Problem $3(60$ points$)$

Marty Ford is an operations analyst for Piedmont Commuter Airlines $($PCA$).$ Recently, Marty was asked to make a recommendation on how many reservations PCA should book on Flight $343-$ a flight from a small regional airport in New England to a major hub at Boston's Logan airport. The plane used on Flight $343$ is a small twin$-$engine turbo$-$prop with $19$ passenger seats available. PCA sells nonrefundable tickets for Flight $343$ for $\backslash (\backslash$$ $150\backslash )$ per seat.

Industry statistics show that for every ticket sold for a commuter flight, a $0\mathrm{.}10$ probability exists that the ticket holder will not be on the flight. Thus, if PCA sells $19$ tickets for this flight, there is a fairly good chance that one or more seats on the plane will be empty. Of course, empty seats represent lost potential revenue to the company. On the other hand, if PCA overbooks this flight and more than $19$ passengers show up$,$ some of them will have to be bumped to a later flight.

To compensate for the inconvenience of being bumped, PCA gives these passengers vouchers for a free meal, a free flight at a later date, and sometimes also pays for them to stay overnight in a hotel near the airport. PCA pays an average of $\backslash (\backslash$$ $325\backslash )($including the cost of lost goodwill$)$ for each passenger that gets bumped. Marty wants to determine if PCA can increase profits by overselling this flight and, if so$,$ how many reservations should be accepted to produce the maximum average profit. To assist in the analysis, Marty analyzed market research data for this flight that reveals the following probability distribution of demand for this flight:

$($a$)(20$ pts$)$ Let $\backslash ($ x $\backslash )$ be the number of reservation accepted, $\backslash ($ x $\backslash$geq $19\backslash ).$ Write down the expected profit of PCA as a function of $\backslash ($ x $\backslash ).$

$($b$)(20$ pts$)$ Use simulation to estimate the expected profit of PCA when $\backslash ($ x$=20\backslash ).$

$($c$)(20$ pts$)$ What is the optimal number of reservations that PCA should accept to maximize the expected profit? $($Use Excel Data Table or Scenario Manager to solve this problem$).$ Please freeze the random numbers in your simulation.