A multi-commodity discrete/continuum model for a traffic equilibrium system

We consider a city with several highly compact central business districts (CBDs). The commuters’ origins are continuously dispersed. The travel demand to each CBD, which is considered to be a distinct commodity of traffic movements, is dependent on the total travel cost to that CBD. The transportation system is divided into two layers: major freeways and a dense network of surface streets. Whereas the major freeway network is modelled according to the conventional discrete-network approach, the dense surface streets are approximated as a continuum. Travellers to each CBD can either travel within the continuum (surface streets) and then transfer to the discrete network (freeways) at an interchange (ramp) before moving to the CBD on the discrete network, or they can travel directly to the CBD within the continuum. Specific travel cost-flow relationships for the two layers of transportation facilities are considered. We develop a traffic equilibrium model for this discrete/continuum transportation system in which, for each origin–destination pair, no traveller can reduce his or her individual travel cost by unilaterally changing routes. The problem is formulated as a simultaneous optimisation programme with two sub-problems. One sub-problem is a traffic assignment problem from the interchanges to the CBD in the discrete network, and the other is a traffic assignment problem within a continuum system with multiple centres (i.e. the interchange points and the CBDs). A Newtonian algorithm based on sensitivity analyses of the two sub-problems is proposed to solve the resultant simultaneous optimisation programme. A numerical example is given to demonstrate the effectiveness of the proposed method.postprin