1. Airline Overbooking
Paul O’Connor
Department of Mathematics and Statistics
University of Limerick
Final Year Project
Supervised by: Prof. James Gleeson
Submitted: April 26, 2015
2. Declaration
This Final Year Project is presented in fulfillment of the requirements. It is entirely my own work,
completed without collaboration with others except my supervisor, Prof. James Gleeson. Where use
has been made of the work of other people it has been fully acknowledged and referenced accordingly.
Signature:
Paul O’Connor
3. Abstract
Revenue Management is a technique used by airline industries to maximize revenue by allocating the
available seats to the right customers at the right price. Overbooking is an airline revenue management
technique that enables airlines to sell more seats than the physical capacity of the plane in order to
account for the fact that some of the passengers may not show-up or cancel their flights on the
departure day. The objective of this project is to develop an overbooking model that determines the
number of overbooked seats.
4. Acknowledgements
This Final Year Project is the result of 9 months work which would not have been possible without
the support of some people.
First of all, I would like thank my supervisor Prof. James Gleeson. His patience, guidance,
suggestions and encouragement made it possible for me to produce this project.
I would also like to thank both my parents, Niall and Bernadette who, have provided this opportunity
for me to attend this college as well as encouraging me at every step along the way. Without their
continuous support I would not have made it this far.
7. Chapter 1
Introduction
1.1 Airline Overbooking
Airlines accept bookings for a flight up to the day of departure.“It is well known that about 10 − 15%
of travelers with confirmed reservations do not show up for their flights without giving prior notice
to airlines (no-shows)” (Suzuki, 2006). Passengers often cancel bookings, they also have the option
of alternative routes and fare prices to their original destination due to the competition that exists
in the air travel market. To compensate for this airlines overbook flights. Overbooking is an airline
revenue management technique that enables airlines to sell more seats than the physical capacity in
order to account for the fact that some of the passengers may not show up or cancel their flights
on the departure day. This technique enables airlines to fly with less empty seats but can also lead
to passengers with confirmed bookings being denied boarding. By flying with less empty seats the
airline will increase the revenue it can earn from that flight, however, a further cost must be paid to
the passengers that are denied boarding for the inconvenience caused to them.
Unlike most major airlines, Ryanair, (Europe’s leading low-cost airline) does not overbook flights.
Klophaus and Polt (2007), reports for instance that flights operated by Lufthansa German Airlines,
4.9 million passengers did not show up in 2005. This corresponds to 12,500 full Boeing 747’s. To
compensate for cancellations and no-shows most airlines overbook their flights and accept bookings
above physical seat capacity. Overbooking allowed Lufthansa to carry more than 570,000 additional
passengers in 2005. Lufthansa credits the practice of selling more tickets for a flight than there are
physical seats for a revenue increase of e105 million in 2005 (denied boarding costs already deducted)
making overbooking not only one of the oldest revenue management techniques applied by Lufthansa
but also one of the most powerful. The examples used in this paper are motivated by the fact that
Ryanair does not overbook its flights. The model developed will overbook a route operated by Ryanair
to investigate if it would increase the revenue generated.
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- Booking Process.png
Figure 1.1: Booking Process (Lawrence et al., 2003)
Figure 1 illustrates the booking process, where the capacity can refer to either the cabin capacity or
the allocation of seats to each booking class. The number of bookings increases as it gets closer to the
departure date. The predicted number of no-shows is constant throughout. This predicted number is
used to set the overbooking limit(the capacity plus number of no-shows). Bookings are accepted up
to this limit, hence the bookings exceed the capacity of the plane. Overbooking to the correct amount
will lead to flying at full capacity. This figure illustrates the perfect situation where by flying at full
capacity will lead to maximum revenue with no denied boardings. This illustrates the points made
by Klophaus. Since Ryanair does not overbook it loses out on the opportunity to fly at maximum
capacity to earn maximum revenue.
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Objectives of this project
The objectives of this project will be:
1. To model the overbooking problem for single-leg single-fare class flight for the deterministic case.
2. To model the overbooking problem for single-leg single-fare class flight for the stochastic case.
3. To develop the 2nd model further for voluntary and involuntary passengers denied boarding.
4. The models will be used to evaluate the increase in profit that can be made by overbooking.
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10. Chapter 2
Literature Review
2.1 History of Airline Overbooking
The airline revenue management technique of overbooking would not be possible without the Dereg-
ulation of the Civil Aeronautics Board (CAB) in 1978. The CAB was set up in 1938 to regulate all
domestic interstate air transport routes as a public utility, setting fares, routes, and schedules. Due to
the CAB airlines had no control over the revenue they earned until all seats were sold. “U.S. deregu-
lation has been part of a greater global airline liberalization trend, especially in Asia, Latin America,
and the European Union” (SmithJr and Cox, 2008). After deregulation, airlines began to use many
airline revenue management techniques. These techniques sought to maximize the amount of revenue
that an airline could earn. Due to the increased competition in the market most airlines had little
control over their fare price. To maximise their revenue airlines wanted to fly at full capacity. To allow
for the number of passengers that cancelled or did not show up, airlines exploited the opportunity of
using overbooking models to earn more revenue.
2.1.1 Literature Review
The pioneering work on Airline Overbooking was done by Beckmann (1958). Beckmann developed a
single leg flight model for overbooking, he used a static model with reservation requests, booking and
initially cancellations that balanced the lost revenue of empty seats with the costs to the airline of
passengers being bumped.
Thompson (1961) extended Beckmann’s model for a two fare class using the cancellation rates for
each class while ignoring the probability distribution of the demand and the no-show rates. His model
determines the overbooking level for a given probability of denied boarding.
Rothstein (1971) viewed the procedure of reservations was as a Markovian sequential decision process.
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He first proposed a mathematical model to analyze the overbooking policy. In (Rothstein, 1985)
he presented a survey of the application of operations research to airline overbooking. The article
analyzed the issues that motivated overbooking and discussed the relevant practices of the air carriers.
Belobaba (1987) conducted a comprehensive review of the aspects in revenue management in the
Airline Industry. He, like Thompson, discussed the problem of overbooking in multiple fare classes
and suggested a heuristic approach to solve the problem.
In (Smith et al., 1992) there is a brief discussion on the overbooking model used by American Airlines.
American Airlines developed different operations research models and implemented the static one-
period model. This model contained additional constraints to ensure that overbooking earned revenue.
Coughlan (1999) explored the idea of passengers that arrive prior to boarding without any confirmed
reservation (called “Go-Shows”). He included this idea in the a multi fare-class model, assuming that
cancellations and number of passengers that show up to be independently normally distributed. More-
over, he assumes that the minimum of the demand and the number of bookings are also independently
normally distributed.
Subramanian et al. (1999) allowed for cancelations and no-shows in a multiple fare class model. They
present two models. In the first model, the cancellation and no-show probabilities do not depend on
the fare classes. They show that the resulting problem can be modeled as a queuing system. In their
second model, they relax the class independence assumption and model a more general problem with
class dependent cancellations and no-shows. Unfortunately, they were unable to solve the dynamic
programming formulation efficiently.
Klophaus and Polt (2007) explored how overbooking could be beneficial to the low-cost airline Ryanair,
who have a no-overbooking policy. They first use a previously developed model to determine the opti-
mal booking limit based on revenue data, variable costs, costs per denied boarding and no-show rates.
They then extend this model to incorporate the costs associated with flying with empty seats. They
find that Ryanair should overbook flights which are in high demand but should be more conservative
when it comes to flights that are not.
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12. Chapter 3
Model Development
3.1 Mathematical Formulation of the Problem
Consider a single-leg single-fare (f) flight having a maximum capacity of C. The airline accepts
bookings up to the day of departure. A passenger who made a booking may not show up on the
departure day. To allow for the number of passengers that do not show up on the day of departure the
airline should overbook the plane. If the number of passengers that show-up (S) is greater than the
capacity of the airplane, customers will be bumped. The term bumped means that a passenger with
a confirmed booking will be denied boarding to this flight. For this inconvenience, the airline has to
pay a compensation cost (X). If the number of passengers that show up during the time of departure
is less than the capacity, the airplane will fly with empty seats resulting in lost revenue. Hence the
objective is to develop a mathematical model that determines the optimal number of overbooked seat
(O) which will maximise the profit (P) by flying the plane at full capacity.
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Variables Variable Description
C Capacity
S Number of passengers that show up
O Number of overbooked seats
β Show-up probability
f Price per seat
X1 Voluntary Compensation cost per passenger
X2 Involuntary Compensation cost per passenger
P Profit
πn Probability that n people show up
n The n passengers than show up
z The number of passenger that volunteer to be bumped
πz Probability that z people volunteer to be bumped
θ Volunteer probability
Table 3.1: Table of Variables
3.2 Assumptions about the model
The model developed in this chapter will consider a single-leg single-fare class flight. The assumptions
made in the model are as follows:
1. All bookings are independent of each other.
2. The capacity of the plane is fixed.
3. Cancellations up to the day of departure will be considered as passengers that do not show up.
4. Passengers that show up for the flight on the day of departure do so independently of each other.
5. The number of passengers that show up follows a binomial distribution.
6. The cost of a passenger being voluntary or involuntary denied boarding is a fixed cost.
7. The number of passengers that volunteer to be bumped follows a binomial distribution.
This model will be effective in determining the excess amount of overbooking to be made by considering
the number of no-shows from a probability distribution. The objective of the model will be to maximise
the profit by flying the plane at full capacity.
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3.3 Graphical Framework
The flow chart below is the idea behind the first model.
Capacity (C)
Overbooked Flight
Involuntary Bumped Passengers
- Loss of Customers
- Compensation Cost (X)
- Empty Seats
- Lost Revenue
Full Flight
- No Lost Revenue
- No Compensation
C=SS<C
S>C
Voluntary Bumped Passengers
- Compensation Cost (X)
Figure 3.1: Graphical Framework
The flowchart represents the 3 different scenarios that occur in the model:
1. If the number of passengers that show up (S) is less than the physical capacity(C) of the plane,
the plane will fly with empty seats resulting in lost revenue for the airline.
2. If the number of passengers that show up (S) is equal to the physical capacity(C) of the plane,
the flight will earn maximum profit(P) and will not have to pay passengers the compensation
cost for being bumped(X).
3. If the number of passengers that show up (S) is greater than the physical capacity(C) of the
plane, the airline will have to pay the bumped passengers the compensation cost for being
bumped(X).
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If the number of passengers that show up exceeds the physical capacity of the plane then passengers
will be denied boarding. At this stage the airline will ask for passengers to volunteer to be denied
boarding. The passengers who volunteer are known as voluntary bumped passengers. If the number
of passengers that volunteer does not reduce the total amount of passengers that showed-up to be
equal to the capacity of the plane then, a certain amount of passengers will be denied boarding. These
passengers are known as involuntary bumped passengers.
3.4 Cost of Overbooking
On 17 February 2005 a new regulation by the European Community (EC) to protect the rights
of air passengers when facing denied boarding and cancellations or long delays of their flights en-
tered into force repealing a weaker regulation dating from 1991. The new EC regulation 261/2004
(Commission For Aviation Regulation, 2015) applies to passengers departing to and from an airport
located in the territory of a member state, on condition that the passengers have a confirmed reser-
vation and show-up at check-in in time.
Passengers who have been denied boarding have the choice between a full refund on the cost of their
ticket and a transfer to their destination either on the first available flight or at a later time, at the
passengers request. In addition, every passenger who is denied boarding is entitled to a minimum
compensation of:
• e125 for all flights of 1,500 km or less where the delay is less than 2 hours,
• e250 for all flights of 1,500 km or less where the delay is greater than 2 hours,
• e200 for all intra-Community flights of more than 1,500 km, and for all other flights between
1,500 and 3,500 km, where the delay is less than 3 hours,
• e400 for all intra-Community flights of more than 1,500 km, and for all other flights between
1,500 and 3,500 km, where the delay is more than 3 hours,
• e400 for all other flights not falling within the categories mentioned above and where the delay
is less than 4 hours,
• e600 for all other flights not falling within the categories mentioned above and where the delay
is more than 4 hours.
When passengers are offered re-routing to their final destination on a alternative flight, the arrival
time of which does not exceed the scheduled arrival time of the flight originally booked, in respect
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of the specifications mentioned above, the air carrier may reduce the compensation by 50 percent.
In comparison to the previous regulation (EEC 295/91), the European Community has significantly
increased the minimum compensation amounts and also extended denied boarding compensation to
non-scheduled flights.
EC regulation 261/2004 requires air carriers, when expecting to deny passengers boarding, to first call
for volunteers to surrender their reservations, in exchange of benefits, instead of denying passengers
boarding against their will. Even before this regulation, airlines point to the great success in luring
passengers off oversold flights with vouchers or money payments, minimizing the number of involuntary
denied boardings. As Ryanair officially does not overbook its flights, there is no Ryanair policy for
compensating passengers denied boarding. Hence, in the following calculation of optimal overbooking
limits, e125 will be used as the estimate for the minimum compensation for voluntary denied boarding
and e250 for involuntary denied boarding.
3.5 First Model
Number of Bookings
The number of bookings is the total Capacity(C) of the plane added to the number of overbooked
passengers (O):
Number of bookings = (C + O).
Number of passengers that show up
If we assume the deterministic scenario that the number of passengers that show up (S) is equal to
the number of bookings multiplied by the show-up probability(β):
S = β(C + O), (3.1)
We are making this assumption as a first attempt to understand the interplay between the variables
of the model. We know it is not realistic and will improve upon it in later versions of the model.
Profit
The profit (P) generated by the flight will depend on the number of passengers that show up. Profit
is generated from the number of bookings multiplied by the fare price (f) less the total cost of
compensation (X) multiplied by the number of people that show up (S) greater than the capacity
(C) of the plane.
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Let’s first consider the scenario that the number of passengers that show up is less then the capacity
of the plane, then the Profit becomes:
P = f(C + O) if S < C.
Here, the opportunity to earn more in revenue by booking more passengers has been lost but we do
not have to pay compensation to any passenger for being bumped.
Now, we shall consider the scenario that the number of passengers that show up is equal to the capacity
of the plane, then profit become:
P = f(C + O) if S = C.
In this scenario the airline will have maximised its profit by flying the plane at full capacity.
Finally, let’s consider that the number of passengers that show up on the day of departure being
greater than the Capacity of the plane. For every passenger greater than the physical capacity of
the plane (denied boarding) the airline will have to pay a compensation cost (X) to each passenger
respectively.
P = f(C + O) − X(S − C) if S > C,
This can be rewritten as:
P = C(f + X) + fO − XS.
Here, the airline will have to pay compensation to each bumped passenger. The airline’s profit will
decrease for every additional passenger the shows up over the physical capacity of the plane.
The profit generated by the model will be a piecewise function as follows:
P(f, C, O, β, X) =
f(C + O) if S ≤ C,
C(f + X) + fO − XS. if S > C.
(3.2)
3.6 Results for first model
Consider the Ryanair flight from Shannon to Manchester. Currently Ryanair operates 303 Boeing 737-
800 aircraft (Ryanair, 2014), each with a capacity of 189. Setting the fare price f=e50, compensation
cost X=e125 from (Citizens Information, 2014), the show up probability β=0.9 and using equation
3.1, we obtain the following result:
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189 = 0.9(189 + O)
O = 21.
Since we want the plane to fly at full capacity, this is the optimal number of overbooked seats in this
scenario. Using this result and using equation 3.2 the number of passengers that show up gave the
following results for profit.
Number that show up(S) Profit (P) Number that show up(S) Profit (P)
171 e9,500 190 e10,430
172 e9,555 191 e10,361
173 e9,611 192 e10,291
174 e9,666 193 e10,222
175 e9,722 194 e10,152
176 e9,777 195 e10,083
177 e9,833 196 e10,013
178 e9,888 197 e9,944
179 e9,944 198 e9,875
180 e10,000 199 e9,805
181 e10,055 200 e9,736
182 e10,111 201 e9,666
183 e10,166 202 e9,597
184 e10,222 203 e9,527
185 e10,277 204 e9,458
186 e10,333 205 e9,388
187 e10,388 206 e9,319
188 e10,444 207 e9,250
189 e10,500 208 e9,180
190 e10,430 209 e9,111
191 e10,361 210 e9,041
Table 3.2: Table of Results
These results are plotted in Figures 3.2 and 3.3.
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If we consider the scenario where the number of passengers that show equal to the capacity of the
plane.
€9,000.00
€9,200.00
€9,400.00
€9,600.00
€9,800.00
€10,000.00
€10,200.00
€10,400.00
€10,600.00
171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
Number of Passengers that Show Up
Profit
Figure 3.2: Profit on the number of passengers that show up for S ≤ C(C=189 here)
In this scenario Ryanair would have earned a maximum profit of e10,500. Note that the profit
increases linearly for
S ≤ C,
this is due the airline earning more profit for each passenger that shows-up, up to the physical capacity
of the plane.
In the scenario where the number of passengers that show up is greater than the capacity of the plane
we obtain the result in Figure 3.3.
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€8,000.00
€8,500.00
€9,000.00
€9,500.00
€10,000.00
€10,500.00
€11,000.00
180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
Number of Passengers that Show Up
Profit
Figure 3.3: Profit on the number of passengers that show up for S > C(C=189 here)
Note that due to having to pay customers compensation for being bumped, Ryanair’s profit decreases
for every person over the physical capacity of the plane (S ≥ C).
3.7 Stochastic Model
Number of Passengers that show up
In the first model we assumed the deterministic scenario. Here we will assume that the number of
passengers that show-up follows a Binomial distribution, the probability (πn) that there are exactly
n show-ups out of the (C + O) bookings is:
πn =
C + O
n
βn
(1 − β)C+O−n
,
where β is the show-up probability for each passenger and we assume that each passenger shows up
independently of each other.
The piecewise profit function becomes:
P(n) =
f(C + O) if n ≤ C
C(f + X) + fO − Xn if n > C
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Here n is the number of passengers that show-up out of (C + O) bookings.
In the first model we assumed that,
S = β(C + O).
Since n ∼Bin(C+O,β) the mean of the distribution is
β(C + O).
This shows that our first model was assuming that exactly the expected number of passengers show
up.
The expected profit on n show ups is
E(P) =
∞
n=0
πnP(n).
The expected profit in the model as follows:
E(P) =
C
n=0
πn[f(C + O)] +
n
n=C+1
πn[C(f + X) + fO − Xn]
3.8 Stochastic Model with denied boarding costs
The first stochastic model has been extended to include the costs associated with denied boarding.
Here, we will assume that the number of passengers that volunteer to be bumped follows a Binomial
distribution, the probability (πz) that there are exactly z volunteers out of the (n) show-ups is:
πz =
n
z
θz
(1 − θ)n−z
,
where θ is the volunteer probability for each individual passenger.
Since z ∼Bin(n,θ) the mean of the distribution is,
θn,
here, we are assuming that exactly z passengers out of the n passengers that show up will volunteer
to be bumped.
This creates three separate scenarios that have to be accounted for by the model.If the number of
passengers that show up is less than the capacity of the plane, the profit on those passengers is,
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P = f(C + O) if n ≤ C.
If the number of passengers that show-up is greater than the capacity of the plane, we will have
passengers who are denied boarding. In this scenario the airline company will ask for volunteers
so that in can minimise the denied boarding costs. The ideal situation will be that the number
of passengers that volunteer will be greater than the excess of passengers over the planes physical
capacity. The profit function becomes,
P = f(C + O) − x1(n − C) if z ≥ n − C.
Here the x1 will be the cost of denied boarding.
If the number of passengers that volunteer to be denied boarding is less than the excess of passengers
over the capacity of the plane we will have involuntary denied boarding. These cost are much higher
and if possible should be avoided. Here the profit is,
P = f(C + O) − x2(n − C − z) − x1(z) if z < n − C.
The piecewise profit function now becomes,
P(n) =
f(C + O) if n ≤ C
f(C + O) − x1(n − C) if z ≥ n − C
f(C + O) − x2(n − C − z) − x1(z) if z < n − C
The expected profit in this model is as follows,
E(P) =
C
n=0
πn[f(C+O)] +
C+z
n=C+1
πn[f(C+O)−x1(n−C)] +
n
n=C+z+1
πn[f(C+O)−x2(n−C−z)−x1(z)].
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3.9 Results for first Stochastic Model
The computer program Matlab was used to solve both stochastic models. The matlab code was
developed in stages, first to find the expected profit for the first stochastic model and then extended
further to include the separate costs associated with voluntary and involuntary denied boarding. The
developed code is explained briefly and attached in the appendix.
Matlab code for Profit
This code finds the profit on each passenger that shows up and plots the output. These results are
similar to those for the first model but the results developed through matlab gave a greater insight into
the revenue from each passenger. We varied the overbooking vector at increments of 10 passengers to
and find the respective profit.
Figure 3.4: Overbooking by 10 passengers
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Figure 3.5: Overbooking by 20 passengers
Figure 3.6: Overbooking by 30 passengers
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Figure 3.7: Overbooking by 40 passengers
As we can see from the figures above the profit linearly increases in figures 3.4 and 3.5 but decreases
at the point of capacity(189 in this case). Here we are trying to find the number of passengers to
overbook by, to fly at full capacity. Profit is a maximum when the plane is at full capacity. Since β
is fixed at 0.9 the maximum profit is achieved by setting the overbooking vector,
O = 21.
Setting the overbooking vector to 21 returns a maximum profit of e10,500.
Code for Step 1
The code for profit was extended to include the binomial distribution for the number of passengers that
show-up to find the expected profit for each overbooked passenger. Here we will vary the overbooking
in increments of 5. The following results we obtained.
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Number of overbooked passengers(O) Expected profit ()
0 e9,450
5 e9,700
10 e9,949
15 e10,178
20 e10,288
25 e10,183
30 e9,928
Table 3.3: Table of Expected Profit
Increase in profit
If Ryanair did not overbook this flight is would have earned a maximum profit of e9,450 (50×189).
We have seen in the deterministic case that if we overbooked by 21 passengers we would have earned
e10,500. This corresponds to an increase of e1,050 in profit. We have also seen that by overbooking
by 20 in the stochastic case returned a maximum profit of e10,288. This again corresponds to an
increase of e838 in profit. This shows that if Ryanair did overbook its flights it could have earned an
additional 8.87% in profit.
Code for Step 2
Step 2 of the Matlab code extends step 1 to create an expected profit vector. Each entry in this vector
corresponds to the expected profit for each overbooked passenger. Here we will vary the overbooking
level again using increments of 10 to find the value that obtains the maximum expected profit.
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Figure 3.8: Expected profit on 10 overbooked passenger
Figure 3.9: Expected profit on 20 overbooked passenger
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Figure 3.10: Expected profit on 30 overbooked passenger
Figure 3.11: Expected profit on 40 overbooked passenger
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As we can see again here that the expected profit increases to a maximum point this time at 20
overbooked passengers. After this point it decrease quite rapidly as the cost of denied boarding are
quite expensive to the airline. The expected profit earned on 20 overbooked passengers is e10,288.
3.10 Results for second Stochastic Model
Code for Step 3
This code extends steps 2 to include the costs associated with voluntary and involuntary denied
boarding. When the Airline identify’s that the number of passengers that have shown up for a
particular flight is over the physical capacity, they then ask for passengers to volunteer to be denied
boarding. As mentioned in section 3.8 the number of passengers that volunteer to be denied boarding
follows a Binomial distribution. I was unable to find data on denied boarding on European Airlines
but was able to find data on Airlines in the United States. Figure 3.12 shows the number of denied
boardings in thousand’s from the year 2000 to 2013.
Figure 3.12: Passenger denied boarding data (US Department for Transport, 2014)
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As we can see from the above figure the number of passengers that volunteer is quite high. The
average number of volunteers over the 13 years was 92.4%. This high figure is a result of Airlines
offering high rewards for these passengers to maintain their customer goodwill and avoid the higher
cost of involuntary denied boarding. In the code developed we will set the θ value equal to 0.924 and
increase the number of overbooking in increments of 10. The code uses the mean of the Binomial
distribution to determine if a passenger will volunteer to be denied boarding.
Figure 3.13: Expected profit on 10 overbooked passengers
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Figure 3.14: Expected profit on 20 overbooked passengers
Figure 3.15: Expected profit on 30 overbooked passengers
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Figure 3.16: Expected profit on 40 overbooked passengers
The results above are identical to those obtained in step 2. At first this lead me to think that there
was an error in the code. As seen above the expected profit increases to a maximum point and then
decreases rapidly due to the cost of denied boarding. Due to setting θ to the high value of 0.924 these
results are in theory correct. For example if we use the values initialized in figure 3.15:
f = 50, C = 189, O = 30, β = 0.9, θ = 0.924, X1 = 125, X2 = 250.
The number of passengers that show up (n) is the mean of binomial distribution in section 3.7,
n = β(C + O),
n = 0.9(189 + 30),
n = 197.1.
Out of the 197 passengers that show-up the model expects that 92.4% of these will volunteer there
seat. Again here we are using mean of the binomial distribution in section 3.8 to determine how many
passengers will volunteer there seat. Here, the number of volunteers will be,
Z = nθ,
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33. Paul O’Connor Airline Overbooking
Z = 197.1(0.924),
Z = 182.12.
Here, out of the 197 passengers that show up 182 of them that will volunteer to be denied boarding.
Due to this high number of volunteers we will not have any passengers being denied boarding invol-
untarily. This is a strong indication that the results obtained are correct. It is also interesting to note
from figure 3.16 that up to the 37th overbooked passenger Ryanair would have earned the same in
profit as it would have if it did not overbook at all. The maximum profit is achieved by overbooking
by 20 passengers returning an expected profit of e10,288.
Due to the high cost associated with involuntary denied boarding I would have expected to reach the
maximum point in less overbooked passengers. The decrease from the maximum point is rapid but
I would have expected this to be much more severe due to the cost of involuntary denied boarding.
Overbooking by 20 and setting the θ value to zero (i.e no passengers volunteer to be denied boarding)
gave a result similar to what would have been expected.
Figure 3.17: Expected profit on 20 overbooked passengers θ = 0
No passengers volunteering to be denied boarding is not entirely trivial, although the incentives that
are offered to passengers by Airlines leads me to think that at least 1 person will except the offer.
These incentives are less of cost to the Airline than a passenger being involuntary denied boarding
and will also help to maintain customer goodwill. Figure 3.17 again shows an increase in expected
30
34. Paul O’Connor Airline Overbooking
profit to maximum a point as seen in figure 3.14. However in figure 3.17 the maximum is achieved at
17 overbooked passengers rather than 20.
Figure 3.17 seemed to be decreasing quite rapidly after the maximum point so I then overbooked by
30 passengers and obtained the results in figure 3.18.
Figure 3.18: Expected profit on 30 overbooked passengers θ = 0
The above result is what I would have expected to achieve as the cost of involuntary denied boarding
are more costly to Ryanair. Due to the high volunteer probability there will no passengers that are
involuntary denied boarding.
31
35. Chapter 4
Conclusion and Discussion
4.1 Discussion
From the outset the project was quite challenging. Airlines regard their data to be highly valuable.
While this is quite understandable as it would lead to alternative Airlines using similar strategies it
was hard to find passenger data. The objective of this project was to develop a model that determines
the number of overbooked seats. I was successful in modeling and obtaining results for the overbooking
problem for single-leg single-fare class flight for the deterministic and stochastic cases. I was successful
in evaluating the increase in profit that can be made by overbooking. I was able to extend the stochastic
model to include the different costs associated with voluntary and involuntary denied boarding and
was able to obtain results for this model.
4.2 Main Results
Ryanair does not overbook its flights. I have modeled this problem and found that Ryanair would
benefit significantly if it overbooked its flights. The code developed is effective at both determining
the number of passengers to overbook by and the maximum expected profit that can be achieved. The
different cost associated with voluntary and involuntary denied boarding are costly to Airlines. As
the value of the incentives offered by most Airlines to passengers is quite high the value used for the
volunteer probability is theoretically correct. This in turn, greatly reduces the number of passengers
to be involuntary denied boarding. The model determines that if the number of passengers that show-
up is greater than the physical capacity of the plane, enough passengers will volunteer their seat and
the plane can then fly at full capacity. The expected profit in the example used corresponded to an
increase in 8.87% in profit that could be obtained if Ryanair overbooked its flights.
32
36. Paul O’Connor Airline Overbooking
4.3 Future Work
To date the first model is effective in determining the number of passengers to overbook by for
the deterministic case. The stochastic model is now fully developed and efffective at determining
the number of passengers to overbook by as well as the expected profit that could be earned. The
objectives going forward would be:
• To extend the stochastic model for multi-fare flight.
• To extend the stochastic model for multi-class flight.
• To extend the model to include passengers that may have been denied boarding on a previous
flight.
Finally, I hope the contents of this project would be helpful to researchers working on Airline overbook-
ing or similar overbooking problems. The performance of the Matlab code developed was effective,
although in future it could be extended further to include more parameters.
33
37. Appendix A
Code for Profit
% Matlab Code to calculate the profit per passenger
% Last Edited 31/3/15
clear
f = 50; % Price per seat
C = 189; % Capacity
Ovect = 0:21; % Overbooking vector
x = 125; % Compensation cost
beta = 0.9; % Show up probability
svect = beta*(C+Ovect); % No of passengers that show up
for i=1:length(Ovect)
if svect(i)<=C
p(i)=f*(C+Ovect(i)); % profit if number of passengers is less than or
% equal to Capacity
else
p(i)=C*(f+x)+(f*Ovect(i))-(x*svect(i)); % profit if the passengers that
% show up is greater than capacity
end %if
end %for
34
38. Paul O’Connor Airline Overbooking
p
plot(svect,p,’:bs’);
ylabel(’Profit’);
xlabel(’Number of Passengers’);
35
39. Paul O’Connor Airline Overbooking
Code for Step1
%Matlab Code for Generating Expected Profit
%Last Edited 31/3/15
% Step 1
clear
f = 50; % Price per seat
C = 189; % Capacity
O = 20; % Number of Overbooked passengers
x1 = 125; % bumped passenger compensation cost
beta = 0.9; % Show up probability
nvect = 0:(C+O); %number of passengers
for i = 1:length(nvect)
n=nvect(i);
Pin(i)=binopdf(n,C+O,beta); % probability that n passengers show up
%out of C+O(bookings) with show up prob beta
if n<=C %if number of passengers that show up is less than Capacity
p(i)=f*(C+O) % returns the profit on those passengers that show up
%less than/ equal the capacity
else % Passengers that show up exceeding capacity
p(i)=C*(f+x1)+(f*O)-(x1*n) % profit on those passengers that show up
%being greater than the capacity including the cost of bumping
end % if
end %for
% Calculate Expected Profit
ExpP= sum(p.*Pin)
36
40. Paul O’Connor Airline Overbooking
Code for Step 2
% Matlab code to find the expected profit on each overbooked passenger
% Create a plot of Exp Profit v Number of Overbooked passengers
% Last Edited 07/04/15
% Step 2
clear
f = 50; % Price per seat
C = 189; % Capacity
x1 = 125; % bumped passenger Compensation cost
beta = 0.9; % Show up probability
Ovect=0:30; % Overbooking vector
tic
for j=1:length(Ovect)
O=Ovect(j);
nvect=0:(C+O);
for i = 1:length(nvect)
n=nvect(i);
Pin(i)=binopdf(n,C+O,beta); % probability that n passengers show
%up out of C+O(bookings) with show up prob beta
if n<=C %if 1
p(i)=f*(C+O); % profit on number of passengers that show up less
% than/ equal the capacity
else
p(i)=C*(f+x1)+(f*O)-(x1*n); % profit on the number of passengers
% that show up being greater than the capacity
37
41. Paul O’Connor Airline Overbooking
end % if 1
end %for nvect
ExpP(j)=sum(p.*Pin) % Calculate the expected profit
end %forj ovect
toc
plot(Ovect,ExpP,’:bs’)
ylabel(’Expected Profit’);
xlabel(’Number of Overbooked Passengers’);
38
42. Paul O’Connor Airline Overbooking
Code for Step 3
% Matlab code to find the expected profit on each overbooked passenger
% Create a plot of Exp Profit v Number of Overbooked passengers
% Include the different cost on Voluntary and Involuntary bumped passengers
% Last Edited 07/04/15
% Step 3
clear
f = 50; % Price per seat
C = 189; % Capacity
x1 = 125; % Voluntary bumped Compensation cost
x2 = 250; % Involuntary bumped Compensation cost
beta = 0.90; % Show up probability S
theta = 0.924; % Volunteer Probability
Ovect=0:21; % Overbooking Vector
tic
for j=1:length(Ovect)
O=Ovect(j);
nvect=0:(C+O);
for i = 1:length(nvect)
n=nvect(i);
Z=(theta*n); %assumption using the mean of the distribution to find Z
Pin(i)=binopdf(n,C+O,beta); % probability that n passengers show up
%out of C+O(bookings) with show up prob beta
if n<=C % number that show up is less than capacity
p(i)=f*(C+O); % profit on number of passengers that show up less
% than/ equal the capacity
39
43. Paul O’Connor Airline Overbooking
else if Z >= n-C % number of volunteers is greater than number of
% passengers that exceed capacity
p(i)=f*(C+O)-x1*(n-C); % profit on the number of passengers less cost
% of passengers volunteers to be bumped x1
else % passengers to be involuntary bumped
p(i)=f*(C+O)-(x1*Z)-x2*(n-C-Z); % Profit on Involuntary Bumped
% Passengers
end % if else
end % if
end %for nvect
ExpP(j)=sum(p.*Pin) % Calculating Expected Profit
end %forj ovect
toc
plot(Ovect,ExpP,’:bs’)
ylabel(’Expected profit’);
xlabel(’Number of overbooked passengers’);
40
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