Most airlines practice overbooking

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Most airlines practice overbooking. That is, they are willing to ma ke more reservations  than they have seats on an airplane. Why would  they  do this?  The basic reason  is simple; on any given flight a few passengers are likely  to  be  "no-shows."  If the airline overbooks slightly, it still may be able to fill the airplane. Of course, this policy  has its risks. If more  passengers arrive to claim their  reservations  than there are seats availa ble, the airline must  "bump" some of  its passengers.  Often this is done by asking for volunteers. If a passenger with a reserved seat is willing to give up his or her seat, the airline typically  will provide incentives of some sort. The fundamenta l trade-off  is whether  the additional  expected  revenue  gained  by flying an airplane that is nearer  to capacity on average is worth  the additional expected cost of  the incentives. To study the overbooking  policy, let us look  at a hypothetical  situation.  Mockingbird  Airlines  has  a small  commuter  airplane  that seats  16 passengers.  The airline uses Lhis jel uu a ruule fur which iL chaq;es $225 for a one-way fare. Every flight has a fixed cost of $900 (for  pilot's salary, fuel, airport fees, and so on). Each passenger  costs Mockingbird  an  additional  $100.

Finally, the no-show rate is 4%. That is, on average approximately 4% of those passengers holding confirmed reservations do not show up. If Mockingbird must bump a passenger, the passenger receives a refund on his or her ticket ($225) plus a $100 voucher toward another ticket.

How many reservations should Mockingbird sell on this airplane? The strategy will be to calculate the expected profit for a given number of reservations. For example, suppose tha t the Mockingbird manager decides to sell 18 reservations. The revenue is $225 times the number of reservations:

R = $225(18)

= $4050

The cost consists of two components. The first is

the cost of flying the plane and hauling the passengers who arrive (but not more than the airplane 's capacity of 16):

C1 = $900 +$100 x M in(Arrivals, 16)

The second component is the cost of refunds and

free tickets that must be issued if 17 or 18 passengers arnve:

C2 = ($225 + $100) x Max( O, Arrivals - 16)

In this expression for C2, the $225 represents the refund for the purchased ticket, and the $100

represents  the cost of the free ticket. The Max ( ) expression calculates the number  of excess passengers who show up (zero if the number  of arrivals is

16 or less.

Questions

1. Find the probability that more than 16 passengers will arrive if Mockingbird sells 17 reservations (Res = 17). Do the same for 18 and 19.

2. Find:

E (RI Res = 16) E(C1IRes = 16)

E(C2IRes = 16)

Finally, calculate

E (Profit j Res = 16) = E( R I Res = 16)

- E(C1IRes = 16) - E(C2 IRes = 16)

3.  Repeat Question 2 for 17, 18, and 19 reservations. What is your conclusion? Should Mockingbird overbook? By how much?