Consistency of total fuzzy relations: New algorithms to detect and repair inconsistent judgments

Abstract

Fuzzy orders, especially T-preorders, can express the vague priority over a list of alternatives. The cardinal consistency of such relations is achieved through the T-transitivity condition, while the ordinal consistency of the final decision at a given threshold α∈(0,1] is not guaranteed by the T-consistency. This paper investigates the problem of ordinal consistency, as the minimum requirement for a reliable judgment, of a partial preference induced by a fuzzy relation at a given level α∈(0,1], so-called α-preference relation. We first define new concepts of T−L-cyclic and T-consistency at a given level α for fuzzy relations. Based on digraph theory, a new methodology is designed for finding the locations of all consistent and inconsistent L-cycles of an α-preference. An algorithm is proposed to eliminate the ordinal inconsistency through the TM-transitivity. Meanwhile, a new T-consistency index is introduced to measure the acceptable consistency level of fuzzy relations. Furthermore, some results concerning the ordinal consistency of T-preorders are applied to design another algorithm for creating a consistent collective judgment based on initial fuzzy assessments of alternatives in a group ranking problem. The proposed method constructs the consistent fuzzy order matrix for each case (expert), then calculates the group consensus by aggregating these matrices. The collective fuzzy order is then checked for consistency level. Finally, a numerical example with a real dataset is given to illustrate the application of the proposed method in a ranking problem.