On multiplicative consistency of interval and fuzzy reciprocal preference relations

•Multiplicative consistency of interval and fuzzy reciprocal preference relations is studied.•Definitions provided by other authors are reviewed and drawbacks are pointed out.•New definition of multiplicative consistency is proposed.•The new definition is invariant under permutation of objects.•The new definition preserves the multiplicative reciprocity of pairwise comparisons.

Extension of Saaty's definition of consistency to interval and fuzzy reciprocal preference relations is studied in the paper. The extensions of the definition to interval and triangular reciprocal preference relations proposed by Wang (2005), Liu (2009), Liu et al. (2014) and Wang (2015a, 2015b) are reviewed and some shortcomings in the definitions are pointed out. Particularly, as was already shown by Wang (2015a, 2015b), the definitions of consistency proposed by Liu (2009) and Liu et al. (2014) are not invariant under permutation of compared objects. Wang's (2015a, 2015b) definitions rectify this drawbacks. However, as is pointed out in this paper, Wang's definitions of consistent interval and triangular reciprocal preference relations do not keep the reciprocity of pairwise comparisons, which is the substance of reciprocal preference relations. In this paper, definitions of consistent interval, triangular and trapezoidal reciprocal preference relations invariant under permutation of compared objects and preserving the reciprocity of pairwise comparisons are introduced. Useful tools for verifying the consistency are proposed and some properties of consistent interval and fuzzy reciprocal preference relations are derived. Furthermore, the new definition of consistency for interval reciprocal preference relations is compared with the definition of consistency proposed by Wang et al. (2005), and numerical examples are provided to illustrate the difference between the consistency definitions.