Discrete-item Inventory Control involving Fixed Setup Costs, Demand Censoring, and Unknown Distribution

Abstract

We investigate a dynamic inventory control problem in which orderings involve fixed setup costs and unsatisfied demands are lost. Stepping off from the traditional literature which stresses the randomness of demand realizations, we move on to tolerate ambiguity. Now even the demand distribution $f\equiv (f(d))_{d=0,1,...}$ itself, out of which random realizations are sampled, can come from a vast and definitely non-singleton set. Lost sales and demand ambiguity would together complicate the problem through censoring, namely, the inability of the firm to observe the lost portion of the demand. Our main policy idea advocates periodically ordering up to seemingly wastefully high levels just to learn and in intervening periods, cleverly exploiting the information gained in these learning periods. By regret, we mean the price paid for ambiguity in long-run average performances. When demand support is unlimited, regret bounds in the orders of $\mathcal{O}(T^{8/9})$ and $\mathcal{O}(T^{(2+\sqrt{2})/4})\simeq \mathcal{O}(T^{0.854})$ can be established for, respectively, the case where $f(0)$ is bounded away from one and the other where the restriction is relaxed with the firm allowed to remove items from the inventory. When demand has a finite support, both bounds could be improved to the order of $\mathcal{O}(T^{2/3}\cdot (\ln T)^{1/2})$. We also propose other policies. % based on the general learning-while-doing idea but the different Kaplan-Meier (KM) estimation. Our simulation demonstrates the merits of the various policy ideas and the hurdles posed by the prospect of $f(0)\longrightarrow 1^-$.