This paper is concerned with robust production planning of a stochastic manufacturing system in which the rate of machine breakdown and repair is much larger than the rate of fluctuation in demand. It is shown that the risk-sensitive production planning problem can be approximated by a limiting problem in which the stochastic machine availability process is replaced by its mean availability. The value function of the limiting problem is shown to be the only viscosity solution to the Isaacs equation associated with a zero-sum, two-player differential game. Near-optimal production plans are constructed from near-optimal controls of the limiting problem. Finally, these results are extended to problems with state constraints.