The benefits of state aggregation with extreme-point weighting for assemble-to-order systems

We provide a new method for solving a very general model of an assemble-toorder system: multiple products, multiple components that may be demanded in different quantities by different products, batch production, random lead times, and lost sales, modeled as a Markov decision process under the discounted cost criterion. A control policy specifies when a batch of components should be produced and whether an arriving demand for each product should be satisfied. As optimal solutions for our model are computationally intractable for even moderately sized systems, we approximate the optimal cost function by reformulating it on an aggregate state space and restricting each aggregate state to be represented by its extreme original states. Our aggregation drastically reduces the value iteration computational burden. We derive an upper bound on the distance between aggregate and optimal solutions. This guarantees that the value iteration algorithm for the original problem initialized with the aggregate solution converges to the optimal solution. We also establish the optimality of a lattice-dependent base-stock and rationing policy in the aggregate problem when certain product and component characteristics are incorporated into the aggregation/disaggregation schemes. This enables us to further alleviate the value iteration computational burden in the aggregate problem by eliminating suboptimal actions. Leveraging all of our results, we can solve the aggregate problem for systems of up to 22 components, with an average distance of 11.09% from the optimal cost in systems of up to 4 components (for which we could solve the original problem to optimality).