An approach to probabilistic hesitant fuzzy risky multiattribute decision making with unknown probability information.

As a useful tool, probabilistic hesitant fuzzy set is an enhanced version for hesitant fuzzy set. It could be used to model the uncertainty very effectively. However, in probabilistic hesitant fuzzy risky multiple attribute decision making problems, the occurrence probabilities of elements in a probabilistic hesitant fuzzy element and the probability of risk status are often difficult to obtain by subjective evaluation of a decision maker. This paper aims to propose two nonlinear programming models for calculating the probabilities of elements in a probabilistic hesitant fuzzy element and the probability of risk status respectively. First, a nonlinear programming model using maximum entropy principle is established for determining the probabilities of elements in a probabilistic hesitant fuzzy element. Second, by introducing the water‐filling theory, we put forward its extension and design a novel mathematical programming model to determine the probability of risk status. Moreover, we have proved that both the two mathematical programming models are convex programming models and their global optimal solutions can be found. Thirdly, the collective overall expected values of alternatives are calculated and the ranking order can be derived. Then, the selection of investment project is investigated, and comparison analysis shows the superiority of the presented approach. [ABSTRACT FROM AUTHOR]