Shadowed Type-2 Fuzzy Sets -Type-2 Fuzzy Sets with shadowed secondary membership functions

Abstract

General Type-2 Fuzzy Sets (GT2 FSs) have been originally proposed to allow for modeling uncertainty associated with the membership grades of Type-1 (T1) FSs. However, because of the computational complexity associated with the processing of GT2 FSs, only their constrained version, the Interval T2 (IT2) FSs, have been widely used. While IT2 FSs allow for fast processing, they lack the expressive power of GT2 FSs when modeling various kinds of uncertainties. In order to combine the best of both types, this paper proposes a novel class of T2 FSs - the Shadowed Type-2 (ST2) FSs. The ST2 FS is a T2 FS with secondary membership functions represented as Shadowed Sets (SSs). Shadowed sets, originally proposed by Pedrycz, are directly induced by the T1 fuzzy membership functions and they are designed to conserve the amount of uncertainty in the original T1 FS. In a similar manner, an ST2 FS is directly induced by a GT2 FS via transforming all the T1 fuzzy secondary membership functions into Shadowed Sets. The resulting ST2 FSs can thus better capture the uncertainty in the original GT2 FSs when compared to the constrained IT2 FSs. Additionally, ST2 FSs offer very efficient computational framework since the secondary membership grades can only attain three values of 0, 1, or completely uncertain (shadowed) grade of [0,1]. This paper introduces the representation, the elementary set-theoretic operations and several methods for type-reduction and defuzzification of ST2 FSs. The modeling capability of ST2 SS was demonstrated on several examples.