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?