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

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

2. Contents • Background • History • What’s the problem? • Solving the basic problem • Making the model more realistic • Conclusion • Finding out more

3. Air Travel in the Good Old Days Only the privileged few – 6000 passengers in the USA in 1926

4. And now … Anyone can go – easyJet carried 30.5 million passengers in 2005

5. 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

6. 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

7. 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!

8. Contents • Background • Solving the basic problem • It’s your turn • Linear programming • Making the model more realistic • Conclusion • Finding out more

9. 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

10. 3 volunteers needed No hard sums!

11. 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

12. 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

13. 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 *

14. 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

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

16. 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 *

17. 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

18. 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

19. Linear Programming Graph

20. Linear Programming Graph

21. 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)

22. 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

23. Contents • Background • Solving the basic problem • Making the model more realistic • Modelling customers • Optimising the price • Conclusion • Finding out more

24. 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

25. Demand Function f(t) e.g. t Departure

26. 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

27. 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

28. 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

29. Number who consider buying Proportion who buy if price is less than or equal to y(t) Price charged at timet Revenue * Maximise * Revenue =

30. 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

31. Optimal Price Departure

32. Contents • Background • Solving the basic problem • Making the problem more realistic • Conclusion • Why just aeroplanes? • Finding out more

33. 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

34. Is this OR? Yes! • OR = Operational Research, the science of better • Using mathematics to model and optimise real world systems

35. 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!

36. Contents • Background • Solving the basic problem • Making the problem more realistic • Conclusion

37. How to Get a Good Deal Book early on an unpopular flight Profit for easyJet in 2007 = £202 million

38. Questions? ?