Statistical characteristics of response consistency parameters in analytic hierarchy process

Using the computer simulation method, we investigate the probability distribution of maximum eigenvalue of pairwise comparison matrix, which has been used as a parameter for measuring the consistency of responses in analytic hierarchy process (AHP). We show that the shape of the distribution of the maximum eigenvalue is different according to the dimension of the matrix. In addition, we cannot find any evidence that the distribution of the Consistency index is a Normal distribution, which has been claimed in the previous literature. Accordingly, we suggest using so-called K-index calculated based on the concept of cumulative distribution function rather than based on that of arithmetic mean because the probabilistic distribution cannot be assumed to be a Normal distribution. We interpret the simulation results by comparing them with the suggestion of Saaty. Our results show that using Saaty's value could be too generous when the dimension of the matrix is 3 and strict over 4. Finally, we propose new criteria for measuring the response consistency in AHP.