Asymptotic Scaling of Optimal Cost and Asymptotic Optimality of Base-Stock Policy in Several Multidimensional Inventory Systems

We consider three classes of inventory systems under long-run average cost: (i) periodic-review systems with lost sales, positive lead times, and a nonstationary demand process; (ii) periodic-review systems for a perishable product with partial backorders and a nonstationary demand process; and (iii) continuous-review systems with fixed lead times, Poisson demand process, and lost sales. The state spaces for these systems are multidimensional, and computations of their optimal control policies/costs are intractable. Because the unit shortage penalty cost is typically much higher than the unit holding cost, we analyze these systems in the regime of large unit penalty cost. When the lead-time demand is unbounded, we establish the asymptotic optimality of the best (modified) base-stock policy and obtain an explicit form solution for the optimal cost rate in each of these systems. This explicit form solution is given in terms of a simple fractile solution of lead-time demand distribution. We also characterize the asymptotic scaling of the optimal cost in the first two systems when the lead-time demand is bounded.