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$1.$ You are the procurement manager of Tacy's. The supplier offers a new style of classical bomber jacket for the winter season. The purchase cost is $$200,$ and the selling price is $$500.$ You consider offering a discount for unsold jackets at the end of the season. The discounted price is $$150.$ The forecasted demand, D$,$ is normally distributed with a mean of $3,000$ units $(\backslash$mu $)$ and a standard deviation of $700$ units $(\backslash$sigma $).$ Assume that the A$/$F ration is $1.$

$($a$)$ You will get a promotion if the total sold jackets is above $4,000$ units. What is the probability that the actual demand will be $4,000$ units or more?

$($b$)$ Selling less than $90\%$ of the average forecast may result in the placement of your position in a lower pay grade. What is the probability of selling more than $90\%?$

$($c$)$ What is the probability that the actual demand is within $25\%$ of the average forecast?

$($d$)$ Whatistheexpectedprofitmaximizingorderquantity,Q$?$

$($e$)$ Assuming Tacy's Manger enforces and orders quantity of Q$=4,500.$ What are the expected lost sales?

$($f$)$ AtQ$=4,500,$ what are the expected sales?

$($g$)$ At Q$=4,500,$ what is the expected leftover inventory?

$($h$)$ At Q$=4,500,$ what is the expected profit?

$($i$)$ AtQ$=4,500,$ what is the expected fill rate?

$($j$)$ AtQ$=4,500,$ what is the stock$-$out probability?