IV$.$ Gak $($Process Capability Index$)$ of capable processes are ones with a ratio$/$index of $1\mathrm{.}33(4$ sigma$)$ or greater. $(25$ points$)$

You are the process improvement manager and have developed a new machine to cut insoles for the company's top$-$of$-$the$-$line running shoes. The company"s goal is no more than $3\mathrm{.}4$ defects per million. The insoles cannot be more than $\frac{+}{-}001$ of an inch from the thickness specification of $\mathrm{.}250.$ The new process has a mean $=\mathrm{.}250$ inch and a standard deviation $=\mathrm{.}0005$ inch,

a$)$ Should you replace the existing machine which has a Colk, of $1,0?$

b$)$ If the insoles can be $\frac{+}{-}\mathrm{.}002"($instead of $\mathrm{.}001")$ from the required $\mathrm{.}250",$ what is new the Cak and would you replace the existing machine?

c$)$ What else can be considered to improve the Cuk?