Validation examples of the analytic hierarchy process and analytic network process

One way to validate a scientific theory is to show that the results predicted by the theory give correct answers; that is, that they match known results. In the Analytic Hierarchy Process (AHP) this usually means finding examples with measures in an already known scale. To validate AHP priority vectors against measures from known scales the measures must first be normalized by dividing by their sum. When the two vectors are the same, or close, then one can say the results of the AHP model have been validated. The AHP and its generalization, the Analytic Network Process (ANP), can be validated at several levels ranging from priority vectors derived from pairwise comparison matrices to the synthesized priorities for a hierarchical model, to the priorities derived for the elements in an ANP network from the limiting supermatrix (perhaps most impressively validated by estimating market share of companies using intangible factors), to the overall results from complex ANP models involving several levels of networks. Many validation examples are presented along with a discussion of the compatibility index that can be used to measure closeness of priority vectors.