Probabilistic picture hesitant fuzzy sets and their application to multi-criteria decision-making

The picture hesitant fuzzy sets (PHFSs), which consider neutral membership degree as well as positive and negative membership degrees, provide decision makers (DMs) a flexible attitude to evaluate criteria values in complex multi-criteria decision-making (MCDM) situations. However, existing MCDM approaches based on PHFSs still have some drawbacks in both evaluation information expression and criteria values fusion. In this paper, our aim is to overcome these shortcomings by proposing new decision-making methods. To achieve this purpose, a new fuzzy information representation tool, called probabilistic picture hesitant fuzzy sets (P-PHFSs), is first introduced by capturing the probability of each element in PHFSs. The characteristic of P-PHFSs is that they provide more freedom to DMs so that criterion values of each alternative can be adequately described. To facilitate the use of P-PHFSs, we define the basic operational rules and comparison method of P-PHFSs. Then we also propose some aggregation operators for P-PHFSs and provide information fusion process. Furthermore, some desirable properties of these operators is discussed, and the relationship between the developed operators and the existing ones is investigated. Based on the proposed operators, two MCDM methods are developed under probabilistic picture hesitant fuzzy environment. Finally, two numerical examples are given to show the application of the developed methods, and a comparison analysis is conducted to demonstrate the effectiveness of the proposed approaches.