Dynamic Inventory Control with Stockout Substitution and Demand Learning

We consider an inventory control problem with multiple products and stockout substitution. The firm knows neither the primary demand distribution for each product nor the customers' substitution probabilities between products a priori, and it needs to learn such information from sales data on the fly. One challenge in this problem is that the firm cannot distinguish between primary demand and substitution (overflow) demand from the sales data of any product, and lost sales are not observable. To circumvent these difficulties, we construct learning stages with each stage consisting of a cyclic exploration scheme and a benchmark exploration interval. The benchmark interval allows us to isolate the primary demand information from the sales data, and then this information is used against the sales data from the cyclic exploration intervals to estimate substitution probabilities. Because raising the inventory level helps obtain primary demand information but hinders substitution demand information, inventory decisions have to be carefully balanced to learn them together. We show that our learning algorithm admits a worst-case regret rate that (almost) matches the theoretical lower bound, and numerical experiments demonstrate that the algorithm performs very well.