Abstract: The rapid growth of e-commerce has spurred the need for efficient urban logistics solutions. Conventional intra-city express delivery networks face challenges like congestion, pollution, and susceptibility to disruptions. One potential solution is the Metro-based Underground Logistics System (MULS), which integrates metro infrastructure with logistics operations, promising cost reduction and operational streamlining. In multi-to-multi networks like intra-city express delivery, aviation, and telecommunications, the hub-and-spoke network shows extensive application potential. However, current research mainly focuses on fully interconnected hub networks, termed the Hub Location Problem (HLP). Some scholars have noted that generating incomplete hub networks may offer advantages, with incomplete hub networks possibly forming circular or tree-like topologies. Apart from incomplete HLP in fully connected networks, limited research exists on incomplete HLP within incomplete underlying networks like subways and railways. Although some past studies have employed hub-and-spoke network designs for MULS systems, applying incomplete HLP methods based on incomplete underlying networks in MULS system design remains relatively unexplored. Moreover, existing research mainly addresses deterministic express delivery demands. However, the random distribution pattern of intra-city express delivery demand requires special consideration in MULS design. Solely addressing deterministic demands may prove inadequate for practical operations and could lead to excessive resource wastage.
This research aims to propose an intra-city metro logistics system utilizing both surface road and metro transportation modes to enhance intra-city express delivery efficiency. We determine metro hub locations from a potential hub set, establish routes between metro hubs, and assign customers to metro hubs. Unlike previous research, we define the Hub Location Problem in an incompletely connected network environment, termed the Incomplete Hub Location Problem (IHLP), to address intra-city metro logistics network planning. The IHLP model proposed in this paper can be directly applied to real metro network structures, enhancing its practicality. Furthermore, we not only consider service time constraints but also incorporate metro transfers into the decision-making process. To address uncertainty in express delivery demand, we develop a distributionally robust optimization model for the IHLP to enhance the robustness of the metro logistics network.
As the proposed distributionally robust optimization problem is a semi-infinite chance-constrained optimization model, we delve into two cases of ambiguous sets. Firstly, we consider probability distributions with zero-mean bounded perturbations and transform the ambiguous chance constraint into operational forms through feasible approximation methods. Secondly, we explore Gaussian perturbation families with partial mean and variance information, enabling us to convert the ambiguous chance constraint into deterministic equivalent forms. The models derived from these two ambiguous set approximations are referred to as the Bounded Perturbation-DRO model and the Gaussian Perturbation-DRO model, respectively.
Finally, we conduct numerical experiments using a portion of the Shanghai metro network as our experimental platform, along with test cases generated from the AP dataset. All numerical experiments are solved using Python calling CPLEX. The optimal values obtained for the Bounded Perturbation-DRO and Gaussian Perturbation-DRO models are 178349.32 and 176817.28, respectively, with the Gaussian Perturbation-DRO model yielding the smallest optimal value. Additionally, sensitivity analysis of parameters like allowed maximum service time, discount factor, and transfer cost reveals their significant impact on network configuration and optimal values. For instance, excessively compressing the allowed maximum service time may lead to increased total operational costs. Furthermore, we compare the proposed distributionally robust optimization with classical robust optimization methods and stochastic optimization methods. The experimental results indicate that compared to classical robust optimization methods, the distributionally robust optimization method effectively avoids overly conservative optimization results, while in comparison to stochastic optimization methods, it incurs a small cost of describing the uncertainty of probability distribution.

Key words: incomplete hub location, underground logistics system, demand uncertainty, integer programming, distributionally robust optimization

摘要： 运用基于地铁的地下物流系统(MULS, Metro-based Underground Logistics System)开展同城快递配送服务,不仅能够提升配送效率,而且能减少快递运输过程中产生的环境污染和缓解其带来的地面交通拥堵压力。为设计高效的同城地铁物流网络,本文综合考虑了地铁网络的非全连通拓扑与换乘操作等关键因素,以总运营成本最小化为优化目标,建立了不完全连接枢纽选址问题(IHLP,Incomplete Hub Location Problem)的整数规划模型。为应对快递需求的不确定性,本研究首先构建了一个基于IHLP的分布鲁棒优化模型,以提升地铁物流网络的鲁棒性。进而,为高效求解该模型,本文探讨了两种非精确集,并分别推导了其等价的可处理逼近形式。为确保所建模型的有效性,本研究采取数值实验进行验证,并深入探讨了关键参数对网络配置的影响规律。实验结果表明:与经典鲁棒优化方法相比,分布式鲁棒优化方法可有效避免产生过于保守的优化结果;与随机优化方法相比,该方法在描述概率分布不确定方面付出了较小的代价。

关键词: 不完全枢纽选址, 地下物流系统, 需求不确定性, 整数规划, 分布鲁棒优化