Entropy Measures for Probabilistic Hesitant Fuzzy Information

The probabilistic hesitant fuzzy set (PHFS), which is remarkable in describing the practical condition, has attracted great attention and been applied to many areas. Although lots of achievements have been obtained, there are also some fields, such as the entropy measures with respect to the uncertainty of the information, have not yet been studied. This paper aims at presenting two kinds of entropy measures for probabilistic hesitant fuzzy elements (PHFEs). First, two membership degree-based entropies for PHFEs inspired by the classical fuzzy entropies are derived. Second, the distance-based entropies for PHFEs which are inversely proportional to the distance measures among the elements and the fuzziest element are proposed. However, it is a pity that the existing distance measures for PHFEs are helpless in the description of the entropies, so a new like-distance measure related to the expectation information of the membership degrees is proposed. Then, these entropies are applied to the decision-making case for "The Belt and Road", and their effectiveness and practicability are verified. Finally, some comparisons among these entropies are made.