Scheduling multiple part-types in an unreliable single machine manufacturing system

Quadratic approximations to the differential cost-to-go function, which yield linear switching curves, have been extensively studied. In this paper, we provide the solution to the partial differential equations associated with the steady-state joint probability density function of the buffer levels for two part-type, single machine flexible manufacturing systems under a linear switching curve (LSC) policy. When there are more than two part-types, we derive the density functions under a prioritized hedging point (PHP) policy by decomposing the multiple part-type problem into a sequence of single part-type problems. The expressions for the steady-state density functions are independent of the cost function. Therefore, for additive cost functions that are non-linear in the buffer levels, we can compute the optimal PHP policy, or the more general optimal LSC policy for two part-type problems.