A gradient method for inconsistency reduction of pairwise comparisons matrices

TL;DR

This paper introduces a gradient-based method for reducing inconsistency in pairwise comparison matrices, proposing new inconsistency indicators and exploring their effects on the consistency process.

Contribution

It presents a novel gradient minimization approach for PC matrix consistency and introduces instant priority vectors and a new family of inconsistency indicators.

Findings

Different inconsistency indicators lead to non-equivalent consistencization procedures

The approach applies to both additive and multiplicative PC matrices

New insights into priority vector definitions and inconsistency measurement

Abstract

We investigate an application of a mathematically robust minimization method -- the gradient method -- to the consistencization problem of a pairwise comparisons (PC) matrix. Our approach sheds new light on the notion of a priority vector and leads naturally to the definition of instant priority vectors. We describe a sample family of inconsistency indicators based on various ways of taking an average value, which extends the inconsistency indicator based on the "$\sup$"- norm. We apply this family of inconsistency indicators both for additive and multiplicative PC matrices to show that the choice of various inconsistency indicators lead to non-equivalent consistencization procedures.