Locating discretionary service facilities, II: Maximizing market size, minimizing inconvenience

Discretionary service facilities are providers of products and/or services that are purchased by customers who are traveling on otherwise preplanned trips such as the daily commute. Optimum location of such facilities requires them to be at or near points in the transportation network having sizable flows of different potential customers. N. Fouska and O. Berman, R. Larson and N. Fouska (BLF) formulate a first version of this problem, assuming that customers would make no deviations, no matter how small, from the preplanned route to visit a discretionary service facility. Here the model is generalized in a number of directions, all sharing the property that the customer may deviate from the preplanned route to visit a discretionary service facility. Three different generalizations are offered, two of which can be solved approximately by greedy heuristics and the third by any approximate or exact method used to solve the p-median problem. We show for those formulations yielding to a greedy heuristic approximate solution, including the formulation in BLF, that the problems are examples of optimizing submodular functions for which the G. Nemhauser, L. Wolsey and M. Fisher bound on the performance of a greedy algorithm holds. In particular, the greedy solution is always within 37% of optimal, and for one of the formulations we prove that the bound is tight.