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AGIFORS--RM Study Group New York City, March 2000 Lawrence R. Weatherford, PhD University of Wyoming Unconstraining Methods

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Outline of Presentation I.Introduction II.Review of Common Unconstraining Methods III.Comparison of Performance IV. Conclusion

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I. Introduction One of the major factors that affects forecast accuracy is the inability to observe the true (unconstrained) demand Improvements in forecast accuracy can translate into substantial revenue increases: Each 10% reduction in forecast error can be worth 2% to 4% in revenues on high demand flights (Lee, MIT Thesis)

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RM forecasting requires a complete system that performs all of the following steps: Collection of historical data Cleaning of data (including outlier editing) Unconstraining of closed observations Estimation of forecast model from historical data Generate forecasts for each future flight Evaluate accuracy of forecasts/provide feedback to users

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II. Review of Common Unconstraining Methods Going to look at the approach taken by 4 commonly used methods: A. Naive unconstraining (or detruncating) B. Pickup unconstraining C. Booking Curve unconstraining D. Projection unconstraining In unconstraining, we consider a class to be “closed” if at a booking period (reading day, DCP) the booking limit doesn’t allow for further bookings.

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A. Naive unconstraining--only use unclosed observations (much more common than you might assume!) For example: Observation #1-9101112 Unclosed 10-12 1 40118726 2 507CC 3 601012C 4 3097521 5 2556314 = (26+21+14)/3 = 20.33

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B. Pickup unconstraining--increment closed observations by the higher of 1) average of the unclosed observations or 2) the actual value for the booking periods that are closed. Observation #1-9101112 Unclosed 10-12 1 40118726 2 507C8C4 3 601012C6 4 3097521 5 2556314 Unclosed3315 Avg Unclosed8.255 For example, using previous data:

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Observation #2 Total for DCP’s 10-12: 7 + 8.25 + 5 = 20.25 Observation #3 Total for DCP’s 10-12: 10 + 12 + 6 = 28 = (26+20.25+28+21+14)/5 = 21.85 Note: just this simple method increases estimate of unconstrained demand from 20.33 to 21.85

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C. Booking Curve unconstraining--divide bookings-in-hand by long- run historical average ratio of bookings-in-hand to bookings at departure Using lots of historical data (not shown), suppose we determine that 75% of the bookings are received by DCP 10, and 85% by DCP 11. Observation #1-9101112 Unclosed 10-12 1 40118726 2 507CC 3 601012C 4 3097521 5 2556314

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Then for this example: Observation #2 Total for DCP’s 10-12: (50+ 7)/.75 - 50 = 26 Observation #3 Total for DCP’s 10-12: (60 + 10 + 12)/.85 - 60 = 36.5 = (26+26+36.5+21+14)/5 = 24.7

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D. Projection unconstraining (statistically known as the EM method)-- uses much more complicated statistics that deal with “censored” observations Basic idea is to iterate at guessing the mean and standard deviation of the pickup from DCP 10-12. First, you use the unclosed observations, then you find the conditional probabilities based on the constrained observations and re-estimate the , . This process continues until the values for , converge. Parameters: # of iterations, convergence limit critieria

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Of course, all of these methods are further complicated by the following: 1) a given leg may be considered “open” and yet be closed to some of the O&D’s flowing over it due to some form of network control (e.g., bid price) 2) a booking class may be considered “open” because it was “open” on the 2 reading days, but could have actually been “closed” in between.

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III. Comparison of Performance Intuitively, it makes sense that more statistically sound procedures like the “Projection” method should do a better job than the “Naïve” method at estimating the true unconstrained demand, but the question is how much better and is it worth the effort? Of course, one of the real problems in performing this analysis is that if one uses real airline data, we never know what the true unconstrained demand is and therefore are not able to accurately compare all 4 methods Leads us to use simulated data--randomly generated “true

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unconstrained demand” and also randomly generated “booking limits” that determine whether or not we observe the true unconstrained demand or some constrained value. Then, we can make an honest evaluation of how much better one method does than another and how close it came to the true unconstrained demand (because we secretly know what that is).

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A. Data Sets We’ll look at 5 different data sets (2 simulated data sets with 1000 observations each, 3 real data sets): 1. Simulated #1, unconstrained mean = 20, % unconstrained varies from 20% to 98% 2. Simulated #2, unconstrained mean = 4, % unconstrained varies from 20 to 98% 3. US Domestic, 14 fare classes, 6 departures, 90 days of data 4. European Continent, 10 fare classes, 14 departures,120 days 5. Carribean flight, 15 fare classes, 7 departures, 45 days

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B. Results from Set #1 Avg improvement of EM over Naïve ranges from 7% to 47%, with an average increase of 21% across the 5 scenarios

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C. Results from Set #2 Avg improvement of EM over Naïve ranges from 10% to 547%, with an average increase of 54% across the 5 scenarios

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D. Results from Set #3 Avg improvement of EM over Pickup ranges from 18% to 1000%, with an average increase of 200% across the 14 fare classes

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E. Results from Set #4 Avg improvement of EM over Pickup ranges from 10% to 200%, with an average increase of 36% across the 10 fare classes

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F. Results from Set #5 Avg improvement of EM over Pickup ranges from 33% to 2000%, with an average increase of 97% across the 15 fare classes

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G. Summary ·On average, using the EM method does much better than using either the Naïve or Pickup approaches (103% improvement in demand estimation across the 3 real data sets) · There is still the issue of what to do when all of your data for a given fare class is constrained, with all 0’s--EM can’t handle that

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Questions? IV. Conclusion The type of unconstrainer you’re using can make a BIG difference.

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