Continuous probability-interval valued fuzzy preference relations and its application in group decision making

Probabilistic hesitant fuzzy set represents the occurrence probabilities of elements. The probabilistic hesitant fuzzy preference relations can more effectively express the hesitant preference information of decision makers. But in the existing research, all of them are based on discrete probability distribution. In order to give decision maker more evaluation space, continuous probability distribution is necessary to be considered. Therefore, in this paper, the continuous probability-interval valued fuzzy set is defined and its probability is represented by a probability density function. A method of converting probabilistic hesitant fuzzy set into continuous probability-interval valued fuzzy set is developed to transform discrete data into continuous data. Then, the continuous probability-interval valued fuzzy preference relations is presented. In order to consider the consistency of continuous probability-interval valued fuzzy preference relations, the multiplication consistent expected preference relations is proposed. The individual consistency index and group consensus index are also presented to determine the consistency level. an‎d then, an algorithm is introduced for checking and improving the individual consistency level and group consensus level. Finally, a numerical example is shown to the effectiveness of proposed algorithm, the comparative analysis is given with the existing methods to show the superiority of this algorithm.