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We consider the following problem in a production-distribution supply chain: a manufacturer is to produce a variety of products. Each product has a due date, and requires an amount of processing time to complete. In addition, it is expected that the finished products are delivered by a transporter with uncertain arrival time. There is a substantial cost incurred if a product cannot be delivered by the transporter. The problem is to determine the optimal production policy by taking into account the probability of the random arrival time of the transporter, so that the expected loss is minimized. We find the specific structures of the optimal policies in different situations.