Multi-Attribute Decision-Making Method Based on Probabilistic Hesitant Fuzzy Entropy

Probabilistic hesitant fuzzy sets (PHFSs) (Zhu in Decision method for research and application based on preference relation, 2014.; Xu and Zhou in Fuzzy Optim Decis Making 16:1–23, 2016) use probabilistic information to express the importance of the membership degree, resulting in a more comprehensive representation of information in the PHFS. However, this also imposes higher technical demands on information processing. In this chapter, we focus on the effective fusion of probability and membership degree information, and presenting two kinds of entropy measures for probabilistic hesitant fuzzy elements (PHFEs). Firstly, two membership degree-based entropies for PHFEs inspired by the classical fuzzy entropies are derived. Secondly, the distance-based entropies for PHFEs which are inversely proportional to the distance measures among the elements and the fuzziest element are proposed. However, since the existing distance measures for PHFEs are helpless in the description of the entropies, 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.