Fuzzistics for interval type-2 fuzzy sets using centroid as measure of uncertainty

How to design an interval type-2 fuzzy set (IT2 FS) from data has been a challenging issue till now. Mendel pointed that the centroid of an IT2 FS may be used as a measure of uncertainty to constraint its footprint of uncertainty (FOU) [1]. How to exact centroid from data has been studied in [1]; however, there exists no method to design an IT2 FS such that its centroid matches the desired one. To fill this gap, this paper will present an approach to obtain FOU parameters of an IT2 FS such that its centroid equals to the desired one. To propose this approach, the centroid requirement was formulated as two equations about all the FOU parameters. The strategy used to obtain FOU parameters satisfying the established equations is to predetermine all the FOU parameters except two of them so that these the established equations can be simplified to two single-variable equations. Then the other two FOU parameters can be obtained by solving these two single-variable equations. Among existing root-finding algorithms, false position algorithm is recommended to solve the established single-variable equations. The overall merits of the proposed approach is its simplicity in the implementation, but also its applicability to an IT2 FS with arbitrary shapes of FOU. In addition, numerical examples are provided to further illustrate how to use the proposed approach to obtain the FOU parameters of an IT2 FS.