AHP rank reversal, normalization and aggregation rules

The authors analyze the Belton and Gear rank reversal problem within an axiomatic framework for deriving consistent weight ratios from pairwise ratio matrices and aggregating weights and ratio matrices. They show that rank reversal in the Analytic Hierarchy Process is avoided when the output of the process is properly redefined as a weight-ratio matrix (rather than a normalized-weight vector) and multiplicative procedures-the geometric mean and the weighted-geometric-mean aggregation rule-which preserve the underlying mathematical structures are used.