Joint scheduling and formation design for modular-vehicle transit service with time-dependent demand

Document Type

Journal Article

Publication Date


Subject Area

mode - other, place - urban, ridership - demand, operations - scheduling, operations - capacity, infrastructure - vehicle


public transit, modular vehicles, scheduling, time-dependent demands


This paper focuses on the operation design of a future public transit service adopting modular vehicles. The unique feature of modular vehicles allows for assembling and disassembling operations along each trip to dynamically adjust the vehicle formation at stations. We propose a mathematical model to determine the optimal scheduling and modular vehicle formation (i.e., capacity adjustment) on a single transit line with time-dependent travel demand. In particular, the limited availability of the modules at stations, as well as the incurred re-balancing cost of the modules during operation, is explicitly considered. The model is originally formulated as a mixed-integer nonlinear program (MINLP). We then apply exact reformulation techniques to recast the MINLP into an equivalent mixed-integer linear program (MILP), which can be solved to its global optimal solution by off-the-shelf solvers. To solve more practical problems with a long planning horizon, we then develop a two-step heuristic that combines a searching algorithm and a rolling horizon approach. The first exact solution method based on MILP reformulation can be used to derive the global optimal solution for the small-size problem, while the heuristic can solve large-size problems efficiently. Numerical studies demonstrate the validity of the formulation and the efficiency of the solution methods. Besides, the experimental results reveal that the timetable and vehicle formations change significantly with the time-dependent demands, and the modular transit services have remarkable advantages in reducing both operator’s and passengers’ costs.


Permission to publish the abstract has been given by Elsevier, copyright remains with them.


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