A Local Relaxation Method for Nonlinear Facility Location Problems

Summary

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 $\mathcal{N}\mathcal{P}$-hard. Consequently, fast algorithms for large problems assured of finding a solution do not exist. Two alternatives to exact algorithms are heuristic algorithms and α-approximation algorithms. The latter esure that a feasible point is found whose objective is no worst than a multiple of α times the optimal objective. However, there has been little success in discovering α-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.

This research was supported by Office of Naval Research grant N00014-02-1-0076 and National Science Foundation CCR-0306662.