Probability Calculation and Element Optimization of Probabilistic Hesitant Fuzzy Preference Relations Based on Expected Consistency

As a generalized fuzzy number, the hesitant fuzzy element (HFE) has received increasing attention. However, it is noted that the occurring probabilities of the elements in the HFE are equal, which is obviously problematic; thus, the preference relations on the HFEs can be inaccurate. To address this issue, this paper proposes the probabilistic hesitant fuzzy preference relations (PHFPRs) based on the probabilistic HFE (PHFE). It seems difficult to provide accurate probabilities that describe the occurring possibilities of the elements in the PHFPRs. This paper further demonstrates the probability calculation method for the PHFPEs based on a proposed variable, which is called the expected consistency. Moreover, the expected consistency index and the judgment principle are designed to evaluate the degree to which the PHFPRs are consistent. For the inconsistent PHFPRs, this paper presents an iterative optimization algorithm to improve their expected consistency by optimizing some elements in the PHFPRs. When the iteration terminates, the consistent PHFPRs and the priorities of the alternatives are identified. Finally, an example that selects a Ph. D. candidate is presented, and the results demonstrate the feasibility and effectiveness of using the PHFPRs and the consistency methods.