Multi-criteria decision-making with probabilistic hesitant fuzzy information based on expected multiplicative consistency

This study presents a multi-criteria decision-making method that considers expected multiplicative consistency and a consensus reaching process with probabilistic hesitant fuzzy information. The concept of expected multiplicative consistent probabilistic hesitant fuzzy preference relation (PHFPR) is defined on the basis of multiplicative transitivity, and a theorem is developed to obtain the score values of complete expected multiplicative consistent PHFPR. Subsequently, a consistency index of individual PHFPR is proposed by using the distance between individual PHFPR and the score values of its complete expected multiplicative consistent PHFPR. When the individual PHFPR consistency level does not meet the expected value, an iteration algorithm is designed to improve its consistency level and obtain an acceptable one. Furthermore, a group consensus index is proposed according to the distance between the individual acceptable multiplicative consistent PHFPR and the score values of collective PHFPR. An iteration algorithm is designed to improve the consensus level when it does not reach the threshold value. Several numerical examples are provided to demonstrate the effectiveness of the proposed method, and a comparative study involving other methods is conducted with the same numerical examples.