Computing a T-transitive lower approximation or opening of a proximity relation

  • Authors:

  • L. Garmendia;A. Salvador;J. Montero

  • Affiliations:

  • Complutense University of Madrid, Spain;Technical University of Madrid, Spain;Complutense University of Madrid, Spain

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2009

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Abstract

Since transitivity is quite often violated even by decision makers that accept transitivity in their preferences as a condition for consistency, a standard approach to deal with intransitive preference elicitations is the search for a close enough transitive preference relation, assuming that such a violation is mainly due to decision maker estimation errors. In some way, the more number of elicitations, the more probable inconsistency is. This is mostly the case within a fuzzy framework, even when the number of alternatives or object to be classified is relatively small. In this paper we propose a fast method to compute a T-indistinguishability from a reflexive and symmetric fuzzy relation, being T any left-continuous t-norm. The computed approximation we propose will take O(n^3) time complexity, where n is the number of elements under consideration, and is expected to produce a T-transitive opening. To the authors' knowledge, there are no other proposed algorithm that computes T-transitive lower approximations or openings while preserving the reflexivity and symmetry properties.