This study addresses a vehicle routing problem (VRP) in which demands are discrete, split delivery is allowed, service time is proportional to the units of delivered products, multiple time windows are provided and the demand of each customer must be delivered in only one time window (this requirement is termed synchronization constraint). We formulate this problem into a three-index vehicle-flow model and a set-covering model. Then, we propose a branch-and-price-and-cut algorithm to solve the problem. We compare our branch-and-price-and-cut algorithm with CPLEX based on 252 randomly generated instances and the computational results demonstrate the effectiveness of our algorithm.

这项研究解决了一个车辆路线选择问题（VRP），其中需求是离散的，允许分开交付，服务时间与所交付产品的单位成比例，提供了多个时间窗口，并且每个客户的需求只能一次交付窗口（此要求称为同步约束）。我们将此问题公式化为三指标车辆流量模型和布景模型。然后，我们提出了一种“分支降价”的算法来解决该问题。我们基于252个随机生成的实例，将我们的分支和价格和割价算法与CPLEX进行了比较，计算结果证明了该算法的有效性。