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TopIntroduction
Practical multi criteria decision making (MCDM) problems are often imprecise due to complexity and diversity in the real world (Zhang et al. 2019). MCDM is a branch of decision-making theory where the goal of an individual is to select the most acceptable alternatives among the feasible ones under some criteria. Due to the increasing complexity of the social-economic environment, a lack of data about the problem domain, or the decision maker's lack of expertise to precisely express their preferences over the considered objects, the preference information provided by decision-makers is frequently uncertain. In such instances, expressing the decision-preference maker's information in terms of probabilistic notions is suitable and convenient.
TOPSIS (Akram et al. 2019, Opricovic and Tzeng 2004) is a useful technique in dealing with multi attribute or multi-criteria decision-making problems in the real world. TOPSIS has been used extensively used for practical MCDM problems problem due to its sound mathematical foundation, simplicity, ease of applicability. It originates from the concept of a displaced ideal point from which the compromise solution has the shortest distance. Hwang and Yoon (1981) proposed that the ranking of alternatives will be based on the shortest distance from the (positive) ideal solution (PIS) and the farthest from the negative ideal solution. OPSIS simultaneously considers the distances to both PIS and NIS, and a preference order is ranked according to their relative closeness, and a combination of these two-distance measure. These salient features make TOPSIS a preferred approach than other existing MCDM approaches. IFSs (Kumar and Garg 2018), interval valued IFSs (Li 2010, Kumar and Garg 2018, Garg and Kumar 2018, Wei et al. 2021), and PFSs (Liang et al. 2018, Biswas and Srakar 2019) have been solved with the help of TOPSIS approach. Applications of TOPSIS approach to MCDM with Pythagorean and hesitant FS were proposed by Zhang and Xu (2014). Hussain and Yang (2018) gave entropy for hesitant fuzzy sets based on Hausdorf metric with construction of hesitant fuzzy TOPSIS.