Notes on order preservation and consistency in AHP

• The ranking error is generalized to the maximal reciprocal error.
• The value of the maximal reciprocal error (MRE) is defined.
• MRE limits the level of Saaty’s consistency index (CI).
• MRE helps to decide whether the Conditions of Order Preservation (COP) are met.
• Provided theorems help to understand the relationship between COP, MRE and CI.

The pairwise comparisons method is a convenient tool used when the relative order among different concepts (alternatives) needs to be determined. One popular implementation of the method is based on solving an eigenvalue problem for the pairwise comparisons matrix. In such cases the ranking result for the principal eigenvector of the pairwise comparisons matrix is adopted, while the eigenvalue is used to determine the index of inconsistency. A lot of research has been devoted to the critical analysis of the eigenvalue based approach. One of them is the work of Bana e Costa and Vansnick (2008). In their work, the authors define the conditions of order preservation (COP) and show that even for sufficiently consistent pairwise comparisons matrices, this condition cannot be met. The presented work defines more precise criteria for determining when the COP is met. To formulate the criteria, an error factor is used describing how far the input to the ranking procedure is from the ranking result. The relationship between the Saaty consistency index and COP is also discussed.