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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! A B C 0 Economy 50 Economy 100 Economy £10,000 £12,500 £5,000 Allocate 0 50economy 100 economy economy Sell 0 50economy 100 economy economy atatat £50 £50 £50 ==£0 =£2,500 £5,000 Sell 50 0 business businessatat£200 £200==£0 £10,000 Total = £10,000 £12,500 £5,000 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 R pe xe pb xb Economy revenue * Maximise * Business revenue Constraints • Constraint 1: the aeroplane has a limited capacity, C xe xb 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 xe , x b 0 More Constraints • Constraint 3: we cannot sell more seats than people want xe de , xb db • 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 R 50 xe 200 xb Economy revenue * Maximise * Business revenue And Constraints … • Constraint 1: aeroplane has limited capacity, C xe xb 100 • Constraint 2: sell positive numbers of seats xe , x b 0 • Constraint 3: can’t sell more seats than demand xe 150, xb 50 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 Function f(t) Departure e.g. f (t ) (gt d ) exp( ht ) t 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 £3 0 I’d like to buy a ticket to Madrid on 2nd June March 2008 I’ve got a meeting in Madrid on 2nd June – I’d better buy a ticket £10 0 Reserve Prices £70 All my friends have booked – I need this ticket May 2008 The meeting’s only a week away – I’d better buy a ticket £20 0 Revenue * Maximise * a Revenue = y (t )f (t )p(t, y (t ))dt b Price charged at time t Number who consider buying Proportion who buy if price is less than or equal to y(t) 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 180 160 140 Price (y(t)) 120 100 80 60 40 20 0 0 Departure 5 10 Days Before Departure (t) 15 20 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 Questions?