User rationality and optimal park-and-ride location under potential demand maximization

The paper develops analytical formulations to gain insight into the optimal location, i.e., the one that maximizes the potential market, and to estimate the potential catchment area of Park and Ride (P&R) facilities. The formulations are based on the assumption that a traveler would use a P&R facility if and only if the corresponding generalized cost is lower than the drive only alternative. The paper considers two different scenarios: a linear city (or a travel corridor), and a two-dimensional city with Euclidean travel. Analytical derivations were obtained for both cases using, as starting point, the necessary condition for P&R use.In the case of the linear city, the paper identifies two breakeven distances (BEDs) of great import to the estimation of the potential P&R market: the (trip) origin BED, i.e., the distance below which a traveler could drive upstream to use the P&R facility to access its downstream destination, and still be better off; and the (trip) destination BED, i.e., the travel distance using transit below which it does not make sense to use P&R. The paper proves that the optimal location of P&R sites is shifted upstream of what seems to be an intuitive solution, i.e., the edge of the congested region, by a distance that depends on the relative values of the origin and destination BEDs.In the two-dimensional city case, the analytical derivations prove that, for a given trip from i to j, the set of feasible locations follows an ellipse-like figure with the trip origin as a focus. These shapes-referred to as limiting functions-depend on variables such as trip distance, transit level of service (LOS), and the like. The analyses indicate that the area enclosed by the limiting functions increases with the transit LOS and trip distance, and so do the corresponding catchment areas. This is because the catchment area is determined by the marginal trip origins, i.e., those for which the P&R facility is just inside the limiting function.In its final section, the paper develops a parabolic approximation to the catchment area for a given P&R site. The approximating parabola is defined by three critical points: the origin BED, and two points that identify the marginal trip origins at the chord of parabola evaluated at the P&R. The numerical experiments indicate that the parabolic approximation provides a fairly good estimate of the catchment area that is easy to produce, conceptually valid, and overcomes the limitations of alternative approaches and rules of thumb used by practitioners and researchers.