A note on the set-theoretical defuzzification

Abstract

The set-theoretic defuzzification (SD) method has been claimed to be a rational defuzzification method which is exclusively based on the fuzzy set theory [J. Valente de Oliveira, Fuzzy Sets and Systems 76 (1995) 63]. However, this paper shows the examples that the method is not applicable with general t-norms and t-conorms. For normal fuzzy sets, we show that the SD values of a normalized fuzzy set are located outside the support of the given fuzzy set with the Min-type t-norm operator. For non-normal fuzzy sets, we show that the defuzzification values of certain kinds of non-normal fuzzy sets are unit value regardless of the support of the fuzzy set. Finally, we show that the SD method does not satisfy two of three basic constraints for rational defuzzification operator, which is proposed by Runkler and Glesner [Proc. 2nd IEEE Internat. Conf. on Fuzzy Systems, p. 1161]. (C) 1998 Elsevier Science B.V. All rights reserved.