JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 89, No. 3, pp. 681-712, JUNE 1996 Hierarchical Production Policies in Stochastic Two-Machine Flowshops with Finite Buffers 1 N. T. FONG 2 AND X. Y. ZHOU 3 Communicated by L. D. Berkovitz Abstract. This paper concerns production planning in manufacturing systems with two unreliable machines in tandem. The problem is formu- lated as a stochastic control problem in which the objective is to mini- mize the expected total cost of production, inventories, and backlogs. Since the sizes of the internal and external buffers are finite, the problem is one with state constraints. As the optimal solutions to this problem are extremely difficult to obtain due to the uncertainty in machine capaci- ties as well as the presence of state constraints, a deterministic limiting problem in which the stochastic machine capacities are replaced by their mean capacities is considered instead. The weak Lipschitz property of the value functions for the original and limiting problems is introduced and proved; a constraint domain approximation approach is developed to show that the value function of the original problem converges to that of the limiting problem as the rate of change in machine states approaches infinity. Asymptotic optimal production policies for the original problem are constructed explicitly from the near-optimal polic- ies of the limiting problem, and the error estimate for the policies con- structed is obtained. Algorithms for constructing these policies are presented. Key Words. Two-machine flowshops, unreliable machines, hierarchical production policy, weak Lipschitz continuity, constraint domain approximations, asymptotic optimal control, convergence rates, error estimates. ~This work was partly supported by CUHK Direct Grant 220500660, RGC Earmarked Grant CUHK 249/94E, and RGC Earmarked Grant CUHK 489/95E. 2PhD Student, Department of Systems Engineering and Engineering Management, Chinese University of Hong Kong, Shatin, Hong Kong. 3Associate Professor, Department of Systems Engineering and Engineering Management, Chinese University of Hong Kong, Shatin, Hong Kong. 681 0022-3239/96/0600-0681509.50/0 9 1996 Plenum Publishing Corporation