Optimality Gap of Constant-Order Policies Decays Exponentially in the Lead Time for Lost Sales Models

Inventory models with lost sales and large lead times have traditionally been considered intractable, due to the curse of dimensionality which arises from having to track the set of orders placed but not yet received (i.e. pipeline vector). Recently, Goldberg et al. (2012) laid the foundations for a new approach to solving these models, by proving that as the lead time grows large (with the other problem parameters fixed), a simple constant-order policy (proposed earlier by Reiman (2004)) is asymptotically optimal. This was quite surprising, as it is exactly this setting (i.e. large lead times) that was previously believed intractable. However, the bounds proven there are impractical, requiring the lead time to be very large before the constant-order policy becomes nearly optimal, e.g. requiring a lead time which is Ω( −2) to ensure a (1 + )-approximation guarantee, and involving a massive prefactor. The authors note that numerical experiments of Zipkin (2008b) suggest that the constant-order policy performs quite well even for small lead times, and pose closing this gap (thus making the results practical) as an open problem. In this work, we make significant progress towards resolving this open problem and closing this gap. In particular, for the infinite-horizon variant of the finite-horizon problem considered by Goldberg et al. (2012), we prove that the optimality gap of the same constant-order policy actually converges exponentially fast to zero, i.e. we prove that a lead time which is O ( log( −1) ) suffices to ensure a (1+ )-approximation guarantee. We demonstrate that the corresponding rate of exponential decay is at least as fast as the exponential rate of convergence of the expected waiting time in a related single-server queue to its steady-state value. We also derive simple and explicit bounds for the optimality gap. For the special case of exponentially distributed demand, we further compute all expressions appearing in our bound in closed form, and numerically evaluate them, demonstrating good performance for a wide range of parameter values. Our main proof technique combines convexity arguments with ideas from queueing theory.