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. It is assumed that demands are random with constant expected rates. Backorders and charge holding and backlogging costs at linear time weighted rates are allowed. 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, an optimal or near-optimal target cyclic schedule is computed. Next, the problem of scheduling the facility after a single disruption perturbs the inventories is studied. The goal is to recover the target cyclic schedule at minimal excess over the average cost of the cyclic schedule. This is formulated as a control problem and a linear recovery policy that is optimal for a large configuration of disruptions is obtained. Finally, safety stocks are selected to minimize the long run average cost of following the target schedule with the recovery policy. It is shown 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. An example that integrates the cyclic schedule, the control policy and the safety stocks is presented.