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I’m not paying that! Mathematical models for setting air fares

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## I’m not paying that! Mathematical models for setting air fares

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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

£100

£30

I fancy a holiday

I’ve got a meeting on 2nd June

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

No hard sums!

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

- 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

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

- 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

- Constraint 3: we cannot sell more seats than people want

- Constraint 4:the number of seats sold is an integer

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 …

- 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

- 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

- 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?

- 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

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

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 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

consider buying

Proportion who buy if price is less

than or equal to y(t)

Price charged

at timet

Revenue* Maximise *

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

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?

Yes!

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

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

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