$6.$ The NPS Corp has a contract with the Navy to deliver $1,000$ fuel pumps for the J$-52$ engines $($used in A$-6,$ A$-4,$ E$/$A$6$ Navy and Marine aircraft$).$ The lifetime of the pump follows a normal distribution with $\backslash$mu $=8$ years, and $\backslash$sigma $=3$ years. The warranty covers the first two$-$year failures and the company will repair the faulty pumps free of charge. Each repair costs the company an average of $$4,000.$ The company has two options to improve the process. Which is the preferred option and what is the maximum that company willing to invest to implement that option?

Option $1$: increase the mean by $1$ year $($i$.$e$.,\backslash$mu $=9,\backslash$sigma $=3)$

Option $2$: decrease the standard deviation by $1$ year $($i$.$e$.,\backslash$mu $=8,\backslash$sigma $=2)$