Multimodal urban transportation systems exhibit complex interactions between components, including users, multimodal transportation facilities, supply side agencies, and operators. Although these interactions are obvious, rigorous quantitative methods for optimizing control variables across modes of transportation on real-world networks are deficient. A social welfare maximization model was established for joint optimization of bus fare, rail transit fare, and congestion tolls for private cars. The authors determined the optimality conditions and the second-order partial derivatives for this optimization problem. Because of the complexity of the multimodal urban transportation system, the objective function of social welfare has no closed form and is extremely expensive to evaluate. The authors therefore proposed a simulation-based framework for evaluation of the objective function and optimization of three decision variables across multiple travel modes. Several metamodels (i.e., mathematical functions that approximate the true shape of an unknown, nonlinear, and complex objective function) were adopted to approximate the highly nonlinear input-output mappings in the urban system. This is the first study to develop a simulation-based method for joint optimization of transit and road network operations. The case study applied the simulation-based optimization framework to the Sino-Singapore ecocity in Tianjin, China, by using VISSUM as the urban systems simulator. An "ecocity" is defined by the authors as "a thriving city that is socially harmonious, environmentally friendly, resource-efficient, and a model for sustainable development." Results show that metamodels can accurately approximate the real objective function and produce good suboptimal and near-optimal solutions. The optimal combination of transit fares and congestion tolls significantly outperform those under two baseline scenarios. The optimal solutions also suggest that extreme transit fares (too high or too low) or congestion tolls are contrary to welfare-maximizing objectives. The proposed method can be applied for joint optimization of other multimodal planning and operational strategies, such as investment and operational decisions across various modes of transportation.
