Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries

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We study a supply chain scheduling problem, where a common due date is assigned to all jobs and the number of jobs in delivery batches is constrained by the batch size. Our goal is to minimize the sum of the weighted number of tardy jobs, the due-date-assignment costs and the batch-delivery costs. We show that some well-known \(\mathcal{NP}\)-hard problems reduce to our problem. Then we propose a pseudo-polynomial algorithm for the problem, establishing that it is \(\mathcal{NP}\)-hard only in the ordinary sense. Finally, we convert the algorithm into an efficient fully polynomial time approximation scheme.