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Im not paying that! Mathematical models for setting air fares

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Contents Background –History –Whats the problem? Solving the basic problem Making the model more realistic Conclusion Finding out more

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

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

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Whats 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

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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 Ive got a meeting on 2 nd June £100

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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!

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Contents Background Solving the basic problem –Its your turn –Linear programming Making the model more realistic Conclusion Finding out more

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Its 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

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3 volunteers needed No hard sums!

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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

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Using equations Assume our airline can charge one of two prices –HIGH price (business class) p b –LOW price (economy class) p e Assume demand is deterministic –We can predict exactly what the demand is for business class d b and economy class d e How many seats should we allocate to economy class to maximise revenue? Write the problem as a set of linear equations

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Revenue We allow x e people to buy economy tickets and x b to buy business class tickets Therefore, revenue on the flight is Business revenue * Maximise * Economy revenue

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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

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

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In Numbers … We allow x e people to buy economy tickets and x b to buy business class tickets Therefore, revenue on the flight is Economy revenue Business revenue * Maximise *

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Constraint 1: aeroplane has limited capacity, C Constraint 2: sell positive numbers of seats Constraint 3: cant sell more seats than demand And Constraints …

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Linear Programming We call x e and x b 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

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Linear Programming Graph

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Solution Fill as many seats as possible with business class passengers Fill up the remaining seats with economy passengers x b = d b, x e = C – x b for d b < C x b = C for d b > C 50 economy, 50 business (Option B)

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But isnt 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

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

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Making the model more realistic: We dont 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

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

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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

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Reserve Prices £3 0 Id like to buy a ticket to Madrid on 2 nd June Ive got a meeting in Madrid on 2 nd June – Id better buy a ticket £10 0 March 2008

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Reserve Prices £70 All my friends have booked – I need this ticket The meetings only a week away – Id better buy a ticket £20 0 May 2008

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

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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

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Optimal Price Departure

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

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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

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

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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!

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Contents Background Solving the basic problem Making the problem more realistic Conclusion

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

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Questions?

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