In airline revenue management, an accurate prediction of cancellations is crucial, since a significant number of bookings are cancelled before departure. Accurate estimation of cancellation behavior is essential for airlines, so that they can allow more reservations on a flight than there is physical capacity (“overbooking”), which is a significant source of revenue. Cancellation probabilities depend in a complex manner on several flight-related and passenger-related attributes. A proportional hazard model is applied to predict the “hazard rates”, i.e. the conditional risks of a reservation being cancelled. These are used to forecast the expected number of cancellations depending on bookings on hand and forecasted future demand. We enhanced the standard maximum-likelihood estimator in order to obtain practicable processing time and memory consumption. The new method provides stable predictions and improves accuracy significantly compared to a time series approach.

1. Predicting cancellation probabilities of airline bookings using a discrete hazard model
Dr. Heiko Schmitz, Dr. Jan Ulbricht, Stephan Würll,
Lufthansa Systems, Berlin, Germany
Motivation
► Large fraction of airline bookings are
cancelled before departure
► Knowing the number of empty seats is
crucial for revenue management
► Overbooking is an important measure
to increase revenue
► Cancellation probability depends on
flight-related and passenger-related
attributes
► Predictions are required in a dynamic
environment (schedule changes, etc.)
► Predictions shall be stable and robust
► Regression models are suitable for
this task
The Model
Goal
Predict the probability that a booking
done at time t will be cancelled at a
later time t‘.
Assumptions and Preconditions
► Sufficient number of observations from
the past are available
► Discrete time
(Data collection points, DCP)
► Model is estimated per OD
(Origin/Destination)
► Sufficient variation in observation data
► GLM with logistic link function
𝐹 𝜂 =
1
1 + exp −𝜂
∈ 0,1 ,
𝜂 = 𝛾0 + 𝛽𝑗 ∙ 𝑥𝑗
► Log-likelihood function
ln𝐿 𝛽 = 𝑦𝑖𝑗ln𝜆 𝑗|𝑥𝑖 + 1 − 𝑦𝑖𝑗 ln 1 − 𝜆 𝑗|𝑥𝑖
𝑡 𝑖
𝑗=1
𝑛
𝑖=1
► Maximum likelihood estimator
𝛽 = argmax ln 𝐿 𝛽
► Fisher scoring
𝛽 𝑘+1 = 𝑍T
∙ 𝑊𝑘 ∙ 𝑍
−1
∙ 𝑍T
∙ 𝑊𝑘 ∙ 𝑦 𝑘 with
weight matrix 𝑊𝑘 = 𝑊 𝛽 𝑘 ,
grand design matrix 𝑍 = 𝑍 𝑥 ,
𝑦 𝑘 = 𝑦 𝑦, 𝛽 𝑘 .
• 𝑑𝑑𝑑𝑑𝑑𝑃(𝑇dddd = 𝑡) =
𝜆(𝑡) 1 − 𝜆(𝑖) , 𝑡 = 0,1, … , 𝑘, 𝑘 + 1
𝑡−1
𝑖=0
Practicalities
Regularization
► Tikhonov regularization
𝑀 → 𝑀 − 𝜀I
avoids numerical problems in inversion
Coding of categorical covariables (e.g. DoW)
► Use effect coding
► No reference category necessary
► Easy extrapolation to unobserved values
New covariate value (e.g. flight on new DoW
can be predicted as “average” of all DoWs
Convergence
► Criterion
ln𝐿 𝛽 𝑘+1 −ln𝐿 𝛽 𝑘
ln𝐿 𝛽 𝑘+1 +0.1
≤ 𝜀
► Usually very fast (10 to 20 iterations)
Hazard Rates and Lifetime
► Discrete hazard rates (of cancellation)
𝜆 𝑡 ≔ 𝑃 𝑇 = 𝑡|𝑇 ≥ 𝑡
► Probability of lifetime (duration) 𝑡
► Lifetime if booking has survived
until today (=r )
𝑃(𝑇 = 𝑡|𝑇 ≥ 𝑟) = 𝜆(𝑡) ∙ 1 − 𝜆 𝑖
𝑡−1
𝑖=𝑟
𝑃(𝑇 = 𝑡) = 𝜆(𝑡) ∙ 1 − 𝜆 𝑖
𝑡−1
𝑖=0
Cancellation
(”Hazard“) at 𝑡
“Survival“
𝑖 = 0, … , 𝑡 − 1
DepartureTodayBooking Cancellation
Timeline
T1 T2
Covariates
Itinerary-related
► Departure DoW
► Departure Time
► Travel Time
► Number of legs
► Number of flight
segments
Customer-related
► Customer segment
► Point of Sale
► Booking DCP
Outlook
► Incorporate tariff information
► Use smooth components
(Tensor B-Splines)
Updating Coefficients
► How do we learn from new
observations?
► Estimate new coefficients 𝛽new
► Mix with old coefficients 𝛽old
𝛽 𝛼 = 1 − 𝛼 𝛽old + 𝛼 𝛽new
► Maximize likelihood on validation data
𝛼∗
= argmax ln𝐿 𝛽 𝛼
today
time
estimation validation
validationestimation
previous update
maximize ln𝐿estimate 𝛽new
Results and Summary
Comparison with old time series method
►Robust and reliable prediction of
cancellation rates
►Modest resource consumption
(memory and CPU)
►Works well in dynamical environment
►Easy to interpret parameters
►Revenue effect: some M$
Market PMAD old PMAD new Diff pp
1 0.372 0.350 2.2
2 0.372 0.349 2.3
3 0.622 0.549 7.3
4 0.654 0.555 9.8
5 0.422 0.389 3.3
Mean 0.530 0.471 5.9
Optimization of Fisher Scoring
Textbook approach
► Each lifetime value is a covariate
► One observation (lifetime 𝑡𝑖 ) is duplicated
into
►𝑡𝑖 − 1 observations of surviving booking
►1 observation of cancellation/survival at 𝑡𝑖
► Design matrix 𝑍 ∈ ℝ 𝑡 𝑖×(𝑘+𝑝)
𝑍 =
𝐈 𝑡1
𝟎 𝑡1×(𝑘−𝑡1)
𝐱1
T
⋮
𝐱1
T
⋮ ⋮ ⋮
𝐈 𝒕 𝒏
𝟎 𝑡 𝑛×(𝑘−𝑡 𝑛)
𝐱 𝑛
T
⋮
𝐱 𝑛
T
= 𝑍1 𝑍2
► Weight matrix 𝑊 ∈ ℝ 𝑡 𝑖× 𝑡 𝑖
𝑊 = diag 𝑤1, … , 𝑤 𝑛 ∈ ℝ 𝑡 𝑖× 𝑡 𝑖
Matrix operations: Typical numbers
► 𝑘 = 100 lifetimes, 𝑝 = 20 covariates,
mean lifetime 𝑡 = 50, 𝑛~50,000 observations
► 𝑍 ∈ ℝ2,500,000×120
, 𝑊 ∈ ℝ2,500,000×2,500,000
► Huge matrices with very regular structure
Use block structure to reduce dimensions
► Matrix algebra yields
𝑍T
∙ 𝑊 ∙ 𝑍 =
𝐷 𝐵
𝐵T
𝐴
;
partitioned with 𝐷 ∈ ℝ 𝑘×𝑘
diagonal,
𝐴 ∈ ℝ 𝑝×𝑝
, 𝐵 ∈ ℝ 𝑘×𝑝
► Sub-matrices are easy to calculate
► Matrices are manageable
► Inversion is easy
𝑍T 𝑊𝑍
−1
= 𝐷−1 0
0 0
+ −𝐷−1
𝐵
I
𝐴 − 𝐵T 𝐷−1 𝐵 −1
−𝐵T 𝐷−1 I
► Similar reduction of dimensions is possible for
𝑍T
∙ 𝑊 ∙ 𝑦 𝑘
► No loss of accuracy (algebraic transformation)
References
■ Dan C. Iliescu,
Customer Based Time-to-Event Models for
Cancellation Behavior: A Revenue Management
Integrated Approach, Proquest, 2011
Contact
heiko.schmitz@LHsystems.com
Regularize!