Recently, machine learning of the branch and bound algorithm has shown promise in approximating competent solutions to NP-hard problems. In this paper, we utilize and comprehensively compare the outcomes of three neural networks--graph convolutional neural network (GCNN), GraphSAGE, and graph attention network (GAT)--to solve the capacitated vehicle routing problem. We train these neural networks to emulate the decision-making process of the computationally expensive Strong Branching strategy. The neural networks are trained on six instances with distinct topologies from the CVRPLIB and evaluated on eight additional instances. Moreover, we reduced the minimum number of vehicles required to solve a CVRP instance to a bin-packing problem, which was addressed in a similar manner. Through rigorous experimentation, we found that this approach can match or improve upon the performance of the branch and bound algorithm with the Strong Branching strategy while requiring significantly less computational time. The source code that corresponds to our research findings and methodology is readily accessible and available for reference at the following web address: \href{https://isotlaboratory.github.io/ml4vrp/}{https://isotlaboratory.github.io/ml4vrp/}.

利用机器学习的分支限界算法，通过比较三个神经网络——图卷积神经网络（GCNN）、GraphSAGE和图注意力网络（GAT）的结果，在解决有容量限制的车辆路径问题方面展现了潜力。通过训练这些神经网络，以模拟计算代价高昂的Strong Branching策略的决策过程。经过严格的实验证明，这种方法可以在较短的计算时间内达到或超过分支限界算法与Strong Branching策略的性能，并且所对应的研究结果和方法的源代码在以下网址中可以轻松获得并进行参考。

关于车辆路径规划分支定界统计学习的研究