Asymptotic Optimality of Simple Policies for Stochastic Inventory Systems with Delivery Lead Time and Purchase Returns

We study a multi-period inventory control problem with a fixed replenishment lead time and stochastic returns. Each unit of sales can be returned within a fixed return window after the purchase time, and the returned unit can be immediately used to fulfill new demands. Demands are integer-valued. We consider both the lost-sales system and the backlogging system. The objective is to find a replenishment policy that maximizes the long-run average profit (equivalently, minimizes the long-run average cost). Due to the presence of stochastic returns, even the optimal policy for the system with zero lead time is already complex. Thus, in this paper, we focus on two simple policies: (1) the “base-stock policy”, which orders to maintain a constant inventory level or position in each period, and (2) the “myopic policy”, which orders to maximize the expected profit for the period when the order arrives. We prove that, in the regime of large unit penalty costs, when demand is bounded, a base-stock policy ordering up to the maximal total demands over the lead times and the myopic policy are optimal; when demand is unbounded, under a large class of distributions, the base-stock policy and the myopic policy are asymptotically optimal. Numerically, when lead times and return window are relatively short, both the best base-stock policy and, in particular, the myopic policy perform close to the true optimal policy, even with small to moderate unit penalty costs. When lead times and the return window get longer, both heuristic policies significantly outperform two service-level-based benchmarks that do not adequately account for the effect of purchase returns.