Facility location with adjacent units: a simple approximation scheme

In this paper, we provide a simple approximation scheme for the optimal objective value for a particular type of location problem. Typically, such problems are solved using the classic set covering formulation. Such a formulation automatically requires data for the constraint matrix and can get too large to implement or too difficult to solve to optimality. The scheme presented in this paper has minimal need for such data. Based on a simple count and with some basic and realistic assumptions about the geometry of the problem, we provide an algebraic formula that gives a close approximation to the optimal objective function value. Our formula can be easily implemented in a spreadsheet or hand-held calculator making it an effective planning tool for practice and also a good pedagogical aid. We illustrate by applying it to a location problem involving individual states in the continental US and collectively to the entire country.