AI 中文总结
针对最大最小覆盖选址阻断问题，提出一种精确嵌套分解算法，利用结构建立并加强阻断割，闭式求解分离问题，显著快于通用双层求解器，可求解30万客户实例。
AI 中文摘要
我们提出了最大最小覆盖选址阻断问题，这是一个双层优化问题，其中领导者阻断一组最小成本的候选位置，使得受预算约束的跟随者的最优覆盖不超过预定目标。每个客户的覆盖由能够为其服务的最不利的开放设施决定。由此产生的跟随者覆盖集函数是非单调且非子模的，因此阻断割的有效性不能从现有框架中继承。我们开发了一种精确的嵌套分解方法，在两层都利用了最大最小覆盖结构。在外层，我们从该结构建立了阻断割的有效性，并通过设施特定的系数加强这些割，这些系数限定了当关键设施被移除时覆盖的损失。我们通过分支-本德斯割方法求解NP困难的跟随者问题，在投影出客户覆盖变量之后，并以闭式形式刻画分离问题的最优对偶解，无需求解线性规划即可在线性时间内生成本德斯最优性割。在基准实例上的计算实验表明，这两个要素都显著提高了方法的性能。在候选位置数量适中的情况下，具有300,000个客户的实例在一小时内可求解至最优。与文献中两个通用双层求解器的比较表明，所提出的方法快一到三个数量级，并且能够求解至最优那些两个求解器都无法闭合的实例。该模型适用于规划者无法控制哪个设施为客户服务的场景，例如公共自动体外除颤器（AED）网络。一项使用弗吉尼亚海滩真实AED数据的网络案例研究说明了该模型。
英文摘要
We introduce the Max-Min Covering Location Blocker Problem, a bilevel optimization problem in which a leader blocks a minimum-cost set of candidate locations so that the optimal coverage of a budget-constrained follower does not exceed a prescribed target. Each customer's coverage is determined by the least favorable open facility that can serve it. The induced follower coverage set function is nonmonotone and nonsubmodular, so the validity of interdiction cuts cannot be inherited from existing frameworks. We develop an exact nested decomposition that exploits the max-min coverage structure at both levels. At the outer level, we establish the validity of interdiction cuts from this structure and strengthen them with facility-specific coefficients bounding the coverage lost when a critical facility is removed. We solve the NP-hard follower problem by branch-and-Benders-cut after projecting out the customer coverage variables, and characterize an optimal dual solution of the separation problem in closed form, generating Benders optimality cuts in linear time without solving a linear program. Computational experiments on benchmark instances show that both ingredients substantially improve the performance of the method. With a moderate number of candidate locations, instances with 300,000 customers are solved to optimality within one hour. A comparison with two general-purpose bilevel solvers from the literature shows the proposed method to be one to three orders of magnitude faster, and to solve to optimality instances that neither of them closes. The model applies wherever the planner cannot control which facility serves a customer, as in public automated external defibrillator (AED) networks. A case study on a network using real AED data from Virginia Beach illustrates the model.
