Please show your work. Numerical answers without any justification are not acceptable.

$$

We have the following information about daily demand for $32$ oz cartons of shelf stable milk at a convenience store:

$$

Average daily demand:

$15$ cartons

Standard deviation of daily demand:

$5$ cartons

Order lead time:

$3$ days

Standard deviation of order lead time:

$1\mathrm{.}5$ days

Item cost:

$$2\mathrm{.}5$ per carton

Cost to place an order:

$$5$ per order

Annual holding cost per carton:

$50\%$ of item cost

$$

$$What is the economic order quantity $($EOQ$)?$ What is the resulting total annual relevant cost?

EOQ $=$

Annual relevant cost $=$

Suppose the store is currently ordering $20$ cases of $9$ cartons at a time $($i$.$e$.,$ the order quantity is $180).$ How does the annual relevant cost of this policy compare with the cost you obtained in part $($a$)?$ Show your work.

$$Annual relevant cost $=$

c$..$ Suppose the store places an order every time the inventory level drops to $100.$ What is the probability of no stock$-$out achieved under this policy?

Probability of no stock$-$out $=$

d$.$ Now assume that the store wants to achieve a probability of no stock$-$out of $99\%($z$=2\mathrm{.}33).$ What is the new reorder point? How much of the reorder point consists of safety stock?

$$

Reorder point $=$

$$Safety stock $=$

e$.$ Using the economic order quantity from part $($a$)$ and the safety stock level from part $($d$),$ compute the annual inventory holding cost this store incurs.

$$

Annual inventory holding cost $=$

f$.$ Because shelf space is very expensive at a convenience store, the store is thinking about raising holding cost from $50\%$ of item cost to a higher percentage. What will be the impact on the economic order quantity? Briefly explain your answer.