A general unified consistency framework of reciprocal preference relations over bipolar Abelian quasi-ordered monoids

Abstract

Diverse categories of reciprocal preference relations (RPRs) have been put forward and extensively employed to quantify preference information. Consistency frameworks, which are profoundly interrelated with algebraic structures of RPRs, play a critical role in decision-making systems with RPRs. This paper introduces a novel concept of bipolar Abelian quasi-ordered monoids (BAQO-monoids) and regards any kind of RPRs as preference matrices over a given BAQO-monoid. Six BAQO-monoids are established to characterize algebraic structures of six types of imprecision-based RPRs: interval multiplicative RPRs, interval additive RPRs, triangular fuzzy multiplicative RPRs, triangular fuzzy additive RPRs, trapezoidal fuzzy multiplicative RPRs and trapezoidal fuzzy additive RPRs. Based on a general BAQO-monoid, a transitivity equation system with the binary mapping and parametric elements is developed. These parametric elements are then identified to propose a general unified consistency framework for RPRs over the BAQO-monoid. A distance-based computational formula is built to acquire inconsistency indices of RPRs over the general BAQO-monoid. The proposed general unified consistency framework and inconsistency measuring model are applied to set up consistency models and to calculate inconsistency indices for the six types of imprecision-based RPRs. Six numerical examples and important properties are provided to validate the proposed consistency frameworks and inconsistency measuring models.