\(p\)-symmetric fuzzy measures. (English) Zbl 1068.28013

Miranda, Pedro; Grabisch, Michel; Gil, Pedro

Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 10, Suppl., 105-123 (2002).

Symmetric fuzzy measures (set functions) depend on the cardinality of the measured subsets only. The authors generalize the concept of symmetry (indistinguishability of single elements of the underlying universe) to the concept of symmetry with \(p\in N\). In this case, the universe is partitioned into \(p\) disjoint classes of indistinguishable elements, and thus the corresponding fuzzy measure (set function) can be understood as a function of \(p\) cardinality variables. In the paper, several properties of \(p\)-symmetric fuzzy measures are discussed. Recall that for a finite universe, the Choquet integral with respect to a symmetric fuzzy measure is just the OWA operator introduced by R. R. Yager [IEEE Trans. Syst. Man. Cybern. 18, No. 1, 183–190 (1988; Zbl 0637.90057)]. The authors have introduced a formula for the Choquet integral with respect to a \(p\)-symmetric fuzzy measure which in a special case can be rewritten into a convex sum of OWA operators acting on sets from the above mentioned partition characterizing the \(p\)-symmetry.

Reviewer: Radko Mesiar (Bratislava)

MSC:

28E10

Fuzzy measure theory

Keywords:

fuzzy measures; Choquet integral; symmetry; OWA operator; ordered weighted aggregation operator

Citations:

Zbl 0637.90057