Rank Reversal in the AHP with Consistent Judgements: A Numerical Study in Single and Group Decision Making

Abstract

In this paper we study, by means of numerical simulations, the influence of some relevant factors on the Rank Reversal phenomenon in the Analytic Hierarchy Process, AHP. We consider both the case of a single decision maker and the case of group decision making. The idea is to focus on a condition which preserves Rank Reversal, RR in the following, and progressively relax it. First, we study how the estimated probability of RR depends on the distribution of the criteria weights and, more precisely, on the entropy of this distribution. In fact, it is known that RR does’nt occur if all the weights are concentrated in a single criterion, i.e. the zero entropy case. We derive an interesting increasing behavior of the RR estimated probability as a function of weights entropy. Second, we focus on the aggregation method of the local weight vectors. Barzilay and Golany proved that the weighted geometric mean preserves from RR. By using the usual weighted arithmetic mean suggested in AHP, on the contrary, RR may occur. Therefore, we use the more general aggregation rule based on the weighted power mean, where the weighted geometric mean and the weighted arithmetic mean are particular cases obtained for the values $p \rightarrow 0$ and $p=1$ of the power parameter p respectively. By studying the RR probability as a function of parameter p, we again obtain a monotonic behavior. Finally, we repeat our study in the case of a group decision making problem and we observe that the estimated probability of RR decreases by aggregating the DMs’ preferences. This fact suggests an inverse relationship between consensus and rank reversal. Note that we assume that all judgements are totally consistent, so that the effect of inconsistency is avoided.