Quay crane scheduling is a key aspect of container terminal operation, which can be regarded as a decision-making process with uncertainty. Each task involves stochastic loading and unloading operation times owing to the existence of uncertainty. In this study, we investigate the unidirectional quay crane scheduling problem for a stochastic processing time, which requires that all the quay cranes move in the same direction either from bow to stern, or vice versa, throughout the planning horizon. The problem is formulated as a two-stage stochastic mixed-integer programming model, where the binary first-stage decision variables correspond to the assignment of tasks to quay cranes, and the mixed-integer second-stage decision variables are related to the generation of detailed schedules. To make the model solvable, we develop an alternative equivalent reformulation with a special structure that involves binary variables in the first stage and continuous variables in the second stage. To solve this reformulated model, an integer L-shaped method is presented for small-size instances, and a simulated annealing algorithm is presented for large-size instances to obtain near-optimal solutions. Numerical experiments show that the integer L-shaped method and simulated annealing algorithm could efficiently solve the unidirectional quay crane scheduling problem with uncertainty. The results also indicate that the stochastic model has distinct advantages in terms of shortening the completion time of vessels and improving the service level of container terminals compared with the expected value problem solutions.

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具有不确定性的单向码头起重机调度问题的随机规划方法

码头起重机调度是集装箱码头运营的一个关键环节，可以看作是一个具有不确定性的决策过程。由于存在不确定性，每个任务都涉及随机加载和卸载操作时间。在这项研究中，我们研究了随机处理时间的单向码头起重机调度问题，这要求所有码头起重机在整个规划范围内从船头到船尾或反之亦然沿同一方向移动。该问题被表述为两阶段随机混合整数规划模型，其中二元第一阶段决策变量对应于码头起重机的任务分配，混合整数第二阶段决策变量与生成详细的时间表。为了使模型可解，我们开发了一种具有特殊结构的替代等效重构，其中涉及第一阶段的二元变量和第二阶段的连续变量。为了解决这个重新制定的模型，针对小尺寸实例提出了一种整数 L 形方法，针对大尺寸实例提出了一种模拟退火算法以获得近乎最优的解决方案。数值实验表明，整数L形方法和模拟退火算法可以有效解决具有不确定性的单向码头起重机调度问题。结果还表明，与期望值问题解决方案相比，随机模型在缩短船舶完工时间和提高集装箱码头服务水平方面具有明显优势。