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.
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 Section 2. 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 be a finite reference set. A hesitant fuzzy set on is in terms of a function when applied to returning a subset of denoted by
where 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 [4]). For a HFE , is called the score function of , where is the number of possible elements in .
For two HFEs and ,(i)if , then (ii)if , then
Definition 3. (see [4]). Let , , and be three HFEs, , and the novel operational rules of HFEs are given as follows:
where .
2.2. BM Operator and HFBM Operator
Definition 4. (see [22]). Assume is a family of positive real number and , then the aggregated mapping
is named BM operator.
Definition 5. (see [14]). Let and . Then the hesitant fuzzy Bonferroni mean (HFBM) operator is expressed as
Definition 6. (see [14]). Let and . Then the hesitant fuzzy weight Bonferroni mean (HFWBM) operator is expressed as
2.3. Archimedean Copula
Definition 7. (see [16]). A two-dimensional mapping is called a copula, if it contents the following properties:(1)(2)(3)
Definition 8. (see [17]). A copula is said to be the Archimedean copula if there exists a continuous and strictly decreasing function from to with , and function from to is defined by
such that for any ,
The AC is propounded based on the following statements: (1) If is strictly increasing on , then we can obtain ; (2) If accords on . Then equation (7) is transformed aswhere the functions and are named strict generator and Archimedean copulas, respectively.
Definition 9. (see [18]). Suppose : be a copula, then co-copula is expressed as
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 , , and be three HFEs, , and the novel operational rules of HFEs are given as follows:
where .
3.2. HFCBM Operator
Based on Definitions 5 and 10, the HFCBM operator can be proposed.
Theorem 1. Let , and . Then the hesitant fuzzy copula Bonferroni mean (HFCBM) operator is expressed as
Proof. Since
Then
Furthermore
Accordingly
Accordingly, we can attain the establishment of Theorem 1.
In what follows, several desired characteristics of HFCBM operator are proved.
Theorem 2. Let , and , then(1)(Idempotency) If , then (2)(Monotonicity) Let , if , so(3)(Boudedness) If ,,
Proof. (1)If , Then Accordingly, the idempotency of the HFCBM operator is proved.(2)Since , , and are decreasing functions and and are increasing functions. Then, we have Then, Furthermore, Moreover, Hence Accordingly, the monotonicity of the HFCBM operator is proved.(3)Let and ; according to the monotonicity, we can acquire According to idempotency, we can acquire Accordingly, we can acquire .
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. Let , and . Then the hesitant fuzzy weight copula Bonferroni mean (HFWCBM) operator is expressed as
where
Proof. The proof of Theorem 3 is similar to Theorem 1, we can utilize the proof of to replace in equation (1).
ThenThenHencewhereSo we can attain the establishment of Theorem 3. □
In what follows, several desired characteristics of HFWCBM operator are proved.
Theorem 4. Let , and , then(1)(Idempotency) If , (2)(Monotonicity) Let ; if ,(3)(Boudedness) If ,,Theorem 4 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 function .
Case 1. When , then
where
where
Specifically, when , then they are the operators proposed by Zhu [14].
Case 2. When with , then
where
where
Case 3. When , then
where
where
Case 4. When with , we have
where
where
Case 5. When , then
where
where
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 are alternatives , n attributes . If DMs provide several values for the alternative under the attribute with anonymity, these values can be considered as a HFE . In the case, if two DMs provide the same value, then the value emerges only once in . In what follows, the specific algorithm for MADM problems under the HF environment will be designed. Step 1: DMs provide their evaluations about the alternative under the attribute , denoted by the hesitant fuzzy elements Step 2: use the HFCBM or HFWCBM to fuse the HFE for Step 3: the score values of are calculated using Definition 2 and compared Step 4: the optimal alternatives are made by ranking
4.1. Empirical Example
Example 1. Selection of Equipment Purchase. Assume that there are four short-term stocks . The following four attributes: , should be considered. : product price; : technology; : after-sales service level; : 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 Table 1
4.2. Sensitivity Analysis
Next, we first analyse the sensitivity of sorting results obtained by diverse operations from AC. In addition, the effect of parameters , and 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 Table 2. 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.
Secondly, we research the effect of parameter in the HFCBM operator on the ultimate orders of alternatives. Table 3 shows the values of score functions with the different when and . Figure 1 shows how the score functions obtained by Case 1 change with the parameter under different values of and . As can be seen from the results of the chart, the score values increase with the increase of for the same alternatives.
(a)
(b)
(c)
(d)
Thirdly, we explore the effect of parameters and in the HFCBM operator on the ranking results of alternatives. The ranking results of alternatives based on diverse values of and can be generated in Table 4 (we select Case 1 as an example and let ). Figure 2 shows the variation of the score function with and . The results indicate that the scores of alternatives by the HFCBM operator increase as the parameter ranges from 0 to 10 and the parameter 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 and to get the desired ranking result.
(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.
From Table 5, we can observe that the orders are marginally diverse, but the optimal selections are all . 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 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 Table 5, 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Sichuan Province Youth Science and Technology Innovation Team (2019JDTD0015), the Scientific Research Project of Department of Education of Sichuan Province (18ZA0273 and 15TD0027), the Scientific Research Project of Neijiang Normal University (18TD08), and Key Project of Sichuan Provincial Department of Education (17ZA0217 and 18ZA0276).