Improvement of the Weights Due to Inconsistent Pairwise Comparisons in the AHP

Abstract

One of the most important problems in the Analytic Hierarchy Process (AHP) is consistency of pairwise comparisons by the decision maker. This study focuses on the comparison methods to be used when the weights of the alternatives and criteria in AHP are inconsistent. In general, the weights in AHP use the principal eigenvector of the pairwise comparison matrix. However, for example, due to the decision maker’s misunderstandings, inconsistencies in pairwise comparisons sometimes arise. The consistency of the pairwise comparison matrix is usually determined using Consistency Index (CI) values. In the traditional AHP, when judged inconsistent, repeating the pairwise comparison is usually recommended. However, if the repeated comparison is arbitrarily performed, the results will not be optimal. In fact, to obtain the overall evaluation of alternatives, we often use inconsistent weights, even given the inconsistencies in the latter. Another method for judging the consistency of the pairwise comparison is to use a directed graph. Cycles in a directed graph represent comparison inconsistencies. Therefore in this paper, based on the principal eigenvalue and cycles in the directed graph of the pairwise comparison matrix, a method of correcting the principal eigenvector taking into consideration consistency is proposed.