Weekly demand for a television is normally distributed with mean $55$ and standard deviation $19.$ The lead time from the plant to your store is $3$ weeks and you order inventory replenishments weekly. You use the order$-$up$-$to model to control inventory.

$1.$ Suppose the order$-$up$-$to level is S$=250.$

$$You are preparing to place an order and note that your inventory level is $111$ and you have $78$ televisions on order. How many televisions will you order? $($Provide your answer as an integer.$)$

The following week, your inventory level is $189$ and you have $64$ televisions on order. How many televisions will you order? $($Provide your answer as an integer.$)$

$2.$ What order$-$up$-$to level would you choose if you want to achieve a $98\mathrm{.}75\%$ in$-$stock probability? $($Provide your answer to two decimal places.$)$

$3.$ Suppose the order$-$up$-$to level is S$=220.$

What is your expected on$-$hand inventory? $($Provide your answer to two decimal places.$)$

What is your expected on$-$order inventory? $($Provide your answer as an integer.$)$

$1.$ Suppose SnowBirds wants to maximize expected profits from the sale of the snowboard.

What is the critical ratio Snowbirds would want to meet? $($report your answer to $3$ decimal places$)$

How many snowboards should they order from the manufacturer? $($report your answer as an integer$)$

$2.$ Suppose SnowBirds orders $4,200$ snowboards.

What are the expected lost sales? $($report your answer to $2$ decimal places$)$

What are the expected sales? $($report your answer to $2$ decimal places$)$

What is the expected leftover inventory? $($report your answer to $2$ decimal places$)$

What is the expected profit? $($report your answer to $2$ decimal places$)$

$3.$ Suppose SnowBirds orders $3,500$ snowboards.

What are the expected lost sales? $($report your answer to $2$ decimal places$)$

What are the expected sales? $($report your answer to $2$ decimal places$)$

How many of these snowboards can SnowBirds expect to sell at the discount price. $($report your answer to $2$ decimal places$)$

$4.$ How many snowboards should they order if they want to minimize their inventory while ensuring that the stockout probability is no greater than $1\%?($report your answer as an integer$)$