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 (α,β) on the fuzzy membership function. An element whose membership grade equals to or is greater than α is put into the positive region, an element whose membership grade equals to or is less than β is put into the negative region, and an element whose membership grade is between β and α 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.