Probabilistic Dual-Hesitant Pythagorean Fuzzy Sets and Their Application in Multi-attribute Group Decision-Making

As modern socioeconomic decision-making problems are becoming more and more complex, it also becomes more and more difficult to appropriately depict decision makers’ cognitive information in decision-making process. In addition, in group decision-making problems, decision makers’ cognition is usually diverse, which makes it more complicated to express the overall preference information. Recently, the dual-hesitant Pythagorean fuzzy sets (DHPFSs) have been proved to be an effective tool to depict decision makers’ evaluation values in multi-attribute group decision-making (MAGDM) procedure. The basic elements of DHPFSs are dual-hesitant Pythagorean fuzzy numbers (DHFNs), which are characterized by some possible membership degrees and non-membership degrees. In a DHFN, all members have the same importance, which indicates that multiple occurrence and appearance of some elements is ignored. Hence, the DHPFSs still have some drawbacks when expressing decision makers’ evaluation information in MAGDM problems. This paper aims at proposing a novel tool to describe decision maker’s evaluation values and apply it in solving MAGDM problems. This paper extends the traditional DHPFSs to probabilistic dual-hesitant Pythagorean fuzzy sets (PDHPFSs), which consider not only multiple membership and non-membership degrees, but also their probabilistic information. Afterward, we investigate the applications of PDHPFSs in MAGDM process. To this end, we first introduce the concept of DHPFSs as well as some related notions, such as operational rules, score function, accuracy function, comparison method, and distance measure. Second, based on the power average and Hamy mean, some aggregation operators for DHPFSs are presented. Properties of these new operators are also discussed. Third, we put forward a novel MAGDM method under PDHPFSs. A novel MAGDM method is developed, and further, we conduct numerical examples to show the performance and advantages of the new method. Results indicate that our method can effectively handle MAGDM problems in reality. In addition, comparative analysis also reveals the advantages of our method. This paper contributed a novel MAGDM method and numerical examples as well as comparative analysis were provided to show the effectiveness and advantages of our proposed method. Our contributions provide decision makers a new manner to determine the optimal alternative in realistic MAGDM problems.