3 Optimal Order - Up - To Quantity

We consider inventory systems of “perishable” goods. Perishable goods cannot be carried from one period to another. Examples of perishable goods include newspaper, Christmas tree, and many grocery items like milk and banana. Here, a period could be one day, or one week, or any time frame that is appropriate in the application context. Suppose that a retailer sells a particular perishable product in a particular market, say, The New York Times at Georgia Tech. Assume the retailer’s selling price of an item is p. At the beginning to the period, the retailer procures the items at a unit price of cv < p. The salvage value of each leftover item is assumed to be sv < cv, which can be negative. (When sv is negative, you pay somebody to get rid of the inventory.) In other words, the per unit holding cost of leftover items is h = −sv. In The New York Times example, one possible set of values for these parameters are: p = $1.00, cv = $.25, and sv = 0. Suppose that the demand during a period is D. If the retailer stocks q items at the beginning of the period, the profit for that period is