Abstract
Pairwise comparison matrices are increasingly used in settings where some pairs are missing. However, there exist few inconsistency indices for similar incomplete data sets and no reasonable measure has an associated threshold. This paper generalises the famous rule of thumb for the acceptable level of inconsistency, proposed by Saaty, to incomplete pairwise comparison matrices. The extension is based on choosing the missing elements such that the maximal eigenvalue of the incomplete matrix is minimised. Consequently, the well-established values of the random index cannot be adopted: the inconsistency of random matrices is found to be the function of matrix size and the number of missing elements, with a nearly linear dependence in the case of the latter variable. Our results can be directly built into decision-making software and used by practitioners as a statistical criterion for accepting or rejecting an incomplete pairwise comparison matrix.
Highlights
Pairwise comparisons form an essential part of many decisionmaking techniques, especially since the appearance of the popular Analytic Hierarchy Process (AHP) methodology [31,32]
He has provided a sharp threshold to decide whether a pairwise comparison matrix has an acceptable level of inconsistency or not
Λmax(A(x)) ≤ λmax(B), CI(A(x)) ≤ CI(B). This implies that the value of the random index RIn, calculated for complete pairwise comparison matrices, cannot be applied in the case of an incomplete pairwise comparison matrix because its consistency index CI is obtained through optimising its level of inconsistency
Summary
Pairwise comparisons form an essential part of many decisionmaking techniques, especially since the appearance of the popular Analytic Hierarchy Process (AHP) methodology [31,32]. He has provided a sharp threshold to decide whether a pairwise comparison matrix has an acceptable level of inconsistency or not. This widely accepted rule of inconsistency has been constructed for the case when all comparisons are known. There is a higher chance that the problem can be solved compared to the usual case when the supervision of the comparisons is asked only after all pairwise comparisons are given This is especially important as these values are often provided by experts who suffer from a lack of time.
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