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 method of computer simulations. Then, a consistency improving method is proposed by constructing an optimization problem, which is solved by particle swarm optimization (PSO). In addition, based on the idea of minimizing the arithmetic-geometric mean difference, an optimization model is established 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 making algorithm is proposed and the case study is carried out to illustrate the proposed methods.