An Alpha-Cut Evaluation of Interval-Valued Fuzzy Sets for Application in Decision Making

In this paper we deal with the problem of evaluating an interval-valued fuzzy set, that is a fuzzy quantity delimited by two (lower and upper) membership functions. The problem of associating this type of set with a real number has been dealt with in different ways. Karnik and Mendel proposed an algorithm for computing the mean of centroids of membership functions that lie within the area delimited by the lower and upper memberships. Nie and Tan choose a simpler way by calculating the centroid of the average of the lower and upper membership functions. In both cases, the value obtained is useful not only in ranking problems but also as a value of defuzzification if the set is the final output of a fuzzy inference system. Since in this last case the obtained set is usually not normal and not convex, the centroid seems to be the only useful defuzzifier. Our purpose is to show that other methods based on alpha-cuts, usually applied in convex type-1 case, can also provide useful answers.