In light of the existing practical applications of the two-dimensional loading on vehicle scheduling and many-to-many supply-demand relationships between suppliers and customers, we address a many-to-many heterogeneous vehicle routing problem with cross-docking and two-dimensional loading constraints (or the heterogeneous 2L-MVRPCD). The newly proposed problem can be regarded as a generalized problem of the many-to-many vehicle routing with cross-docking (MVRPCD) and capacitated vehicle routing problem with two-dimensional loading constraints (2L-CVRP). To solve small-scale 2L-MVRPCDs, a mixed integer linear programming (MILP) model is developed, whereas two hybrid optimization heuristic algorithms are proposed to solve large-scale 2L-MVRPCDs. The first heuristic incorporates a basic adaptive large neighborhood search (ALNS) algorithm and a new best-fit-skyline (BFS) packing heuristic, while the second heuristic, a Tabu-based ALNS (ALNS/TS), extends the first one by embedding an insert-tabu strategy to enhance the intensity and diversity of search. Wide-ranging instances with various many-to-many supply-demand scenarios and different loading configurations are employed to verify the efficacy of the proposed MILP model and two heuristics. Numerical results show that small-scale heterogeneous 2L-MVRPCD instances with and without the rotation constraints can be solved to optimality by commercial solvers, and the proposed heuristics can achieve high-quality solutions within a reasonable computational time for large-scale instances. Meanwhile, the comparison between the proposed heuristics and existing methods verifies the effectiveness and applicability of the proposed heuristics for solving MVRPCD and 2L-CVRP.

鉴于二维装载在车辆调度上的现有实际应用以及供应商和客户之间的多对多供需关系，我们解决了具有交叉对接和双向的多对多异构车辆路径问题。尺寸载荷约束（或异构 2L-MVRPCD）。新提出的问题可以看作是具有交叉对接的多对多车辆路径问题 (MVRPCD) 和具有二维装载约束的有能力车辆路径问题 (2L-CVRP) 的一般化问题。为了解决小型 2L-MVRPCD，开发了混合整数线性规划 (MILP) 模型，同时提出了两种混合优化启发式算法来解决大型 2L-MVRPCD。第一个启发式结合了基本的自适应大邻域搜索 (ALNS) 算法和新的最佳拟合天际线 (BFS) 打包启发式，而第二个启发式是基于 Tabu 的 ALNS (ALNS/TS)，通过嵌入扩展了第一个一种插入禁忌策略，以增强搜索的强度和多样性。采用具有各种多对多供需场景和不同加载配置的广泛实例来验证所提出的 MILP 模型和两种启发式方法的有效性。数值结果表明，具有和不具有旋转约束的小规模异构 2L-MVRPCD 实例可以通过商业求解器求解到最优，并且所提出的启发式可以在合理的计算时间内为大规模实例实现高质量的解决方案。同时，