On the decomposition problem of fuzzy sets

Abstract

In this paper, we consider an important problem that is frequently encountered and yet almost universally ignored both in the application and in the theoretic studies of fuzzy sets. We refer to this problem as the decomposition of fuzzy sets. In nearly all applications of fuzzy sets theory (e.g., fuzzy decision making, fuzzy control, fuzzy forecasting, etc.), the output is, in general, a fuzzy set; this is then defuzzified before it is applied. Although defuzzification may be necessary if a crisp quantity is definitely needed, when a fuzzy output is appropriate, desired or required in an application, then the defuzzification step becomes unnecessary. Yet, in this case, one finds that it is some times difficult to correctly interpret a fuzzy quantity or identify the true meaning of this fuzzy output using a family of predefined fuzzy sets. Thus, how to match the fuzzy output and the predefined fuzzy sets is the problem of concern in this paper. To address this problem, we first present and review some definitions and then proceed with an investigation of the preliminary properties of these concepts, complementing them with illustrative examples. Finally, we propose some algorithms for the solution of this problem.