Customer order scheduling on a single machine with family setup times: Complexity and algorithms

We consider a situation where C customers each order various quantities (possibly zero in some cases) of products from P different families, which can be produced on a continuously available machine in any sequence (requiring a setup whenever production switches from one family to another). We assume that the time needed for a setup depends only on the family to be produced immediately after it, and we follow the item availability model (which implies that all units are ready for dispatch as soon as they are produced). However, an order is shipped only when all units required by a customer are ready. The time from the start (time zero) to the completion of a customer order is called the order lead time. The problem, which restates the original description of the customer order scheduling problem, entails finding a production schedule that will minimize the total order lead time. While this problem has received some attention in the literature, its complexity status has remained vexingly open. In this note, we show for the first time that the problem is strongly NP-hard. We proceed to give dynamic programming based exact solution algorithms for the general problem and a special case (where C is fixed). These algorithms allow us to solve small instances of the problem and understand the problem complexity more fully. In particular, the solution of the special case shows that the problem is solvable in polynomial time when C is fixed. 2006 Elsevier Inc. All rights reserved.