Control and optimal control of assemble to order manufacturing systems under heavy traffic

Abstract

We do a heavy traffic analysis of the optimal control of a classical manufacturing and inventory process, called Assemble-to-Order. Demand consists of one or more final products, each requiring either one or several of each of various part types. The intervals between demands are random and might occur either singly or in batches. The part types are produced by dedicated processors, with random production times. Unneeded parts of each type are stored in a finite buffer. When a demand for a final product arrives, if all needed parts are available, the product is assembled and delivered. Otherwise, the demand can either be lost or backlogged. Control is exercised by idling the processors or possibly via external supply sources. Thorough analyses of a great variety of problem formulations of the optimal control problem are given. The general limit problem is what is called a singular control problem, or a combination of the singular and impulsive control problems. New issues in heavy traffic analysis arise. The processes might not be tight in the Skorohod topology and non-classical questions can arise in the computation of reflection directions. Extensions to unreliable processors are given. The basic results assert the convergence of the optimal costs for the physical problem to that for the weak sense limit problem, and that the weak sense limit optimal control problem provides good approximations to the physical problem. Numerical methods for solving the limit problem are available and show the possible improvement in performance. The resulting optimal controls are quite reasonable, but are not of the threshold type