A concrete and rational approach for building type-2 fuzzy subsethood and similarity measures via a generalized foundational model

Abstract

Subsethood and similarity between fuzzy sets have always been intensely studied concepts in fuzzy set theory (FST). However, researches on subsethood and similarity for truly general type-2 fuzzy sets (T2FSs) have been comparatively scarce, because of the intrinsic difficulties of directly dealing with the secondary membership functions of very general nature. While the advent of the α-plane/z-slice representation by Mendel and his colleagues as well as by Wagner and Hagras has led to progress in confronting this challenge, there remains quite a number of limitations and unsolved issues. The contribution of this article is to utilize a generalized foundational model (introduced in Ngan, 2018) to construct T2FS subsethood and similarity measures as rationally, concretely and systematically as feasible, such that (i) these T2FS measures are applicable to truly general type-2 fuzzy sets, that (ii) the actions of these measures can be very simply understood, analyzed and even customized by the T2FS users, and that (iii) these T2FS measures can process and output results that appropriately maintain and reflect the high degree of fuzziness involved in T2FSs. Last but not least, for applications, (iv) these measures will be demonstrated on multiple criteria decision making and pattern recognition problems, and (v) in a brief sketch, we will illustrate how the generalized-foundational-model-based method of building T2FS subsethood and similarity measures can be adapted to building other T2FS measures that embrace the advantages described in (i), (ii) and (iii).