A probability distribution and convergence of the consistency index u in the analytic hierarchy process

Abstract

In 1977, T.L. Saaty developed the analytic hierarchy process, which is widely applied in decision sciences. For each level of the hierarchy, a pairwise comparison matrix  is formed and a weight vector is derived by solving the eigenvector problem of the matrix Â. Saaty defined u  λ max − n n − 1 as the consistency index, which is a measure of the consistency or reliability of information contained in the comparison matrix, where λ max is the principal eigenvalue of  and n is the number of objects in a level of hierarchy. However, there has been little success in obtaining the probability distribution of u, because it is mathematically difficult to derive the analytic form of such a distribution function. This paper presents a numerical method to calculate the probability distribution function of u under the assumption that the error terms are independently and identically distributed with the lognormal distribution. The numerical result of such a probability distribution of u for σ = 1, n = 3, 4, …, 10, which are very often used in practice, can be found in the Appendix. This paper also points out via an example the difficulty in proving or disproving Saaty's conjecture: the consistency index u converges. The paper gives a necessary and sufficient condition that the convergence of u is almost certain under the assumption that the error terms ϵ ij are independently and identically distributed.