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Replies: 1   Last Post: Jul 5, 1996 2:22 PM

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 Peter Blaskiewicz Posts: 124 Registered: 12/6/04
Posted: Jul 3, 1996 11:02 PM

One possible mini-project that can be used to illustrate the expected
value of a discrete distribution would be to have the class look at why
airlines overbook. Here's a rough sketch of a problem that would
require everyone to contribute a chunk in order for the class as a
whole to find a solution.

Set-up: An airline runs a jet with such-and-so many seats (say, 125)
between these two cities. Suppose that each of the tickets cost a
certain fixed amount (say, \$159) (an unrealistic assumption that all
seats cost the same; you're welcome to refine this) and the ticket is
fully refundable. Past experience informs the airline that there is
a certain likelihood (say 5%) that any given ticketholder won't show up.
(Another unrealistic assumption; here, independence - no families, etc.,
travel together.) If the airline overbooks and has to deny boarding
to confirmed ticketholders, then they give the bumped passenger a
compensation package (cash, meal, hotel, guaranteed seat on next plane,...)
worth \$x (say, \$400). So, if more people show up than the jet has seats,
the airline loses \$x per bumpee, and if fewer people show up, the
airline loses (\$159 or whatever) on each empty seat.

Give individuals or small groups a certain number of confirmed
ticketholders, different numbers to each person/group, and ask questions
such as these:
- How many should show up for the airline to make the most off the flight?
- If a certain number t tickets are sold, what is the probability
distribution for the number of folks who actually show up (easy)
- If X represents the possible values of the airline's 'loss' on the
flight, what values can X take on? What is the probability dist of X?
(And what does 'loss' mean here?) (easy, but definitely less so)
Note: The individual/small group is still working with only the
single number of tickets sold that you assigned.
- If that many tickets are sold, what is the airline's expected 'loss'
on the flight? What does this mean?

Have all the individuals/groups pool their results and decide how many
tickets the airline should sell for this flight in order to maximize
their revenue.

Thus, overbooking. (In case you ever wondered.)