Part $1$: Please choose TRUE or FALSE for the statement.

In the newsvendor problem, safety stock reduction from aggregating customer demand from different locations is larger when the demand correlation between locations is higher. $($True$/$False$)$

In the EOQ problem with production, the average inventory level is lower than that in the standard EOQ problem where the seller purchases the material instead of producing it$.($True$/$False$)$

In the EOQ problem, if there is a positive lead time when demand is not random, the optimal order quantity is higher than that without a lead time. $($True$/$False$)$

Part $2$: Newsvendor Model

Problem $2\mathrm{.}1$

Demand for T$-$shirts is normally distributed with a mean $1,000$ and a standard deviation $200.$ Cost of one T$-$shirt is $$10,$ the selling price of one T$-$shirt is $$15,$ each unsold T$-$shirt is worth $$8.$

$4.$ What is the under$-$stocking cost $Cu?$

$5.$ What is the over$-$stocking cost Co$?$

$6.$ What is the optimal initial order quantity if the demand is given by $N(1,000,200)?$

Suppose we have a second ordering opportunity when the demand exceeds our initial order. The second order prevents lost sales because we can simply choose a second order quantity to ensure that all demand is satisfied. However, we need to pay an extra $20$ percent premium over the regular price for those $T$ shirt purchased from the second order. Given this new opportunity,

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$7.$ What is the under$-$stocking cost $Cu$ for the initial ordering decision?

$8.$ What is the over$-$stocking cost Co for the initial ordering decision?

$9.$ What is the optimal initial order quantity if the demand is given by $N(1,000,200)?$

Problem $2\mathrm{.}2$

There are $200$ seats on a Seattle$-$Boston flight. Suppose the ticket price is $$475$ on average and the number of passengers who reserve a seat but do not show up for departure is normally distributed with a mean $30$ and a standard deviation $15.$ You decide to overbook the flight and estimate that the average loss from a passenger who will have to be "bumped" $($if the number of passengers exceeds the number of seats$)$ is $$800.$

$10.$ What is the maximum number of reservations that should be accepted? You must show your calculations to get full marks.

$11.$ Suppose you allow $220$ reservations. What is the probability that you will have to deal with bumped passengers? $($HINT: Use NORMDIST$($x$,$mean,standard$\_$dev,TRUE$)$ in Excel as the normal CDF for x$.)$

$12.$ Now assume there is no overbooking, and the high fare is $$675$ and the low fare is $$375.$ Demand for the low fare is abundant while demand for the high fare is normally distributed with a mean of $80$ and a standard deviation $35.$ What is the probability of selling $200$ reservations if you set the optimal protection level, that is$,$ the number of seats that you keep for the full$-$fare customers?