b$)$

Question $4(30$ points$)$

A flight operator sells tickets for a flight with a capacity of $150$ seats. The operator overbooks the flight to account for no$-$shows. Due to historical data, they know that the number of no$-$shows follows a uniform distribution between $2$ and $7$ passengers.

a$)$ The operator must compensate a passenger who is bumped with $$500.$ The regular price for each seat is $$200,$ and you can assume there is enough demand to fill all $150$ seats. If a passenger is a no$-$show and the flight isn't overbooked, the operator can call a passenger from the waitlist to fill the seat, but since this is last$-$minute, the operator only charges $$50$ for the seat. What is the optimal number of overbooked seats?

$C_u=200-50=$$$150,C_o=500$

$S{L}^{**}=\frac{c_u}{C_u+C_o}=\frac{150}{150+500}=0\mathrm{.}23$

$Q^{**}=a+(b-a)S{L}^{**}=2+5**\mathrm{.}23=3\mathrm{.}15$

b$)$ Suppose the cost of overbooking increases to $$800$ per bumped passenger. How does this change the optimal number of tickets to sell? Explain the reasoning.

$C_o=800$

$S{L}^{**}=\frac{C_u}{C_u+C_o}=\frac{150}{150+800}=0\mathrm{.}1$

$Q^{**}=a+(b-a)S{L}^{**}=2+5**\mathrm{.}16=2\mathrm{.}8$

We should overbook fewer seats.

c$)$ The operator wants to minimize the probability of filling a no$-$show seat to $60\%.$ What number of tickets should they sell to achieve this goal?

$S{L}^{**}=0\mathrm{.}6$

$Q^{**}=a+(b-a)S{L}^{**}=2+5**\mathrm{.}6=5$

d$)$ The operator wants to minimize the probability of a seat being assigned to a wait$-$listed customer to no more than $5\%.$ What number of tickets should they sell to achieve this goal?

$S{L}^{**}=1-0\mathrm{.}05=0\mathrm{.}95$

$,Q^{**}=a+(b-a)S{L}^{**}=2+5**\mathrm{.}95=6\mathrm{.}75$