$2.$ You have been the inventory manager at Malmart Distribution Center. You have estimated that the annual demand for passenger car tires is $200,000$ units. Placing an order costs $\backslash (\backslash$$ $300\backslash ).$ Cost to hold a tire is $\backslash (\backslash$$ $3\backslash )$ per year. Cost per tire is $\backslash (\backslash$$ $70\backslash ).$

$($a$)$ How much $($Q$*)$ and how often do you need to order and fulfill the demand?

Q$*$: $\backslash ($ T$^\{*\}\backslash )$ :

$($b$)$ At the end of your first year of managing, you realize that the demand is variable. However, the average annual demand is still $200,000$ with a standard deviation of $5\mathrm{.}6$ tires per day. The tires supplier cannot deliver the products on the same day. The average lead time is now $20$ days with a standard deviation of $4\mathrm{.}1$ days. You consider a $\backslash (95\backslash \%\backslash )$ Service Level to satisfy customers. What is the Reorder Point $($ROP$)?$

$\backslash ($ R O P: $\backslash )$

$($c$)$ You also want to test OUT model. The review process, $\backslash ($ T $\backslash ),$ is $30$ days. Assume that the onhand inventory is $15,000$ tires. What is the OUT inventory level and the Order Quantity $($Q$)?($Hint: Use the information in Part b to find the value for the remaining parameters.$)$