The three-machine proportionate flow shop problem with unequal machine speeds

In flow shop scheduling it is usually assumed that the processing times of the operations belonging to the same job are unrelated. But when a job corresponds to producing a certain quantity of some good, then it is likely that the processing times are related to this quantity, and hence are constant except for some factor that depends on the speed of the machine. In this paper, we consider this special case of the flow shop problem, which we call the proportionate flow shop problem with unequal machine speeds. This is a special case of the flow shop problem with ordered processing times that has been studied by Smtih, Panwalkar, and Dudek. Their results imply that makespan minimization is easy if the first or last machine is slowest; if the second machine is slowest, then there exists an optimum schedule that is V-shaped. We provide an algorithm that solves this problem (and the more general problem with ordered processing times) to optimality in pseudopolynomial time, and we show that this is best possible by establishing NP-hardness in the ordinary sense.