Enhancing Consistency of Pairwise Comparisons on the Base of Linear Algebraic Equations

Abstract

A problem of improving consistency of pairwise comparisons matrices in application to ranking given alternatives is considered in the paper. But it can be shown that consistency is not the only issue as to the quality of pairwise comparisons. Given an arbitrary positive square matrix, we can obtain an ideally consistent pairwise comparison matrix with the same Perronian vector. Therefore, the quality of experts’ judgements is an issue of great importance as well.Technically, an approach to improving consistency of pairwise comparisons on the basis of solving a linear algebraic equations system is suggested. The system contains two groups of equations. One of them represents experts’ judgments, and the other is related to demands of cardinal consistency. Such a system can be over- or maybe underdetermined, and it typically can be inconsistent. Then a pseudo-solution can be obtained by means of pseudo-inverse Moore-Penrose matrix.For improving the quality of pairwise comparisons, it appears urgent to take into account reliabilities of certain judgements by giving them appropriate weight coefficients.Some numerical examples are provided in the paper. The first is a simple basic example without any serious inconsistencies. The second illustrates as to treat incomplete pairwise comparison matrices. And the latest illustrates possible expert’s manipulation, when an expert wants to secure the winning of a certain alternative whereas they don’t want to postulate the advantage of this alternative implicitly, and this results in the order violation. It is illustrated how introducing weight coefficients of equations can help counteract such manipulations.