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I’m not paying that! Mathematical models for setting air fares. Contents. Background History What’s the problem? Solving the basic problem Making the model more realistic Conclusion Finding out more. Air Travel in the Good Old Days.

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contents
Contents
  • Background
    • History
    • What’s the problem?
  • Solving the basic problem
  • Making the model more realistic
  • Conclusion
  • Finding out more
air travel in the good old days
Air Travel in the Good Old Days

Only the privileged few – 6000 passengers in the USA in 1926

and now
And now …

Anyone can go – easyJet carried 30.5 million passengers in 2005

what s the problem
What’s 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
make them pay
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

£30

I fancy a holiday

I’ve got a meeting on 2nd June

clever pricing
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!
contents8
Contents
  • Background
  • Solving the basic problem
    • It’s your turn
    • Linear programming
  • Making the model more realistic
  • Conclusion
  • Finding out more
it s your turn
It’s 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
slide10

3 volunteers needed

No hard sums!

slide11

A

B

C

0 Economy

50 Economy

100 Economy

£12,500

£5,000

£10,000

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

using equations
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
revenue

Economy revenue

Business revenue

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

* Maximise *

constraints
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
more constraints
More Constraints
  • Constraint 3: we cannot sell more seats than people want
  • Constraint 4:the number of seats sold is an integer
in numbers

Economy revenue

Business revenue

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

* Maximise *

and constraints
And Constraints …
  • Constraint 1: aeroplane has limited capacity, C
  • Constraint 2: sell positive numbers of seats
  • Constraint 3: can’t sell more seats than demand
linear programming
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
solution
Solution
  • Fill as many seats as possible with business class passengers
  • Fill up the remaining seats with economy passengers

xb = db, xe = C – xbfor db < C

xb = C for db > C

50 economy, 50 business (Option B)

but isn t this easy
But isn’t 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
contents23
Contents
  • Background
  • Solving the basic problem
  • Making the model more realistic
    • Modelling customers
    • Optimising the price
  • Conclusion
  • Finding out more
making the model more realistic
Making the model more realistic:
  • We don’t 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
demand function
Demand Function

f(t)

e.g.

t

Departure

reserve prices
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
reserve prices27
Reserve Prices

I’d like to buy a ticket to Madrid on 2nd June

£30

March 2008

£100

I’ve got a meeting in Madrid on 2nd June – I’d better buy a ticket

reserve prices28
Reserve Prices

All my friends have booked – I need this ticket

£70

May 2008

£200

The meeting’s only a week away – I’d better buy a ticket

revenue29

Number who

consider buying

Proportion who buy if price is less

than or equal to y(t)

Price charged

at timet

Revenue

* Maximise *

Revenue =

maximising revenue
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
optimal price
Optimal Price

Departure

contents32
Contents
  • Background
  • Solving the basic problem
  • Making the problem more realistic
  • Conclusion
    • Why just aeroplanes?
  • Finding out more
why just aeroplanes
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
is this or
Is this OR?

Yes!

  • OR = Operational Research, the science of better
    • Using mathematics to model and optimise real world systems
is this or35
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!
contents36
Contents
  • Background
  • Solving the basic problem
  • Making the problem more realistic
  • Conclusion
how to get a good deal
How to Get a Good Deal

Book early on an unpopular flight

Profit for easyJet in 2007 = £202 million