H-Rank Consensus Models for Fuzzy Preference Relations Considering Eliminating Rank Violations

Abstract

Fuzzy preference relations (FPRs) have been widely used when ranking alternatives using pairwise comparison. However, when the FPRs have rank violations, the ranking results tend to contradict the decision makers’ (DMs) preferences. To date, no studies have examined the elimination of these FPR rank violations. Further, traditional consensus models utilizing distance measures between individual preferences are not suitable to manage classification-based consensus problems. To overcome these problems, this article develops several optimization models to address rank violations and h-rank consensus issues. First, the conditions that satisfy the FPRs’ preservation of order preferences (POPs) are analyzed and a system of constraints derived to ensure that the POPs are explicitly controlled by the optimization model, after which a mixed integer linear optimization model is developed to assist DMs to satisfy both the POP conditions and acceptable consistency. Finally, a linear model for the h-rank ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h \geq 2$</tex-math></inline-formula> ) consensus reaching process is designed to ensure that each individual FPR also satisfies the POP conditions and acceptable consistency. The feasibility of the proposed models is illustrated using numerical examples, with extensive comparisons further validating the usefulness of the proposed models for group decision-making problems.