Procurement planning with batch ordering under periodic buyback contract

This paper deals with the deterministic single-item procurement planning problem with batch ordering under the buyback contract. We assume a buyback contract with returns of unused products at the end of each period to the supplier and we consider a piecewise procurement cost structure composed by a fixed ordering cost, a variable procurement cost and a fixed cost per batch replenished, incurred by the retailer. This problem is studied under various hypotheses : in the first, the acquisition is made in only full batches, in the second, there is no specific assumption on the acquisition quantity, in the third, the procurement must be in only full batches with a possibility of lost sales, and in the fourth, there is just the option of lost sales for the general case. For these four cases, several optimality properties and polynomial time algorithms are proposed.