Discretionary service facilities are providers of products and/or services that are purchased by customers who are traveling on otherwise pre-planned 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. In [1] a first version of this problem was formulated, assuming that customers would make no deviations, no matter how small, from the pre-planned 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 pre-planned 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 pmedian problem. It is shown for those formulations yielding to a greedy heuristic approximate solution, including the formulation in [1], that the problems are examples of optimizing submodular functions for which the Nemhauser et. al. [51 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 it is proved that the bound is tight.Revised November 1992