Coby Inn is a hotel that offers low$-$price rooms with free breakfast.$$One of the popular items in its dinner menu is barbecue ribs. Coby Inn orders frozen ribs from a major distributor at the cost of $$9$ per pound. The distributor charges the hotel $$60$ for delivery and shipment cost. Coby Inn stores frozen ribs in its freezer with the inventory carrying rate of $20$ percent. Coby Inn estimates that telling a guest that they are out of ribs and the guest needs to order another item from the menu results in $$3$ loss of profit. The data shows that the daily demand for ribs is Normally distributed with mean $50$ pounds and standard deviation $15$ pounds. The hotel manager realized that, while she has the flexibility of placing an order to distributor of rib at any time, the distributor often misses its delivery schedule. Looking at the past deliveries, the manager finds that the time duration from when the order was placed until the order was delivered has mean of $5$ days with standard deviation of $2$ days $($assume $360$ days in a year$).$

$*$How much ribs the hotel should order each time and when should it place its orders to minimize the total average annual holding, shipment, and shortage cost?

$*$What is the average time a pound of rib stays in the freezer before it is cooked and served to a guest?

$*$What is the optimal ordering policy if Coby Inn wants to reduce the fraction of its guests who must choose another dish $($because rib is not available$)$ to $5$ percent? What is the safety inventory under this policy?

$*$Suppose Coby Inn has a limited space in its freezer and thus it cannot order more than $500$ pounds each time. How does this change your answer to Part $($a$)?$