$10\mathrm{.}22.$ An $\stackrel{}{x}$ chart is used to maintain current control of a process. The cost parameters are $a_1=$$$2,a_2=$$$0\mathrm{.}50,a_3=$

$$50,a_3^{\text{'}}=$$$75,$ and $a_4=$$$200.$ A single assignable cause occurs, with magnitude $=1,$ and the run length of the

process in control is exponentially distributed with mean $100h.$ It requires $0\mathrm{.}05h$ to sample and test, and $1h$ to

locate the assignable cause.

a$.$ Evaluate the cost of the arbitrary $\stackrel{}{x}$ chart design $n=5,k=3,$ and $h=0\mathrm{.}5.$

b$.$ Find the economically optimum design.