Integral Aggregations of Continuous Probabilistic Hesitant Fuzzy Sets

Probabilistic hesitant fuzzy sets (PHFSs) add the probability value corresponding to each degree of membership on the basis of hesitant fuzzy sets, so as to express the initial decision information given by experts more accurately and comprehensively. In this article, we mainly study how to integrate large-scare probabilistic hesitant fuzzy information more efficiently. We first discuss some basic operation laws of probabilistic hesitant fuzzy numbers, based on which the concepts of continuous PHFSs and continuous probabilistic hesitant fuzzy functions (c-PHFFs) are defined. They are the main objects of our research. We further explore definite integrals of the c-PHFFs and their related properties. They have direct and powerful applications in continuous probabilistic hesitant fuzzy environments, and lay the foundation for subsequent theoretical analysis. Based on the weight density function, we finally get the weighted-integral operator of continuous probabilistic hesitant fuzzy information. Then, some important properties of this integration operator are studied, including normalization, monotonicity, boundedness, etc. We are also devoted to revealing the inner connection between the continuous probabilistic hesitant fuzzy weighted-integral operator and the probabilistic hesitant fuzzy weighted averaging operator, the latter is usually used when dealing with discrete information. At last, we state why it is necessary to introduce a novel aggregation method based on continuous probabilistic hesitant fuzzy definite integrals, and in turn provide an application of the proposed method to prove its validity and rationality.