Hesitant fuzzy ELECTRE II approach: A new way to handle multi-criteria decision making problems
Introduction
As an effective framework, multi-criteria decision making (MCDM) has always been used to evaluate a finite number of decision alternatives having multiple criteria. It has thus been widely applied to diverse scientific fields [11], [31], [32], [33], [46], [65] such as manufacturing systems, environmental impact assessment and location selection. In many real-world problems, it is difficult for decision makers (DMs) to give their assessments on performance ratings and criteria weights with precise values [26]. The fuzzy set [72] has been found to be particularly suitable to describe the ambiguities when one evaluates decision options for the MCDM problems. Accordingly, various fuzzy MCDM methods [6], [14], [42], [43], [44], [58] based on fuzzy sets, interval-valued fuzzy sets and intuitionistic fuzzy sets have been suggested to handle fuzzy information.
Torra and Narukawa have noticed that the previous fuzzy sets and their extensions are unsuitable to describe the situation that the membership degree of an element to a given set has a few different values. For example, two DMs cannot reach a final agreement when they discuss the membership degree of an element x to a set A, because one wants to assign 0.5 and the other 0.6. To address the issue, they proposed the concept of hesitant fuzzy set (HFS) [61], [62], an extension of fuzzy set theory, and made a detailed discussion on the similarities and differences between HFSs and other generalized fuzzy sets [2], [17], [39]. Furthermore, for the above example, the membership degree can be represented by a hesitant fuzzy element (HFE) {0.5, 0.6} [68], [70].
HFS has attracted a lot of attention recently [4], [12], [13], [45], [47], [68], [69], [70], [71]. Studies have been performed on aggregation operators [68], [69], [74] as well as on distance and similarity measures [70], [36] for HFSs. As an alternative way to depict uncertainties, the HFS provides an intuitive description on the difference of opinions among group members since it avoids aggregation [61]. As one knows, in the usual MCDM methods it is necessary to aggregate the DMs’ opinions for each alternative under the given criteria. Consequently, only a set of average criteria can be obtained, implying a valid common decision; that is, these aggregation methods neglect the differences among the individual DMs’ opinions.
As a major category of MCDM, outranking [1], [19], [23], [29], [49], [50], [60] can be used to select which alternative is preferable, incomparable or indifferent by a pairwise comparison of alternative under each criterion. Among the outranking methods, the ELECTRE (ELimination Et Choix Traduisant la REalité) method is the most popular one, whose main idea is the proper utilization of the outranking relations [66]. Since ELECTRE I [5], [52], the first version of the ELECTRTE method, was introduced, the ELECTRE approach has evolved into a number of variants, including ELECTRE II, III and IV as well as ELECTRE-A and ELECTRE TRI methods, which constitute a family of ELECTRE methods, see Ref. [21] for a review. The ELECTRE method has been further developed to treat groups with imprecise information on parameter values [15], [16], to solve inconsistencies among constraints on the parameters [40], [41], to assist a group of DMs with different value systems [35], and to incorporate the ideas of concordance and discordance for group ranking problems [20], etc. In addition, the ELECTRE method [28], [34], [51], [57] has been applied to project selection [8], transportation [53], [54] and environment management [55].
Intensive efforts have been made to deal with various types of fuzzy MCDM problems within the framework of ELECTRE methods. For example, Hatami and Tavana [25] proposed the extended ELECTRE I method to take account of the uncertain linguistic assessments, and further applied an integrated fuzzy group ELECTRE method to safety and health assessments in hazardous waste recycling facilities [26]. Wu and Chen [67] adopted a similar approach to solve the MCDM problems under Atanassov’s intuitionistic fuzzy environments. Vahdani et al. [63], [64] performed an extension of the ELECTRE I for multi-criteria group decision making problems on the basis of intuitionistic fuzzy sets and interval-valued fuzzy sets.
The ELECTRE I method is suitable to construct a partial prioritization and to choose a set of promising alternatives [25]. Different from ELECTRE I, ELECTRE II [24], [48] is the first of ELECTRE methods especially designed to handle the ranking problems. The ELECTRE II method considers several concordance and discordance levels, which can be used to construct two embedded outranking relations (i.e., strong and weak outranking relations). With these relations, the strong and weak graphs can be depicted and the ranking of alternatives is finally derived.
The present work is devoted to proposing an extended ELECTRE II method within the context of hesitant fuzzy circumstances, called hesitant fuzzy ELECTRE II (HF-ELECTRE II) method, in which the difference of opinions among group members is taken into account by HFSs. The formulation of the HF-ELECTRE II method contains a construction of strong and weak outranking relations between alternatives that is based on a classification of different types of hesitant fuzzy concordance and discordance sets. The structure of the paper is as follows: In Section 2, basic concepts associated with HFSs and the ELECTRE methods are introduced. In Section 3, we outline the proposed HF-ELECTRE II approach. Numerical examples are presented in Section 4. In Section 5, we compare the ranking results derived by the HF-ELECTRE II method with those derived from the aggregation operators and the fuzzy group ELECTRE I methods. To better handle MCDM, a decision supporting system formulated on the basis of HF-ELECTRE II method is proposed to aid the DMs to make decisions. Section 6 presents conclusions and future research challenges.
Section snippets
Preliminaries
Basic concepts related to HFSs and the ELECTRE methods are introduced below.
The HF-ELECTRE II method
In this section, we combine the idea of HFSs with ELECTRE II to formulate a new approach, named as hesitant fuzzy ELECTRE II (HF-ELECTRE II) here, to solve a MCDM problem under hesitant fuzzy environment. The formulation is a two-stage process, i.e., the construction and exploitation of one or several outranking relation(s). The construction is by means of the score function and the deviation function that give different types of concordance and discordance sets and the corresponding indices.
Numerical example
Below, we use a numerical example to illustrate the details of the HF-ELECTRE II method: Example 2 A battery industry involved in the recycling process desires to select a suitable third-party reverse logistics provider (3PRLP) to perform the reverse logistics activities. A committee of three DMs has been formed to select the most suitable 3PRLP. They evaluate the performance of each 3PRLP from the following seven aspects in selection of the potential 3PRLPs Ai (i = 1, 2, …, 5): (1) C1: quality; (2) C2:[30], [59]
Comparison with the aggregation operator based approach
In Ref. [68] the aggregation operators approach has been suggested to aggregate hesitant fuzzy information, where the ranking of projects is got by computing the score functions. To facilitate a comparison with our HF-ELECTRE II approach, we consider here the example used in Ref. [68]. The detailed calculation process and the ranking are presented in Appendix A.
The ranking deduced with the HF-ELECTRE II approach is found to be consistent with that derived from the aggregation operators
Conclusions
By extending the ELECTRE II method that can incorporate the concept of HFS, which is utilized to denote uncertainties caused by the DMs with several possible values, we have proposed a new approach, named HF-ELECTRE II approach, for solving the MCDM problems. To formulate the new approach, we have first defined various types of hesitant fuzzy concordance and discordance sets by the score function and the deviation function. We have then derived hesitant fuzzy concordance and discordance
Acknowledgements
The authors thank Editor-in-chief and the anonymous reviewers for their helpful comments and suggestions, which have led to an improved version of this paper. The work was supported by the National Natural Science Foundation of China (Nos. 71071161 and 61273209).
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