$10\mathrm{.}20.$ An $\stackrel{}{x}$ chart is used to maintain current control of

a process. A single assignable cause of magnitude

$2$ occurs, and the time that the process remains

in control is an exponential random variable with

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

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

gate 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 $0\mathrm{.}05h,$ and it takes $2h$ 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 $10\mathrm{.}31.$

$10\mathrm{.}21.$ Consider the cost information given in Exercise

$10\mathrm{.}20.$ Suppose that the process model represented

by equation $10\mathrm{.}31$ is appropriate. It requires $2h$ to

investigate a false alarm, the profit per hour of oper$-$

ating in the in$-$control state is $$500,$ and it costs $$25$

to eliminate the assignable cause. Evaluate the cost

of the arbitrary control chart design $n=5,k=3,$

and $h=1.$