Value functions and trade-offs associated with the analytic hierarchy process composition law

In this article, we provide a fresh bridge between the original formulation of the analytic hierarchy process (AHP) and the more classical theory of value functions. We focus on the hierarchic composition law (HCL) as a device for aggregation. Multi-attribute value functions can be similarly described. We derive a class of fundamental functional equations which relate the priorities construct of the AHP to a particular value function. It is therefore possible to associate a class of trade-off or indifference curves to the HCL by this technique. Our results indicate that the number of alternatives being compared in the AHP is the main cause of rank reversal. We show that different numbers of alternatives determine different trade-off curve systems, and hence necessarily, different value functions. Two classical cases of rank reversal, on account of alternative addition and deletion, respectively, are reexamined for illustration. The rank reversal can be resolved in both cases by this methodology. The results obtained here establish a link between the original AHP formulation using the HCL and multi-attribute value function theory, with promise of more positive mutual benefit between the topics in the future. In particular, the value function class associated with the HCL has a number of desirable features.