While many time-dependent network design problems can be formulated as time-indexed formulations with strong relaxations, the size of these formulations depends on the discretization of the time horizon and can become prohibitively large. The recently-developed dynamic discretization discovery (DDD) method allows many time-dependent problems to become more tractable by iteratively solving instances of the problem on smaller networks where each node has its own discrete set of departure times. However, in the current implementation of DDD, all arcs departing a common node share the same set of departure times. This causes DDD to be ineffective for solving problems where all near-optimal solutions require many distinct departure times at the majority of the high-degree nodes in the network. Region-based networks are one such structure that often leads to many high-degree nodes, and their increasing popularity underscores the importance of tailoring solution methods for these networks. To improve methods for solving problems that require many departure times at nodes, we develop a DDD framework where the set of departure times is determined on the arc level rather than the node level. We apply this arc-based DDD method to instances of the service network design problem (SND). We show that an arc-based approach is particularly advantageous when instances arise from region-based networks, and when candidate paths are fixed in the base graph for each commodity. Moreover, our algorithm builds upon the existing DDD framework and achieves these improvements with only benign modifications to the original implementation.

翻译：许多时间依赖网络设计问题可转化为具有强松弛性质的时间索引公式，但其规模取决于时间范围离散化程度，往往导致计算量过大。最新发展的动态离散化发现方法通过迭代求解较小网络上的问题实例——其中每个节点拥有独立的离港时间离散集——使许多时间依赖问题更易处理。然而现有DDD实现中，从同一节点出发的所有弧共享相同的离港时间集。这导致当网络中度数较高的节点在大多数近最优解中均需大量不同离港时间时，DDD方法难以有效求解。区域网络这类结构常包含大量高节点度，其日益普及凸显了针对此类网络定制求解方法的重要性。为改进需要节点大量离港时间的问题求解方法，我们开发了一个DDD框架，其中离港时间集在弧层面而非节点层面确定。我们将该基于弧的DDD方法应用于服务网络设计问题实例，证明当实例源于区域网络且各商品候选路径在基图中固定时，基于弧的方法具有显著优势。此外，我们的算法基于现有DDD框架构建，仅需对原始实现进行细微修改即可实现这些改进。