Capacitated Two - Stage Multi - Item Productioni 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 O(n 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).