An Extended Type-Reduction Method for General Type-2 Fuzzy Sets

A centroid type-reduction strategy for computing the centroids of type-2 fuzzy sets based on decomposed <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-planes was proposed by Liu. However, it cannot be applied to type-2 fuzzy sets with concave secondary membership functions. In this paper, we extend the Liu's method so that the centroids of type-2 fuzzy sets with concave secondary membership functions can be derived. For each decomposed <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-plane, we convert it into a group of interval type-2 fuzzy sets. The union of the centroids of its member interval type-2 fuzzy sets constitutes the centroid of the <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> -plane. Then, the weighted union of the centroids of the decomposed <inline-formula><tex-math notation="LaTeX">$\alpha$ </tex-math></inline-formula>-planes becomes the centroid type-reduced set of the original type-2 fuzzy set. When dealing with type-2 fuzzy sets with convex secondary membership functions, our proposed method is reduced to the Liu's method.