Optimal means for continuous processes in series

We discuss the problem of determining the means of a set of processes in series. Each process generates a random quality characteristic that in turn has lower and upper specification limits. Depending on the value of the quality characteristic, an item can be reworked, scrapped or forwarded to the next process. An item is reworked at the same stage. The processes are continuously running, hence we develop the "long term" probabilities of meeting specifications, and of violating each limit. These are used to construct the profit function to be maximized. We present a recursive form of the profit function that yields a very efficient method for determining the means. The method relies on solving single stage problems. Next, we turn our attention to the single stage problem and show that if the quality characteristics are normally distributed, then a local optimum is also global. Finally, we present a very fast solution method for this problem.