An Overview and Comparison of Axiomatization Structures Regarding Inconsistency Indices' Properties in Pairwise Comparisons Methods

TL;DR

This paper reviews and compares various axiomatic frameworks for inconsistency indices in pairwise comparison methods, highlighting recent developments and suggesting future research directions.

Contribution

It provides a comprehensive overview of axiomatization structures for inconsistency indices and compares their properties, addressing fragmentation in the literature.

Findings

Multiple axiomatic frameworks are analyzed and compared.

Recent advancements in axiomatization are summarized.

Future research directions are proposed.

Abstract

Mathematical analysis of the analytic hierarchy process (AHP) led to the development of a mathematical function, usually called the inconsistency index, which has the center role in measuring the inconsistency of the judgements in AHP. Inconsistency index is a mathematical function which maps every pairwise comparison matrix (PCM) into a real number. An inconsistency index can be considered more trustworthy when it satisfies a set of suitable properties. Therefore, the research community has been trying to postulate a set of desirable rules (axioms, properties) for inconsistency indices. Subsequently, many axiomatic frameworks for these functions have been suggested independently, however, the literature on the topic is fragmented and missing a broader framework. Therefore, the objective of this article is twofold. Firstly, we provide a comprehensive review of the advancements in the axiomatization of inconsistency indices' properties during the last decade. Secondly, we provide a comparison and discussion of the aforementioned axiomatic structures along with directions of the future research.