On Deriving the Spectrum of Augmented Pairwise Comparison Matrices

Abstract

This paper concerns with the spectral properties of some pairwise comparison matrices(PCM) used in the decision making method called the analytic hierarchy process(AHP). By repeating any column and the corresponding row of a simple perturbed PCMthe so called augmented PCM is introduced. The principal eigenvector for such abordered matrix is developed in an explicit form and its components are studied todetermine a possible rank reversal between a transitive and such an augmented PCMs.Intervals over which such rank reversals occur are also established. The results aredemonstrated through a sample example.