Principles of demand management Airline yield management Determining the booking limits. » A simple problem» Stochastic gradients for general problems

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1 Demand Management Principles of demand management Airline yield management Determining the booking limits» A simple problem» Stochastic gradients for general problems

2 Principles of demand management Issues:» Customers are not all the same» The highest paying customers will not fill up available capacity.» The lowest paying customers will not cover the full costs of operations.» Marginal costs may be much lower than average costs.

3 Principles of demand management From microeconomics: Consumers surplus Price A single price leaves a lot of money on the table. The airlines would like to capture some of this. P Revenue Q Quantity

4 Principles of demand management Price We would like to charge a range of prices to capture the diversity of price elasticities. Quantity

5 Principles of demand management Modes of control:» Primal Limit access to the system (now) Reservations (future)» Dual Pricing Service (waiting in line)» Informational Advertising Promotions» Combinations: Coupons - provide discounts (dual) with restricted services (primal) to customers who hold coupons (informational).

6 Principles of demand management Market differentiating characteristics used in practice:» Advance notice (pre-booking, reservations)» Length of stay, stay over Saturday, etc.» Willingness to accept nonrefundable tickets» Service Additional amenities (first class, business class) Pre-board privileges (frequent flier status)

7 Principles of demand management Applications of demand management» Airlines» Hotels» niversities (with rolling admissions)

8 Lecture outline - Demand Management Principles of demand management Airline yield management Determining the booking limits» A simple problem» Stochastic gradients for general problems

9 Airline yield management Overview» Old system: Airlines establish a fare (or two, such as coach and first class) Sell seats at the fare until all seats are sold or the flight departs. Because of no-shows, airlines may sell too many seats to increase load factors.» Issue: Business travelers are willing to pay much more, but want the flexibility of making plans at the last minute. Different customers want different products measured in terms of price and service.» Solution: Offer blocks of seats at increasingly higher prices. The lower demand for higher priced seats has the effect of making seats available for last-minute business travelers.

10 Airline yield management Booking limits» Blocks of seats are offered at different prices: $1700 $1425 $1150 $910 $850 $775 $525

11 Airline yield management Consider a two-class problem:

12 Airline yield management What price to charge? Quantity Aircraft capacity A high price gets the business travelers, but squeezes out the low-priced segment. But there are not enough business people to fill the plane. A lower price attracts the personal travelers, but squeezes out the business people. P Price

13 Airline yield management Different customer classes vary in terms of their booking process. Arrival rate Time

14 Airline yield management Demand management techniques» Informational Advertising services Discounts» Dual (pricing) Setting the price for seats Pricing differentiated services» Primal (booking limits) We have to limit the number of seats we sell at each fare.

15 Airline yield management Consider a two-class problem: Lost customers: Cheap seats Expensive seats How do we decide if we picked the right booking limit?

16 Airline yield management Challenge question:» What happens when the allotment of cheap seats is set too high? Too low?

17 Airline yield management Notation: Activity variables : J D f p Set of ( y) p.d.f. of demand for fare class. Decision variables : x Limit (upper bound) on number of reservations allowed for fare class fare classes. 1is the "highest" fare class.. Number of reservations made for fare class. Parameters : C Capacity of the aircraft (random) Demand for fare class Fare (price) for fare class ( p p 1 )

18 Airline yield management Obective: We would like to find the booking limits that solve : max 1... J E J p x subect to limits on how much we can book. We can think of as a single flight, and the expectation is a summation over many flights. We cannot maximize the profits for a single flight, but we would like to maximize profits over many flights.

19 Airline yield management Booking limits:» Separable booking limits- Booking limit limits reservations for fare class alone:» Nested booking limits - Booking limit limits fare class and all lower fare classes. aircraft capacity C x J C x m m 1 1

20 Airline yield management A heuristic derivation for booking limits: Start with a separable (non-nested) policy: Let: P ( x) Prob[ D x] y0 ( y) dy This means that if we set a booking limit at we could have x f exceeded it is 1 P ( ). Now let:, the probability that EMSR ( ) The expected marginal seat revenue for fare class The value an additional seat allocated for fare class. p 1 P ( ).

21 Airline yield management ) ( P 1- ) ( Prob ) F(, E : means that This Otherwise 0 ) ( ) F(, Now consider the derivative : } ), ( min{ ), ( where : subect to : ), ( max ) ( We start by expressing the obective function : J p D p D p D p F C F E F

22 Airline yield management We can formulate the problem as an unconstrained problem by relaxing the constraint: L F (, ) max EF(, ) C J L F (, ) At optimality, we expect to find 0. This means that: Or: p 1-P ( ) 0 p 1-P ( ) for each. What is the economic interpretation of this equation?

23 Airline yield management The EMSR algorithm: For a given value of, find the booking limit : P( ) 1- Given, we can find Now we have to pick so that p p p ( ) C ( ) so that this is satisfied. Since is a scalar, ust have to try different values.

24 Lecture outline - Demand Management Principles of demand management Airline yield management Determining the booking limits» A simple problem» Stochastic gradients for general problems

25 Stochastic gradient algorithms Nested booking limits: Fare Class» A sample booking process: Demand D() Booking limit Total reservations Fare class reservations

26 Stochastic gradient algorithms Nested booking limits: Start by defining the cumulative demand: Dˆ, Total reservations made of customer classes and "lower." 1 min D Dˆ, 1 Note: Dˆ (, ) Dˆ (, ) is the number of fare class reservations that have been accepted. What are we implicitly assuming about the callin process? Time

27 Stochastic gradient algorithms J D D p F F E F ), ( ˆ ), ( ˆ ), ( where : ), ( max ) ( to solve : again want we problem, For this 1 But how do we find the gradient?

28 Stochastic gradient algorithms Consider a two class system... We start with C. 1 ˆ ˆ ˆ F (, ) p D(, ) D(, ) p D(, ) p min D Dˆ, Dˆ (, ) p Dˆ (, ) Let's now assume that with our low "vacationers" fare p, D 2 2. In this case, we have: F (, ) p min D, p p min D, p p min D, p

29 Stochastic gradient algorithms Taking the derivative with respect to gives: F (, ) {X} 2 pi 1 D ( ) (Remember: I 1 if event X is true, and 0 otherwise. E I is then the probability that X is true.) Taking expectations gives: F (, ) E p1prob[ D1 1 2] p2 2 Recall that C. Setting the derivative equal to zero, we get: 1 2 pprob[ D C ] p p {X}

30 Stochastic gradient algorithms When the problem gets more complicated, we do not get such neat results. Instead, we can resort to our stochastic gradient algorithms. max E F(, ) max E p D (, ) D (, ) J ˆ ˆ 1 Assume we have an initial vector using the stochastic gradient iteration: k 1 k k k k F(, ) 0. We can find the best value of

31 Stochastic gradient algorithms Consider the change in a booking limit:» Assume we change from 4 =40 to 4 =41. Fare Demand Booking Total Fare class Class D() limit reservations reservations (41) 40(41) 25(26) (80) 80(80) 40(39) Change (120) 120(120) 40(40) (150) 147(147) 27(27) Change in profits is p 4 -p 3. What if 3 was not binding?

32 Stochastic gradient algorithms Consider the change in a booking limit:» What if 3 is not binding, but 2 is? Fare Demand Booking Total Fare class Class D() limit reservations reservations (41) 40(41) 25(26) Change (80) 74(75) 34(34) (120) 120(120) 46(45) (150) 147(147) 27(27) Now the change in profits is p 4 -p 2.

33 Stochastic gradient algorithms Consider the change in a booking limit:» What if nothing else is binding? Fare Demand Booking Total Fare class Class D() limit reservations reservations (41) 40(41) 25(26) Change (80) 74(75) 34(34) (120) 114(115) 40(40) (150) 141(142) 27(27) Now the change in profits is p 4.

34 Stochastic gradient algorithms Consider the change in a booking limit:» What if 4 is not binding, but 2 is? Fare Demand Booking Total Fare class Class D() limit reservations reservations Change (41) 35(35) 20(20) (80) 69(69) 34(34) (120) 120(120) 51(51) (150) 147(147) 27(27) Now there is no change in profits.

35 Stochastic gradient algorithms So we have a formula for the gradient when increasing i: Let: X pi If booking limit i is binding. 0 Otherwise. X If iis binding, and is binding, and i is the first class p whose booking limit is binding. 0 If no other booking limit for a higher fare class is binding. We can now express the gradient using: F (, ) i X X Since X and X depend on the sample realization of demands, they are random variables..

36 Booking limits Seats Booking limit Fare 5 Fare 4 Fare 3 Fare 2 Fare Iteration Iteration

37 Higher demand Booking for limits fare class Iteration Iteration Fare 5 Fare 4 Fare 3 Fare 2 Fare 1 Booking Seats limit

38 Lecture outline - Demand Management Issues in demand management

39 Extensions The no-show problem» How can we manage this? The fairness issue» Paying different amounts for the same seat» In what way are seats different? Managing complexity» 1000 markets» 30,000 fares!