Defuzzification of Uncertain Fuzzy Sets

At the present time, the only deficiency in developing efficient realizations of general type-2 fuzzy logic systems are effective defuzzification procedures for general fuzzy valued fuzzy sets, since the common defuzzification procedures (like the exhaustive centroid method and the

α

-cut strategy) require them to be discrete in two dimensions. We propose to limit the discretization only to the primary domain, which is a dimension of elements, and to obtain a convex and normal centroid fuzzy set (conditions for this are given in a corresponding theorem). Our main contribution to this chapter are exact and approximate formulae and procedures for the extended centroid of triangular, trapezoidal, Gaussian and asymmetric-Gaussian fuzzyvalued fuzzy sets. Additionally, this chapter provides conditions under which centroids preserve triangular, trapezoidal or Gaussian shapes of membership functions. Since our results are still based on the KM iterative procedure for interval type-reduction, we recall basic defuzzification methods for intervalvalued fuzzy sets. To make the following discussion complete, we leave proofs of propositions and theorems in this chapter instead of referring the reader to appendices.