Probabilistic consistency of stochastic multiplicative comparison matrices based on Monte Carlo simulation

Abstract

Stochastic multiplicative comparison matrices (SMCMs) are widely accepted due to their flexibility in measuring various types of decision-makers' uncertain preferences by treating the judgment as a random variable. However, few existing consistency indices for SMCMs account for the influence of uncertainty of SMCMs on consistency. This may lead to some contradictions in consistency measurement for SMCMs. This paper proposes a probabilistic consistency index (PCI) considering the uncertainty of SMCMs from a statistical perspective. The index satisfies the property of invariance under the permutation of the labels of objects. The influence of uncertainty on the PCI is discussed in detail by numerical simulation. In addition, the perfect consistency and acceptable consistency of SMCMs based on the PCI are introduced, and a general threshold determination algorithm for acceptable consistency of SMCMs based on Monte Carlo simulation is designed considering the uncertainty, the distribution and the order of an SMCM. Subsequently, a consistency improvement method for SMCMs based on stochastic programming is proposed, which preserves the original preference of decision-makers to the greatest extent. Finally, examples are given to illustrate the significance of the consistency verification by using the PCI, and the most likely preference structure is used as the ranking for SMCMs.