Optimization methods to estimate alternatives in AHP: The classification with respect to the dependence of irrelevant alternatives

This paper focuses on specific rank reversal phenomena in optimization methods (the least squares method, the chi-square method, etc.) designed to derive preference weights of alternatives from pairwise comparison matrices in the Analytic Hierarchy Process. It is preferable that the most irrelevant alternative had no effect on the ranking of the other alternatives. Unfortunately, it appears that, for many methods, most irrelevant alternatives tend to dictate the rank order of all the remaining alternatives. Respectively, adding some irrelevant alternative may turn the most important alternative into an unimportant one and conversely. We classify the optimization methods with respect to the dependence of irrelevant alternatives and specify all possible “dictatorial” methods, which provide the absolute dictate of very irrelevant alternatives, and all methods, which are free from the dictate of such alternatives. For the dictatorial methods, we propose “weight function” modifications, which prevent the influence of irrelevant alternatives. We show that without the modification, “dictatorial” methods can add confusion and false recommendations in the decision-making process even in the most ordinary decision-making situations.