The synthesis of the AHP as a well-posed mathematical problem and matrix norms appropriate for sensitivity analysis via condition number

The analytic hierarchy process (AHP) is a decision-making method, which has as its greatest criticism the rank reversal effect. This paper formulates the fourth step of the AHP (synthesis) as a 'well-posed' mathematical problem. A theorem guarantees the existence of the square condensed original formulation for the AHP. This means that any decision problem modelled by AHP with a different number of alternatives and criteria can be condensed into a model with an equal number of alternatives and criteria without loss of condensed information. This condensed formulation can be better conditioned than the original rectangular formulation of the AHP. The square condensed equivalent formulation is also a 'well-posed' mathematical problem. The concepts are applied to two practical cases from the literature, and sensitivity analysis is performed. Four classical matrix norms are reformulated to obtain theoretical bounds for the error estimate closer to actual error.