An arithmetic–geometric mean difference based approach to consistency index and priority vector of pairwise comparison matrices

Consistency measure and prioritization elicitation are two important issues for pairwise comparison matrices (PCMs) originating from the analytic hierarchy process (AHP). In this study, a novel consistency index is reported using the difference between arithmetic mean and geometric mean, which is called the arithmetic–geometric mean difference consistency index (AGMD-CI). Some interesting properties of AGMD-CI are investigated and the thresholds of acceptable consistency are determined using the simulation method. Then, a consistency improving method is proposed by constructing an optimization model, which is solved using particle swarm optimization (PSO) algorithm. In addition, based on the idea of minimizing the arithmetic–geometric mean difference, an optimization based method is developed to obtain the priority vector from PCMs. The analytical solution of the proposed model is further derived. It is found that the proposed prioritization method is equivalent to the logarithmic least squares method. Finally, a novel decision algorithm is proposed and the case study is carried out to illustrate the proposed methods. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2025.