Aggregation-based extensions of fuzzy measures

Authors:
Anna Kolesárová;Andrea Stupňanová;Juliana Beganová
Affiliations:
Institute of Information Engineering, Automation and Mathematics Slovak University of Technology, 812 37 Bratislava, Slovakia;Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia;Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia
Venue:
Fuzzy Sets and Systems
Year:
2012

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Abstract

We present a method extending fuzzy measures on N={1,...,n} (represented as Boolean utility functions) to n-ary aggregation functions (utility functions) by means of a suitable n-ary aggregation function and the Mobius transform of the considered fuzzy measure. The method generalizes the well-known Lovasz and Owen extensions of nondecreasing pseudo-Boolean functions linked to fuzzy measures. All n-ary aggregation functions suitable for the proposed construction are completely characterized, including, among others, all n-ary copulas. Associative extended aggregation functions applicable in the case of an arbitrary arity are also completely characterized.