Capacitated Two-Stage Multi-Item Production/Inventory Model with Joint Setup Costs

We analyze a continuous-time, two-stage production/inventory system. In the first stage, a common intermediate product is produced in batches, and possibly stored. In the second phase, the intermediate product is fabricated into n distinct finished products. Several finished products may be included in a single production batch of limited capacity to exploit economies of scale. We propose a planning methodology to address the combined problem of joint setup costs and capacity limits per setup. We restrict ourselves to a class of replenishment strategies with the following properties: a replenishment strategy specifies a collection of families subsets of items covering all end-items, if an item belongs to several families a specific fraction of its sales is assigned to each family. Each time the inventory of one item in a family is replenished, the inventories of all other items in the family are replenished as well. We derive a simple roughly 0n log n algorithm that results in a strategy whose long-run average cost comes within a few percentage points of a lower bound for the minimum achievable cost within the above class of strategies.