On extension of multiplicative consistency to interval fuzzy preference relations

Various definitions of consistency for interval fuzzy preference relations have been proposed in the literature. The aim of this paper is to review the definitions of multiplicative consistency based on the extension of Tanino’s multiplicative-transitivity property and to point out their drawbacks. In particular, some of the definitions proposed in the literature are not invariant under permutation of objects in interval fuzzy preference relations and some of them violate reciprocity of the pairwise comparisons of objects, which is not acceptable. The weak form of multiplicative-consistency defined by Xu and Chen (Eur J Oper Res 184:266–280, 2008) is approved as the only one appropriate among all definitions examined in the paper. Further, a new definition of multiplicative consistency that is much stronger than the definition proposed by Xu and Chen (Eur J Oper Res 184:266–280, 2008) is introduced. Tools for verifying both the multiplicative consistency defined in this paper and the multiplicative weak consistency defined by Xu and Chen (Eur J Oper Res 184:266–280, 2008) are proposed, and some interesting properties of both multiplicatively consistent and multiplicatively weakly consistent interval fuzzy preference relations are demonstrated. Finally, numerical examples are provided in order to illustrate and compare both types of consistency.