Robust berth allocation and scheduling under uncertain arrival and processing times

Port operations frequently face substantial uncertainty in arrival and processing times due to adverse weather, equipment failures, and upstream delays. These uncertainties complicate berth allocation and vessel scheduling and often render deterministic or fully specified stochastic models fragile. In this study, we investigate a novel berth allocation and scheduling problem under uncertainty that jointly optimises assignment and sequencing of vessels by explicitly considering uncertain vessel arrival and processing times. The objective is to minimise total operating cost and the worst-case expected overtime. We formulate the problem as a two-stage distributionally robust optimisation (DRO) model. The first-stage decisions determine berth assignments prior to the realisation of uncertainty, while the second-stage decisions schedule vessels afterward. Our DRO model accounts for uncertain vessel arrival and processing times using their limited distributional information. Specifically, we construct an ambiguity set based on empirical means, mean absolute deviation information, and probability bounds by leveraging historical data, which captures both the central tendency and variability of the uncertain parameters. To effectively solve the DRO model, we first reformulate it into a tractable equivalent mixed-integer linear program by exploiting structural properties of the problem. We then develop a tailored logic-based Benders decomposition (LBBD) algorithm, enhanced by proposed analytically strengthened optimality cuts. In addition, we develop a scenario-based robust optimisation model (RO-BAP) as a conservative benchmark to examine the trade-off between worst-case protection and out-of-sample performance. Extensive computational results confirm the effectiveness of the proposed DRO model and the LBBD algorithm. Notably, the proposed DRO approach outperforms stochastic programming, deterministic BAP, and scenario-based RO-BAP in terms of average out-of-sample performance, particularly when distributional perturbations are introduced. Compared with SAA, the proposed DRO approach reduces the 99th percentile delay by about 9% on average under perturbed distributions, while maintaining lower conservatism than RO-BAP. These results demonstrate that implementing the proposed DRO scheme can better hedge against tail risk and maintain service performance under uncertainty.