Remarks on the inconsistency measure of the Analytic Hierarchy Process

Abstract

The Analytic Hierarchy Process (AHP) is a popular Multi-Criteria Decision Making (MCDM) method. The workability of AHP made it suitable for solving complicated and elusive decision problems that subsequently led to its widespread applications in highly diverse fields. However, AHP has also received criticisms on various fronts, one of which is the rank reversal problem. When a replica of an existing alternative is introduced in the Multi-Criteria Decision (MCD) setting, it sometimes causes rank order reversal among alternatives. However, the addition of a replica of an alternative in the MCD setting is not limited to the rank reversal problem, but it also affects the inconsistency measure computed for the decision-maker (DM). An empirical study was conducted using AHP to measure the changes in the inconsistency of the DM on a well-defined and familiar MCD problem. The results indicate that when a replica is added to a pair-wise comparison matrix, the inconsistency of the DM reduces. It is found that there are two sources of inconstancy in a pair-wise preference matrix. One is intransitivity and another is the limitation of the 1–9 ratio scale. It is found that an inconsistency up to 50% is purely because of limitations of the ratio scale and higher inconsistencies are purely because of intransitivity in preferences defined by the DM. Therefore, the DMs should review and revise their preferences when their inconsistency exceeds 50%. This 50% threshold is also useful in deciding whether to apply a prediction algorithm to identify near consistent matrices. If the inconsistency of a matrix is above 50%, the prediction algorithms used to improve the consistency cannot be applied on the original inconsistent matrix because the source of inconsistency is intransitivity which means that the DM either does not have complete information about the problem or has not attended to the problem carefully.