基于边的连续p-中值问题及其与物流分区的关联
The Edge-based Contiguous p-median Problem with Connections to Logistics Districting
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中文总结 AI 辅助
本文提出ECpM问题，构建两种建模连续性的二元规划模型，经测试SPC约束模型求解速度最高提升17倍，且在EBD问题中表现优于传统割集模型。
中文摘要 AI 辅助
本文提出了基于边的连续p-中值（ECpM）问题，旨在将网络中的道路划分为给定数量的紧凑且连续的区域。研究引入了两种二元规划模型，均纳入了网络距离。第一种模型需要指数级数量的基于割集的约束来建模连续性，结合通常仅生成少量此类约束的分离方案，构成了分支-割（B&C）算法。第二种模型利用多项式数量的最短路径约束来建模连续性，可通过现成求解器求解。在节点数超2700、边数近3400的道路网络上对各求解方法进行测试，涉及含超960万个二元变量的模型。通过标准分支定界法求解基于最短路径连续性（SPC）约束的模型，相比基于割集的B&C实现，计算时间最高可提速17倍。此外，SPC约束被证实是基于边的p-中值（EpM）模型的超有效不等式（即未显式要求连续性的模型），意味着它们可排除整数可行解及该简化问题的部分（非全部）最优解。最后，本文探究了ECpM与基于边的分区（EBD）问题的结构关联，EBD需额外满足工作平衡准则。现有基于割集连续性约束的模型在测试实例中均无法在12小时内找到可行解，而基于SPC的EBD模型可将多数实例求解至最优。
英文摘要
This paper introduces the edge-based contiguous p-median (ECpM) problem to partition the roads in a network into a given number of compact and contiguous territories. Two binary programming models are introduced, both of which incorporate a network distance. The first model requires an exponential number of cut set-based constraints to model contiguity; it is paired with a separation scheme that usually generates only a small number of these constraints, namely, a branch-and-cut (B&C) algorithm. The second model utilizes a polynomial number of shortest-path constraints to model contiguity and can be solved with off-the-shelf solvers. The respective solution approaches are tested on road networks with over 2,700 nodes and close to 3,400 edges, yielding models with over 9.6 million binary variables. Solving the model based on shortest path contiguity (SPC) constraints via standard branch and bound attains speedups in computational time of up to 17x relative to the cut set-based B&C implementation. In addition, the SPC constraints are demonstrated to be supervalid inequalities of the edge-based p-median (EpM) model (i.e., for which contiguity is not explicitly required), meaning that they may cut off integer-feasible solutions and some, but not all, of the optimal solutions of this simpler problem. Finally, the paper explores structural insights and connections between ECpM and the edge-based districting (EBD) problem, which enforces an additional work balance criterion. An existing model that utilizes cut set-based contiguity constraints was unable to find a feasible solution within 12 hours for any of the tested instances, while an SPC-based EBD model was able to solve most of these to optimality.
