A column generation approach for the crane scheduling with sidekick in a perpendicular automated yard block

In a new container terminal design, the perpendicular block is equipped with a Rail Mounted Gantry crane (RMG) and a sidekick Ground Trolley (GT). The RMG can handle containers by itself, or utilize the sidekick GT for container delivery. Unlike the traditional crane scheduling problem (CSP), the makespan is not merely determined by the job sequence, but also affected by the GT allocation and the RMG-GT synchronization. This novel optimization problem is defined as the crane scheduling problem with sidekick (CSP-S). It can be formulated as a mixed integer programming model to explicitly determine the GT allocation and the handover points. When the choice of handover points is remodeled as an implicit decision, the CSP-S can be transformed into an arc-based surrogate model. Based on the solution patterns of the surrogate model, it is found that the RMG-GT working plan can be decomposed into smaller “segments”. Each segment consists of two intertwined paths of the RMG and the GT respectively. A column generation approach is proposed, where the segments can be generated by a multi-attribute labeling algorithm based on the surrogate model. This solution approach can optimally solve the traditional CSP, and outperforms all benchmark methods under the CSP-S. This can be attributed to the multi-attribute labeling algorithm, which accurately controls the RMG-GT synchronization. Numerical experiments showed that the column generation approach could be well implemented under the rolling-horizon manner in practice. It improved the “Nearest Job” strategy and the TSP-based method by more than 10 % and 5 % respectively under the CSP-S. Experiments also revealed that the RMG-GT system would benefit from a faster GT, but the marginal contribution of the GT speed was gradually reducing.