In this note, we give a complete solution to a scheduling problem for a two-part-type, single-machine, flexible manufacturing system, with finite-capacity buffers. Backlogged and rejected requests incur a cost which must be minimized over an infinite time interval. If buffer capacities were infinite, the well-known cmu rule would have solved the problem. In this note, we find the optimal policy for the finite-capacity case and give a computation procedure and some illustrative examples.