Technical Note - Optimal Control of an Assembly System with Multiple Stages and Multiple Demand Classes

We consider an assembly system with multiple stages, multiple items, and multiple customer classes. The system consists of m production facilities, each producing a different item. Items are produced in variable batch sizes, one batch at a time, with exponentially distributed batch production times. Demand from each class takes place continuously over time according to a compound Poisson process. At each decision epoch, we must determine whether or not to produce an item and should demand from a particular class arise whether or not to satisfy it from existing inventory, if any is available. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. In contrast to systems with exogenous and deterministic production leadtimes, we show that the optimal production policy for each item is a state-dependent base-stock policy with the base-stock level non-increasing in the inventory level of items that are downstream and non-decreasing in the inventory level of all other items. For inventory allocation, we show that the optimal policy is a multi-level state-dependent rationing policy with the rationing level for each demand class non-increasing in the inventory level of all non-end items. We show how the optimal control problem can be reformulated in terms of echelon inventory and how, under this reformulation, the problem can be essentially reduced to the optimal control of a single stage assemble-to order system. We discuss various extensions, including to systems with more general production time distributions. We also discuss the performance of common heuristics that use fixed thresholds to determine production and inventory allocation.