Archimedean Copula-Based Hesitant Fuzzy Information Aggregation Operators for Multiple Attribute Decision Making
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.
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)  which is proposed by Zadeh and their various extensions, including the intuitionistic fuzzy set (IFS) , interval-valued intuitionistic fuzzy set (IVIFS) , hesitant fuzzy set (HFS) [4, 5], Pythagorean fuzzy set (PFS) , neutrosophic set (NS) , and so on.
In the numerous extensions of the FS, IFS as one of the most important, was introduced by Atanassov . 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  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  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.  developed some hesitant fuzzy aggregation operators based on Frank operations, such as HFFWA operator, HFFOWA operator, and so on. Yu et al.  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.  proposed the generalized hesitant fuzzy harmonic mean operators including GHFWHM operator, GHFOWHM operator, and GHFHHM operator. Li and Chen  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  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.  studied a new computational model for unbalanced linguistic variables. Chen et al.  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 Section 2. 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.
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 ). Let be a finite reference set. A hesitant fuzzy set on in terms of a function when applied to returns a subset of denoted bywhere is a collection of numbers from , indicating the possible membership degrees of to . We call a hesitant fuzzy element (HFE) and the set of all HFEs.
To compare the HFEs, the comparison laws are defined as follows .
Definition 2 (see ). For a HFE , is called the score function of , where is the number of possible elements in .
For two HFEs and , If , then ; If , then .
2.2. Copulas and Cocopulas
Definition 3 (see ). A two-dimensional function is called a copula, if the following conditions are met:(1), (2)where and .
Definition 4 (see ). A copula is named as an Archimedean copula, if there is a strictly decreasing and continuous function with , and from to is defined as follows:For all , we haveIf is strictly increasing on , and coincides with on , then is written as and the function is called a strict generator and is called a strict Archimedean copula.
Definition 5 (see ). Let be a copula, and the cocopula is introduced as follows:If is a strict Archimedean copula, is also changed to beIn order to introduce some new operations based on copulas and cocopulas mentioned above, the following conclusion is given firstly.
Theorem 1. For , then .
Proof. If , then . As is strictly decreasing and ,So,We haveThus, Theorem 1 holds.
Definition 6. Let ; the algebra operations based on copula and cocopula are defined as follows:It is easy to verify that and satisfy associative law, that is, for ,
Theorem 2. For , , we have , .
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. Let , , and be three HFEs and ; the novel operational rules of HFEs are given as follows:From the above definition, the following conclusions can be easily drawn.
Theorem 3. Let , , and be three HFEs and ; then, we have
The algorithms can be used to fuse the HF information and investigate their ideal properties which is the focus of the following sections.
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. Let be a set of n HFEs and be a function on , ; then, .
Definition 9. Let , be the weight vector of with and . Archimedean copula-based hesitant fuzzy weighted averaging operator (AC-HFWA) is defined as follows:
Theorem 4. Let , be the weight vector of with and ; then,
Theorem 5. Let , be the weight vector of with and ; then,(1)(Idempotency) If , .(2)(Monotonicity) Let ; if ,(3)(Boundedness) If and ,
Proof. (1).(2)If , and . Then, and . So, .(3)Suppose and .Therefore, , , for all and .
Since is strictly decreasing, is also strictly decreasing.
Then, , and soThat is, .
Therefore, , .
Definition 10. Let , be the weight vector of with and . The Archimedean copula-based generalized hesitant fuzzy averaging operator (AC-GHFWA) is given byEspecially, when , the AC-GHFWA operator becomes the AC-HFWA operator.
The following theorems are easily obtained from Theorem 4 and the operations of HFEs.
Theorem 6. Let , be the weight vector of with and ; then,Similar to Theorem 5, the properties of AC-GHFWA can be obtained easily.
Theorem 7. Let , be the weight vector of with and ; then,(1)(Idempotency) If , .(2)(Monotonicity) Let ; if ,(3)(Boundedness) If and ,
3.3. Different Forms of AC-HFWA
We can see from Theorem 4 that some specific AC-HFWAs can be obtained when is assigned different generators. Case 1. If , then . So, , . Specifically, when , then , , and the AC-HFWA becomes the following: They are the HF operators defined by Xia and Xu . Case 2. If , then . So, , . Case 3. If , then . So, , . Case 4. If , then . So, , . Case 5. If , then . So , .
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 generator takes different functions.
Definition 11. Let , be the weight vector of with and . The Archimedean copula-based hesitant fuzzy weighted geometric operator (AC-HFWG) is defined as follows:
Theorem 8. Let , be the weight vector of with and ; then,
Theorem 9. Let , be the weight vector of with and ; then,(1)If , .(2)Let ; if ,(3)If and ,
Proof. Suppose and .
Since is strictly decreasing, is also strictly decreasing.
, , so .
Definition 12. Let , be the weight vector of with and ; then, the generalized hesitant fuzzy weighted geometric operator based on Archimedean copulas (AC-CHFWG) is defined as follows:Especially, when , the AC-GHFWG operator becomes the AC-HFWG operator.
Theorem 10. Let , be the weight vector of with and ; then,
Theorem 11. Let , be the weight vector of with and ; then,(1)If , .(2)Let ; if ,(3)If and ,
4.2. Different Forms of AC-HFWG Operators
We can see from Theorem 8 that some specific AC-HFWGs can be obtained when is assigned different generators. Case 1. If , then . So, , . Specifically, when and , then and , and the AC-HFWG operator reduces to HFWG and GHFWG : Case 2. If , then . So, , . Case 3. If , then . So, , . Case 4. If , then . So, , . Case 5. If , then . So, , .
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 generator . In this section, we will introduce the properties of the AC-HFWA and AC-HFWG operator regarding to the parameter κ.
Theorem 12. Let be the generator function of copula, and it takes five cases proposed in Section 4.2; then is an increasing function of , is an decreasing function of , and .