Schedule Reliability in Liner Shipping Timetable Design: A Convex Programming Approach

Container liner shipping is the primary mode of moving manufactured products across continents. Partly due to inherent uncertainties at sea and ports, the liner shipping industry has long had a notorious reputation of schedule delays and unreliable on-time performance. This paper formulates a new approach to incorporate schedule reliability targets in liner shipping timetable design, to balance bunker consumption, time taken for the voyage, and schedule delays. We first model a surrogate problem using a copositive program through a moment decomposition approach and solve it as a convex semidefinite programming relaxation. We next incorporate schedule reliability targets implicitly by exploiting the optimality condition of this surrogate model. We use this approach to analyze the trade-offs between the bunker consumption and the schedule reliability targets for each port call. Furthermore, we derive the optimal speed of the vessel in each leg of the schedule to control for total bunker consumption. Surprisingly, the analytical analysis shows that, if a shipping line has the freedom to choose its preferred berthing times at all ports, it is optimal to set design sailing speeds equal at all legs to maximize schedule reliability. We validate our model using data from a Daily Maersk service, and demonstrate that our schedule can achieve an even higher schedule reliability than the innovative Daily Maersk service schedule, which attained more than 98% reliability in practice, with 11.4% lower bunker consumption. In comparison with a common schedule design heuristic, our model can design service schedules that improve the reliability performance by at least five percentage points, consuming the same amount of bunker. For the same reliability target (of 80%) in schedule design, our model can help reduce bunker consumption by about 11.6% for an 11-week schedule. The savings in bunker consumption can be much more substantial when an ocean carrier aims for higher schedule reliability.