An approach to the comparison of fuzzy subsets with an alpha -cut dependent index

Abstract

An approach to the comparison of fuzzy subsets or fuzzy numbers is proposed and studied. A function-type index J/sub ij/( alpha ), indicating a degree of dominance of a fuzzy subset A/sub i/ over the other A/sub j/, is derived from an alpha -cut of the difference of the two subsets compared. This index can be converted to a single-valued version F/sub ij//sup 0/ by way of a alpha -weighted averaging over alpha for a quick reference to the conclusion of comparison. Linguistic descriptions of the results of comparison and a fuzzy ordering relation are discussed. For the case of trapezoidal membership functions the shapes of J/sub ij/( alpha ) are classified and the formulas to calculate J/sub ij//sup 0/ are obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>