The single-item lot-sizing problem with immediate lost sales

We introduce a profit maximization version of the well-known Wagner–Whitin model for the deterministic uncapacitated single-item lot-sizing problem with lost sales. Demand canot be backlogged, but it does not have to be satisfied either. Costs and selling prices are assumed to be time-variant, differentiating our model from previous models with lost sales. Production quantities and levels of lost sales in different periods represent a twofold decision problem. We first transform the total profit function into a special total cost function. We then prove several properties of an optimal solution. A forward recursive dynamic programming algorithm is developed to solve the problem optimally in O(T²) time, where T denotes the number of periods in the problem horizon. The proposed algorithm can solve problems of sizes up to 400 periods in less than a second on a 500 MHz Pentium® III processor.