Traffic assignment in a congested discrete/ continuous transportation system

Consider a city where all workplaces are concentrated in a highly compact central business district (CBD) and the commuters' homes are continuously dispersed over the residential area surrounding the CBD. The transportation facilities in the city comprise a discrete freeway network and a two-dimensional continuum of dense surface streets. The freeway network is assumed to be superimposed on the continuum and connected with it at a limited number of points (freeway ramps). During the morning peak-hours, the commuters traveling to work have a choice between two routes to the CBD: traveling along the minor access roads to enter the freeways and then proceeding along the freeway to the CBD, or using only minor street roads straight to the CBD. Given this transportation system, it is important to estimate the total trips using each freeway ramp and their spatial distribution. In this paper, an optimization model is developed to deal with this traffic assignment problem, provided that each commuter seeks to minimize his individual congested travel time. The model is formulated by intergrating the conventional network and continuum equilibrium models and its dual formation is derived. A dual-based solution method is developed using the finite element technique, and illustrated with a numerical example for a hypothetical city.