Transitive full covers of incomplete preference relations

When a decision maker is asked to compare a set of alternatives, it may happen that the information provided is incomplete because she has no time to compare all the options or is unable to compare some alternatives against others. This contribution departs from an incomplete fuzzy weak preference relation by completing it on a consistent way with the known information. Herein the original notion of fuzzy transitivity, min-transitivity, is considered. If the decision maker is assumed to be coherent, i.e. under the assumption that the known preferences satisfy transitivity, a complete transitive preference relation that preserves the information given by the coherent decision maker is derived. In the case of the given preference values violate transitivity, a degree of transitivity is defined, and an algorithm is presented to provide the most coherent preference relation that preserves the original information according to that degree of transitivity.

Automated summary

This study completes an incomplete fuzzy weak preference relation in a consistent way to derive a preference relation that preserves the original information. A complete transitive preference relation is obtained if the decision maker is coherent, while a degree of transitivity is defined to provide the most coherent preference relation when preference values violate transitivity.