Profit-earning facility location

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Abstract
We consider opening facilities in order to gain a profit. We are given a set of demand points, and we must open some set of facilities such that every demand may be satisfied from a local facility and the total profit gained in this process is maximized. This contrasts with previous work on facility location and k-center problems, where opening a facility incurred a cost. The profit gained by opening a facility is a function of the amount of demand the facility satisfies. We model the dependence of profit on demand by creating many different possible facilities at each location, each of which provides a certain profit if opened and requires at least a certain amount of demand in order to open. Our model captures problem instances where profits may be positive or negative, and also instances where it is not necessary to satisfy every demand. Our algorithms provide the optimum total profit, while stretching the definition of locality by a constant and violating the required demands by a constant. We prove that without this stretch, the problem becomes NP-Hard to approximate.

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