1. © Michael O. Ball Competitive Analysis of Online Booking: Dynamic Policies Michael Ball, Huina Gao, Itir Karaesman R. H. Smith School of Business University of Maryland and Maurice Queyranne Sauder School of Business University of British Columbia

2. © Michael O. Ball Motivation: Assemble-to-Order Products Some particular features: Multiple capacity constraints Time dimension – production capacity and components replenish over time Time dimension – product differentiation based on delivery time No low before high components products production capacity Very simple case: 1 key component 2 products

3. © Michael O. Ball Competitive Analysis Algorithmalgorithminput Evil DesignerstreamAdversary i 1, i 2, i 3, … Competitive Ratio = Min input streams {(alg performance)/(best performance)} “Traditional” revenue management analysis has assumed: Demand can be forecast reasonably well Risk neutrality Are these valid??

4. © Michael O. Ball Sample Result Flight has 95 available seats, three fare classes: $1,000, $750, $500 Policy that guarantees at least 63% of the max possible revenue: –Protect 15 high fare seats –Protect 35 seats for two higher fare classes (i.e. sell at most 60 lowest fare seats)

5. © Michael O. Ball Two-Fare Analysis n = number of seats f1 = higher fare; f2 = lower fare. r = f2/f1 = discount ratio. Key quantity: b(r) = 1 / (2 – r) Policy intuition: Always best to accept any high fare (H) that comes along  Adversary will start off with low fare requests (L): Question – how many to accept before only H’s are accepted??

6. © Michael O. Ball Basic Trade Off L L L L L L L L L L L L L H H H H H H H H H L L L L L L L L Stop accepting L’s available seats Accept too few L’s  adversary will only send additional L’s Accept too many L’s  adversary will only send additional H’s Note: must accept first order, o.w. adversary will stop after submitting one order with algorithm performance = 0.

7. © Michael O. Ball Best Two-Fare Policy L L L L L L L L L L L L L L L L L L L Proposal: protect (1 – b(r)) n = (1 – 1/(2 – r)) n … assume for the moment that b(r) n is integer.. L L L L L L L L L L L L L H H H H H All H’s after stopping  Performance = [f2 b(r) n + f1 (1 – b(r)) n] / (f1 n) = b(r) All L’s after stopping  Performance = (f2 b(r) n ) / (f2 n) = b(r)

8. © Michael O. Ball More General Cases q = total order quantity accepted f = price n f 4 = f min f3f3 f2f2 f 1 =f max protection levels:  (f 3 )  (f 2 )  (f 1 )

9. © Michael O. Ball m Fare Classes Define:  = m -  {i=2,m} f i / f i-1 Theorem: For the continuous m-fare problem, no booking policy, deterministic or random, has a competitive ratio larger than 1 / . Define:  i = (n /  ) (i -  {j=1,i} f j+1 / f j ) Theorem: For the continuous m-fare problem, the protection level policy using protection levels  i achieves a competitive ratio of at least 1 / . e.g. f 1 = 1000, f 2 = 750, f 3 = 500, 1 /   63 %

10. © Michael O. Ball Approach to alternate type of policy q = total order quantity accepted f = price n f 4 = f min f3f3 f2f2 f 1 =f max protection levels:  (f 3 )  (f 2 )  (f 1 ) Optimal adversary strategy

11. © Michael O. Ball Order Quantity Control Fcn q = total order quantity accepted f = price n P’(q)

12. © Michael O. Ball Order Quantity Control Policy q = total order quantity accepted f = price n Order quantity control policy: after q orders have been accepted, accept next order if price of order > P’(q)

13. © Michael O. Ball Order Quantity Control Policy q = total order quantity accepted f = price n Order quantity control policy: after q orders have been accepted, accept next order if price of order > P’(q) Accept!! Reject!! or

14. © Michael O. Ball Numerical Experiments Compare against another approach that does not require a-priori demand information: Van Ryzin & McGill, “Revenue Management without Forecasting or Optimization: an Adaptive Algorithm for Determining Airline Seat Protection Levels” Two categories of demand generation: –Stationary – demand in each fare class is normally distributed –Non-stationary – for a given day/flight, demand is normally distributed but mean demand jumps between one of three mean pairs with a certain probability: 48 65 30 65 48 65 30 65.9.05.1.05.1.9

15. © Michael O. Ball Comparison: capacity: 100 High: fare -- $2000, mean demand -- 15 Low: fare -- $1000, mean demand -- 81 Van Ryzin & McGill Robust Low before High Random Ave diff: -$7210 -$10140

16. © Michael O. Ball Comparison: capacity: 100 High: fare -- $2000, mean demand -- 15 Low: fare -- $1500, mean demand -- 81 Van Ryzin & McGill Robust Low before High Random Ave diff: -$1760 -$6130

17. © Michael O. Ball Comparison: capacity: 100 High: fare -- $2000, mean demand -- 48 Low: fare -- $1000, mean demand -- 48 Van Ryzin & McGill Robust Low before High Random Ave diff: +$ 2750 +$6550

18. © Michael O. Ball Comparison of protection levels Van Ryzin & McGill Robust Low before High Random

19. © Michael O. Ball Comparison: capacity: 100 High: fare -- $2000, mean demand -- 65 Low: fare -- $1500, mean demand -- 31 Van Ryzin & McGill Robust Low before High Random Ave diff: +$1620 +$8130

20. © Michael O. Ball Comparison: capacity: 100 High: fare -- $2000, mean demand -- 81 Low: fare -- $1000, mean demand -- 15 Van Ryzin & McGill Robust Low before High Random Ave diff: -$650 +$7320

21. © Michael O. Ball Comparison: capacity: 100 Non-stationary demand: High: fare -- $2000, mean demand -- 65 48 30 Low: fare -- $1000, mean demand -- 30 48 65 Van Ryzin & McGill Robust Low before High Random Ave diff: +$4170 +$8080

22. © Michael O. Ball Comparison of protection levels Van Ryzin & McGill Robust Low before High Random

23. © Michael O. Ball Comparison: capacity: 100 Non-stationary demand: High: fare -- $2000, mean demand -- 65 48 30 Low: fare -- $1500, mean demand -- 30 48 65 Van Ryzin & McGill Robust Low before High Random Ave diff: +$1150 +$8740

24. © Michael O. Ball Observations Random vs low-before-high makes a difference – for random, lower protection levels seem better. Robust policy works well across range of demand scenarios – demand distribution most useful in cases of unbalanced demand.

25. © Michael O. Ball Dynamic Policy Motivation: Taking advantage of an inferior adversary nb(r)n Optimal Adversary Policies: Start low After threshold: Stay low Or jump to high and stay high

26. © Michael O. Ball Dynamic Policy Motivation: Taking advantage of an inferior adversary nb(r)n

27. © Michael O. Ball Dynamic Policy Motivation: Taking advantage of an inferior adversary nb(r)nn’n’b(r) Having 2 (or more) high’s “in the bank” leads to a problem on a smaller n  threshold can be smaller & overall guarantee is improved.

28. © Michael O. Ball Dynamically Revising Threshold orig threshold: n b(r) = n / (2-r) let: h’ = # high fare request accepted so far  = h’/n  = (  f 1 + (1 -  ) f 2 ) / f 2 revised threshold: n (1 + (1 – r)  /r ) / (1 +  (1 – r) )

29. © Michael O. Ball Dynamic Policy: numerical test h’ ” dynamic guarantee ’ static guarantee 165.78.6766.67.67 562.30.6966.67.67 1058.60.7166.67.67 2050.00.7566.67.67 3042.42.7966.67.67 4035.29.8266.67.67 5028.57.8666.67.67 n = 100; r =.5

30. © Michael O. Ball Final Thoughts Many implicit and explicit assumptions in airline revenue management problems – relaxing these can lead to novel problems. Online algorithm approach shows significant promise: –No risk neutrality assumption. –Demand information not required. –Policies are practical and seem to work well across range of demand distributions.