. An # chart is used to maintain current control of a process. A single assignable cause of magnitude 20 occurs, and the time that the process remains in control is an exponential randoma variable with mean 100 h. Suppose that sampling costs are $0.50 per sample and $0.10 per unit, it costs $5 to investigate a false alarm,

$2.50 to find the assignable cause, and

$100 is the penalty cost per hour to oper-

ate in the out-of-control state. The time required to collect and evaluate a sample is 0.05 h, and it takes 2 h to locate the assignable cause. Assume that the process is allowed to continue operating during searches for the assignable cause.

(a) What is the cost associated with the arbitrary control chart design n = 5, k=3, and h =1?

(b) Find the control chart design that minimizes the cost function given by equation 9-31.