An inclusion measure between general type-2 fuzzy sets

The inclusion measure indicates the degree that a fuzzy set is contained in another fuzzy set and can be used in many fields. However, the inclusion measure between type-2 fuzzy sets has received little attention. Hence in this paper, we propose an inclusion measure for general type-2 fuzzy sets. Firstly, we select an axiomatic definition for the new inclusion measure. Then, according to the selected axiomatic definition, we propose a computation formula by considering the FOU and the secondary membership function of the general type-2 fuzzy set. Finally, we present two examples to explain its calculation and validate its performance. The results show that the proposed inclusion measure is reasonable and reliable for general type-2 fuzzy sets.

I. Introduction

Over the past four decades, fuzzy sets, which let us model uncertainties in a way that other techniques may be unable, have been perfectly developed and widely used in control and other fields. However, general fuzzy sets can not model directly some linguistic uncertainties because of their totally crisp membership functions. For that, Zadeh [1] extended general fuzzy sets (type-l fuzzy sets) to type- 2 fuzzy sets, which are characterized by 3-D membership functions and can minimize the effects of uncertainties in a more effective way. From then on, more studies were explored by numerous researchers. For example, Mizumoto and Tanaka [2] studied the properties of membership grades of type-2 fuzzy sets. Nieminen [3] analyzed the algebraic structures of type-2 fuzzy sets. Dubois and Prade [4] studied the set theoretic operations of type-2 fuzzy sets. Mendel et al. [5]–[7] established a complete type-2 FLS theory. Today, type-2 fuzzy sets have been further studied and widely used in many areas [8].