The ordinal consistency of a fuzzy preference relation

In this paper, we first analyze the rationality of weak transitivity of a fuzzy preference relation defined by Tanino [Fuzzy preference orderings in group decision making, Fuzzy Sets and Systems 12 (1984) 117–131]. We then propose a revised definition of weak transitivity (we call it ordinal consistency). We propose the ordinal consistency index (OCI) to measure the degree of ordinal consistency of a fuzzy preference relation, which is to count the unreasonable 3-cycles in a directed graph that represents the fuzzy preference relation. Afterwards, a procedure to compute the order consistency index and to locate each cycle, as well as to find the inconsistent judgments in the fuzzy preference relation is proposed. In order to repair the inconsistency of a fuzzy preference relation, an algorithm is developed to find and remove 3-cycles in the graph. The algorithm eliminates 3-cycles in a graph more effectively and the proposed method for improving consistency method aims to preserve the initial preference information of the decision maker. Furthermore, the method can be used not only for a strict fuzzy preference relation, but also for non-strict fuzzy preference relation. Finally, we provide some examples to show the effectiveness and validity of the proposed method.