Size independence of consistency index for pairwise comparison matrices in analytic hierarchy process

Pairwise comparisons are fundamental in the analytic hierarchy process. Various consistency indices have been proposed to assess inconsistencies in these comparisons. Since Saaty first proposed his consistency index, the assessment of the degree of consistency in pairwise comparison matrices has remained an open and hot topic in the study of the analytic hierarchy process. The consistency indices CI and CR proposed by Saaty are defined using the principal eigenvalue of the pairwise comparison matrix. In our previous study, we introduced an alternative index derived from the relationship between the coefficient of the characteristic polynomial and the consistency of comparisons.
Saaty proposed a fixed threshold of 0.1 for CI or CR as a guideline for an acceptable level of consistency, regardless of the matrix size. However, whether this threshold represents an equivalent level of consistency across different matrix sizes, that is, across different numbers of evaluation items, remains unclear. This study analysed the relationship between consistency and matrix size by examining pairwise comparison matrices constructed from subsets of evaluation items. Based on this analysis, we propose the fundamental property to be satisfied by a size-independent consistency index.
Furthermore, we refine our previously proposed index to ensure that it satisfies this property, demonstrating that it coincides with the existing consistency index. Finally, we visualise the relationship between the matrix size and consistency index values using randomly generated pairwise comparison matrices, thereby providing insights into the impact of matrix size on consistency evaluation.