Inventory control in directed networks: a note on linear costs

We consider periodic review inventory control problems in directed networks, primary examples of which are distribution systems and assembly systems. External demand could occur at each node. When inventory is insufficient to meet requirements at a node, a portion of this demand is backordered and the remaining is lost. External demands, as well as lead times for inventory purchase, assembly, and transportation, are stochastic. In each period, linear sales revenues and the following costs, all linear, are charged at each node: (a) inventory purchase/assembly/transfer cost to receive inventory, (b) holding cost, (c) backorder cost, and (d) lost sales cost. When the objective function of interest is a discounted sum of profits over a finite planning horizon, it is shown that the sales prices and the inventory purchase/assembly/transfer cost parameters can be assumed to be zero without loss of generality. The result is proved for every realization of demands and lead times. Some extensions to these results are discussed. During this process, we also generalize the concept of echelon inventories to directed networks.