Scheduling the production of several items with random demands in a single facility

Consider the problem of scheduling the production of several items in a single facility that can produce only one item at a time. This problem occurs since it is often economic to produce several items in a single facility. The objective is to reduce the long run average holding, backorder and setup costs. We assume that demands are random with constant expected rates. We allow backorders and charge holding and backlogging costs at linear time weighted rates. Items are produced at continuous constant rates. Setup times and setup costs are item dependent constants. These parameters, however, are independent of the order of setups. A real-time scheduling tool is developed in three steps. First, with demands replaced by their expectations, we compute an optimal or near-optimal target cyclic schedule. Next, we study the problem of scheduling the facility after a single disruption perturbs the investories. The goal is to recover the target cyclic schedule at minimal excess over the average cost of the cyclic schedule. We formulate this as a control problem and obtain a linear recovery policy that is optimal for a large configuration of disruptions. Finally, we select safety stocks to minimize the long run average cost of following the target schedule with the recovery policy. We show that optimal safety stocks are unique and have the property that in the long run the proportion of time that an item is in stock is the ratio of backorder to holding plus backorder cost. We present an example that integrates the cyclic schedule, the control policy and the safety stocks.