Representing the strengths and directions of pairwise comparisons

We consider the problem of finding weights that well represent a set of pairwise multiplicative comparisons of a set of objects (as in the analytic hierarchy process and other methods). Our main contribution is a method for deriving such weights that takes into consideration not only the strengths of the pairwise comparisons, but also their directions. For example, if the comparison directions satisfy transitivity, then the weights produced by our method also satisfy transitivity (this is not always true for other methods). We also present a set of reasonable axioms for which our method is the (essentially) unique solution. Our method and axioms are closely related to those of Cook and Kress. Our method, like theirs, reduces to solving a linear program (hence it is different from the approach used in the AHP). For the special case that the comparison directions satisfy transivity, our method is quite simple and reduces to performing a forward pass as in the critical path method.