Maximizing network utility while considering proportional fairness for rail transit systems: Jointly optimizing passenger allocation and vehicle schedules

To deal with the contradiction between oversaturated demand and the restricted transport infrastructure and capacity during peak periods in a rail transit system, several passenger flow control methods that include soft pricing incentive, rigorous physical barrier and so on are widely used in current scenarios. However, the traffic disorderliness at the bottleneck and the control unfairness for the flows on specific stations may occur with those methods in the rail transit corridor. To alleviate those problems, this paper focus on the passenger allocation and matching with the corresponding schedules by considering the fairness of passengers and cost of operation from a macro perspective, where the passenger trip plan can be collected via a reservation platform in advance. A passenger utility maximization model that programs passenger allocation is proposed firstly, and we theoretically prove that there is an optimal traffic flow allocation with maximum utility under the oversaturated flow condition for any network. The model is solved by the Newton iteration algorithm and further proven to effectively make a tradeoff between efficiency and fairness. Furthermore, the transportation network utility maximization model is constructed by jointly optimizing the passenger allocation and vehicle schedules, thereby saving the cost of vehicle operation while guaranteeing the fairness of passenger allocation macroscopically. The model is solved by Multi-tree depth-first search algorithm based on Lagrangian relaxation with a subgradient in an iterative manner. A real-world case from the Batong subway line in Beijing is studied to validate the proposed models and algorithms. The results show that the model can jointly optimize the passenger allocation and vehicle schedules effectively in both an economical and fair manner, and the passengers can pass through the bottleneck in an orderly way