On additive consistency of interval fuzzy preference relations

- Additive consistency of interval fuzzy preference relations is studied.
- Definitions provided by other authors are reviewed and drawbacks are pointed out.
- New definitions of additive consistency and additive weak consistency are proposed.
- The new definitions are invariant under permutation of objects.
- The new definitions preserve the additive reciprocity of pairwise comparisons.

Various definitions of consistent interval fuzzy preference relations have been proposed in the literature. This paper aims to review the definitions of additive consistency based on the interval extension of Tanino's additive-transitivity property. It is demonstrated that some of the definitions are not invariant under permutation of objects in interval fuzzy preference relations and some of them violate the reciprocity of pairwise comparisons of objects, which is not acceptable. Afterwards, two new definitions of consistency, additive consistency and additive weak consistency, based on a proper interval extension of Tanino's additive-transitivity property are proposed. Both definitions are invariant under permutation of objects and do not violate the additive reciprocity of pairwise comparisons in interval fuzzy preference relations. Further, useful tools for verifying the consistency are given, and some interesting properties of both additively consistent and additively weakly consistent interval fuzzy preference relations are demonstrated. The new definitions of consistency are compared with the reviewed definitions on several illustrative examples and on a real-life decision-making problem.