Defuzzification of fuzzy intervals

Abstract

Abstract This paper deals with defuzzification of fuzzy intervals classified as regular or irregular. Based on the defuzzification method used, a fuzzy interval is regular if its membership function is maximized at its defuzzified value. Otherwise the fuzzy interval is irregular. Three methods for defuzzification are presented in this paper: (1) mean-of-maxima (MOM) method; (2) center-of-area (COA) method; and (3) fuzzy mean (FM) method. Fuzzy intervals with symmetrical membership function are regular, and all three defuzzification methods give the same results. For any regular fuzzy interval based on the COA method its defuzzified value is equal to the middle point of the corresponding ‘mean value’ interval. For irregular fuzzy intervals, two approaches are proposed. One is to apply the defuzzification method but relax the reasonable but not necessary (RNN) requirement. Another is to apply the RNN requirement but relax the defuzzification method. This paper shows that the regularity of a fuzzy interval is preserved after applying an arithmetic operation with a nonzero real number. The formulas for calculating the defuzzified value of the arithmetic operations between a regular fuzzy interval and a real number are derived. This paper also shows that the regularity of two fuzzy intervals based on the COA method is preserved after applying addition or subtraction operation between them. The formulas for calculating the defuzzified value of the sum and the difference of regular fuzzy intervals are derived. The properties of fuzzy intervals discussed in this paper can be taken into consideration before or when a defuzzification method is selected.