Simultaneous Optimization Formulation of a Discrete–Continuous Transportation System

Consider a city with a highly compact central business district (CBD) in which commuters’ origins are continuously dispersed. The travel demand is dependent on the total travel cost to the CBD. The transportation system is divided into two layers: major freeways and dense surface streets. Whereas the major freeway network is modeled according to the conventional discrete network approach, the dense surface streets are approximated as a continuum. Travelers to the CBD either travel on the continuum (surface streets) and then exchange to the discrete network (freeways) at an interchange (ramp) before moving to the CBD on the discrete network, or they travel directly to the CBD on the continuum. Specific travel cost–flow relationships for the two layers of transportation facilities are considered. A traffic equilibrium model is developed for this discrete–continuous transportation system in which for a particular origin no traveler can reduce his or her individual travel cost to the CBD by unilaterally changing routes. The problem is formulated as a simultaneous optimization program with two subproblems. One subproblem is a traffic assignment problem from the interchanges to the CBD in the discrete network, and the other is a traffic assignment problem with multiple centers (i.e., the interchange points and the CBD) in the continuous system. A Newtonian algorithm based on the sensitivity analyses of the two subproblems is proposed to solve the resultant simultaneous optimization program. A numerical example is given to demonstrate the effectiveness of the proposed methodology.