Hierarchical production controls in a stochastic two-machine flowshop with a finite internal buffer

This paper presents an asymptotic analysis of hierarchical production planning in a manufacturing system with two tandem machines that are subject to breakdown and repair. The system produces a single product, whose rate of demand over time is given to be 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 machines is assumed to be finite. As the rates of change in machines' states approach infinity, the analysis results in a limiting problem in which the stochastic machine capacity is replaced by the average capacity. The value function for the original problem is shown to converge to the value function of the limiting problem. Both open-loop and feedback controls for the original problem are constructed from near-optimal controls of the limiting problem in a way which guarantees their asymptotic optimality as the rates of changes in machines' states become large in comparison to the discount rate. The convergence rate of the value function for the original problem to that of the limiting problem together with the error estimate for the constructed asymptotic optimal controls are obtained. In addition, the constructed feedback control is compared to the Kanban control policy for the stochastic two-machine flowshop.