Balancing Pairwise Comparison Matrices by Transitive Matrices

We discuss the development and use of a recursive rank-one residue iteration (triple R-I) to balancing pairwise comparison matrices (PCMs). This class of positive matrices is in the center of interest of a widely used multi-criteria decision making method called analytic hierarchy process (AHP). To find a series of the ‘best’ transitive matrix approximations to the original PCM the Newton-Kantorovich (N-K) method is employed for the solution to the formulated nonlinear problem. Applying a useful choice for the update in the iteration, we show that the matrix balancing problem can be transformed to minimizing the Frobenius norm. Convergence proofs for this scaling algorithm are given. A comprehensive numerical example is included to illustrate the useful features to measuring and reducing perturbation errors and inconsistency of a PCM as a result of the respondents’ judgments on the pairwise comparisons.