Multi-shift drayage planning for batches of containers: A Branch-and-Benders-and-Price algorithm

This paper investigates a multi-shift drayage planning problem arising from container truck transportation across multiple terminals within a port area. We consider it at the tactical planning level and determine the optimal truck workload for each shift. The main distinction between our problem and others is the incorporation of multiple shift planning, handling container transportation requests—each consisting of a batch of containers with the same origin and destination—and accounting for their completion times. A mixed integer programming model is proposed to minimize total transportation completion time. To solve large-scale instances, we develop a Branch-and-Benders-and-Price algorithm. This approach not only decomposes the problem into a series of manageable sub-problems but also divides the workload determination into two tractable steps: one for assigning workloads to shifts and another for verifying the feasibility of these assignments. Unlike the common Branch and Price, our approach maintains a subset of variables as integers while allowing the remaining variables to be continuous, significantly improving the lower bound and enabling obtaining optimal solutions efficiently. We validate the proposed approach via random instances and real-world cases. The results demonstrate that our approach outperforms a solver and a Branch and Price. We also apply our method to a real-world case involving Roll-On/Roll-Off terminal cargo transfer, which shares key similarities with the problem at hand, thereby further broadening the scope of our approach’s applicability. And, sensitivity tests are conducted to demonstrate the robustness of our approach against variations in problem settings.