This article presents a linear programming formulation to solve the network design problem using the link transmission model (LTM) as the underlying traffic flow model. The original LTM was adapted by incorporating link-sending and receiving flows using linear inequalities. Furthermore, route choice was relaxed, and transfer flow variables were used to model vehicles’ routing decisions within the network. The objective function of the linear program aimed to minimize the total difference between the cumulative vehicle numbers (CVN) at the upstream and at the downstream boundaries of each link subject to flow-conservation and budget constraints. CVN were represented using transfer flows from connected links. The resulting formulation is a linear program that represents a dynamic system optimum traffic flow pattern, embedding the LTM’s network loading procedure. In contrast to the single-destination system optimum dynamic traffic assignment, based on the cell transmission model, the proposed formulation requires considerably fewer decision variables, thus potentially providing a more scalable approach. The proposed formulation was implemented on an example network to illustrate the behavior of the model, and was compared with the cell-based formulation. We show that the model describes free-flow and congested traffic flow states accurately in terms of shock-wave propagation, queuing of vehicles, and optimal total system travel time.
Observations on the linear programming formulation of the single reflector design problem
