In this paper, we introduce a mixed integer quadratic formulation for the congested variant of the partial set covering location problem, which involves determining a subset of facility locations to open and efficiently allocating customers to these facilities to minimize the combined costs of facility opening and congestion while ensuring target coverage. To enhance the resilience of the solution against demand fluctuations, we address the case under uncertain customer demand using $\Gamma$-robustness. We formulate the deterministic problem and its robust counterpart as mixed-integer quadratic problems. We investigate the effect of the protection level in adapted instances from the literature to provide critical insights into how sensitive the planning is to the protection level. Moreover, since the size of the robust counterpart grows with the number of customers, which could be significant in real-world contexts, we propose the use of Benders decomposition to effectively reduce the number of variables by projecting out of the master problem all the variables dependent on the number of customers. We illustrate how to incorporate our Benders approach within a mixed-integer second-order cone programming (MISOCP) solver, addressing explicitly all the ingredients that are instrumental for its success. We discuss single-tree and multi-tree approaches and introduce a perturbation technique to deal with the degeneracy of the Benders subproblem efficiently. Our tailored Benders approaches outperform the perspective reformulation solved using the state-of-the-art MISOCP solver Gurobi on adapted instances from the literature.

翻译：本文针对部分集覆盖选址问题的拥塞变体，提出混合整数二次规划模型，旨在通过确定待开放设施子集并优化客户分配方案，在确保目标覆盖率的同时最小化设施开放与拥塞的联合成本。为增强解对需求波动的鲁棒性，我们采用Γ-鲁棒性方法处理客户需求不确定情形。将确定性模型及其鲁棒对应问题构建为混合整数二次规划问题，通过文献改编算例探究保护水平的影响，揭示规划方案对保护水平的敏感程度。由于鲁棒对应问题的规模随客户数量增长（现实场景中可能非常庞大），我们提出采用Benders分解法，通过将依赖客户数量的变量投影出主问题，有效降低变量规模。详细阐述如何将Benders方法集成至混合整数二阶锥规划（MISOCP）求解器中，明确给出所有关键成功要素。讨论单树与多树求解策略，并引入扰动技术高效处理Benders子问题的退化现象。在文献改编算例上的实验表明，我们定制的Benders方法优于采用最先进MISOCP求解器Gurobi实现的视角重构方法。