From Introduction to Statistical Quality Control CH $10$

$10\mathrm{.}30.$ An $\stackrel{}{x}$ chart is used to maintain current control of a

process. A single assignable cause of magnitude $2$TABLE $10$E$.11$

Chemical Product Viscosity

occurs, and the time that the process remains in con$-$

trol is an exponential random variable with mean

$100$ h $.$ Suppose that sampling costs are $$0\mathrm{.}50$ per

sample and $$0\mathrm{.}10$ per unit, it costs $$5$ to investigate a

false alarm, $$2\mathrm{.}50$ to find the assignable cause, and

$$100$ is the penalty cost per hour to operate in the out$-$

of$-$control state. The time required to collect and

evaluate a sample is $0\mathrm{.}05$ 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 $10\mathrm{.}31$or

$E(A)=V_0-E(L)$

where

$E(L)=\frac{a_1+a_{2}n}{h}$

$+\frac{a_4[\frac{h}{1-}-+gn+D]+a_3+a_3^{\text{'}}\frac{e^{-h}}{1-e^{-\frac{}{h}}}}{\frac{1}{}+\frac{h}{1-}-+gn+D}$