Una metodología difusa para aproximar conjuntos difusos de la recta real mediante números difusos

Zusammenfassung

In this Memory we study two open problems in the context of fuzzy sets, introduced by L.A. Zadeh in 1965 as a mathematical way of expressing the uncertainty that surrounds us in the real world. On the one hand, we present an approximation operator that associates a unique normal fuzzy number to each fuzzy set in the interval [0, 1]. The need and the great applicability of this operator is justified by the fact that the main mathematical and statistical techniques that have been proposed up to now (regression, distances, etc.) make exclusively use of fuzzy numbers. Such techniques cannot be applied in general contexts where the input data are arbitrary fuzzy sets, since they require the special algebraic and geometric properties that, concretely, fuzzy numbers verify. In this way, the proposed operator is able to transform the fuzzy input data, in a reasonable way, into fuzzy numbers to work with afterwards. This operator depends on a wide collection of initial parameters that give it great ductility from its very conception. Additionally, the main properties that this operator satisfies are shown, among which we highlight the study of its fixed points and the minimizing property that it verifies. On the other hand, we introduce a definition of overlap index in the framework of type-2 fuzzy sets. We had observed that, after the introduction of overlap functions and their subsequent success in applications to different research problems, some authors had tried to extend this notion to the field of type-1 fuzzy sets, which had been achieved inspired by Zadeh’s consistency index. Thus, it was interesting to address the case of type-2 fuzzy sets, which are able to model uncertainty situations through a finer algebraic structure where type-1 fuzzy sets themselves do not reach. In this way, we introduce the main conditions to be verified by an overlap index on type-2 fuzzy sets, we study its first properties and we show large families of examples of this class of indices, relating the different levels of fuzzy structures. Furthermore, we discuss the normality condition for this class of indices and we show alternatives to the one presented here. Finally, we illustrate how to employ overlap indices of type (2, 0) and (2, 1) to implement inferential algorithms for type-2 fuzzy interpolative fuzzy systems, such that a conclusion can be drawn from fuzzy rules and a fact, all of them expressed as type-2 fuzzy sets.