Cross-dockingis a useful concept used by many companies to control the product flow. It enables the transshipment process of products from suppliers to customers. This research thus extends the benefit of cross-docking with reverse logistics, since return process management has become an important field in various businesses. The vehicle routing problem in a distribution network is considered to be an integrated model, namely the vehicle routing problem with reverse cross-docking (VRP-RCD). This study develops a mathematical model to minimize the costs of moving products in a four-level supply chain network that involves suppliers, cross-dock, customers, and outlets. A matheuristic based on an adaptive large neighborhood search (ALNS) algorithm and a set partitioning formulation is introduced to solve benchmark instances. We compare the results against those obtained by optimization software, as well as other algorithms such as ALNS, a hybrid algorithm based on large neighborhood search and simulated annealing (LNS-SA), and ALNS-SA. Experimental results show the competitiveness of the matheuristic that is able to obtain all optimal solutions for small instances within shorter computational times. For larger instances, the matheuristic outperforms the other algorithms using the same computational times. Finally, we analyze the importance of the set partitioning formulation and the different operators.

交叉对接是许多公司用来控制产品流的有用概念。它支持产品从供应商到客户的转运过程。因此，这项研究扩展了与逆向物流交叉对接的好处，因为退货流程管理已成为各种业务的重要领域。配电网络中的车辆路径问题被认为是一个集成模型，即具有反向交叉对接的车辆路径问题（VRP-RCD）。本研究开发了一个数学模型，以最大限度地减少在四级供应链网络中移动产品的成本，该网络涉及供应商、交叉码头、客户和网点。引入了基于自适应大邻域搜索 (ALNS) 算法和集合分区公式的数学算法来求解基准实例。我们将结果与优化软件以及其他算法（如 ALNS、基于大型邻域搜索和模拟退火的混合算法 (LNS-SA) 和 ALNS-SA）获得的结果进行比较。实验结果显示了数学算法的竞争力，它能够在更短的计算时间内获得小实例的所有最优解。对于较大的实例，数学算法优于使用相同计算时间的其他算法。最后，我们分析了集合划分公式和不同算子的重要性。实验结果显示了数学算法的竞争力，它能够在更短的计算时间内获得小实例的所有最优解。对于较大的实例，数学算法优于使用相同计算时间的其他算法。最后，我们分析了集合划分公式和不同算子的重要性。实验结果显示了数学算法的竞争力，它能够在更短的计算时间内获得小实例的所有最优解。对于较大的实例，数学算法优于使用相同计算时间的其他算法。最后，我们分析了集合划分公式和不同算子的重要性。