Geometric and Algebraic Characterizations of Infeasibility of Rank Reversal in the Saaty and Belton–Gear Frameworks

We investigate the infeasibility of rank reversal in the Analytic Hierarchy Process under the Saaty and Belton–Gear frameworks. In the Saaty framework, we propose an algebraic characterization and show that it is equivalent to the geometric characterization in previous studies. We then extend both characterizations to the Belton–Gear framework and prove that the infeasibility of rank reversal is identical across the two frameworks. Our analysis further reveals that the feasibility of rank reversal is an intrinsic property of the original matrix, whereas its realization additionally depends on the introduced option, thereby reconciling seemingly contradictory findings reported in the literature. Several examples are provided to illustrate the theoretical results.