The Capacitated Location Routing Problem is an important planning and routing problem in logistics, which generalizes the capacitated vehicle routing problem and the uncapacitated facility location problem. In this problem, we are given a set of depots and a set of customers where each depot has an opening cost and each customer has a demand. The goal is to open some depots and route capacitated vehicles from the opened depots to satisfy all customers' demand, while minimizing the total cost. In this paper, we propose a $4.169$-approximation algorithm for this problem, improving the best-known $4.38$-approximation ratio. Moreover, if the demand of each customer is allowed to be delivered by multiple tours, we propose a more refined $4.091$-approximation algorithm. Experimental study on benchmark instances shows that the quality of our computed solutions is better than that of the previous algorithm and is also much closer to optimality than the provable approximation factor.

翻译：容量限制选址路径问题是物流领域中一个重要的规划与路径问题，它推广了容量限制车辆路径问题与无容量限制设施选址问题。在该问题中，给定一组仓库和一组客户，每个仓库有开设成本，每个客户有需求量。目标是在开设部分仓库后，从已开设的仓库派出容量受限的车辆来满足所有客户的需求，同时最小化总成本。本文针对该问题提出了一个$4.169$近似算法，改进了当前已知的$4.38$近似比。此外，若允许每个客户的需求由多条路径共同配送，我们进一步提出了一个更精细的$4.091$近似算法。在基准算例上的实验研究表明，我们算法所得解的质量优于先前算法，并且其实际表现比可证明的近似比更接近最优解。