Problem $2(18$ points$)$ MicroApple is a manufacturer of personal computers. It currently manufactures a single model $$ the MacinDOS $$ whose weekly demand is normally distributed with an average of $200$ and standard deviation of $60.$ MicroApple orders the disk drives for the MacinDOS $(1$ per computer$)$ from an outside supplier at a cost of $$30$ each. The administrative costs for placing an order total $$30.$ The annual holding cost is $$6$ per drive. If MicroApple stocks out of disk drives, production is halted. Because of the seriousness of stock$-$outs, management wants to keep enough safety stock to maintain a $99\%$ service level.

The supplier now is offering two shipping options. With option $1,$ the lead time would be $2$ days. For each order, the shipping cost charged to MicroApple would be $$100,$ regardless of how many drives are shipped. With option $2,$ the lead time would be $1$ day, but for each order, the expedited shipping cost charged to MicroApple would be $$200,$ regardless of how many drives are shipped. Assume that the manufacturing facility works $7$ days a week $(365$ days a year$).$ There are $52$ weeks in a year.

$1.(4$ points$)$ For each option, what is the optimal batch size to use?

$2.(4$ points$)$ For each option, how much safety stock should MicroApple use?$3.(6$ points$)$ For each option, how much does MicroApple pay for purchasing, ordering, and holding inventory on average each year?

$4.(2$ points$)$ Based on your answers in parts $14,$ which option should MicroApple select and what is the associated $($Q$,$ R$)$ policy?

$5.(2$ points$)$ Using the shipping option and policy in part $5,$ on average, how long does a disk driver stay in MicroApple$$s warehouse before being used to produce MacinDOS? $($Hint: use Little$$s law.$)$