Demand for vanilla ice cream at a small ice cream shop can be approximated by a Normal distribution with a mean of $27$ litres per week and a standard deviation of $2\mathrm{.}0$ litres per week. The ice cream is purchased from an ice cream producer. The store manager desires a lead time service level of $90$ percent. Lead time from the producer is three days. The store is open seven days a week. Refer: z$-$values table.

a$.$ If the EOQ$/$ROP model is used for ordering the ice cream from the producer, what ROP would be consistent with the desired lead time service level? $($Round the final answer to $2$ decimal places.$)$

ROP

$29\mathrm{.}56$

litres

b$.$ If a fixed$-$interval model is used instead, what order quantity should be used if the order interval is $9$ days and $9$ litres are on hand and none are on order at the time of order? $($Do not round the intermediate calculations. Round the final answer to the nearest whole number.$)$

Quantity

$234$

litres

c$.$ Suppose that the manager is using the EOQ$/$ROP model described in part a$.$ One day after placing an order with the producer, the manager receives a call from the producer saying that the order will be delayed because of problems at the producer$$s plant. The producer promises to have the order there in two days. After hanging up$,$ the manager checks the inventory of vanilla ice cream and finds that $1$ litres have been sold since the order was placed. Assuming that the producer$$s promise is valid, what is the probability that the store will run out of vanilla ice cream before the shipment arrives? $($Round your intermediate calculations to $2$ decimal places and the final answer to $4$ decimal places.$)$

Probability