This paper establishes a criterion that measures approximately the average net income of a process under surveillance of an X chart when the process is subject to random shifts in the process mean. The quality control rule assumed is that an assignable cause is looked for whenever a point falls outside the control limits. The criterion is for the case in which it is assumed that the process is not shut down while the search for the assignable cause is in progress, nor is the cost of adjustment or repair and the cost of bringing the process back into a state of control after the assignable cause is discovered charged to the control chart program. The paper shows how to determine the sample size, the interval between samples, and the control limits that will yield approximately maximum average net income. Numerical examples of optimum design are studied to see how variation in the various risk and cost factors affects the optimum.
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