A continuous analysis framework for the solution of location–allocation problems with dense demand

Location–allocation problems arise in several contexts, including supply chain and data mining. In its most common interpretation, the basic problem consists of optimally locating facilities and allocating customers to facilities so as to minimize the total cost. The standard approach to solving location–allocation problems is to model alternative location sites and customers as discrete entities. Many problem instances in practice involve dense demand data and uncertainties about the cost and locations of the potential sites. The use of discrete models is often inappropriate in such cases. This paper presents an alternative methodology where the market demand is modeled as a continuous density function and the resulting formulation is solved by means of calculus techniques. The methodology prioritizes the allocation decisions rather than location decisions, which is the common practice in the location literature. The solution algorithm proposed in this framework is a local search heuristic (steepest-descent algorithm) and is applicable to problems where the allocation decisions are in the form of polygons, e.g., with Euclidean distances. Extensive computational experiments confirm the efficiency of the proposed methodology.