Notes on discrepancy in the pairwise comparisons method

The pairwise comparisonsmethod 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 the principal eigenvector of the pairwise comparisonmatrix is adopted, whilst the eigenvalue is used to determine the index of inconsistency. Studies on the eigenvalue based inconsistency index have led the author to propose a more general index of discrepancy. This describes how far the input to the ranking procedure is from the ranking result. It may help to detect, localize and quantify possible problems related to the input data and the ranking procedure. Some of its properties those related to the eigenvalue based inconsistency index and the condition of order preservation are presented as appropriate theorems.