In this paper we introduce an a priori variant of the capacitated vehicle routing problem and provide a constant factor approximation algorithm, when the demands of customers are independent and identically distributed Bernoulli experiments. In the capacitated vehicle routing problem (CVRP) a vehicle, starting at a depot, must visit a set of customers to deliver the requested quantity of some item. The vehicle has capacity k, which is the maximum number of items that the vehicle can carry at any time, but it can always return to the depot to restock. The objective is to find the shortest tour subject to these constraints. In the a priori CVRP with unit demands the vehicle has to visit a set of active customers, drawn from some distribution. Every active customer has a demand of one. The goal is to find a master tour, which is a feasible solution to the deterministic CVRP, i.e. where all customers are active. Then, given a set of active customers, the tour is shortcut to only visit those customers. The cost of the tour is the expected cost with respect to the distribution of active customers. We consider the model, where every customer is independently active with the same probability.

Let N be the number of customers. We provide an algorithm, which takes as input an a priori TSP solution with approximation factor γ, and gives a solution to the a priori CVRP with unit demands, whose cost is at most $(1+k/N+\gamma )$ times the value of the optimal solution. Specifically, this gives an expected 5.5−approximation by using the 3.5−approximation to the a priori TSP from Van Ee and Sitters [17] and a deterministic 8.5−approximation by using the deterministic 6.5−approximation from Van Zuylen [18].

在本文中，当客户的需求是独立且分布均匀的伯努利实验时，我们介绍了有能力车辆路径问题的先验变体，并提供了恒定因子近似算法。在严重的车辆路线问题（CVRP）中，从仓库开始的车辆必须拜访一组客户才能交付所需数量的某些物品。车辆的容量为k，这是车辆随时可以携带的最大物品数量，但始终可以返回到仓库进行补货。目的是找到受这些限制的最短行程。在具有单位需求的先验CVRP中，车辆必须拜访一些经分配的活跃客户。每个活跃的客户都有一个需求。目的是找到一个大师之旅，这是确定性CVRP（即所有客户活跃的地方）的可行解决方案。然后，在给定一组活跃客户的情况下，游览是仅拜访那些客户的捷径。巡回费用是与活跃客户分配有关的预期费用。我们考虑该模型，其中每个客户以相同的概率独立活跃。

令N为客户数量。我们提供了一种算法，该算法以具有近似因子γ的先验TSP解作为输入，并给出了具有最大单位成本的先验CVRP的解。$（\mathrm{1个}+ķ/ñ+\gamma ）$乘以最优解的值。具体来说，通过使用Van Ee和Sitters [17]的先验TSP的3.5逼近，可以得出期望的5.5逼近；使用Van Zuylen [18]的确定性6.5逼近，可以得出确定的8.5逼近。