Visualizing and rectifying different inconsistencies for fuzzy reciprocal preference relations

Abstract

Consistency is related with reasonableness. This paper proposes a graphical method to visualize and rectify different inconsistencies for fuzzy reciprocal preference relations (FPRs), and the consistency includes the ordinal and cardinal consistencies. Ordinal inconsistency implies cardinal inconsistency, but not vice versa, and cardinal consistency of FPRs includes additive consistency and multiplicative consistency. Therefore, there are four types of inconsistencies, includes: (1) the additive inconsistency; (2) the multiplicative inconsistency; (3) the ordinal and additive inconsistencies, and (4) the ordinal and multiplicative inconsistencies. First, Gower plots are proposed to visualize the ordinal and cardinal inconsistencies on a two-dimensional plane. Then, some optimization models are formulated to rectify the four different types of inconsistences, respectively. Some examples are illustrated to show how the models work, and comparisons with the existing methods are also offered to show the advantages of the proposed methods.