Note on “The breaking of additively reciprocal property of fuzzy preference relations and its implication to decision making under uncertainty”

Abstract

A recent article by Liu et al. [8] devised measurement models to calculate additive consistency indices and weak transitivity indices for fuzzy preference relations (FPRs) with additive reciprocity breaking (ARB), and developed some decision models, such as the equivalency between FPRs with ARB and inconsistent FPRs with additive reciprocity, the optimization based weak transitivity improving model and priority vector derivation models. This paper utilizes six numerical illustrations to show that the measurement models are technically incorrect and sometimes the decision models are invalid or create contradictory results. New frameworks are proposed to define and measure additive consistency and weak transitivity for FPRs with ARB. Computational formulas are brought forward to acquire interval-valued or intuitionistic fuzzy priority vectors from additively consistent FPRs with ARB, and a linear programming model is built to ascertain interval-valued or intuitionistic fuzzy priority vectors from inconsistent FPRs with ARB.