Joint optimization of train timetabling and rolling stock circulation planning with flexible train composition for metro lines

To better satisfy spatiotemporally uneven passenger demand, we investigate a flexible train composition strategy for metro operations, allowing train units to be coupled or decoupled at intermediate and terminal stations. We formulate a mixed-integer nonlinear program to jointly optimize train timetables and rolling stock circulation, aiming to minimize the number of required train units and stranded passengers while maximizing onboard occupancy. The model is reformulated as a mixed-integer linear program, and a tailored Lagrangian relaxation algorithm is developed to enhance computational efficiency by decomposing the problem into train-scheduling and passenger-flow subproblems. Numerical experiments on a small case and Shanghai Metro Line 16 demonstrate that flexible compositions outperform fixed ones, though their advantage declines as (de)coupling time increases. With a 30-s (de)coupling time, the number of required train units and stranded passengers decreases while onboard occupancy increases. The Lagrangian approach consistently delivers high-quality solutions with reduced computation time.