Probabilistic analysis on three-level distribution systems

We consider the inventory-routing problem for a three-level distribution system consisting of a single outside vendor, a single warehouse and many geographically dispersed retailers. Each retailer faces external demands for a single item which arise at a deterministic, retailer specific rate. Inventory holding cost is charged at both the warehouse and the retailers. All shipments are delivered by a fleet of homogeneous vehicles of limited capacity. We develop a lower bound on the long run average cost over any feasible policies. We use this lower bound to show that an effective strategy, in which all shipments are delivered from the vendor to the retailers not to pass the warehouse, is at least radic2 asymptotic optimal, and with a high probability, the strategy has a higher asymptotic optimality than the strategy to pass the warehouse. Particularly, if the probability distribution of the retailer demand rates allows for perfect packing, then the strategy is almost surely 100% asymptotic optimal and better than the strategy to pass the warehouse. Our results also show that the strategy not to pass the warehouse as well as the strategy to pass the warehouse would not work very well in three-level distribution systems with limited number of retailers or with many retailers but the perfect packing ratio allowed by the retailer demand rates is small. Further we provide a numerical example to show that in some conditions the strategy to pass the warehouse is better than the strategy not to pass the warehouse. We then can conclude that a hybrid strategy, i.e., combing the strategy not to pass the warehouse with the strategy to pass the warehouse, should be used in three-level distribution systems with limited number of retailers or with many retailers but the perfect packing ratio allowed by the retailer demand rates is small.