Local minimum adjustment for the consensus model with distribution linguistic preference relations considering preference reliability

Preference reliability analysis for preference relations reflected by the ordinal consistency and the cardinal consistency was intensively studied. The ordinal consistency is a minimum condition to guarantee that the decision maker’s pairwise comparison preferences between any triple alternatives are transitive. These issues for group decision-making with distribution linguistic preference relations have not been well addressed. This paper aims to provide a consensus model with distribution linguistic preference relations that controls both the cardinal consistency and the ordinal consistency. To do so, a new ordinal consistency for the distribution linguistic preference relations is defined, which is based on the transformation from the distribution linguistic preference relations to its expectation-based matrixes and then to its numerical scale matrixes. Two mixed integer linear programming models are respectively designed to eliminate the ordinal inconsistencies, and to simultaneously control the ordinal and cardinal consistencies so as to ensure the rationality of individual preference relations. Then, a consensus optimization model is developed in which only the individuals with consensus levels lower than the given threshold are allowed to modify their preferences. Preference reliability is also analyzed in the proposed consensus model. Finally, some classical examples are given to illustrate the proposed models. Comparative analysis validates the feasibility and effectiveness of the proposed approaches.