In this paper, a variant of the vehicle routing problem with mixed backhauls (VRPMB) is presented, i.e. goods have to be delivered from a central depot to linehaul customers, and, at the same time, goods have to be picked up from backhaul customers and brought to the depot. Both types of customers can be visited in mixed sequences. The goods to be delivered or picked up are three-dimensional (cuboid) items. Hence, in addition to a routing plan, a feasible packing plan for each tour has to be provided considering a number of loading constraints. The resulting problem is the vehicle routing problem with three-dimensional loading constraints and mixed backhauls (3L-VRPMB). The simultaneous transport of linehaul and backhaul items presents a particular challenge of the problem. We consider two different loading variants in order to avoid any reloading during the tour: (i) rear loading with separate linehaul and backhaul sections and (ii) loading at a long side. In order to solve the problem, we propose a hybrid metaheuristic consisting of a reactive tabu search for the routing problem and different packing heuristics for the loading problem. Numerical experiments are reported with benchmark instances from the literature for the one-dimensional VRPMB to examine the performance of the routing algorithm and with newly generated instances for the 3L-VRPMB.

中文翻译：

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具有三维载荷约束和混合回程的车辆路径问题的混合算法

在本文中，提出了混合回程（VRPMB）车辆路径问题的一个变体，即货物必须从中央仓库交付给长途客户，同时必须从回程客户那里取货并带到仓库。可以混合顺序访问这两种类型的客户。要交付或提取的货物是三维（长方体）物品。因此，除了路线计划外，还必须考虑到许多装载限制，为每次旅行提供可行的打包计划。由此产生的问题是具有三维负载约束和混合回程 (3L-VRPMB) 的车辆路径问题。线路和回程项目的同时传输提出了该问题的特殊挑战。我们考虑了两种不同的装载方式，以避免在旅行期间重新装载：(i) 使用单独的线路和回程部分进行后装载和 (ii) 长边装载。为了解决这个问题，我们提出了一种混合元启发式算法，包括针对路由问题的反应禁忌搜索和针对加载问题的不同打包启发式算法。使用一维 VRPMB 文献中的基准实例报告了数值实验，以检查路由算法的性能，以及新生成的 3L-VRPMB 实例。我们提出了一种混合元启发式算法，包括针对路由问题的反应禁忌搜索和针对加载问题的不同打包启发式算法。使用一维 VRPMB 文献中的基准实例报告了数值实验，以检查路由算法的性能，以及新生成的 3L-VRPMB 实例。我们提出了一种混合元启发式算法，包括针对路由问题的反应禁忌搜索和针对加载问题的不同打包启发式算法。使用一维 VRPMB 文献中的基准实例报告了数值实验，以检查路由算法的性能，以及新生成的 3L-VRPMB 实例。