Flow-shop scheduling in the metal industry

In the metal industry, the processing of a part requires first the heating of the ingot, and when the ingot has reached the desired temperature, it is then launched in the shop and processed by the machines. In a flow-shop environment, a part cools down and its processing time on each machine is increased, to a point where it may be necessary to stop the operations on that part, and reheat it before processing can be resumed. This introduces a state dependence in the processing times. Since the scheduling problem requires the knowledge of the processing times of the jobs, thus the processing times themselves depend on the schedule. We consider the problem of scheduling the jobs if the processing times are schedule dependent. We review the main results in the literature in a deterministic environment