Scheduling ordered open shops

We examine special cases of the open-shop problem to further define the boundary between “easy” and “hard” versions of the problem. A polynomial algorithm to minimize the sum of completion times in a preemptive open shop with ordered jobs is presented. A modification of this algorithm also minimizes tardiness in an agreeable open shop. We then show that minimizing makespan in an ordered three-machine open shop is NP-complete, which implies many other open-shop problems are also NP-complete. Finally, we show that slightly different assumptions can change a “hard” problem to an “easy” one. This is illustrated by the problem of minimizing makespan in a three-machine open shop with ordered machines, which is NP-complete. However, by further assuming the job with longest processing time on machine one is different from the job having the longest processing time on the second machine, a polynomial algorithm exists.