Exploration of maximum-to-minimum shifts in generic methods for departure time choice models

Vickrey’s seminal departure time choice model is based on a penalty function which is a linear combination of travel time, earliness, and lateness. The original model depicts a single link from a single origin to a single destination, serving homogeneous travelers by a deterministic point-queue regime. Numerous variants of the basic model, relaxing one or more of these assumptions, have been used in a wide range of contexts. The equilibrium solution of the basic model can be computed directly by exact formula. Specific convergent methods have been proposed for certain variants. One of the troubling challenges in this model is the need for a generic iterative numeric approach, that may address complex models in which departure time choice is embedded. Natural candidates were shown to fail even on the basic model. In this paper we explore a fairly naive approach, where, in each iteration, demand is shifted from the maximum cost time interval to the minimum cost time interval. Results for the basic model are promising, demonstrating that, with a fixed shift, solutions converge to a deviation which is proportional to the shift size and that semi-adaptive or adaptive shift size may offer convergence to any desirable level of approximation of the exact equilibrium.