On the compatibility between defuzzification and fuzzy arithmetic operations

Abstract

Since the introduction of the extension principle by Zadeh, the arithmetic of fuzzy numbers has gained importance both from the theoretical and the practical points of view. For the former, many works were accomplished on the topological level as well as on the parametrization level in order to improve the foundation of the theory and to simplify the performance of different combination operations. For the latter, in many practical applications, the need for a permanent switch from a fuzzy representation to a numerical representation is patent. This transformation is usually carried out by the defuzzification process. This paper addresses some theoretical results about some invariance properties concerning the relationships between the defuzzification outcomes and the arithmetic of fuzzy numbers. One of the benefits of such analysis is the fact that when the matter is the determination of the defuzzified value pertaining to the result of some manipulation of fuzzy quantities, the explicit determination of the resulting fuzzy set (or distribution) can be obviated, while the process may be restricted to a standard computation over single values corresponding to defuzzified initial inputs.