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We address a practical problem faced by many firms. The problem is deciding on the production levels for a product that has a very short selling season. The firm has a full period to produce and meet a lumpy demand which occurs at the end of the period. The product is no longer demanded after the end of the period. A constant production rate which minimizes average unit cost may increase holding costs. Varying the production rate at discrete points in time may increase production costs but may also decrease holding costs. In addition, allowing changes in the production rate enables the incorporation of forecast revisions into the production plan. Therefore, the best production plan depends on the flexibility of the production system and on the holding cost. In this paper, we formulate and solve a model of this production planning problem. Two models are developed to deal with two types of the average unit cost function. Numerical examples are used to illustrate the results of the model.