Modifying inconsistent comparison matrix in analytic hierarchy process: A heuristic approach

Test of consistency is a critical step in the AHP methodology. When a pairwise comparison matrix fails to satisfy the consistency requirement, a decision maker needs to make revisions. To aid the decision maker's revising process, several approaches identify changes to the consistency requirement with respect to changes to a single entry in the original inconsistent matrix [P.T. Harker, Derivatives of the Perron root of a positive reciprocal matrix: with application to the analytic hierarchy process, Applied Mathematics and Computation, 22, 217–232 (1987); T.L. Saaty, Decision-making with the AHP: Why is the principal eigenvector necessary, European Journal of Operational Research, 145, 85–91 (2003)]. Instead of revising single entries, Xu and Wei [Z. Xu and C. Wei, A consistency improving method in the analytic hierarchy process, European Journal of Operational Research, V.116, 443–449 (1999)] derived a consistent matrix by an auto-adaptive process based on the original inconsistent matrix. In this paper, we develop a heuristic approach that auto-generates a consistent matrix based on the original inconsistent matrix. Expressing the inconsistent matrix in terms of a deviation matrix, an iterative process adjusts the deviation matrix to improve the consistency ratio, while preserving most of the original comparison information. We show that the proposed method is able to preserve more original comparison information than Xu and Wei [Z. Xu and C. Wei, A consistency improving method in the analytic hierarchy process, European Journal of Operational Research, V.116, 443–449 (1999)]. It is also shown that the heuristic approach can be used to examine the effects of revising a sub-bloc as well as revising a single entry of the original matrix.