Efficient optimization of train timetable and rolling stock circulation plans for urban rail transit lines under a flexible train composition mode

This paper studies the dynamic co-optimization of timetables and rolling-stock plans for urban rail transit under a novel flexible train composition mode. To accommodate pronounced spatio-temporal variability in passenger demand, we allow coupling and decoupling of convoys not only at the depot but also at intermediate stations equipped with storage sidings, so that the size of the rolling stock assigned to each train service can be adapted in real time. We formulate a mixed-integer nonlinear program that links train traffic dynamics with passenger flow dynamics and minimizes a weighted sum of passenger waiting time and operating cost subject to capacity and feasibility constraints. The nonlinearities are handled by linearizing the constraint set and by applying a tightened piecewise McCormick relaxation for the bilinear terms in the objective. The resulting mixed-integer linear program is embedded in a rolling-horizon framework to deliver implementable decisions within operational time limits. Numerical experiments on a synthetic line and on a real-world instance from Beijing Metro Line 9, both compared against the traditional depot-only flexible train composition strategy (TFTC), show that the proposed approach improves capacity utilization and reduces costs while keeping passenger waiting time at comparable levels. In particular, relative to TFTC, the proposed method achieves about a 6% reduction in operating cost with similar aggregate waiting times on Line 9, and in the synthetic case it lowers stranded waiting time by nearly 14% while reducing operation cost by more than 8%. These results indicate that online flexible composition can materially enhance operational efficiency in urban rail systems.