Airlines overbook planes because they know some passengers will not turn up. Sup

Question

Airlines overbook planes because they know some passengers will not turn up. Suppose that the chance of a passenger not turning up is 0.024 so the chance of turning up is 0.976. You book seats on an airbus with 520 seats but accept 530 bookings for a particular flight. You would ideally like exactly 520 to turn up but it may well be less than this (in which case you lose some money since you are required to put them on an alternative flight) or more than this (in which case you will have to “bump them” off the flight and compensate). The key issue then is how many passengers turn up. Suppose that the revenue on each seat is $450 and the compensation costs you $300. What profits will you be likely to make?

Explanation / Answer

Here the probability of passenger not turing up = 0.024

Pr(Turning up) = 0.976

so Let say X are the number of passengers that are turning up.

Total number of bookings = 530

expected customers too take seats = 530 * 0.976 = 517.28

Standard deviation of customers to take seats = sqrt (530 * 0.976 * 0.024) = 3.5235

Now We have to find expected profits that airlines will make.

Here Profit P(X)= 520 * 450 [ where X >= 520]

= 450 X - 300 * (520 - X) where X < 520

So we E(P) = Pr(X 520) * 520 * 450 +Pr( X < 520) * E[450 X - 300 * (520 - X) ]

so here Pr( X < 520) = NORMAL (X < 519.5 ; 517.28; 3.5235 ) [ Using continuity correction factor]

Z = (519.5 - 517.28)/ 3.5235 = 0.63

Pr( X < 520) = NORMAL (X < 519.5 ; 517.28; 3.5235 ) = Pr(Z < 0.63) = 0.7357

so Pr(X 520) = 1 - 0.7357 = 0.2643

as we know

E(P) = Pr(X 520) * 520 * 450 +Pr( X < 520) * E[450 X - 300 * (520 - X) ]

E(p) = 0.2643 * 520 * 450 + 0.7357 * [ 450 * E(X) - 300 * E(520 -X)]

E(p) = 0.2634 * 520 * 450 + 0.7357 * (450 * 517.28 - 300 * (520 - 517.28))

E(P) = 61635.6 + 170652.972

E(P) = $ 232288.572

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