Constrained location of competitive facilities in the plane

In a competitive business environment where products are not distinguishable, facility location plays an important role in an organization's success. This paper examines a firm's problem of selecting the locations in the plane for a set of new facilities such that market capture is maximized across all of the firm's facilities (both new and pre-existing). Customers are assumed to divide their demand among all competing facilities according to a utility function that considers facility attractiveness (measured by facility capacity for satisfying demand) and customer-facility distance. The level of customer demand is assumed to be a function of the facility configuration. Three types of constraints are introduced, involving facility capacity, forbidden regions for new facility location, and a budget function. Two solution algorithms are devised, one based on branch-and-bound methods and the other based on penalty functions. Computational testing is presented, comparing these two algorithms along with a general-purpose nonlinear solver.