Bi-level programming for continuous network design of comprehensive transportation system based on external optimization

To determine the optimal investments on road network of comprehensive transportation system, a bi-level programming model for continuous network design problem was employed. At the upper level problem, planner makes investment decision in links and hubs of a comprehensive transportation system to minimize the total times costs plus investment costs and the external costs such as environment pollution, land use and energy exhaustion. At the lower level, users choose their paths in accordance with deterministic user equilibrium. We design an algorithm based on extremal optimization to solve the upper level problem, in which only investments on those links and hubs with high marginal cost are updated randomly at each step to combine the merits of gradient-based methods and intelligent heuristics. Numerical comparison was made on a grid network with 9 nodes and 14 links. The result shows that the algorithm is competitive with gradient-based and simulated annealing methods.