Revisiting inconsistent judgments for incomplete fuzzy linguistic preference relations: Algorithms to identify and rectify ordinal inconsistencies

Abstract

Fuzzy linguistic preference relations (FLPRs), which decision makers (DMs) use to express their pairwise comparison information, are suitable for modeling the qualitative aspects of problems. Ordinal consistency is the minimum condition that DM’s preference values arenot self-contradictory. In this paper, we first define and study the ordinal consistency of the incomplete FLPRs. A novel algorithm is developed to judge whether an incomplete FLPR is ordinally consistent. The proposed algorithm can also identify all cycles of length 3 to n in the digraph of an incomplete FLPR. Based on the proposed algorithm and two rules, another algorithm is proposed to rectify the inconsistency of an incomplete FLPR and improve it so that it is ordinally consistent. Furthermore, the proposed algorithms can also be used to identify and rectify the ordinal inconsistency of the complete FLPRs. Finally, two examples are given, and comparisons with the existing methods are also provided to demonstrate the effectiveness and validity of the proposed methods.