Probabilistic hesitant fuzzy multiple criteria decision-making with triangular norm based similarity and entropy measures

Existing probabilistic hesitant fuzzy set (PHFS) measures are constructed using two information measures: hesitancy and unwrapped probabilities. We argue that unifying these semantic terms in PHFS information theory is not logical. We introduce a new class of information measures for PHFSs, which address the logical wrapping of hesitant fuzzy sets (HFS) and probability. We propose several similarity measures for these sets that use the Triangular norm operator. We consider the relationship between measures of entropy and similarity and represent the axiomatic definition of PHFS entropy measures. Finally, we use case studies to demonstrate applications of these information measures. We describe two multiple-criteria decision-making algorithms. The last step is devoted to PHFS ranking procedures: one based on the score function of alternatives and the other based on the relative closeness of alternatives. This contribution describes new information measures and uses case studies to illustrate how they can be applied to decision-making processes.