Problem $12-21($Algo$)$

Demand for walnut fudge ice cream at the Sweet Cream Dairy can be approximated by a normal distribution with a mean of $20$ gallons per week and a standard deviation of $3\mathrm{.}0$ gallons per week. The new manager desires a service level of $90$ percent. Lead time is two days, and the dairy is open seven days a week. $($Hint: Work in terms of weeks.$)$

Use Table B and Table B$1.$

a$-1.$ If an ROP model is used, what ROP would be consistent with the desired service level? $($Do not round intermediate calculations. Round your final answer to $2$ decimal places.$)$

a$-2.$ How many days of supply are on hand at the ROP, assuming average demand? $($Do not round intermediate calculations. Round your final answer to $2$ decimal places.$)$

b$-1.$ If a fixed$-$interval model is used instead of an ROP model, what order size would be needed for the $90$ percent service level, with an order interval of $5$ days and a supply of $8$ gallons on hand at the order time? $($Do not round intermediate calculations. Round your final answer to the nearest whole number.$)$

b$-2.$ What is the probability of experiencing a stockout before this order arrives? $($Do not round intermediate calculations. Round your final answer to the nearest whole percent.$)$

c$.$ Suppose the manager is using the ROP model described in part a$.$ One day after placing an order with the supplier, the manager receives a call from the supplier that the order will be delayed because of problems at the supplier$$s plant. The supplier promises to have the order there in two days. After hanging up$,$ the manager checks the supply of walnut fudge ice cream and finds that $2$ gallons have been sold since the order was placed. Assuming the supplier$$s promise is valid, what is the probability that the dairy will run out of this flavor before the shipment arrives? $($Do not round intermediate calculations. Round your final answer to the nearest whole percent.$)$