Hesitant Fuzzy Multiple Criteria Decision Analysis Based on TODIM

The TODIM is a valuable technique for solving classical MCDM problems in case of considering the decision maker’s psychological behavior. One main goal of this chapter is to introduce the measured functions-based hesitant fuzzy TODIM technique to deal with the behavioral MCDM problem under hesitant fuzzy environments. The main advantages of this technique are that (1) it can handle the MCDM problems in which the ratings of alternatives with respect to each criterion are represented by HFEs or IVHFEs and (2) it can take the decision maker’s psychological behavior into account. Another aim of this chapter is to present the hesitant trapezoidal fuzzy TODIM method with a closeness index-based ranking approach to handle MCGDM problems in which decision data is expressed as comparative linguistic expressions based on HTrFNs. This proposed method first transforms comparative linguistic expressions into HTrFNs for carrying out computing with word processes. Then, a closeness index-based ranking method is proposed for comparing the magnitude of HTrFNs. By using such a ranking method, the dominance values of alternatives over others for each expert are calculated. Next, a nonlinear programming model is established to derive the dominance values of alternatives over others for the group and correspondingly the optimal ranking order of alternatives is determined. The classical TODIM method originally proposed by Gomes and Lima (1991, 1992) is a discrete multiple criteria decision analysis method based on prospect theory (Kahneman and Tversky 1979) and has been proven to be a valuable tool for solving the classical MCDM problems in case of considering the decision maker’s psychological 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 decision maker’s behavioral characteristics such as reference dependence and loss aversion; and then the ranking orders of alternatives can be obtained by calculating the overall prospect value of each alternative. The classical TODIM method has been extensively applied in various fields of decision making, such as the selection of the destination of natural gas (Gomes et al. 2009), the evaluation of residential properties (Gomes and Rangel 2009), and the oil © Springer International Publishing Switzerland 2017 X. Zhang and Z. Xu, Hesitant Fuzzy Methods for Multiple Criteria Decision Analysis, Studies in Fuzziness and Soft Computing 345, DOI 10.1007/978-3-319-42001-1_2 31 spill response problem (Passos et al. 2014), etc. Considering the fact that in some real-world situations the relationships among criteria are interdependent, Gomes et al. (2013) developed a method combining Choquet integral and the classical TODIM to handle the MCDM problems with criteria interactions. Owing to the fact that in many situations crisp data are inadequate or insufficient to model the real-world decision making problems, the fuzzy set and its extensions are more appropriate to model human judgments. This realization has motivated many researchers to extend the classical TODIM method for dealing with the MCDM problems under various fuzzy environments. For instance, considering the decision data assessed by TFNs or TrFNs, Krohling and de Souza (2012) developed a fuzzy extension of TODIM (named F-TODIM) for solving the fuzzy MCDM problems. Fan et al. (2013) proposed another extension of TODIM (named H-TODIM) to deal with the hybrid MCDM problems with three forms of criteria values (crisp numbers, interval numbers and fuzzy numbers). More recently, Lourenzutti and Krohling (2013) also presented a generalization of the TODIM method (named IF-RTODIM) which considers intuitionistic fuzzy information and an underlying random vector. Although the existing TODIM methods can solve effectively the classical MCDM problems or fuzzy MCDM problems in case of considering the decision maker’s psychological behavior, they fail to handle such MCDM problems under hesitant fuzzy environment. The MCDM problems with HFEs and/or IVHFEs have recently received increasing attentions and many corresponding MCDM methods (Farhadinia 2013; Liao and Xu 2013; Xu and Zhang 2013; Zhang 2013) have also been developed, but none of them can be used to solve the hesitant fuzzy MCDM problems in case of considering the decision maker’s psychological behavior. To this end, Zhang and Xu (2014a) extended the classical TODIM method to solve the hesitant fuzzy MCDM problems in case of considering the decision maker’s psychological behavior. In this approach, two novel ranking functions are developed for comparing the magnitude of HFEs and IVHFEs, which are more reasonable and effective compared with the existing ranking functions. Then, the prospect values of each alternative related to the others are calculated based on novel ranking functions and distance measures. By aggregating these prospect values, the overall prospect value of each alternative is further obtained and the ranking of alternatives is also obtained. Finally, Zhang and Xu (2014a) provided a decision making problem that concerns the evaluation and ranking of the service quality among domestic airlines to illustrate the validity and applicability of this approach. On the other hand, Zhang et al. (2016) proposed a new concept of HTrFN which is an extension of HFE and is well enough to represent the uncertainty and vagueness of comparative linguistic expressions. The HTrFNs benefited from the superiority of both TrFNs and HFEs can be directly applied in MCDM and MCGDM. To handle the MCGDM problems in which the decision data are expressed by comparative linguistic expressions based on HTrFNs, Zhang and Liu (2016) developed a hesitant trapezoidal fuzzy TODIM method with a closeness index-based ranking approach. This proposed method first transforms comparative linguistic expressions into HTrFNs for carrying out computing with word processes. Then, a closeness index-based ranking method is proposed for comparing the magnitude of HTrFNs. By using the closeness index-based ranking method of 32 2 Hesitant Fuzzy Multiple Criteria Decision Analysis Based on TODIM