Aggregation Operators Defined by k-Order Additive/Maxitive Fuzzy Measures

Abstract

It is well known that the Choquet and Sugeno integrals w.r.t. a discrete fuzzy measure can be considered as aggregation operators. In this paper, we study in detail two special cases of symmetric fuzzy measures, i.e. fuzzy measures of which the values only depend upon the cardinality of the arguments. The first case is that of a symmetric k-order additive fuzzy measure, i.e. a fuzzy measure of which the Möbius transform vanishes in sets with cardinality greater than k. A new increasing sequence of binomial OWA operators is introduced. It is recalled that weighted sums of aggregation operators, with possibly negative weights, may also lead to aggregation operators. The Choquet integral w.r.t. a symmetric k-order additive fuzzy measure is then characterized as such a weighted sum of the first k binomial OWA operators. The second case is that of a symmetric k-order maxitive fuzzy measure, i.e. a fuzzy measure of which the possibilistic Möbius transform vanishes in sets with cardinality greater than k. The Sugeno integral w.r.t. a symmetric k-order maxitive fuzzy measure is then characterized as a weighted maximum of the last k order statistics.