Hierarchical Investment and Production Decisions in Stochastic Manufacturing Systems

This paper presents an asymptotic analysis of hierarchical investment and production decisions in a manufacturing system with machines subject to breakdown and repair. The demand facing the system is assumed to be a given constant. The production capacity can be increased by purchasing a new machine at a fixed cost at some time in the future. The control variables are a pair of a Markov time to purchase the new machine and a production plan. The rate of change in machine states is assumed to be much larger than the rate of discounting of costs. This gives rise to a limiting problem in which the stochastic machine availability is replaced by the equilibrium mean availability. The value function for the original problem converges to the value function of the limiting problem. Moreover, three different methods are developed for constructing controls for the original problem from the optimal controls of the limiting problem in a way which guarantees their asymptotic optimality. The convergence rate of the value function for the original problem to that of the limiting problem is also found. This helps in providing error estimates for the constructed asymptotically optimal controls