Download reserve price

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
Only the privileged few – 6000 passengers in the USA in 1926
And now …
Anyone can go – easyJet carried 30.5 million passengers in 2005
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!
• 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
£30
I fancy a holiday
I’ve got a meeting
on 2nd June
£100
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!
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!
• 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
3 volunteers needed
No hard sums!
A
B
C
0 Economy
50 Economy
100 Economy
£10,000
£12,500
£5,000
Allocate 0
50economy
100
economy
economy
Sell 0
50economy
100
economy
economy
atatat
£50
£50
£50
==£0
=£2,500
£5,000
Sell 50
0 business
businessatat£200
£200==£0
£10,000
Total = £10,000
£12,500
£5,000
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
• We allow xe people to buy economy tickets and xb to
buy business class tickets
• Therefore, revenue on the flight is
R  pe xe  pb xb
Economy revenue
* Maximise *
Business revenue
Constraints
• Constraint 1: the aeroplane has a limited capacity, C
xe  xb  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
xe , x b  0
More Constraints
• Constraint 3: we cannot sell more seats than people want
xe  de , xb  db
• Constraint 4: the number of seats sold is an integer
In Numbers …
• We allow xe people to buy economy tickets and xb to
buy business class tickets
• Therefore, revenue on the flight is
R  50 xe  200 xb
Economy revenue
* Maximise *
Business revenue
And Constraints …
• Constraint 1: aeroplane has limited capacity, C
xe  xb  100
• Constraint 2: sell positive numbers of seats
xe , x b  0
• Constraint 3: can’t sell more seats than demand
xe  150, xb  50
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
Linear Programming Graph
Linear Programming Graph
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 < C
xb = C for db > C
50 economy, 50 business (Option B)
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
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:
• 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
f(t)
Departure
e.g. f (t )  (gt  d ) exp( ht )
t
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 Prices
£3
0
I’d like to buy a ticket to
Madrid on 2nd June
March 2008
I’ve got a meeting in
Madrid on 2nd June –
I’d better buy a ticket
£10
0
Reserve Prices
£70
All my friends have
booked – I need this
ticket
May 2008
The meeting’s only a
week away – I’d better
buy a ticket
£20
0
Revenue
* Maximise *
a
Revenue =
 y (t )f (t )p(t, y (t ))dt
b
Price charged
at time t
Number who
consider buying
Proportion who buy if
price is less
than or equal to y(t)
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
180
160
140
Price (y(t))
120
100
80
60
40
20
0
0
Departure
5
10
Days Before Departure (t)
15
20
Contents
• Background
• Solving the basic problem
• Making the problem more realistic
• Conclusion
– Why just aeroplanes?
• Finding out more
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?
• OR = Operational Research, the science of better
– Using mathematics to model and optimise real world
systems
Yes!
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!
Contents
• Background
• Solving the basic problem
• Making the problem more realistic
• Conclusion
How to Get a Good Deal
Book early on an
unpopular flight
Questions?