This paper investigates a robust optimization problem concerning the integration of fleet deployment and empty container repositioning in a shipping line network, where a fleet of vessels is dispatched to transport both laden and empty containers, aiming to fulfill a predetermined set of requests over a defined time horizon. The sizes of customer demands are uncertain and are characterized by a budgeted uncertainty set. This study aims to ascertain the vessel types assigned to each shipping route, the routing of laden containers, and the repositioning of empty containers in a manner that minimizes the total cost. Simultaneously, it ensures the feasibility of all transportation plans for any realization of demand within the uncertainty set. We introduce a path-based two-stage robust formulation for addressing the problem. In the first stage, the assignment of vessel types to each shipping route is determined, and the second stage focuses on establishing the routing of laden containers and repositioning of empty containers under a worst-case scenario. We propose the Column-and-Constraint Generation algorithm for solving the proposed robust formulation. To address large-scale size instances, we propose an acceleration technique, i.e., the piece-wise affine policy, which reduces the dimensions of the uncertainty set while maintaining a bounded compromise in solution quality. Comprehensive numerical experiments derived from real-world industries, such as the Shanghai port and CMA CGM, are conducted to validate the proposed formulation and solution methodologies.

中文翻译：

---

需求不确定性下的班轮船队部署和空箱重新定位：一种稳健的优化方法


本文研究了一个稳健的优化问题，该问题涉及在航运公司网络中集成船队部署和空箱重新定位，其中一支船队被派遣运输满载集装箱和空箱，旨在在定义的时间范围内满足一组预定的请求。客户需求的大小是不确定的，其特点是预算的不确定性集。本研究旨在确定分配给每条运输路线的船舶类型、满载集装箱的路线以及以最小化总成本的方式重新定位空集装箱。同时，它确保了所有运输计划在不确定性范围内实现需求的可行性。我们引入了一种基于路径的两阶段稳健公式来解决这个问题。在第一阶段，确定每条航线的船舶类型分配，第二阶段侧重于在最坏情况下确定满载集装箱的路线和空集装箱的重新定位。我们提出了 Column-and-Constraint Generation 算法来求解所提出的稳健公式。为了解决大规模实例的问题，我们提出了一种加速技术，即分段仿射策略，它减少了不确定性集的维度，同时保持了解决方案质量的有限折衷。进行了来自实际行业（如上海港和 CMA CGM）的综合数值实验，以验证所提出的公式和求解方法。