Revisiting fuzzy set operations: A rational approach for designing set operators for type-2 fuzzy sets and type-2 like fuzzy sets

Created by Zadeh in 1965, Fuzzy Set Theory (FST) has been continuously developed in the past five decades and has established itself one of the preeminent approximate reasoning methodologies. In the literature, there exists multitudes of differing approaches for specifying FST's foundational building blocks (such as the various versions of fuzzy set and arithmetic operators) that it has become challenging for an FST user (e.g. an otherwise proficient engineer who is not an expert in fuzzy set theory) to select or tailor-make fuzzy operators truly appropriate for solving his/her problems. In a recent work (Ngan, 2017a), within the type-1 fuzzy setting, a framework termed probabilistic linguistic computing (PLC) has been proposed to empower such type of FST users to understand and dissect fuzzy operators available in the literature, and to tailor-make their own fuzzy operators to solve their own application problems. In this article, we will revisit this work and extend it to the general type-2 fuzzy setting, as well as to other type-2-like fuzzy settings. Among the contributions are: (i) we demonstrate that the generalized PLC framework can indeed provide a concrete, accessible pathway for the FST users to understand and tailor-make their own set operators in the type-2 and type-2-like fuzzy settings - this is distinct from many other theoretical studies within the type-2 fuzzy set (T2FS) literature which primarily focus on deep technical developments while offering virtually no accessible guidelines for the FST users to use T2FS in their applications; (ii) the set operators constructed from the proposed approach are computationally simple and efficient; and (iii) the proposed approach offers a straightforward way to understand the logical relations between set operators for the general type-2 fuzzy sets and those for type-2-like fuzzy sets.