An almost robust optimization model for integrated berth allocation and quay crane assignment problem

The integrated berth allocation and quay crane assignment problem is an important issue for the operations management in container terminals. This issue primarily considers the assignment of berthing time, position, and the number of quay cranes in each time segment to ships that must be discharged and loaded at terminals. This study examines such a problem by considering uncertainties in the late arrival of ships and inflation of container quantity. Based on historical data, we first divide the uncertainty set into K non-overlapping full-dimensional clusters via K-means clustering, and the weight of each cluster is calculated. Then, we formulate an almost robust model by introducing the weighted max penalty function with the objective of minimizing the total cost, which is caused by the deviations from the expected berthing location and departure time. The concept of robustness index is introduced to investigate the trade-off between the changes in the objective value and the penalty violation. A decomposition method, which contains a deterministic master problem and a stochastic subproblem, is proposed to solve the problem. In each iteration, the subproblem checks the master problem under different realizations, adds scenarios, and cuts into the master problem if needed. Numerical experiments demonstrate that (i) the proposed method can solve the model efficiently, (ii) the robustness index shows that a significant improvement in objective can be achieved at the expense of a small amount of penalty, (iii) the proposed model can handle uncertainties better than the deterministic, fully robust, and worst-case models in terms of total expected cost, total vessel delays, and utilization rates of the berth and quay crane, and (iv) the proposed method becomes more attractive compared with the first come first served approach as the congestion situation or uncertainty degree increase.