Online ordering rules for the multi-period newsvendor problem with quantity discounts

In this paper, we study the multi-period newsvendor problem with quantity discounts and stationary demand, where order quantities need to be decided sequentially over a finite or infinite horizon without making statistical assumption on demands. The Weak Aggregating Algorithm (WAA), which is an online learning method of prediction with expert advice, is applied to the problem. We first consider the all-unit discount case, in which the reduced unit cost is applied to all units in an order. We present explicit online ordering rules by using the fixed-stock strategy as expert advice during the application of WAA. A modified gain function implying convoluted contracts between the newsvendor and his supplier is defined to obtain some theoretical guarantees, which ensure that the newsvendor’s average gains are almost as large as those from the best expert advice for a sufficiently large horizon. Further, we generalize the results to the multi-level incremental discount case, in which there are different unit costs for different quantity ranges. We also extend the results to cases where order quantities are integer-valued, and provide correspondingly online ordering rules and theoretical guarantees. Finally, numerical experiments are performed and show that the average gains obtained by the proposed online ordering rules are comparable to those offered by the best experts in hindsight. The results also indicate that the cumulative gains achieved by the ordering rules for the problem with all-unit discount are larger than those achieved by the ordering rules for the problem with incremental discount; and order quantities increase with the discount level for the all-unit discount while decrease for the incremental discount. The results obtained in this paper can provide competitive online ordering rules for industry managers who need to place long-term continuous orders for perishable products when the demand distribution is unknown.