We consider a production/inventory system consisting of one production line and multiple products. Finished goods are kept in stock to serve stochastic demand. Demand is fulfilled immediately if there is an item of the requested product in stock and otherwise it is backordered and fulfilled later. The production line is modeled as a non-preemptive single server and the objective is to minimize the sum of the average inventory holding costs and backordering costs. We investigate the structure of the optimal production policy, propose a new scheduling policy, and develop a method for calculating base stock levels under an arbitrary but given scheduling policy. The performance of the various production policies is evaluated in extensive numerical experiments.

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