Operations and integrations of probabilistic hesitant fuzzy information in decision making

In recent years, the hesitant fuzzy set (HFS) has been widely applied in practical decision making processes. As a matter of fact, it indeed describes the thoughts of experts better because of a better tolerance. But there is a big defect in its applications-the serious loss of information. To improve the HFS, the probabilistic hesitant fuzzy set (P-HFS) has been put forward. It adds the probability to the HFS and can retain more information than the HFS. However, the P-HFS is still not perfect. In this paper, the definitions of the P-HFS and the probabilistic hesitant fuzzy element (P-HFE) are improved at first. After that, the properties and operations of the improved P-HFEs are studied, and some corresponding aggregation operators are given. Furthermore, the concept of P-HFEs with the continuous form is proposed, some operations and distance measures for them are given. Finally, the integrations of the improved P-HFEs are used to deal with a practical case concerning the automotive industry safety evaluation.