In this research, we have formulated a Two-Stage Distributionally Robust Optimization (TDRO) model within the context of a mean–variance ambiguity set, specifically designed to address the challenges in the Integrated Berth Allocation and Quay Crane Assignment Problem (BACAP). A key consideration in this study is the inherent uncertainty associated with ships’ arrival times. During the initial stage, we derive a baseline schedule governing berth allocation and quay crane assignment. Anticipating potential disruptions arising from uncertain arrival delays, the second stage is meticulously formulated to determine the worst-case expectation of adjustment costs within the mean–variance ambiguity set. Subsequently, we undertake an equivalent transformation, converting the general TDRO model into a Two-Stage Robust Second-Order Cone Programming (TRO-SOCP) model. This transformation facilitates the application of the Column and Constraint Generation (C&CG) algorithm, ensuring the derivation of an exact solution. To address the computational intricacies associated with second-order cone programming, we propose two enhancement strategies for upper and lower bounds, aimed at expediting the solution process. Additionally, to contend with large-scale instances, we introduce a refinement and approximation method, transforming the TDRO model into a Mixed-Integer Programming (MIP) model. Furthermore, extensive numerical experiments are executed on both synthetic and real-life instances to validate the superior performance of our model and algorithms. In terms of the total cost, the TDRO model demonstrates superior performance compared with Two-Stage Stochastic Programming (TSP) and Two-Stage Robust Optimization (TRO) models.

中文翻译：

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优化综合泊位分配和码头起重机分配：一种分布式稳健方法


在这项研究中，我们在均值-方差模糊集的背景下制定了一个两阶段分布稳健优化 （TDRO） 模型，专门用于解决综合泊位分配和码头起重机分配问题 （BACAP） 中的挑战。本研究的一个关键考虑因素是与船舶到达时间相关的固有不确定性。在初始阶段，我们得出一个管理泊位分配和码头起重机分配的基线时间表。预测到不确定的到达延迟可能造成的中断，第二阶段经过精心制定，以确定均值-方差模糊集内调整成本的最坏情况预期。随后，我们进行了等效的转换，将通用 TDRO 模型转换为两阶段鲁棒二阶锥规划 （TRO-SOCP） 模型。这种转换促进了列和约束生成 （C&CG） 算法的应用，从而确保推导出精确的解决方案。为了解决与二阶锥规划相关的计算复杂性，我们提出了两种上限和下限的增强策略，旨在加快求解过程。此外，为了应对大规模实例，我们引入了一种细化和近似方法，将 TDRO 模型转换为混合整数规划 （MIP） 模型。此外，在合成和真实实例上执行了大量的数值实验，以验证我们的模型和算法的卓越性能。在总成本方面，与两阶段随机规划 （TSP） 和两阶段稳健优化 （TRO） 模型相比，TDRO 模型表现出卓越的性能。