On a Descriptive Analytic Hierarchy Process (D-AHP) for modeling the legitimacy of rank reversal

In this paper we propose a descriptive extension of a conventional AHP, called D-AHP, such that the rank reversal phenomena are legitimately observed and explanatory. In general, the main causes of rank reversal are violation of transitivity and/or change in decision making structure. In AHP these causes correspond to inconsistency in pairwise comparison and change in hierarchical structure, respectively. Without these causes, AHP should not lead to rank reversal. But if we use inappropriate normalization procedure such that the entries sum to 1, the method will lead to rank reversal even when the rank should be preserved. Concerned with normalization procedure of importance of alternatives with respect to each criterion, we propose to add a hypothetical alternative such that it gives aspiration level for each criterion, and the scale is determined to normalize the eigenvectors so that the entry for this hypothetical alternative is equal to 1 rather than the entries summing up to 1. The relative importance of each criterion is evaluated as follows: If the average importance of all alternatives in the set is far from 1 under a criterion, the weighting coefficient for this criterion is increased. Furthermore, the criterion which gives larger consistency index can be regarded that the decision maker's preference is fuzzy under this criterion. Thus, the importance for such criterion is assigned to be relatively lower. Some numerical examples obtained by D-AHP are included, which could explain the legitimacy of rank reversal.