A new defuzzification method and ranking method for type-2 fuzzy numbers

The type-2 fuzzy number (T2 FN) is a special type-2 fuzzy set whose all secondary membership functions are fuzzy numbers. This paper proposes a new defuzzification method and ranking method for T2 FNs. The defuzzification method can be expressed in a closed-form formula or can be divided into two computational steps. The first step computes the centroid of each secondary membership function to obtain a type-reduced set using the method derived from generalizing the existing alpha-cut defuzzification methods by introducing a parameter that controls the importance of secondary membership degrees, and the second step calculates the centroid of the type-reduced set. The defuzzified values are used to provide position information of T2 FNs. Secondly, we define a function to calculate the area enclosed by the image of the type-reduced set and the coordinate axes, and prove that this function is continuous. The values of this function are used to provide shape information of T2 FNs. Finally, to rank T2 FNs, we define a family of quasi- orders based on their position and shape information, and illustrate the validity and rationality of these quasi-orders through examples.