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Im not paying that!Mathematical models for
setting air fares
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Contents

• Background
• History
• Whats the problem?
• Solving the basic problem
• Making the model more realistic
• Conclusion
• Finding out more

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Air Travel in the Good Old Days
Only the privileged few 6000 passengers in the
USA in 1926
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And now
Anyone can go easyJet carried 30.5 million
passengers in 2005
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Whats the problem?

• Different people will pay different amounts for
  an airline ticket
• Business people want flexibility
• Rich people want comfort
• The rest of us just want to get somewhere
• You can sell seats for more money close to
  departure

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Make them pay!

• Charge the same price for every seat and you miss
  out on money or people
• Too high only the rich people or the business
  people will buy
• Too low airline misses out on the extra cash
  that rich people might have paid

100
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I fancy a holiday
Ive got a meeting on 2nd June
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Clever Pricing

• Clever pricing will make the airline more money
• What fares to offer and when
• How many seats to sell at each fare
• Most airlines have a team of analysts working
  full time on setting fares
• Turnover for easyJet in 2007 was 1.8 billion so
  a few percent makes lots of money!

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Contents

• Background
• Solving the basic problem
• Its your turn
• Linear programming
• Making the model more realistic
• Conclusion
• Finding out more

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Its your turn!

• Imagine that you are in charge of selling tickets
  on the London to Edinburgh flight
• How many tickets should you allocate to economy
  passengers?

• Capacity of plane 100 seats
• 150 people want to buy economy seats
• 50 people want to buy business class seats
• Economy tickets cost 50
• Business class tickets cost 200

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3 volunteers needed No hard sums!
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A
B
C
0 Economy
50 Economy
100 Economy
10,000
12,500
5,000
Allocate 50 economy Sell 50 economy at 50
2,500 Sell 50 business at 200 10,000 Total
12,500
Allocate 100 economy Sell 100 economy at 50
5,000 Sell 0 business at 200 0 Total 5,000
Allocate 0 economy Sell 0 economy at 50
0 Sell 50 business at 200 10,000 Total
10,000
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Using equations

• Assume our airline can charge one of two prices
• HIGH price (business class) pb
• LOW price (economy class) pe
• Assume demand is deterministic
• We can predict exactly what the demand is for
  business class db and economy class de
• How many seats should we allocate to economy
  class to maximise revenue?
• Write the problem as a set of linear equations

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Revenue

• We allow xe people to buy economy tickets and xb
  to buy business class tickets
• Therefore, revenue on the flight is

Maximise
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Constraints

• Constraint 1 the aeroplane has a limited
  capacity, C
• i.e. the total number of seats sold must be less
  than the capacity of the aircraft
• Constraint 2 we can only sell positive numbers
  of seats

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More Constraints

• Constraint 3 we cannot sell more seats than
  people want

• Constraint 4 the number of seats sold is an
  integer

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In Numbers

• We allow xe people to buy economy tickets and xb
  to buy business class tickets
• Therefore, revenue on the flight is

Maximise
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And Constraints

• Constraint 1 aeroplane has limited capacity, C
• Constraint 2 sell positive numbers of seats
• Constraint 3 cant sell more seats than demand

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Linear Programming

• We call xe and xb our decision variables, because
  these are the two variables we can influence
• We call R our objective function, which we are
  trying to maximise subject to the constraints
• Our constraints and our objective function are
  all linear equations, and so we can use a
  technique called linear programming to solve the
  problem

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Linear Programming Graph
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Linear Programming Graph
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Solution

• Fill as many seats as possible with business
  class passengers
• Fill up the remaining seats with economy
  passengers

xb db, xe C xb for db lt C xb C for db gt C
50 economy, 50 business (Option B)
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But isnt this easy?

• If we know exactly how many people will want to
  book seats at each price, we can solve it
• This is the deterministic case
• In reality demand is random
• We assumed that demands for the different fares
  were independent
• Some passengers might not care how they fly or
  how much they pay
• We ignored time
• The amount people will pay varies with time to
  departure

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Contents

• Background
• Solving the basic problem
• Making the model more realistic
• Modelling customers
• Optimising the price
• Conclusion
• Finding out more

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Making the model more realistic

• We dont know exactly what the demand for seats
  is
• - Use a probability distribution for demand
  
• Price paid depends only on time left until
  departure or number of bookings made so far
• Price increases as approach departure
• Fares are higher on busy flights
• Model buying behaviour, then find optimal prices

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Demand Function
f(t)
e.g.
t
Departure
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Reserve Prices

• Each customer has a reserve price for the ticket
• Maximum amount they are prepared to pay
• The population has a distribution of reserve
  prices y(t), written as p(t, y(t))
• Depends on time to departure t

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Reserve Prices
Id like to buy a ticket to Madrid on 2nd June
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March 2008
100
Ive got a meeting in Madrid on 2nd June Id
better buy a ticket
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Reserve Prices
All my friends have booked I need this ticket
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May 2008
200
The meetings only a week away Id better buy a
ticket
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Revenue
Maximise
Revenue
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Maximising Revenue

• Aim Maximise revenue over the whole selling
  period, without overfilling the aircraft
• Decision variable price function, y(t)
• Use calculus of variations to find the optimal
  functional form for y(t)
• Take account of the capacity constraint by using
  Lagrangian multipliers

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Optimal Price
Departure
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Contents

• Background
• Solving the basic problem
• Making the problem more realistic
• Conclusion
• Why just aeroplanes?
• Finding out more

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Why Just Aeroplanes?

• Many industries face the same problem as airlines
• Hotels maximise revenue from a fixed number of
  rooms no revenue if a room is not being used
• Cinemas maximise revenue from a fixed number of
  seats no revenue from an empty seat
• Easter eggs maximise revenue from a fixed
  number of eggs limited profit after Easter

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Is this OR?
Yes!

• OR Operational Research, the science of better
• Using mathematics to model and optimise real
  world systems

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Is this OR?

• OR Operational Research, the science of better
• Using mathematics to model and optimise real
  world systems
• Other examples of OR
• Investigating strategies for treating
  tuberculosis and HIV in Africa
• Reducing waiting lists in the NHS
• Optimising the set up of a Formula 1 car
• Improving the efficiency of the Tube!

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Contents

• Background
• Solving the basic problem
• Making the problem more realistic
• Conclusion

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How to Get a Good Deal
Book early on an unpopular flight
Profit for easyJet in 2007 202 million
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Questions?
?