Spacial and Objective Decompositions for Very Large SCAPs

This paper reconsiders the single commodity allocation problem (SCAP) for disaster recovery, which determines where and how to stockpile a commodity before a disaster and how to route the commodity once the disaster has hit. It shows how to scale the SCAP algorithm proposed in [1] to a geographical area with up to 1,000 storage locations (over a million decision variables). More precisely, the paper shows that spatial and objective decompositions are instrumental in solving SCAP problems at the state scale (e.g., for the state of Florida). The practical benefits of these decompositions are demonstrated on large-scale hurricane disaster scenarios generated by Los Alamos National Laboratory using state-of-the-art disaster simulation tools. 1 Background and Motivation Every year, considerable human and monetary resources are spent to prepare for, and recover from, seasonal hurricanes. Existing procedures rely on the experience of policy makers but they are often ad-hoc and do not exploit recent progress in optimization to address natural disasters more effectively. Our earlier research [1] demonstrated the benefits of optimization technology to meet the population needs and to reduce storage and transportation costs by using a two-stage stochastic optimization problems with explicit scenarios generated by the National Hurricane Center (NHC) of the National Weather Service in the United States. However, only disasters with up to 100 storage locations (city scale) were considered, although large-scale planning may require as many as 1,000 storage locations (state scale). Indeed, the start-of-the-art algorithm in [1], and its underlying MIP model, have difficulties scaling to problems with 250 storage locations and runs out of memory on larger instances. This paper shows how to scale the approach for disasters at the state scale using spatial and objective decompositions. The spatial decomposition performs a geographic clustering of the repositories and aggregates the flows across the clusters, thus reducing the number of decision variables considerably. The objective decomposition applies when the SCAP objective function is lexicographic: It separates the decisions taken for meeting the demands and reducing travel time. Experimental results demonstrate the benefits of both approaches on large and very large instances respectively. Given the sizes and complexity of the models considered here, our results are purely empirical: Their practicability is demonstrated by showing improvements on the practice in the field. Note also that the resulting approaches are now deployed and are activated each time a hurricane of category 3 or above threatens the coast of the United States. The rest of the paper is organized as follows. Section 2 of this paper reviews related work. Section 3 presents a mathematical formulation of the SCAP and Section 4 reviews the approach presented in [1]. Sections 5 and 6 present the novel decomposition techniques. Section 7 reports the experimental results and Section 8 concludes the paper.