Consistency in Positive Reciprocal Matrices: An Improvement in Measurement Methods

Abstract

Consistency estimation of decision-makers' judgments in decision-making processes is fundamental to generating agreements and making decisions. Analytic hierarchy process (AHP) is a widely used method to solve this type of problem, enabling evaluation of the consistency of judgments emitted by the decision-makers through the maximum eigenvalue of the matrix of judgments. In addition to the consistence index originally proposed in AHP, different indexes have been proposed in the literature, which use the minimum element of consistency. These indices that solve some of the original consistency index problems, present others that may question their usefulness. The purpose of this paper is to propose a new index, as an improvement of the previous indexes. Among other characteristics, this new index is intuitive and easy to use, is bounded in the interval [0,1], proposes a critical value to accept or reject matrices, that depends on the size of the pairwise-comparison matrix, and can be extended to another type of pairwise-comparison scales. In addition, the probability distribution for the new index is defined, which enables calculating the probability of a matrix being consistent, as a function of the critical acceptance value.