Summary: Let demand for a product be a random variable, which is independently and identically distributed in successive periods. Consider a firm producing this commodity subject to the following costs: the direct cost of production is proportional to output; the cost of changing the rate of production is proportional to the size of the change with possibly a different cost coefficient applying to upward and downward shifts; the cost of storage is proportional to the size of the stock at the beginning of the period, and the cost of (penalty for) shortage is proportional to the size of the shortage. Let unfilled demand be carried over into the next period. The problem is to find the optimal rate of production conditional on the rate of production in the last period and on the current level of stock. This paper discusses the nature of the solution, but leaves open all questions concerning the most efficient methods of computation.