Inventory Replenishment Scheduling to Minimize the Number of Vehicles

We deal with a vendor managed inventory system in which one supplier supplies an item to multiple retailers. Customer demand arrives at each retailer at a constant rate, and each retailer has a maximal replenishment interval. A fleet of vehicles are required by the supplier to replenish the retailers. It is assumed that one vehicle can replenish only one retailer per period and shortages are not allowed. The goal is to determine the replenishment scheduling during infinite horizon that minimize the number of vehicles. By analyzing the existing condition of the feasible scheduling of multiple vehicles, we develop an approximation algorithm, and prove the approximation ratio of this algorithm to be 2.