A Local Relaxation Method for Nonlinear Facility Location Problems

Abstract

A common problem that arises is the number and placement of facilities such as warehouses, manufacturing plants, stores, sensors, etc., needed to provide service to a region. Typically, the greater the number of facilities, the cheaper the cost of providing the service but the higher the capital cost of providing the facilities. The location of a facility is usually limited to a number of fixed locations. Consequently, when modeling such problems, binary variables are introduced to indicate whether or not a facility exists at a particular location. The resulting problem may then be modeled as a discrete optimization problem and could, in theory, be solved by general purpose algorithms for such problems. However, even in the case of a linear objective, such problems are NP-hard. Consequently, fast algorithms for large problems assured of finding a solution do not exist. Two alternatives to exact algorithms are heuristic algorithms and alpha-approximation algorithms. The latter ensure that a feasible point is found whose objective is no worst than a multiple of a times the optimal objective. However, there has been little success in discovering a-approximation algorithms [Hoc97] when the problem has a nonlinear objective. Here we discuss a generic heuristic approach that exploits the fact that the number of facilities is usually small compared to the number of locations. It also takes advantage of the notion that moving a facility to a neighboring location has a much smaller impact on the cost of service compared to that of moving it to a distant location. A specific form of this algorithm is then applied to the problem of optimizing the placement of substations in an electrical network.