Integrated Optimization of Transit Networks with Schedule- and Frequency-Based Services Subject to the Bounded Stochastic User Equilibrium

In many European metropolitan areas, the urban transit system is a mixture of schedule- and frequency-based services. This study proposes an integrated transit frequency and schedule design problem (ITFSDP), where frequencies and schedules are simultaneously determined, and develops a biobjective model for the ITFSDP to minimize operation costs and total passenger-perceived generalized travel cost. Meanwhile, the passengers’ route choice behavior is described by the bounded stochastic user equilibrium (BSUE). The in-vehicle congestion effect is represented using a set of constraints that differ in terms of the sitting and standing costs as sitting and standing passengers perceive crowding differently. This set of constraints captures the realistic behavioral feature that having occupied a seat, users remain seated at subsequent stops in the same vehicle. The problem is formulated as a mixed integer nonlinear programming problem, which is subsequently linearized to a mixed integer linear programming problem and solved using a branch and bound algorithm. A column generation and reduction phase is embedded in the solution algorithm to obtain the bounded choice set according to the BSUE constraints. Experiments are conducted to illustrate the model’s properties and the performance of the solution method. In particular, we demonstrate a Braess-like paradoxical phenomenon in the context of transit scheduling and highlight that well-synchronized transit services can deteriorate the network performance in terms of the total passengers’ generalized travel cost when considering passenger congestion costs because of crowding.