Scheduling on a proportionate flowshop to minimise total late work

We study a scheduling problem to minimise total late work, i.e. each job is penalised according to the duration of its parts scheduled after its due-date. The machine setting is an m-machine proportionate flow shop. Two versions of the problem are studied: (i) the case that total late work refers to the last operation of the job (i.e. the operation performed on the last machine of the flow shop); (ii) the case that total late work refers to all the operations (on all machines). Both versions are known to be NP-hard. We prove a crucial property of an optimal schedule, and consequently introduce efficient pseudo-polynomial dynamic programming algorithms for the two versions. The dynamic programming algorithms are tested numerically and proved to perform well on large size instances.