Inconsistency in the analytic hierarchy process under data uncertainties

Abstract

In this paper, we study the inconsistency of matrices in multicriteria group decision making under the assumption of uncertainty in their entries. The study binds together an $n$-dimensional probabilistic product measure $P$ and the Koczkodaj indicator $K_{n}$ of order $n$ used in pairwise comparisons. In the case of $n$ greater than 3, it starts with a new definition of $K_{n}$ based on the Steiner triple system of order $n$. It is of interest that the new indicator requires only $O \left( n ^{2} \right)$ evaluations, while the old one $O \left( n ^{3} \right)$. Our study results in analytic formulae for the probabilities of events $K_{n} \leq \varepsilon$ corresponding to uniform, exponential, and normal distribution functions, whenever the remainder of $n$ divided by 6 equals 1 or 3. For the remaining values of $n$, we present analytic formulae for their strict bounds. The formulae lead to much more efficient algorithms when compared with Monte Carlo simulations. This unexpected feature is illustrated by a few numerical experiments.