregate the values of the alternatives, taking Case 1 as an example. The results are listed in Tables5 and 6. Figures 4 and 5 show the variation of the score function with κ and θ.Table 5
The ordering results obtained by AC-GHFWA in Case 1.
ParameterκParameterθμϒ1μϒ2μϒ3μϒ4Ranking order110.56120.60090.51780.6524ϒ4≻ϒ2≻ϒ1≻ϒ350.63170.68060.57220.7313ϒ4≻ϒ2≻ϒ1≻ϒ3100.67010.72360.60790.7745ϒ4≻ϒ2≻ϒ1≻ϒ3510.67570.73560.60450.7893ϒ4≻ϒ2≻ϒ1≻ϒ350.68520.74420.61420.7972ϒ4≻ϒ2≻ϒ1≻ϒ3100.68920.74790.61860.8005ϒ4≻ϒ2≻ϒ1≻ϒ31010.70810.76880.63860.8199ϒ4≻ϒ2≻ϒ1≻ϒ350.71120.77120.64200.8219ϒ4≻ϒ2≻ϒ1≻ϒ3100.71260.77230.64350.8227ϒ4≻ϒ2≻ϒ1≻ϒ3Table 6
The ordering results obtained by AC-GHFWG in Case 1.
ParameterκParameterθμϒ1μϒ2μϒ3μϒ4Ranking order110.47440.46250.46610.5131ϒ4≻ϒ1≻ϒ3≻ϒ250.39620.37060.41710.4083ϒ3≻ϒ4≻ϒ1≻ϒ2100.35620.32620.38080.3607ϒ3≻ϒ4≻ϒ1≻ϒ2510.35230.32220.38360.3636ϒ3≻ϒ4≻ϒ1≻ϒ250.34170.31240.37460.3527ϒ3≻ϒ4≻ϒ1≻ϒ2100.33720.30830.37060.3481ϒ3≻ϒ4≻ϒ1≻ϒ21010.32000.28700.35170.3266ϒ3≻ϒ4≻ϒ1≻ϒ250.31650.28410.34840.3234ϒ3≻ϒ4≻ϒ1≻ϒ2100.31500.28290.34700.3220ϒ3≻ϒ4≻ϒ1≻ϒ2Figure 4
The score functions obtained by AC-GHFWA in Case 1.
(a)(b)(c)(d)Figure 5
The score functions obtained by AC-GHFWG in Case 1.
(a)(b)(c)(d)The results indicate that the scores of alternatives by the AC-GHFWA operator increase as the parameterκ goes from 0 to 10 and the parameter θ ranges from 0 to 10, and the ranking of alternatives has not changed. But the score values of alternatives reduce, and the order of alternatives obtained by the AC-GHFWG operator changes significantly.


## 5.2.1. The Effect of Parameters on the Results

We take Case 1 as an example to analyse the effect of parameter changes on the result. The results are shown in Table2 and Figure 1.Table 2
The ordering results obtained by AC-HFWA in Case 1.
Parameterκμϒ1μϒ2μϒ3μϒ4Ranking order10.56120.60090.51780.6524ϒ4≻ϒ2≻ϒ1≻ϒ31.20.57200.61600.52490.6678ϒ4≻ϒ2≻ϒ1≻ϒ320.60840.66210.54980.7156ϒ4≻ϒ2≻ϒ1≻ϒ32.50.62580.68230.56260.7365ϒ4≻ϒ2≻ϒ1≻ϒ330.63990.69810.57360.7525ϒ4≻ϒ2≻ϒ1≻ϒ350.67570.73560.60450.7893ϒ4≻ϒ2≻ϒ1≻ϒ3Figure 1
The score values obtained by AC-HFWA in Case 1.It is easy to see from Table2 that the scores of alternatives by the AC-HFWA operator increase as the value of the parameter κ ranges from 1 to 5. This is consistent with Theorem 12. But the order has not changed, and the best alternative is consistent.


## 5.2.2. The Effect of Different Types of Generators on the Results in AC-HFWA

The ordering results of alternatives using other generators proposed in present work are listed in Table3. Figure 2 shows more intuitively how the score function varies with parameters.Table 3
The ordering results obtained by AC-HFWA in Cases 2–5.
FunctionsParameterλμϒ1μϒ2μϒ3μϒ4Ranking orderCase 210.60590.66810.54250.7256ϒ4≻ϒ2≻ϒ1≻ϒ320.64110.71100.56560.7694ϒ4≻ϒ2≻ϒ1≻ϒ330.66500.73510.58510.7932ϒ4≻ϒ2≻ϒ1≻ϒ3Case 310.57100.61430.52410.6665ϒ4≻ϒ2≻ϒ1≻ϒ320.58150.62780.53110.6806ϒ4≻ϒ2≻ϒ1≻ϒ330.59220.64100.53860.6943ϒ4≻ϒ2≻ϒ1≻ϒ3Case 40.250.56660.60840.52120.6604ϒ4≻ϒ2≻ϒ1≻ϒ30.50.57390.61880.52560.6717ϒ4≻ϒ2≻ϒ1≻ϒ30.750.58490.63490.53200.6894ϒ4≻ϒ2≻ϒ1≻ϒ3Case 510.56120.60090.51780.6524ϒ4≻ϒ2≻ϒ1≻ϒ320.58300.62790.53370.6781ϒ4≻ϒ2≻ϒ1≻ϒ330.60220.64940.54840.6996ϒ4≻ϒ2≻ϒ1≻ϒ3Figure 2
The score functions obtained by AC-HFWA.
(a)(b)(c)(d)(e)The results show that the order of alternatives obtained by applying different generators and parameters is different, the ranking order of alternatives is almost the same, and the desirable one is alwaysϒ4, which indicates that AC-HFWA operator is highly stable.


## 5.2.3. The Effect of Different Types of Generators on the Results in AC-HFWG

In step 2, if we utilize the AC-HFWG operator instead of the AC-HFWA operator to calculate the values of the alternatives, the results are shown in Table4.Table 4
The ordering results obtained by AC-HFWG.
FunctionsParameterλμϒ1μϒ2μϒ3μϒ4Ranking orderCase 11.10.46870.45410.46270.5042ϒ4≻ϒ1≻ϒ3≻ϒ220.42470.39700.43580.4437ϒ4≻ϒ3≻ϒ1≻ϒ280.32490.29640.36050.3365ϒ3≻ϒ4≻ϒ1≻ϒ2Case 210.43190.40100.43890.4573ϒ4≻ϒ3≻ϒ1≻ϒ220.39620.36010.41590.4147ϒ3≻ϒ4≻ϒ1≻ϒ230.36990.33460.39820.3858ϒ3≻ϒ4≻ϒ1≻ϒ2Case 310.46420.44820.45990.5257ϒ4≻ϒ1≻ϒ3≻ϒ230.44170.41900.44620.5212ϒ4≻ϒ3≻ϒ1≻ϒ280.39040.36280.41240.4036ϒ3≻ϒ4≻ϒ1≻ϒ2Case 4−0.90.48660.48030.47320.5317ϒ4≻ϒ1≻ϒ2≻ϒ30.250.46890.45470.46270.5051ϒ4≻ϒ1≻ϒ3≻ϒ20.750.45070.42900.45110.4804ϒ4≻ϒ3≻ϒ1≻ϒ2Case 510.47440.46250.46610.5210ϒ4≻ϒ1≻ϒ3≻ϒ220.45030.42950.45260.5157ϒ4≻ϒ3≻ϒ1≻ϒ280.36450.33990.39220.3750ϒ3≻ϒ4≻ϒ1≻ϒ2Figure3 shows the variation of the values with parameter κ. We can find that the ranking of the alternatives may change when κ changes in the AC-HFWG operator. With the increase in κ, the ranking results change from ϒ4≻ϒ1≻ϒ3≻ϒ2 to ϒ4≻ϒ3≻ϒ1≻ϒ2 and finally settle at ϒ3≻ϒ4≻ϒ1≻ϒ2, which indicates that AC-HFWG has certain stability. In the decision process, DMs can confirm the value of κ in accordance with their preferences.Figure 3
The score functions obtained by AC-HFWG.
(a)(b)(c)(d)(e)Through the analysis of Tables3 and 4, we can find that the score values calculated by the AC-HFWG operator decrease with the increase of parameter κ. For the same generator and parameter κ, the values obtained by the AC-HFWA operator are always greater than those obtained by the AC-HFWG operator, which is consistent with Theorem 12.


## 5.2.4. The Effect of Parameterθ on the Results in AC-GHFWA and AC-GHFWG

We employ the AC-GHFWA operator and AC-GHFWG operator to aggregate the values of the alternatives, taking Case 1 as an example. The results are listed in Tables5 and 6. Figures 4 and 5 show the variation of the score function with κ and θ.Table 5
The ordering results obtained by AC-GHFWA in Case 1.
ParameterκParameterθμϒ1μϒ2μϒ3μϒ4Ranking order110.56120.60090.51780.6524ϒ4≻ϒ2≻ϒ1≻ϒ350.63170.68060.57220.7313ϒ4≻ϒ2≻ϒ1≻ϒ3100.67010.72360.60790.7745ϒ4≻ϒ2≻ϒ1≻ϒ3510.67570.73560.60450.7893ϒ4≻ϒ2≻ϒ1≻ϒ350.68520.74420.61420.7972ϒ4≻ϒ2≻ϒ1≻ϒ3100.68920.74790.61860.8005ϒ4≻ϒ2≻ϒ1≻ϒ31010.70810.76880.63860.8199ϒ4≻ϒ2≻ϒ1≻ϒ350.71120.77120.64200.8219ϒ4≻ϒ2≻ϒ1≻ϒ3100.71260.77230.64350.8227ϒ4≻ϒ2≻ϒ1≻ϒ3Table 6
The ordering results obtained by AC-GHFWG in Case 1.
ParameterκParameterθμϒ1μϒ2μϒ3μϒ4Ranking order110.47440.46250.46610.5131ϒ4≻ϒ1≻ϒ3≻ϒ250.39620.37060.41710.4083ϒ3≻ϒ4≻ϒ1≻ϒ2100.35620.32620.38080.3607ϒ3≻ϒ4≻ϒ1≻ϒ2510.35230.32220.38360.3636ϒ3≻ϒ4≻ϒ1≻ϒ250.34170.31240.37460.3527ϒ3≻ϒ4≻ϒ1≻ϒ2100.33720.30830.37060.3481ϒ3≻ϒ4≻ϒ1≻ϒ21010.32000.28700.35170.3266ϒ3≻ϒ4≻ϒ1≻ϒ250.31650.28410.34840.3234ϒ3≻ϒ4≻ϒ1≻ϒ2100.31500.28290.34700.3220ϒ3≻ϒ4≻ϒ1≻ϒ2Figure 4
The score functions obtained by AC-GHFWA in Case 1.
(a)(b)(c)(d)Figure 5
The score functions obtained by AC-GHFWG in Case 1.
(a)(b)(c)(d)The results indicate that the scores of alternatives by the AC-GHFWA operator increase as the parameterκ goes from 0 to 10 and the parameter θ ranges from 0 to 10, and the ranking of alternatives has not changed. But the score values of alternatives reduce, and the order of alternatives obtained by the AC-GHFWG operator changes significantly.


## 5.3. Comparisons with Existing Approach

The above results indicate the effectiveness of our method, which can solve MADM problems. To further prove the validity of our method, this section compares the existing methods with our proposed method. The previous methods include hesitant fuzzy weighted averaging (HFWA) operator (or HFWG operator) by Xia and Xu [13], hesitant fuzzy Bonferroni means (HFBM) operator by Zhu and Xu [32], and hesitant fuzzy Frank weighted average (HFFWA) operator (or HFFWG operator) by Qin et al. [22]. Using the data in reference [22] and different operators, Table 7 is obtained. The order of these operators is A6≻A2≻A3≻A5≻A1≻A4, except for the HFBM operator which is A6≻A2≻A3≻A1≻A5≻A4.(1)
From the previous argument, it is obvious to find that Xia’s operators [13] and Qin’s operators [22] are special cases of our operators. When κ=1 in Case 1 of our proposed operators, AC-HFWA reduces to HFWA and AC-HFWG reduces to HFWG. When κ=1 in Case 3 of our proposed operators, AC-HFWA becomes HFFWA and AC-HFWG becomes HFFWG. The operational rules of Xia’s method are algebraic t-norm and algebraic t-conorm, and the operational rules of Qin’s method are Frank t-norm and Frank t-conorm, which are all special forms of copula and cocopula. Therefore, our proposed operators are more general. Furthermore, the proposed approach will supply more choice for DMs in real MADM problems.(2)
The results of Zhu’s operator [32] are basically the same as ours, but Zhu’s calculations were more complicated. Although we all introduce parameters, which can resize the aggregate value on account of actual decisions, the regularity of our parameter changes is stronger than Zhu’s method. The AC-HFWA operator has an ideal property of monotonically increasing with respect to the parameter, and the AC-HFWG operator has an ideal property of monotonically decreasing with respect to the parameter, which provides a basis for DMs to select appropriate values according to their risk appetite. If the DM is risk preference, we can choose the largest parameter possible, and if the DM is risk aversion, we can choose the smallest parameter possible.Table 7
The scores obtained by different operators.
OperatorParameterμA1μA2μA3μA4μA5μA6HFWA [13]None0.47220.60980.59880.43700.54790.6969HFWG [13]None0.40330.50640.48960.33540.46110.6472HFBM [32]p=2,q=10.53720.57580.55760.39730.52750.7072HFFWA [22]λ=20.36910.53220.47820.37590.47080.6031HFFWG [22]λ=20.54190.63350.62320.47120.60290.7475AC-HFWA (Case 2)λ=10.50690.66470.59850.49160.58880.7274AC-HFWG (Case 2)λ=10.37060.46160.45740.28730.41580.6308


## 6. Conclusions

From the above analysis, while copulas and cocopulas are described as different functions with different parameters, there are many different HF information aggregation operators, which can be considered as a reflection of DM’s preferences. These operators include specific cases which enable us to select the one that best fits with our interests in respective decision environments. This is the main advantage of these operators. Of course, they also have some shortcomings, which provide us with ideas for the following work.We will apply the operators to deal with hesitant fuzzy linguistic [33], hesitant Pythagorean fuzzy sets [34], and multiple attribute group decision making [35] in the future. In order to reduce computation, we will also consider to simplify the operation of HFS and redefine the distance between HFEs [36] and score function [37]. Besides, the application of those operators to different decision-making methods will be developed, such as pattern recognition, information retrieval, and data mining.The operators studied in this paper are based on the known weights. How to use the information of the data itself to determine the weight is also our next work. Liu and Liu [38] studied the generalized intuitional trapezoid fuzzy power averaging operator, which is the basis of our introduction of power mean operators into hesitant fuzzy sets. Wang and Li [39] developed power Bonferroni mean (PBM) operator to integrate Pythagorean fuzzy information, which gives us the inspiration to research new aggregation operators by combining copulas with PBM. Wu et al. [40] extended the best-worst method (BWM) to interval type-2 fuzzy sets, which inspires us to also consider using BWM in more fuzzy environments to determine weights.



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