Towards a More Subtle (and Hopefully More Adequate) Fuzzy "And"-Operation: Normalization-Invariant Multi-Input Aggregation Operators

Authors

Yusuf Güven, Istanbul Technical UniversityFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

6-1-2024

Comments

Technical Report: UTEP-CS-24-34

Abstract

Many reasonable conditions have been formulated for a fuzzy "and"-operation: idempotency, commutativity, associativity, etc. It is known that the only "and"-operation that satisfies all these conditions is minimum, but minimum is not the most adequate description of expert's "and", and it often does not lead to the best control or the best decision. Many other more adequate "and"-operations (t-norms) have been proposed and effectively used, but they do not satisfy the natural idempotency condition. In this paper, we show that a small relaxation of the usual description of "and"-operations leads to the possibility of non-minimum idempotent operations. We also show that another natural condition -- of normalization invariance -- uniquely determines the resulting "and"-operation. This new "and"-operation is not only more intuitive, it leads to better application results.