Catastrophic events such as hurricanes, earthquakes or floods require emergency responders to rapidly distribute emergency relief supplies to protect the health and lives of victims. In this paper we develop a simulation and optimization framework for managing the logistics of distributing relief supplies in a multi-tier supply network. The simulation model captures optimized stocking of relief supplies, distribution operations at federal or state-operated staging facilities, demand uncertainty, and the dynamic progression of disaster response operations. We apply robust optimization techniques to develop optimized stocking policies and dispatch of relief supplies between staging facilities and points of distribution. The simulation framework accommodates a wide range of disaster scenarios and stressors, and helps assess the efficacy of response plans and policies for better disaster response.