The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment
Introduction
Hesitant fuzzy set (HFS) was originally introduced by Torra [23] and is characterized by a membership function which is represented by a set of possible values. The motivation of introducing HFSs originated from the difficulty of establishing the membership degree, when defining the membership degree of an element to a set, the difficulty is not caused by a margin of error (as in intuitionistic fuzzy set), or some possibility distribution (as in type-2 fuzzy set) on the possible values, but by a hesitation among a set of possible values. Later, Xia et al. [27] gave an example to illustrate this situation: two decision makers (DMs) discuss the membership of x into A, and one wants to assign 0.5 and the other 0.6; and considering these two DMs as a whole (i.e., an organization), thus the organization has a hesitation about the membership of x into A between 0.5 and 0.6, which can be represented by a hesitant fuzzy element (HFE) {0.5, 0.6}. HFEs have been found to be highly useful in handling the decision making problems when the DMs have some hesitations among several possible memberships for an element to a set.
In recent years, many scholars discussed the HFSs’ basic operators and their properties [23], the aggregating operators [25], [26], [27], [32], [33], correlation coefficients [3], and the distance and similarity measures of HFEs [20], [29], [28]. These research works have made great contributions to enrich hesitant fuzzy theory and have been applied to various fields, such as cluster analysis [3], [6], [31], and mainly in the decision making fields [7], [20], [25], [27], [29], [30]. For example, Farhadinia [7] proposed a decision making method based on a novel ranking approach of HFEs to solve the multi-criteria decision making (MCDM) problems with hesitant fuzzy information. Xu and Zhang [30] put forward a hybrid approach combining TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) and the maximizing deviation to handle the MCDM problems in which the evaluation information is expressed by HFEs and the information about criteria weights is incomplete. Liao and Xu [16] proposed an extended VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method to solve the MCDM problems with hesitant fuzzy information. Zhang [32] developed a method based on the hesitant fuzzy power aggregation operators for multi-criteria group decision making (MCGDM) with hesitant fuzzy information. In addition, based on the HFSs, Rodriguez et al. [22] proposed the concept of hesitant fuzzy linguistic term set to provide a linguistic and computational basis to increase the richness of linguistic elicitation based on the fuzzy linguistic approach and the use of context-free grammars by using comparative terms. Chen et al. [3], [4] also generalized the concept of HFS to propose the concepts of interval-valued hesitant fuzzy set (IVHFS). Qian et al. [21] extended HFSs by using intuitionstic fuzzy sets and referred to them as generalized hesitant fuzzy sets, which is fit for the situations where the DMs have a hesitation among several possible memberships under uncertainty.
As mentioned previously, some decision making methods have been proposed to solve the MCDM problems with hesitant fuzzy information, but it is necessary to point out that these methods are based on the strict assumption regarding complete rationality of the DMs. However, many excellent papers involving behavior experiments [1], [2], [14], [24] have shown that the DM is bounded rational in decision making process and the behavior of the DM which plays an important role in decision analysis should be considered in decision making process. Consequently, in case of considering the DM’s psychological behavior, how to solve the MCDM problem with hesitant fuzzy information is a valuable research topic, which is just the focus of this study.
The TODIM (an acronym in Portuguese of interactive and multi-criteria decision making) method, proposed by Gomes and Lima in 1991 [9], [8] in 1992, is a discrete multi-criteria method based on prospect theory [14] and has been proven to be a valuable tool for solving the MCDM problem considering the DM’s behavior. In the classical TODIM approach, the prospect value function is first built to measure the dominance degree of each alternative over the others, which reflects the DM’s behavioral characteristic such as reference dependence and loss aversion, and then the overall value of each alternative is calculated and whereby the ranking of alternatives can be obtained. Afterwards, the TODIM method has been extensively applied in various fields of decision making, such as selection of the destination of natural gas [13], evaluation of residential properties [12], and oil spill response [19]. On the other hand, considering the fact that in some cases, the relationships among criteria are interdependent, Gomes et al. [10], [11] developed a method combining Choquet integral and the TODIM to handle the MCDM problems with criteria interactions. Although the TODIM method can solve well the decision making problems with crisp numbers, in many situations crisp data are inadequate or insufficient to model the real-life decision problems; Instead, the fuzzy set and its extensions are more appropriate to model human judgments. This realization has motivated many researchers to extend the TODIM method for dealing with the decision making problems under various fuzzy environments. For instance, considering the decision information assessed by triangular fuzzy numbers or trapezoidal fuzzy numbers, Krohling and de Souza [15] developed a fuzzy extension of TODIM, named F-TODIM, for solving the MCDM problems under fuzzy environments. Fan et al. [5] proposed another extension of TODIM (H-TODIM) to deal with the hybrid MCDM problems with three forms of criterion values (crisp numbers, interval numbers and fuzzy numbers). More recently, Lourenzutti and Krohling [18] also presented a generalization of the TODIM method (IF-RTODIM) which considers intuitionistic fuzzy information and an underlying random vector. In the IF-RTODIM approach, through the use of the risk function, the DM has a certain freedom in the definition of the estimator that will be used to rank the alternatives. These capabilities make the model much more realistic and flexible to be used by the DM. Apparently, the TODIM method is a valuable tool for solving the classical MCDM problems considering the DM’s behavior and its extensions can also effectively solve the MCDM problems under various fuzzy environments, but none of studies have used this method to handle the MCDM problems with hesitant fuzzy information. Can the TODIM approach be generalized to deal with the MCDM problems under hesitant fuzzy environments? In this paper, we try to answer this question.
The remainder of this paper is organized as follows: In Section 2, we review some concepts related to HFSs, IVHFSs and the classical TODIM approach. In Section 3, we propose two new measured functions for comparing the magnitude of HFEs and interval-valued HFEs (IVHFEs), respectively. In Section 4, we put forward a generalization of the TODIM method which simultaneously considers hesitant fuzzy information and the DM’s behavior. Section 5 employs a decision making problem that concerns the evaluation and ranking of the service qualities among domestic airlines to demonstrate the applicability and the implementation process of the proposed methodology and the paper finishes with some concluding remarks in Section 6.
Section snippets
Preliminaries
In this section, we first recall some basic concepts, such as HFEs and IVHFEs, and their basic operations and distance measures, and then introduce the classical TODIM approach, which will be used in the next sections.
Novel measured functions related to hesitant fuzzy information
In practical fuzzy decision making process, the ranking of fuzzy information plays an important role in solving the fuzzy multi-criteria decision making (FMCDM) problems while the ranking method is essentially based on measured functions of fuzzy information which map fuzzy information into real numbers. In general, the measured functions of fuzzy information can be classified into two categories: algorithmic approaches and non-algorithmic approaches. In the non-algorithmic approaches of fuzzy
The study of TODIM under hesitant fuzzy environment
In this section, we will present a generalization of the TODIM approach (HF-TODIM) for handling the MCDM under hesitant fuzzy scenario. For this, we first give the description of the MCDM problem under hesitant fuzzy context. Then, we extend the TODIM method to solve this MCDM problem. At length, we introduce an algorithm for the HF-TODIM approach.
Illustration example
In this section, we now consider a decision making problem that concerns the evaluation and ranking of the service quality among domestic airlines (adapted from [16]) to demonstrate the applicability and the implementation process of our proposed approach.
Conclusions
TODIM is a helpful tool for solving the MCDM problems, particularly in a situation where the DM’s behavior is taken into account, but it cannot be used to directly handle the MCDM problems with fuzzy information. Considering the fact that the HFS, characterized by a membership function represented by a set of possible values, is a new effective tool to express human’s hesitancy in daily life, therefore, in this paper, we have proposed a new method, namely, the HF-TODIM method, for solving the
Acknowledgements
The authors are very grateful to the anonymous reviewers and the editor for their insightful and constructive comments and suggestions that 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), the Fundamental Research Funds for the Central Universities (No. CXZZ13_0139) and the excellent Ph.D. thesis Foundation of Southeast University (No. YBJJ1339).
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