On the centroid of a general type-2 fuzzy set with monotonically increasing second membership functions

Abstract

In a type-2 fuzzy logic system, the centroid computation of a general type-2 fuzzy set (T2 FS) is one of the most important type reduction methods. Nearly all of the existing algorithms are based on the α-plane representation or z-slice representation. Therefore, all of these algorithms can only approximately compute the centroid of a general T2 FS and cannot obtain the analytic expression of the centroid of a general T2 FS. The main objective of this study is to obtain the analytic expression of the centroid of a general T2 FS with monotonically increasing second membership functions based on a discrete universe of discourse. First, assuming that the second membership functions of a general T2 FS is linear and monotonically increasing, a method is proposed to compute its centroid. Second, an approach to calculate the centroid of a general T2 FS is also analyzed, if the second membership functions are only monotonically increasing. Finally, two numerical examples are given to demonstrate the calculation processes of the proposed method in this study. Notably, the proposed method can also be employed to obtain the analytic expression of the centroid of a general T2 FS with monotonically decreasing second membership functions.