Fuzzy entropy and hesitancy entropy in probabilistic hesitant fuzzy information and their applications

In recent years, the concept of probabilistic hesitant fuzzy sets (P-HFSs) has been proposed as an improvement of hesitant fuzzy sets (HFSs), which has the advantage of representing deeper uncertainty. In order to fully measure the uncertainty for probabilistic hesitant fuzzy information, in this study, the axiomatic definition of fuzzy entropy of individual elements in weak probabilistic hesitant fuzzy elements (WPHFEs) is proposed according to the distribution characteristics of fuzziness, and the axiomatic definitions of fuzzy entropy and hesitancy entropy of WPHFEs are established as well. On this basis, the axiomatic definition of overall entropy for WPHFEs is studied by combining fuzzy entropy and hesitancy entropy. Meanwhile, a new ranking method and decision-making method are given based on the proposed entropies. Finally, the practicability and effectiveness of the proposed methods are illustrated by an example and two comparative analyses.