Inventory Control involving Unknown Censored Demand and Convex Inventory Costs

We study inventory control involving lost sales and hence censored demand. In a long-run average framework, the demand distribution is largely unknown. As long as the stationary inventory costs are truly convex to the extent that the second lost item costs strictly more than the first lost item, the regret would be $\Omega(T^{2/3})$. We propose a policy that deliberately orders up to very high levels in designated learning periods and in the remaining doing periods, uses base-stock levels tailored to empirical distributions formed over the learning periods. A matching $O(T^{2/3})$ upper bound can be achieved by the policy. When inventory costs assume the special $V$-shaped form in which all holding cost rates are the same and so are all the lost cost rates, the bound could be extended to the case where items are nonperishable. Our numerical study indicates the $V$-shape restriction to be potentially relaxable. It also confirms the advantage of our policy over a slew of others from the literature