On multiplicative priority rating methods for the AHP

Recently, several alternative variants to the original Analytic Hierarchy Process (AHP) have been proposed. Most of these sought to resolve some of the theoretical problems associated with the original AHP, which uses an additive preference aggregation. In this paper, we take a close look at the multiplicative ratings method, which has recently received growing attention. The interest in the multiplicative AHP (MAHP) is motivated by the fact that, in contrast with the original AHP, it precludes certain types of rank reversals as the composite priority ratings continue to follow a ratio scale, even after normalization. The purpose of this paper is threefold. First, we derive and discuss several interesting properties of the MAHP that have eluded attention in previous studies. Second, we argue that these properties of the MAHP are interesting not only for mathematical reasons but also on behavioral grounds. We show how the MAHP offers a more flexible preference modeling framework, while still preserving the ratio scale property, by relaxing the "constant returns to scale" assumption made in previous research. Third, we use simulation experiments to explore the extent to which the theoretical differences between the original AHP (additive AHP) play out computationally for various different types of preference structures, enabling us to assess whether the MAHP is merely an interesting theoretical construct, or can in fact make a substantial difference in terms of the rankings and ratings of the alternatives and rank reversals between the alternatives.