Strategy for supplier's direct delivery to minimize the number of vehicles

We present a replenishment scheduling problem to minimize the number of vehicles, when the retailers face a periodic demand and the supplier adopts direct delivery strategy. First, we prove a sufficient condition of one vehicle to replenish several retailers. Based on this sufficient condition, we design a subsets dividing strategy, which divides the set of all the retailers into several subsets and ensures one vehicle can replenish the entire retailers in any one subset. Further, we prove the number of vehicles which is given by our strategy would not exceed 2.29 times of the minimal number of vehicles.