An Infeasibility Condition for Rank Reversal in the Analytic Hierarchy Process

The analytic hierarchy process (AHP) is one of the most widely applied tools in Multi-Criteria Decision Making (MCDM), yet it is often criticized for rank reversal when a new option is introduced. We present an infeasibility condition under which rank reversal is theoretically impossible. The condition serves as a sufficient criterion for rank stability, clarifying when the overall ranking remains invariant despite the addition of new alternatives. If the condition is not satisfied, rank reversal may occur, explaining why it appears in some applications but not in others.