A Novel Consistency Index CI-G: Recruiting Compatibility Index G for Consistency Analysis

Consistency indices quantify the degree of transitivity and proportionality violations in a pairwise comparison matrix (PCM), forming a cornerstone of the Analytic Hierarchy Process (AHP) and Analytic Network Process (ANP). Several methods have been proposed to compute consistency, including those based on the maximum eigenvalue, dot product, Jaccard index, and the Bose index. However, these methods often overlook two critical aspects: (i) vector projection or directional alignment, and (ii) the weight or importance of individual elements within a pointwise evaluative structure. The first limitation is particularly impactful. Adjustments made during the consistency improvement process affect the final priority vector disproportionately when heavily weighted elements are involved. Although consistency may improve numerically through such adjustments, the resulting priority vector can deviate significantly, especially when the true vector is known. This indicates that approaches neglecting projection and weighting considerations may yield internally consistent yet externally incompatible vectors, thereby compromising the validity of the analysis. This study builds on the idea that consistency and compatibility are intrinsically related; they are two sides of the same coin and should be considered complementary. To address these limitations, it introduces a novel metric, the Consistency Index G (CI-G) based on the compatibility index G. This measure evaluates how well the columns of a PCM align with its principal eigenvector, using CI-G as a diagnostic component. The proposed approach not only refines consistency measurement but also enhances the accuracy and reliability of derived priorities.