Marginal optimization method to improve the inconsistent comparison matrix in the analytic hierarchy process

Abstract

To improve the inconsistency in the analytic hierarchy process (AHP), a new method based on marginal optimization theory is proposed. During the improving process, this paper regards the reduction of consistency ratio (CR) as benefit, and the maximum modification compared to the original pairwise comparison matrix (PCM) as cost, then the improvement of consistency is transformed to a benefit/cost analysis problem. According to the maximal marginal effect principle, the elements of PCM are modified by a fixed increment (or decrement) step by step till the consistency ratio becomes acceptable, which can ensure minimum adjustment to the original PCM so that the decision makers' judgment is preserved as much as possible. The correctness of the proposed method is proved mathematically by theorem. Firstly, the marginal benefit/cost ratio is calculated for each single element of the PCM when it has been modified by a fixed increment (or decrement). Then, modification to the element with the maximum marginal benefit/cost ratio is accepted. Next, the marginal benefit/cost ratio is calculated again upon the revised matrix, and followed by choosing the modification to the element with the maximum marginal benefit/cost ratio. The process of calculating marginal effect and choosing the best modified element is repeated for each revised matrix till acceptable consistency is reached, i.e., CR<0.1. Finally, illustrative examples show the proposed method is more effective and better in preserving the original comparison information than existing methods.