In this paper we consider a segment of a supply chain comprising an inventory and a transportation system that cooperate in the fulfillment of stochastic customer orders. The inventory is operated under a discrete time (r, s, q) policy with backorders. The transportation system consists of an in-house transportation capacity which can be extended by costly external transportation capacity (such as a third-party logistics provider). We show that in a system of this kind stock-outs and the resulting accumulation of inventory backorders introduces volatility in the workload of the transportation process. Geunes and Zeng (2001) have shown for a base-stock system, that backordering decreases the variability of transportation orders. Our findings show that in inventory systems with order cycles longer than one period the opposite is true. In both cases, however, inventory decisions and transportation decision must be taken simultaneously. We present a procedure to compute the probability distribution of the number of transportation orders and the resulting excess transportation requirements or rather transportation costs. We show that the increase of transportation costs resulting from a safety stock reduction may offset the change of the inventory costs. This effect may have a significant impact on general optimality statements for multi-echelon inventory systems. 
Integrated optimization of safety stock and transportation capacity