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Top1. Introduction
Multi-criteria decision making (MCDM) is a useful way for decision making under conflict criteria. The MCDM methods assist to find the best alternative from a set of candidate alternative with considering conflicting criteria (Hashemi et al., 2013; Mousavi et al., 2014). One of the well-known classical MCDM methods is the technique for order of preference by similarity to ideal solution (TOPSIS). In past, classical MCDM methods were offered under precise environments, while in reality decision environments have vagueness and uncertainty (e.g., Vahdani et al., 2014; Gitinavard et al., 2016a, 2016b). Under many conditions, crisp data are inadequate to model real-life situations. The fuzzy sets theory is a powerful tool to deal with uncertainty which derives from real-world situations; in other words, the evaluation ratings and criteria weights are presented by fuzzy sets. TOPSIS method as well-known MCDM approach has been developed under fuzzy environment for group decision-making problems in last decade. Furthermore, a fuzzy TOPSIS approach based on subjective and objective weights presented by Wang and Lee (2009). Vahdani et al. (2013) introduced a modified TOPSIS method under interval-valued fuzzy sets, which could reflect both subjective judgments and objective information in real situations. Moreover, entropy method has been used in many case of TOPSIS method. Also, Yue (2011) developed TOPSIS method to compute the weight of each DM by interval fuzzy numbers for group decision-making problems.
The entropy method has been developed in recent years in fuzzy environments. For instance, Hwang et al. (2011) presented an entropy method based on Sugeno integral with an interval type-2 fuzzy (IT2F) environment. Zamri and Abdullah (2013) introduced a linguistic variable in IT2F entropy weight of a decision-making method. They used the distance concept and developed it to entropy method under an IT2F environment. Furthermore, Hafezalkotob and Hafezalkotob (2016) proposed a fuzzy entropy method based on Shannon entropy concept. They used defuzzification method at first of calculation entropy method.
To review the recent research in fuzzy MCDM area, Yu et al. (2017) proposed a satisfactory degree method by nonlinear programing for solving multi-attribute decision-making problems, in which ratings of alternatives on attributes were expressed via interval-valued intuitionistic fuzzy sets. Zhu and Li (2016) introduced a new axiomatical entropy for intuitionistic fuzzy sets. Also, Li and Ren (2015) developed an effective method for solving multi-attributes decision-making problems and a new ranking function in an intuitionistic fuzzy environment. Moreover, Li (2011) expressed a new closeness coefficient based nonlinear programing model for solving multi-attribute decision-making problems. Furthermore, Li (2014) published a book in the case of decision and game theory in the management with intuitionistic fuzzy sets.