Lower bounds for the head–body–tail problem on parallel machines: A computational study of the multiprocessor flow shop

The multiprocessor flow-shop is the generalization of the flow-shop in which each machine is replaced by a set of identical machines. As finding a minimum-length schedule is NP-hard, we set out to find good lower and upper bounds. The lower bounds are based on relaxation of the capacities of all machine sets except one. This results in a parallel-machine scheduling problem with release dates and delivery times, for which we derive a number of lower bounds. We pay special attention to the time complexity of algorithms for computing these bounds. To obtain the upper bounds a constructive algorithm in subsequent stages is used. We present an experimental comparison of the various lower and upper bounds for the multiprocessor flow-shop problem.