Two Proofs and One Algorithm Related to the Analytic Hierarchy Process

Abstract

36 years ago, Thomas Saaty introduced a new mathematical methodology, called Analytic Hierarchy Process (AHP), regarding the decision-making processes. The methodology was widely applied by Saaty and by other authors in the different human activity areas, like planning, business, education, healthcare, etc. but, in general, in the area of management. In this paper, we provide two new proofs for well-known statement that the maximal eigenvalue ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is equal to $n$ for the eigenvector problem $Aw=\lambda w$, where $A$ is, so-called, the consistent matrix of pairwise comparisons of type $n\times n$ ($n$$\ge$ 2) with the solution vector $w$ that represents the probability components of disjoint events. Moreover, we suggest an algorithm for the determination of the eigenvalue problem solution $Aw=nw$ as well as the corresponding flowchart. The algorithm for arbitrary consistent matrix $A$ can be simply programmed and used.