Key words: Analytic Hierarchy Process (AHP), Analytic Network Process (ANP), supermatrix, fundamental scale, pairwise comparison, priority, inconsistency

Abstract

The Analytic Hierarchy and Network Processes perform paired comparisons to derive priorities. It is shown in three ways that it is necessary that these priorities take the form of the principal eigenvector of the matrix of paired comparisons when it is inconsistent. One way is to show that if the priorities generated in any way are used to weight the corresponding rows of the matrixes that are then summed, they must reproduce themselves and this leads to the principal eigenvector. The second way is to obtain the priorities by considering all order dominance and use Cesaro summability to show that the limit is the principal eigenvector. The third and final way uses perturbation of the judgments of a consistent matrix, whose priorities are trivially known to be the principal right eigenvector, to obtain the same kind of solution for an inconsistent matrix with small inconsistency. It is also shown how to derive the fundamental scale of the AHP from the solution (in the real domain) of the generalization of the eigenvalue problem to Fredholm's equation of the second kind and its corresponding stimulus response first order approximation. A discussion of scales of measurement, of relative scales and of the absence of a unique unit and an absolute zero is included. Finally, the paper contains a brief discussion of rank preservation and reversal with and without conditional independence.

INTRODUCTION

People have thought written books about and taught us that the only way to measure things is by having a scale that has a zero and a unit and applying such a scale to objects concrete or abstract. But that is not true. Our genetics have endowed us with the ability to make comparisons, and from such comparisons we can also derive scales that have no predetermined zero and unit.