A general unified framework for interval pairwise comparison matrices

TL;DR

This paper introduces a comprehensive unified framework for Interval Pairwise Comparison Matrices using Abelian linearly ordered groups, enabling consistent comparison across various types like multiplicative, additive, and fuzzy matrices.

Contribution

It generalizes consistency and indeterminacy measures for interval matrices within a unified algebraic framework, allowing cross-type comparison via isomorphisms.

Findings

Unified approach encompasses multiplicative, additive, and fuzzy matrices.

Provides new consistency and indeterminacy indices based on a generalized distance.

Enables comparison of different matrix types on a common coordinate system.

Abstract

Interval Pairwise Comparison Matrices have been widely used to account for uncertain statements concerning the preferences of decision makers. Several approaches have been proposed in the literature, such as multiplicative and fuzzy interval matrices. In this paper, we propose a general unified approach to Interval Pairwise Comparison Matrices, based on Abelian linearly ordered groups. In this framework, we generalize some consistency conditions provided for multiplicative and/or fuzzy interval pairwise comparison matrices and provide inclusion relations between them. Then, we provide a concept of distance between intervals that, together with a notion of mean defined over real continuous Abelian linearly ordered groups, allows us to provide a consistency index and an indeterminacy index. In this way, by means of suitable isomorphisms between Abelian linearly ordered groups, we will be able to compare the inconsistency and the indeterminacy of different kinds of Interval Pairwise Comparison Matrices, e.g. multiplicative, additive, and fuzzy, on a unique Cartesian coordinate system.