An axiomatic approach to approximation-consistency of triangular fuzzy reciprocal preference relations

Under the assumption of rational economics, the consistency of judgments is one of the important issues in multiple criteria decision making (MCDM) methods. We propose three axiomatic properties of the consistent judgments in relative measurements being considered from the perspective of strict logic and rationality: (A1) the reciprocal property of pairwise comparisons, (A2) the invariance of consistency with respect to permutations of alternatives, and (A3) the robustness of the ranking of alternatives. These properties are further applied to analyze the consistency of the judgments expressed by positive real numbers and triangular fuzzy numbers, respectively. The inconsistency of comparison matrices encountered in the Analytic Hierarchy Process (AHP) can be considered to be the case of weakening the axiomatic property (A3). Fuzzifying the judgments is due to the weakening of axiomatic property (A1). A method of weakening the axiomatic properties for triangular fuzzy reciprocal matrices is proposed and a new concept of approximation-consistency is put forward to distinguish from the typical consistency. By considering the permutations of alternatives, an illustrative example is presented to show the use of the proposed procedures for solving the decision making problems with triangular fuzzy reciprocal preference relations.