Faster method to check decision data consistency under uncertainty
Inconsistency in the analytic hierarchy process under data uncertainties
Computational Engineering, Finance, and Science
Summary
When people make group decisions using many criteria, they compare options pair-by-pair, but the data can be uncertain and inconsistent. The authors studied how to measure this inconsistency when the numbers used have random variations. They created a new way to calculate a common inconsistency measure that is much faster for larger problems. Their formulas can predict how likely it is for data to be consistent under various randomness assumptions without slow simulations. This helps check decision data reliability more efficiently.
What this means in practice
- •For business analysts: Assess consistency of pairwise comparison data faster in group decision-making tools under uncertain inputs.
- •For risk assessment teams: Evaluate the likelihood of consistent decision data for complex criteria under different uncertainty models analytically.
Tested on simulated data.
Authors
Przemysław Rutka, Ryszard Smarzewski
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.