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

An approach to the comparison of fuzzy subsets or fuzzy numbers is proposed and studied. A function-type index J_ij(α), indicating a degree of dominance of a fuzzy subset A_i over the other A_j, is derived from an α-cut of the difference of the two subsets compared. This index can be converted to a single-valued version F_ij⁰ by way of a α-weighted averaging over α 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_ij(α) are classified and the formulas to calculate J_ij⁰ are obtained