A note: maximizing the weighted number of Just-in-Time jobs for a given job sequence

We study a single machine scheduling problem to maximize the weighted number of Just-in-Time jobs, i.e., jobs which are completed exactly at their due-dates. We focus on the case that the job sequence is given. A pseudo-polynomial solution algorithm has been published for this problem, and we introduce here an efficient polynomial time dynamic programming algorithm. The new algorithm is tested numerically, and the results are compared to those obtained by the old algorithm. While the running times required by both algorithms for solving large instances are extremely small, it appears that the new algorithm performs better in two aspects: (1) The resulting running time is independent of the actual density of the due-dates, (2) The required memory is significantly reduced. An extension to a two-machine flowshop is provided, and this case is shown to remain polynomially solvable.