Optimal scheduling of ships in inland waterway with serial locks

Locks built on inland waterways enable ships to overcome water level differences. However, potential congestion at locks poses challenges in scheduling problems for inland shipping. This study designs an optimal plan for multiple ships passing through serial locks on both the main channel and tributaries of an inland waterway. Two novel mixed-integer linear programming (MILP) methods, one based on constraint analysis and one on bisection search, are proposed to solve the problem. Moreover, a dynamic linearization method is developed, which strategically simplifies the scheduling problem in its early stages to reduce solution times. Computational experiments based on a realistic case in China are conducted. The results show that the constraint analysis-based MILP method outperforms the bisection search-based MILP method for small- and medium-scale instances, while the bisection search-based MILP method is more effective for large-scale instances. Additionally, a dynamic linearization method improves computational efficiency while ensuring high solution quality. For 40-ship instances, the dynamic linearization method finds 67% more optimal solutions than the constraint analysis-based MILP method while requiring only 40% of the time. Subsequently, we compare optimal ship speed between scenarios considering and disregarding potential congestion at locks and find that shorter legs support a larger range of speed adjustment, and that integrated optimization tends to adjust ship speed during shorter legs. Furthermore, sensitivity analyses are conducted to gain managerial insights.