An  chart is sued to maintain current control of a process. A single assignable cause of magnitude 2σ occurs, and the time that the process remains in control is an exponential random variable with mean 100 hr. 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 $00 is the penalty cost per hour to operate in the out-of-control state. The time required to collect and evaluate a sample is 0.05 hr, and it takes 2 hr to locate the assignable cause. Assume that the process is allowed to continue operating during searches for the assignable cause.
 = 0.01/hr or 1/ = 100hr;  = 2.0
a1 = $0.50/sample; a2 = $0.10/unit; a'3 = $5.00; a3 = $2.50; a4 = $100/hr g = 0.05hr/sample; D = 2hr
(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 10.31.