Measures of Probabilistic Interval-Valued Intuitionistic Hesitant Fuzzy Sets and the Application in Reducing Excessive Medical Examinations

Distance and similarity measures act as the links from the expression of fuzzy information to various application fields such as decision making, cluster analysis, medical diagnosis, and so on. Some scholars have paid attention to improving the approaches of defining the distance and similarity measures. One of the most common practical ways to update the measures is to add an item about hesitancy into the traditional definitions of them. This way is very effective to promote the performances of decision making. However, there are still some drawbacks in the existing distance and similarity measures even though some of them have contained the hesitant information. In this paper, we first define a probabilistic interval-valued intuitionistic hesitant fuzzy set (PIVIHFS) as an extended mathematical expression of fuzzy sets. Afterward, the axiomatic concepts are given for the distance, similarity, and entropy measures of PIVIHFSs. Then, we investigate the relationships among the new distance, similarity, and entropy measures. Besides, we establish some new measure models deduced by the axiomatic concepts of PIVIHFSs. After that, we compare the performances of the new distance measure models and several related existing ones, such as Hamming distance measure and Euclidean distance measure, so do the similarity measures. Finally, we illustrate their applicability concerning the medical diagnosis, which can be used to reduce unnecessary medical examinations.