Optimization model for reducing inconsistency of pairwise comparison matrices with minimal change

Consistency has a major impact on the quality of the decision using processes that are based on pairwise comparison. Techniques to reduce inconsistency in pairwise comparison matrices have been widely studied. These methods propose changes to the decision maker’s judgments so that the result is a less inconsistent matrix that corresponds a predefined acceptance rate. It is important not only to reduce inconsistency, but also to stay close to the decision-maker’s original judgment i.e. minimize the deviation. Hence, the problem can be formulated as finding the closest and least inconsistent matrix. Some of the optimization methods interpret the distance between pairwise comparison matrices as the number of differing elements in the same position (i.e. Hamming distance). In this paper, we propose a method that defines the distance between the original and suggested matrices as the absolute sum of the differences in the same positions (i.e. Manhattan distance). Our model considers two objectives at the same time: minimizing the distance and maximizing the reduction of the inconsistency. We point out that combining the two objectives (distance and consistency) in our model avoids that optimal solutions for one factor differ in the other factor not included in the objective. Finally, we compare our model with other similar methods, examining the consistency, the distance of both approaches from the original judgements, and the differences in the priority weight vectors generated from the proposed matrices.