Decision-theoretic three-way approximations of fuzzy sets

A three-way, three-valued, or three-region approximation of a fuzzy set is constructed from a pair of thresholds (alpha,beta) on the fuzzy membership function. An element whose membership grade equals to or is greater than alpha is put into the positive region, an element whose membership grade equals to or is less than beta is put into the negative region, and an element whose membership grade is between beta and alpha is put into the boundary region. A fundamental issue is the determination and interpretation of the required pair of thresholds. In the framework of shadowed sets (i.e., an example of three-way approximations of fuzzy sets), Pedrycz provides an analytic solution to computing the thresholds by searching for a balance of uncertainty introduced by the three regions. To gain further insights into three-way approximations of fuzzy sets, we introduce an alternative decision-theoretic formulation in which the required thresholds are computed by minimizing decision cost. (C) 2014 Elsevier Inc. All rights reserved.