Optimal pricing and investment in a multi-modal city - Introducing a macroscopic network design problem based on the MFD

Improving the performance of an existing transportation system is a challenge for engineers and policy makers as many dimensions and system design variables are interacting. In this paper, we propose the three-dimensional macroscopic fundamental diagram network design problem (3D-MFD-NDP). It is a strategic macroscopic tool to identify the directions of decision making in a multimodal transportation system, where the provision of roads and public transport services are interacting with costs for cars and public transport services in the performance of the entire road surface transportation system. The 3D-MFD-NDP models their effects aggregated at the network level and does not locate all measures to the road network. The objective function of the introduced 3D-MFD-NDP is minimizing the total travel time, while the design variables of the problem are the user costs for cars and bus tickets, the bus headway, the share of dedicated bus lanes and the length of the road network. The advantage of the 3D-MFD-NDP compared to existing approaches is that it is formulated as a mathematical program with equilibrium constraints (MPEC) that allows a fast closed-form solution instead of being simulation-based, usually computing many details not required in strategic decision making. We apply the 3DMFD-NDP to the greater area of Zurich to study two different problems. First, we investigate how the current network performance can be increased by pricing and investment measures. Despite difficulties in identifying reliable cost information for the provision of roads, we find that substantial travel time savings are possible, especially when limiting car use by restricting its space and increasing its costs. Second, we investigate the response to a 20% population growth in urban and suburban regions with car and public transport prices as well as bus frequency as the free design variables. We find that the system can accommodate the population growth, but as the system costs are shared among more users, the costs per trip are lessened, which attenuates the steering effect of prices.