The vector assignment p-median problem (VAPMP) and the ordered p-median problem (OMP) are important extensions of the classic p-median problem. The VAPMP extends the p-median problem by allowing assignment of a demand to multiple facilities, and a wide variety of multi-assignment and backup location problems are special cases of this problem. The OMP optimizes a weighted sum of service distances according to their relative ranks among all demands. The OMP is well known as it represents a generalization of both the p-median and the p-center problems. In this article, a new model is developed which extends both the VAPMP and OMP problems. In addition, beyond median, center, and vector assignment, this new model can resolve problems where the system objective involves maximizing distance. The new model also gives rise to meaningful special-case problems, such as a "reliable p-center" problem. Different integer linear programming (ILP) formulations of the new problem are presented and tested. It is demonstrated that an efficient formulation for a special case of the VAOMP problem can solve medium sized problems optimally in a reasonable amount of time.