Deriving the AHP 1-9 scale from first principles

Thomas Saaty
Joseph M. Katz Graduate School of Business
University of Pittsburgh
United States

Publication date: Aug, 2001

Journal: Proceedings of the Sixth International Symposium on the Analytic Hierarchy Process
- Pages: 397-401

Abstract: We demonstrate how the integers 1 to 9 used in the Fundamental Scale of the AHP to represent pairwise comparison judgments can be derived from stimulus-response theory. The conditions required for the stability of the eigenvector of priorities, known from the mathematics literature, are briefly mentioned. These conditions require that the elements being compared be homogeneous. This limits the upper value of the scale to 9. They also require that the number of elements compared be small. It is widely known that both of these conditions are intrinsic to the way in which our brains actually operate. A brief discussion is given about two ways to deal with a large number of elements, both included in the AHP protocol.

Keywords: Eigenvector, Eigenfunction, Consistency, Stability, Homogeneity