Make-to-order policies for a stochastic lot-sizing problem using overtime

We address a stochastic single item production system in a make-to-order environment where customer orders receive a promised delivery date upon arrival. Capacity is reserved for the production of the item but can be extended by working overtime. The problem consists of determining the optimal size of a production lot, so that delivery promises are met on time at the expense of minimal average costs. These include setup costs, holding costs for orders that are finished before their promised delivery date and penalty costs for orders that are not satisfied on time and are therefore backordered. Also, the use of capacity beyond the regularly available incurs overtime costs. Given that the optimal policy is likely to be too complex in most practical situations, we propose four different strategies for obtaining production lot sizes. Three of the rules are based on the well-known (s, S) and (R, S) policies for make-to-stock problems while the fourth strategy is motivated by a rule originally derived for a similar problem with unlimited production capacity. An extensive numerical study indicates the conditions under which each lot-sizing rule shows the best performance.