For deteriorating items with seasonal demand, a supplier usually requests that the buyer (retailer) prepays a fraction of the acquisition cost as a deposit. The expiration date of a deteriorating item is an important factor in a buyer's purchase decision. Despite its importance, relatively little attention has been paid to the effects of the expiration date; the versions of economic order quantity models that are available consider fixed deterioration rates. This paper considers a more realistic situation where the deterioration rate of a product gradually increases as the expiration date approaches. In this paper, the optimal cycle time and the cycle fraction of no shortages are the decision variables that minimize the total cost. The total annual relevant cost is shown to be strictly pseudo-convex for each of the decision variables, which simplifies the search for the global solution to a local minimum. This paper provides an improvement on earlier work, as it provides an optimal rather than a near-optimal solution. Several numerical examples are provided to illustrate the behaviour of the model and to highlight some managerial insights.