Fuzzy Entropy: A More Comprehensible Perspective for Interval Shadowed Sets of Fuzzy Sets

Shadowed sets, proposed by Pedrycz, provide a three-way approximation scheme for transforming the universe of a fuzzy set into three disjoint areas, i.e., elevated area, reduced area, and shadow area. To calculate a pair of decision-making thresholds, an analytic method was proposed by solving a minimization problem of the uncertainty arising from the three areas. However, some uncertainty will be lost in the process of constructing the shadowed set model using Pedrycz's method. Moreover, few references on how to measure the uncertainty of shadow sets exist. In this article, a comprehensible method for measuring the fuzzy entropy of a shadowed set, i.e., interval fuzzy entropy, is defined. Based on the interval fuzzy entropy, a new shadowed set model, namely, interval shadowed sets, is proposed. Compared with Pedrycz's model, the main difference is that the range of the shadow area in this model is [beta, alpha] (0 <= beta < alpha <= 1) while not [0, 1]. By solving a fuzzy entropy loss-minimization problem, a pair of optimal thresholds, alpha and beta, can be obtained. Finally, the results of the instance analysis of different types of representative membership functions and many experiments show that the fuzzy entropy loss of the interval shadowed set is lower than that of the traditional shadowed set of a fuzzy set. These results enrich shadowed set theory from a new perspective.