Single-period inventory model with a delayed incentive option for selling excess inventory

The single-period problem (SPP) is to find the order quantity which maximizes the expected profit in a single period probabilistic demand framework. We extend the SPP to a case where a retailer uses delayed incentives in the form of cash mail-in rebates to sell remaining inventory that did not sell at the regular price. The advantage of cash mail-in rebates is that not all consumers will redeem them. We address three cases: (1) rebate value is predetermined and the order quantity is a decision variable; (2) order quantity is a decision variable and rebate value is set to the value needed to sell all excess inventory, and (3) order quantity is a decision variable and the rebate may be used to sell part of or all excess inventory. In the third case, any inventory remaining after the rebate offer is salvaged at reduced price.