AGIFORSRM Study Group

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Unconstraining Methods

• AGIFORS--RM Study Group
• New York City, March 2000
• Lawrence R. Weatherford, PhDUniversity of Wyoming

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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 doesnt allow for further
bookings.
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A. Naive unconstraining--only use unclosed
observations (much more common than you might
assume!) For example
lt---Bookings by DCP---gt Observation
1-9 10 11 12 Unclosed 10-12 1
40 11 8 7 26 2 50 7 C C 3
60 10 12 C 4 30 9 7 5 21 5
25 5 6 3 14 ? (262114)/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
1-9 10 11 12 Unclosed 10-12 1
40 11 8 7 26 2 50 7 C8 C4 3
60 10 12 C6 4 30 9 7 5 21 5
25 5 6 3 14 Unclosed 33 15 Avg
Unclosed 8.25 5 For example, using
previous data
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Observation 2 Total for DCPs 10-12 7 8.25
5 20.25 Observation 3 Total for DCPs 10-12
10 12 6 28 ? (2620.25282114)/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. lt---Bookings by
DCP---gt Observation 1-9 10 11 12 Unclosed
10-12 1 40 11 8 7 26 2 50 7 C C 3
60 10 12 C 4 30 9 7 5 21 5 25 5 6 3 14
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Then for this example Observation 2 Total for
DCPs 10-12 (50 7)/.75 - 50 26 Observation 3
Total for DCPs 10-12 (60 10 12)/.85 - 60
36.5 ? (262636.52114)/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 ODs 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 Well 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 0s--EM cant handle that
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IV. Conclusion The type of unconstrainer youre
using can make a BIG difference.
Questions?