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A single production facility is dedicated to producing one type of product with completed units going directly into inventory. The demand for the product is governed by a Poisson process and is supplied directly to inventory when available, or is backordered until it is produced by the production facility. Relevant costs are a linear inventory holding cost, a linear backorder cost, a fixed setup cost for initiating a production run. The objective is to find a control policy to minimize the expected cost per time unit. It is shown that the problem may be modeled as an M/D/1 queueing system with the optimal policy being a two-critical-number policy. Cost expressions are derived as functions of the policy parameters, and based on a convexity property of these cost expressions, an efficient search procedure is proposed for finding the optimal policy. In computational tests that were performed on an IBM 360/65, optimal policies were computed in less than 1/8 second of CPU time.
Gavish et al. (1980) studied this question.