Hierarchical Production Control in a Stochastic N-Machine Flowshop with Limited Buffers1

This paper is concerned with an asymptotic analysis of hierarchical production planning in a stochastic manufacturing system consisting of machines that are subject to breakdown and repair. The system produces a single product whose rate of demand over time is constant. The problem is formulated as a continuous-time dynamic programming problem in which the objective is to minimize the cost of production, inventories, and backlogs. The size of the buffer between the two adjacent machines is assumed to have lower and upper bound constraints. As the rates of change of the machine states approach infinity, the analysis results in a limiting problem in which the stochastic machine availability is replaced by its equilibrium mean availability. The value function for the original problem converges to the value function of the limiting problem. A method of ‘‘shrinking,’’ ‘‘total lifting,’’ and ‘‘modification’’ is introduced in order to construct near-optimal controls for the original problem by using near-optimal controls of the limiting problem. The convergence rate of the value function for the original problem to 1This research was supported in part by NSERC Grant A4619, ONR Grant N00014-96-1- 0263, and faculty research grants from the University of Texas at Dallas and the University of Georgia.that of the limiting problem is established. This helps in providing an error estimate for the constructed open-loop asymptotic optimal control. Q 2000 Academic Press