Production smoothing by balancing capacity utilization and advance orders

This paper focuses on advance orders and continuous-time production smoothing under uncertain demands. Similar to the classical newsboy problem, the demands may represent items that quickly become obsolete, spoil or have a future that is uncertain beyond a single period or selling season. The exact demand for items is unknown prior to the end of the selling season and may exceed the available capacity. To handle the uncertainty, initial inventories are accumulated by advance ordering or contracting out at a lower cost relative to the manufacturer's production cost. In contrast to the classical newsboy problem, the manufacturer's capacity is used to smooth inaccuracy in demand estimation during the selling season. The objective is to determine both the advance order quantity to be delivered by the beginning of the selling season and the production rates over the selling season to minimize advance order costs and expected inventory shortage or surplus costs as well as production costs during the season. The maximum principle is employed to study the problem. As a result, closed-form optimal solutions are derived for various production conditions. The sensitivity analysis shows that these solutions do not always depend on the demand shape. An example illustrates the approach.