Continuous-Review (R, nQ) Policies for Inventory Systems with Dual Delivery Modes

We study a continuous-review, infinite-horizon inventory system with compound Poisson demand and dual sourcing/delivery modes. Ordering from either source/mode incurs a fixed cost and the expedited mode provides a shorter lead time than the regular mode. As the optimal ordering policy is unknown, while expected to be very complicated, we propose a class of simple policies called single-index ( R, nQ ) policies—when ordering from each mode, based on the inventory position, the system follows an ( R, nQ )-type of policy. We provide an exact procedure to compute the expected long-run average cost. Specifically, we first analyze the steady-state distribution of the inventory position, which is found to be no longer uniform in general as in the classic ( R, nQ ) inventory system where only one delivery mode is available. We then develop a recursive procedure, which overcomes the order-crossing effect, to determine the steady-state distribution of the inventory level. Two simple heuristics for computing near-optimal policy parameters are provided. For a special case where ordering from the regular mode incurs no fixed cost and follows a base-stock policy, we derive closed-form solution bounds by applying normal approximation. To assess the performance of the single-index ( R, nQ ) policy, we further study a more complicated class of policies called dual-index ( R, nQ ) policies and numerically illustrate that the simpler single-index policy performs close to the dual-index policy. Finally, the performance of the single-index policy is also shown comparable to the policy computed via dynamic programming.