Validity and inconsistency in the analytic hierarchy process

A mathematically justifiable method of measurement is needed to enable us to make all the tradeoffs that are necessary among the myriads of intangible criteria alongside the tangible ones in order to surface the best of several alternatives in a decision. The analytic hierarchy process (AHP) and its generalization to dependence and feedback, the analytic network process (ANP) are methods that provide us with a meaningful way to measure and combine tangible and intangible criteria in any decision. Paired comparisons of the alternatives of a decision with respect to each criterion and also of the criteria with respect to a higher goal, form the underlying approach of the AHP/ANP. By making all possible paired comparisons there is redundancy in the information provided and the more judgments one makes the less consistent they are. However, redundancy is needed to improve the validity of the outcome particularly when intangibles are involved. This paper addresses the question of how many elements comparisons should be limited to in order to control increase in inconsistency yet sufficiently large to enable capture validity that can be improved by proposed changes in judgments using a gradient method to which the decision maker is both sensitive and agreeable to a certain extent. It is because of such a limit that a decision problem must be decomposed into several clusters of elements with a pivot from one to another to connect their measurements into a single scale. An example is provided to illustrate this idea. Finally it is shown that it is possible to extend the idea of consistency and validity from a group of elements compared in a matrix to the elements in an entire decision structure.