The Distance Of The Probability Hesitant Fuzzy Set And It's Application

The probability hesitant fuzzy set is a generalized form of fuzzy set,which allows the membership degree of objects to take multiple possible values in [0,1],and provides a probability for each value,reflecting the decision maker's psychology of hesitation and preference inconsistency in the decision making process.Distance measure is an important research content of the probabilistic hesitant fuzzy set.It can be used to estimate the similarity between two probabilistic hesitant fuzzy sets.It has been widely used in pattern recognition,cluster analysis,multi-attribute group decision making and other fields.In this paper,we summarize and analyze the existing distance measures of probability the hesitant fuzzy set,propose several new distance measures of the probability hesitant fuzzy set,and apply them to multi-attribute group decision making and attribute reduction of probability hesitant fuzzy information system.The main contents and innovations of this paper are summarized as follows.(1)Aim at existing distance measures of the probabilistic hesitant fuzzy set,which requires reordering the elements of probabilistic hesitant fuzzy elements,and needs to supplement the elements of the probabilistic hesitant fuzzy element so that their length is the same,an improved distance measure of the probability hesitant fuzzy set is proposed.This new distance measure does not need to reorder the elements of the probability hesitant fuzzy elements,and does not need the same length of probability hesitant fuzzy elements.Then it is proved that the value of this distance measure is between [0,1],and this new distance measure has the properties of symmetry and trigonometric inequality.(2)Based on the new distance measure of the probabilistic hesitant fuzzy set,a decision method of the probabilistic hesitant fuzzy multi-attribute TOPSIS decision method with unknown attribute weight is proposed.This decision making method uses the new distance to measure the relative proximity of the scheme to the ideal solution and the negative ideal solution.Combined with the idea of the maximum deviation method,the specific steps of solving the attributes weight model and TOPSIS method are given.Then,the effectiveness and rationality of the decision making method are analyzed with an example.(3)On account of the new distance measure of the probabilistic hesitant fuzzy set,the attribute reduction of the probabilistic hesitant fuzzy information system is discussed.First of all,the probabilistic hesitant fuzzy information system is defined.At the same time,the fuzzy tolerance relation between objects in the probabilistic hesitant fuzzy information system is given by using the proposed distance measure,and the properties of the relation are discussed.Then,the upper and lower approximations of decision classes and related properties in theprobabilistic hesitant fuzzy information system are given.Next,the fuzzy similarity relation is defined,and the properties of the fuzzy similarity relation are studied.The distribution reduction are proposed.In order to obtain positive region reduction and distribution reduction,use the heuristic reduction algorithm to give the algorithm of positive region reduction and distribution reduction.Finally,the process of positive region reduction and distribution reduction algorithm are systematically illustrated by some examples.