Optimal Control of Assemble-to-Order Systems

We consider the optimal control of an assemble-to-order (ATO) system with m components, a single end-product, and n customer classes. Demand from each class occurs continuously over time according to a Poisson process. Components are produced in separate production facilities, each with a finite production rate and exponentially distributed production times. Components can be stocked ahead of demand but incur a holding cost. When an order arises, it can be satisfied only if all m components are available in stock. Otherwise, it must either be backordered or rejected if backorders are not allowed. A backordered or rejected demand incurs a shortage cost (either a backorder or a lost sales cost), which may vary by demand class. At any time, the system manager must decide which components to produce and, if there is on-hand inventory, which orders from those pending to satisfy. We formulate the problem as a Markov decision process and characterize the structure of the optimal policy. We show that the optimal production policy for each component is a state-dependent base-stock policy, where the base-stock level for each component is non-decreasing in the inventory level of other components. We show that the optimal inventory allocation is a state-dependent multi-level rationing policy where the component rationing level for each class is non-decreasing in the inventory level of other components. Using numerical results, we compare the performance of the optimal policy to simple heuristics. For cases where it is possible to compare the heuristics to the optimal policy, we find the heuristics to perform well.