  We consider Assemble-to-Order (ATO) inventory systems with a general Bill of
Materials and general deterministic lead times. Unsatisfied demands are always
backlogged. We apply a four-step asymptotic framework to develop inventory
policies for minimizing the long-run average expected total inventory cost. Our
approach features a multi-stage Stochastic Program (SP) to establish a lower
bound on the inventory cost and determine parameter values for inventory
control. Our replenishment policy deviates from the conventional constant base
stock policies to accommodate non-identical lead times. Our component
allocation policy differentiates demands based on backlog costs, Bill of
Materials, and component availabilities. We prove that our policy is
asymptotically optimal on the diffusion scale, that is, as the longest lead
time grows, the percentage difference between the average cost under our policy
and its lower bound converges to zero. In developing these results, we
formulate a broad Stochastic Tracking Model and prove general convergence
results from which the asymptotic optimality of our policy follows as
specialized corollaries.
