Smart Fares: What if we sold transit fares like cell phone minutes?
Im not paying that! Mathematical models for setting air fares.
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Transcript of Im not paying that! Mathematical models for setting air fares.
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!
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
A
0 Economy
B
50 Economy
C
100 Economy
£10,000 £12,500 £5,000
Allocate 0 economy
Sell 0 economy at £50 = £0
Sell 50 business at £200 = £10,000
Total = £10,000
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
bbee xpxpR
Business revenue
* Maximise *
Economy revenue
• 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
Constraints
Cxx be
0, be xx
More Constraints• Constraint 3: we cannot sell more seats than people
want
bbee dxdx ,
• 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
be xxR 20050
Economy revenue Business revenue
* Maximise *
• Constraint 1: aeroplane has limited capacity, C
• Constraint 2: sell positive numbers of seats
• Constraint 3: can’t sell more seats than demand
And Constraints …
100 be xx
0, be xx
50 ,150 be xx
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 Functionf(t)
tDeparture
)exp()()( htdgttf e.g.
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
£30
I’d like to buy a ticket to Madrid on
2nd June
I’ve got a meeting in Madrid on 2nd
June – I’d better buy a ticket
£100
March 2008
Reserve Prices
£70All my friends have
booked – I need this ticket
The meeting’s only a week away – I’d better buy a ticket
£200
May 2008
Revenue
a
b
dttytptfty ))(,()()(
Proportion who buy if price is less
than or equal to y(t)
Number whoconsider buying
Price chargedat time t
Revenue =
* Maximise *
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
0
20
40
60
80
100
120
140
160
180
0 5 10 15 20
Days Before Departure (t)
Pri
ce (
y(t)
)
Departure
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
Profit for e
asyJet in 2007 = £202 m
illion
Questions?