Demand for walnut fudge ice cream at the Sweet Cream Dairy can be approximated by a normal distribution with a mean of $21$ gallons per week and a standard deviation of $3\mathrm{.}5$ gallons per week. The new manager desires a service level of $90$ percent. Lead time is $3$ days, and the dairy is open seven days a week. $($Hint: Work in terms of weeks.$)$

For this problem, we consider both continuous inventory system and periodic inventory system.

If we use periodic inventory system, the time from delivery of an order until next review is $10$ days $($Note: this is reported in days$).$

What is the corresponding z$-$value for the desired service level? $($keep three decimal places$)$

How long is the lead time $($in fractions of a week$)?$

How much is the average demand during a week?

How much is the standard deviation of the demand during the week?

If we use continuous inventory system, what safety stock is consistent with the desired service level?

If we use continuous inventory system, what ROP is consistent with the desired service level?

If we use periodic inventory system, what safety stock is consistent with the desired service level?

If we use periodic inventory system, what is the quantity to order when the supply at hand is $8$ gallons?