The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets

Pythagorean fuzzy sets (PFSs) as a new generalization of fuzzy sets (FSs) can handle uncertain information more flexibly in the process of decision making. In our real life, we also may encounter a hesitant fuzzy environment. In view of the effective tool of hesitant fuzzy sets (HFSs) for expressing the hesitant situation, we introduce HFSs into PFSs and extend the existing research work of PFSs. Concretely speaking, this paper considers that the membership degree and the non-membership degree of PFSs are expressed as hesitant fuzzy elements. First, we propose a new concept of hesitant Pythagorean fuzzy sets (HPFSs) by combining PFSs with HFSs. It provides a new semantic interpretation for our evaluation. Meanwhile, the properties and the operators of HPFSs are studied in detail. For the sake of application, we focus on investigating the normalization method and the distance measures of HPFSs in advance. Then, we explore the application of HPFSs to multi-criteria decision making (MCDM) by employing the technique for order preference by similarity to ideal solution (TOPSIS) method. A new extension of TOPSIS method is further designed in the context of MCDM with HPFSs. Finally, an example of the energy project selection is presented to elaborate on the performance of our approach.