Benchmarking filter-based demand estimates for airline revenue management

Abstract

In recent years, revenue management research developed increasingly complex demand forecasts to model customer choice. While the resulting systems should easily outperform their predecessors, it appears difficult to achieve substantial improvement in practice. At the same time, interest in robust revenue maximization is growing. From this arises the challenge of creating versatile and computationally efficient approaches to estimate demand and quantify demand uncertainty. Motivated by this challenge, this paper introduces and benchmarks two filter-based demand estimators: the unscented Kalman filter and the particle filter. It documents a computational study, which is set in the airline industry and compares the estimators’ efficiency to that of sequential estimation and maximum-likelihood estimation. We quantify estimator efficiency through the posterior Cramér–Rao bound and compare revenue performance to the revenue opportunity. Both indicate that unscented Kalman filter and maximum-likelihood estimation outperform the alternatives. In addition, the Kalman filter requires comparatively little computational effort to update and quantifies demand uncertainty.

Full text

RESEARCH PAPER
Benchmarking filter-based demand estimates for airline
revenue management
Philipp Bartke
1
•Natalia Kliewer
2
•Catherine Cleophas
3
Received: 16 January 2016 / Accepted: 12 May 2017 / Published online: 31 May 2017
ÓSpringer-Verlag Berlin Heidelberg and EURO - The Association of European Operational Research
Societies 2017
Abstract In recent years, revenue management research developed increasingly
complex demand forecasts to model customer choice. While the resulting systems
should easily outperform their predecessors, it appears difficult to achieve sub-
stantial improvement in practice. At the same time, interest in robust revenue
maximization is growing. From this arises the challenge of creating versatile and
computationally efficient approaches to estimate demand and quantify demand
uncertainty. Motivated by this challenge, this paper introduces and benchmarks two
filter-based demand estimators: the unscented Kalman filter and the particle filter. It
documents a computational study, which is set in the airline industry and compares
the estimators’ efficiency to that of sequential estimation and maximum-likelihood
estimation. We quantify estimator efficiency through the posterior Crame
´r–Rao
bound and compare revenue performance to the revenue opportunity. Both indicate
that unscented Kalman filter and maximum-likelihood estimation outperform the
alternatives. In addition, the Kalman filter requires comparatively little computa-
tional effort to update and quantifies demand uncertainty.
Keywords Revenue management Demand estimation Uncertainty Kalman
filter Particle filter Simulation
&Catherine Cleophas
catherine.cleophas@ada.rwth-aachen.de
Philipp Bartke
philipp.bartke@gmail.com
Natalia Kliewer
natalia.kliewer@fu-berlin.de
1
Information Systems Department, Freie Universita
¨t Berlin, Berlin, Germany
2
Information Systems Department, Freie Universita
¨t Berlin, Berlin, Germany
3
School of Business and Economics, RWTH Aachen University, Aachen, Germany
123
EURO J Transp Logist (2018) 7:57–88
https://doi.org/10.1007/s13676-017-0109-4

1 Introduction
Revenue management aims to optimally allocate a fixed capacity to exploit valuable
demand. Common models maximize revenue by optimizing inventory controls
based on demand estimates derived from historical data (Talluri and van Ryzin
2005). The quality of these demand estimates is crucial for the overall performance
of the revenue management system (Besbes and Zeevi 2009).
Modern revenue management accounts for demand dependencies, where
customers’ choices depend on the offered products. However, the complex models
required to represent such choices create challenges for estimation: revenue
management research has to find computationally efficient approaches to estimate
the relevant demand parameters from scarce and noisy data.
Furthermore, interest in robust revenue maximization is growing. However,
related approaches as proposed by Lan et al. (2008) and Perakis and Roels (2010)
require an indication of the uncertainty of the resulting estimates. Beyond robust
revenue management, such an indication would also benefit revenue management
analysts working to complement automated systems as described in Mukhopadhyay
et al. (2007). The more uncertain demand estimates are, the more likely the system
will benefit from manual input.
Finally, the variety of implementations found in practice calls for versatile
estimation methods. Here, we define a versatile method as being independent of a
given customer choice model. As there is no single best-practice demand model,
individual airlines select a model depending on business needs and available
resources. Therefore, estimation methods that are tailored towards a specific model
are of limited practical appeal. Versatile approaches can be implemented in a
variety of application cases, without requiring the firm to overthrow of the existing
revenue management system. For example, a versatile estimation method can
improve the forecast accuracy while maintaining a given optimization model. This
reduces the cost of implementing the desired improvement, making it more likely to
be profitable. This motivates us to describe the proposed estimation methods in a
general way to ensure versatility, while exemplifying their application on a specific
case as informed by an industry partner.
In this paper, we adapt the unscented Kalman and the particle filter estimation
methods for revenue management. Both filter-based methods are versatile, as they
can estimate parameters for a wide range of demand models. The Kalman filter, in
particular, is computationally efficient, as it allows for incremental estimation
updates. By estimating the covariance of demand, both approaches indicate the
degree of uncertainty.
We differentiate several terms with regard to demand forecasting. Demand,
denotes the general concept of customers wanting to buy tickets offered by the
airline. A demand model describes how customers’ choices are modeled, be that via
a multinomial logit model or as a set of independent requests per product. Any
demand model includes parameters; the particular demand parameter values
describe the actual demand expected in one particular sales period. For instance, the
number of requests expected to arrive per product can be the parameter value for an
independent demand model. A demand estimate is an estimated set of demand
58 P. Bartke et al.
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parameter values. When demand meets offers, we assume that a demand function
can compute the resulting bookings. Here, we differentiate observed bookings
reported in the past and expected bookings computed by the demand function.
To measure performance, we benchmark the filter-based estimates on sequential
estimation and maximum-likelihood estimation. Following the industry partner’s
indication, we chose sequential estimation to represent a simple, straight-forward
estimation method that is common in the industry. Its appeal lies in its simplicity;
we implement it as a minimal benchmark. Maximum-likelihood estimation is the
most popular approach to estimate the parameter values for a statistical model that
are most likely to explain the observation history (Scholz 2004). In terms of
estimator efficiency, this approach represents is a very strong benchmark. However,
it requires both more input data and more computing power than filter-based
approaches. Therefore, filter-based methods would be the preferred option if they
achieved a similar estimator efficiency.
We measure estimator efficiency in terms of the posterior Crame
´r–Rao bound. In
addition, we evaluate revenue performance by comparing the results to a hindsight
view of revenue opportunity. To ensure both stable laboratory conditions and a
sufficiently realistic setting, we benchmark approaches by implementing them in the
industry simulation system REMATE. This lets us test them on simulation scenarios
calibrated via empirical data as provided by a European network carrier.
To summarize, this paper seeks answers to the following questions: can filter-
based approaches match or even outperform the common sequential estimation or
maximum-likelihood estimation? Furthermore, does a resulting gain in estimation
efficiency correspond to a similar increase in revenue?
This paper is organized as follows: the next section reviews research on demand
estimates for revenue management. It also provides a theoretical background
introducing filter-based estimates. Section 3presents a generic airline revenue
management model. Section 4introduces the analyzed estimation approaches.
Subsequently, Sect. 5documents the simulation study and its results. The final
section discusses our findings and concludes with an outlook on future research.
2 Related research and theoretical background
This section first summarizes related research on estimating demand for revenue
management. Subsequently, it presents research on robust revenue management,
which motivates our consideration of approaches that quantify demand uncertainty.
Finally, it provides theoretical background on filter-based estimation.
2.1 Estimating demand for revenue management
Estimating demand from censored observations, as a major task of revenue
management, is considered summarily by Araman and Caldentey (2011). Research
in this area is further motivated by Besbes and Zeevi (2009), who focus on the
revenue gap that results from imperfect demand knowledge. Keskin and Zeevi
(2014) extend this research for policies that suffer from incomplete learning.
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Azadeh et al. (2014) present a taxonomy of unconstraining methods. They
categorize them by type and application area, considering classical methods, such
as pick-up, as well as more sophisticated methods, such as expectation maximiza-
tion. Weatherford (2016) also considers unconstraining methods, albeit from a
historical perspective. Following the emphasis of this research, the methods
compared here account for unconstraining by modeling the effect of product
availability on expected bookings.
Maximum-likelihood estimation is presented in Vulcano et al. (2012) and
Stefanescu (2009). Vulcano et al. (2012) consider customers arriving according to a
Poisson process and choosing products according to a multinomial-logit model.
They estimate primary demand, as observable if all products are offered, to create a
simplified expectation maximization procedure for both the arrival rate and for
product valuations. In practice, assuming primary demand is rather limiting, as it
can rarely be observed. Stefanescu (2009) foregoes choice modeling in favor of a
multivariate Gaussian distribution, relying on demand correlation to account for
time and inter-product dependence. Implicitly, this results in a very specific, not
versatile demand model, which is, i.a., not applicable to the example used in the
simulation study. maximum-likelihood estimation requires the complete data set to
update. As it allows for incremental updates, the Kalman filter adaptation presented
here provides a computationally more efficient alternative.
Nonparametric approaches enable versatile demand estimation by abandoning a
priori assumptions about model characteristics, such as multinomial choice. For
example, Farias et al. (2013) present a promising nonparametric approach to
estimate revenue from demand segments as defined by customers’ preferences over
a list of products. van Ryzin and Vulcano (2015) also propose to characterize
customer segments by preference lists. A similar approach is described in Haensel
and Koole (2011). Preference lists can, in principle, emulate any demand model
given a sufficient number of customer groups. In practice, the number of customer
groups has to be rather small to achieve stable estimation results. This would seem
to limit the demand models that can be considered.
The filter-based approaches introduced here can be adapted to estimate the
required parameter values for an existing demand model. We exemplify this through
the hybrid demand model described in Fiig et al. (2009). This demand model feeds
into an optimization approach that is already well-accepted in the industry. In
contrast, the approach described by van Ryzin and Vulcano (2015) is not suitable to
estimate parameter values for given demand models. While acknowledging that
nonparametric approaches are promising for a complete overhaul of the revenue
management system, we, therefore, neglect them in the benchmarking study.
Robust revenue maximization experiences growing interest and motivates our
search for approaches that can quantify the uncertainty of estimates. As an early
example, van Ryzin and McGill (2000) consider an adaptive approach to optimize
seat protection limits for individual flights. Ball and Queyranne (2009) formulate an
online algorithm that accounts for competitive aspects in network models. Lan et al.
(2008) extends this research by assuming given lower and upper bounds of demand.
Perakis and Roels (2010) suggest employing maximin and minimax regret criteria
for booking limits under interval uncertainty.
60 P. Bartke et al.
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2.2 Filter-based estimation approaches
Kalman filters iteratively estimate a system’s hidden state from indirect and noisy
observations. Kalman (1960) considers a state-space model, where the new system
state is a linear function of the previous state. Past observations only influence the
current estimate through the prior distribution of the system state. As this
distribution is Gaussian, it suffices to keep track of its mean and the covariance
matrix. This renders Kalman filters computationally efficient.
However, when aligning the observation function with the demand model, a
linear observation function would severely limit permissible demand models.
Additionally, a Gaussian distribution is not well-suited to approximate non-negative
and integer demand. This drives us to adapt the unscented Kalman filter.
The unscented Kalman filter was developed to model non-linear state evolution
and observation functions (Julier and Uhlmann 1997). It is based on an alternative
parameterization of the normal distribution, the so-called unscented transform.
While conceptually simple and computationally efficient, it outperforms other
methods overcoming the linearity restriction.
Kalman filter equations also form the antetype for Bayesian update equations as
employed by Lobo and Boyd (2003). The authors consider actively estimating a
linear demand model with an intercept and one coefficient corresponding to price.
Talluri and van Ryzin (2005, p. 458ff.) mention Kalman filter estimates for time-
series forecasting, but do not explicate how such estimates could be computed for
revenue management. Carvalho and Puterman (2015) employ Kalman filters as a
heuristic to develop a one-step-look-ahead strategy based on a second degree Taylor
expansion of future revenue. Kwon et al. (2009) use a Kalman filter to estimate
demand parameters for competing service providers. They assume that demand is
independent of offered alternatives, but depends linearly on past and current market
prices and evolves according to a random walk over a single, continuous sales
period. Li et al. (2009) and Chung et al. (2012) extend this model to allow for a
more general demand evolution. They highlight the notion of a state-space model of
dynamic pricing and demand estimation and use a Markov chain Monte Carlo
technique for parameter estimation.
Here, we extend research relying on Kalman filter equations such as Lobo and
Boyd (2003) and Carvalho and Puterman (2015). We adapt the idea of demand
evolving in the form of an auto-regressive process from Li et al. (2009) and Chung
et al. (2012) to consecutive sales periods. One of the simplest forms of an auto-
regressive process is the random walk, in which the demand parameter values in
period tare those of period t1 plus a random variable. If the variance of the
random variable is small, the values in period tcan be easily predictable from the
values of period t1. Otherwise, they can be almost unpredictable.
The particle filter is a more general extension of the Kalman filter, in that it does
not assume a specific form of the belief function. Instead, it approximates the belief
function by a discrete set of points in the parameter space, termed particles. While
its accuracy increases with the number of particles, so does the computational effort.
For an infinite number of particles, the particle filter approaches the minimum mean
squared error; the approximated posterior density converges to the real posterior
Benchmarking filter-based demand estimates for... 61
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density (Gordon et al. 1993). However, the best finite number of particles can only
be found experimentally.
The particle filter is closely related to Monte Carlo integration with importance
sampling and also known as sequential importance resampling filter. The earliest
reference we are aware of is Mu
¨ller (1991), who proposes a particle filter with
rejection sampling to estimate the parameters in general dynamic models. Gordon
et al. (1993) and Kitagawa (1996) independently propose the importance resam-
pling filter employed here. Doucet et al. (2000) review a variety of particle filter
methods and develop a general framework. The adaptation of the particle filter
described here follows the framework of Doucet et al. (2000), but approximates the
importance function locally via a multivariate Gaussian distribution.
3 The airline revenue management model
Revenue management systems differ in nomenclature, models, and algorithms. To
highlight the versatility of the filter-based approaches, we consider generic airline
revenue management as illustrated in Fig. 1. To explicate the generic concepts, we
show how they apply to exemplary demand models and the optimization approach
implemented in the computational study. The system includes four components:
forecast, optimization, inventory and historical data. It interfaces with the market by
controlling the offered products and, thereby, the resulting bookings. This section
explicates relevant aspects of forecast and optimization to highlight both the
Fig. 1 Revenue management model
62 P. Bartke et al.
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challenges and the motivation of estimating demand for airline revenue
management.
The notation listed in Table 1formalizes the airline revenue management model
described here. Products are offered throughout one sales period t; at the end of
period t, the left-over capacity perishes. For each new period t, the forecast predicts
demand by estimating the model’s parameter values and their future evolution.
Based on the result, the optimization computes inventory controls to maximize
expected revenue. Finally, the inventory accepts bookings.
For each period t, the set of inventory controls Atdefines the availability of each
product iincluded in the overall set of products I. For leg-based airline revenue
management, a product represents a particular fare class on a particular flight. For
network-based airline revenue management, a product represents a particular fare
class on a set of flights, which together form a travel itinerary. Each product icomes
at a fare fi2F. For leg-based models, each booking requires one seat on a single
flight. For network-based models, each booking requires one seat per flight included
in the booked travel itinerary. Here, we model this by considering that the number
of already accepted bookings, combined with a limited capacity, limits the set of
feasible inventory controls. For example, when there are only three seats left on a
flight, inventory controls that allow for four bookings on this flight are not feasible.
Table 1 Notation describing the revenue management model
tCurrent sales period
s2f1;...;Tg Time slices of the sales period
ISet of products, indexed by i
FSet of fares fifor all products i2I
B0Set of currently accepted bookings
B0!ASet of feasible functions calculating inventory controls from bookings B0
BtSet of bookings btisobserved per product i2I, during time slice sof sales period t
AtSet of inventory controls atisobserved per product i2I, during time slice sof sales
period t
Ati Subset of At, including only inventory controls observed for product i2I
a
tðB0ÞFunction calculating optimal inventory controls for sales period tgiven bookings B0
AtSet of optimal inventory controls in sales period t
XtSet of demand parameters values valid during sales period t
^
XtSet of demand estimates computed for during sales period t
H(A,X) Demand function; describes how demand parameter values Xinteract with inventory
controls Ato produce expected bookings
^
BtSet of bookings ^
btisexpected for product i2I, during time slice sof sales period t
^
Bti Subset of ^
Bt, including only bookings expected for product i2I
etNð0;QÞMultivariate Gaussian random variable with zero mean and covariance matrix Q, used
to describe demand evolution from period tto period tþ1
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3.1 Forecast
First, the demand forecast has to estimate demand parameter values from observed
bookings. Based on the resulting estimates, it predicts the future evolution of
demand. Thus, the demand estimators considered here are part of the forecast. To
model the interdependence of inventory controls A, demand model parameters X
and expected bookings ^
B, we formalize a demand function HðA;XÞ¼ ^
B. It states
that ^
Bare the bookings expected to result when a demand model with parameter
values Xencounters the offers described by A.
Here, Aand ^
Bare sets, which can, e.g., be indexed per product or slice of the
sales period. Size and interpretation of parameter set Xdepend on the implemented
demand model. For example, let product idescribe a fare of 100 Euro for an
economy seat on a flight from Hamburg to Munich at 7 a.m. on April 15, 2016.
Furthermore, let a parameter value xi2Xindicate that five customers are interested
in booking this product. Let inventory control ai2Aoffer four units of this product.
Then, the expected bookings ^
biwould result from a function HðA;XÞ¼
minfai;xig¼minf4;5g¼4.
In the example above, H(A,X) models products’ expected bookings as
independent of other offers. For such independent demand models, Xindicates
the expected bookings per offered product. However, modern revenue management
frequently models dependent demand. Thus, expected bookings for product
ipotentially depend on the availability of all other products, as described by the
set A. For the example above, a customer may consider to buy an economy seat for
100 Euro on the flight departing at 7 a.m., or an economy seat for 150 Euro on the
flight departing at 8 a.m.. In this case, at least some components of Xhave to
describe the demand dependency. For example, Xmay include a parameter that
indicates the customers’ price-sensitivity or their likelihood to substitute one
product for another.
Revenue management cannot observe the true demand parameter values. For
example, customers do not announce their price-sensitivity—they simply refuse to
book if no product with a sufficiently low price is offered. Therefore, to predict the
demand parameter values for period tþ1, Xtþ1, the forecast considers the set of
historical bookings Bt0and the set of historical inventory controls At0for all previous
periods t021;...;t
fg
. For each of these past periods, HðAt0;Xt0Þpredicted expected
bookings ^
Bt0. Comparing previously expected bookings to observed bookings Bt0
ties inventory controls At0and observed bookings ^
Bt0to the demand parameter
values Xt0. The forecast regards the previous parameter values Xt0as conditional on
At0and Bt0. For the example above, assume the previous inventory controls had
allowed for four bookings of the 100 Euro tickets for the 7 a.m. flight. If only three
bookings were observed, the number of customer interested in booking this flight at
this fare, given this particular set of inventory controls, could not have been larger
than three. Based on this assumption, the forecast computes estimates, e.g. via
maximum-likelihood estimation or the conditional mean.
When the demand model is complex or inventory controls constrained the
possible bookings, multiple approximations of Xt0can lead to identical expected
64 P. Bartke et al.
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bookings ^
Bt0. For the example, there may have been only three customers interested
in traveling at 7 a.m., or the 8 a.m. flight may have drawn demand from the 7 a.m.
flight. Therefore, Bt0¼HðAt0;Xt0Þcannot be simply inverted to estimate Xt0. Instead,
the forecast needs to observe bookings for diverse inventory controls to create a
sufficiently accurate demand estimate.
Demand continuously varies over time. Auto-regressive processes are useful to
model such gradually evolving time series. For ease of exposition, we assume the
demand parameter values to follow an auto-regressive process of order 1 (AR(1)),
which represents the simplest model of this class. In an AR(1)-process, the value at
period tþ1 equals the value at period tplus a random, additive error term. This
allows demand to evolve without a particular trend or pattern. Trends or patterns,
such as seasonality, which require higher order models, are beyond the scope of this
paper.
When demand follows an AR(1)-process, a multivariate Gaussian random
variable etwith zero mean and a covariance matrix Qdescribes the evolution from
demand parameter values Xtto Xtþ1as
Xtþ1¼XtþetetNð0;QÞ:ð1Þ
If a parameter’s range has to be constrained to provide meaningful inputs for the
demand model, we assume a truncated normal distribution of et. For example, let
one of the components of Xtdescribe the overall demand volume. In that case,
negative values are not meaningful and should be avoided. This can be achieved by
truncating the corresponding component of etat Xt, which ensures that
Xtþet0.
We model the number of expected bookings for sales period tand product ivia a
Poisson distribution with means ^
Bti. This distribution is censored if the set of
inventory controls for sales period tand product i,Ati , constrain bookings. For the
example above, inventory controls prohibit the number of bookings at 100 Euro on
the considered flight to exceed four.
Computing optimal inventory controls A
tþ1to maximize expected revenue
requires a forecast of expected bookings, ^
Btþ1¼HðAtþ1;Xtþ1Þ. This forecast must
describe the outcome of each feasible set of inventory controls. Inventory controls
are feasible if they do not allow bookings to exceed capacity.
Exemplary Demand Model To exemplify the working of the demand function,
consider the following example. Let an airline sell tickets for a single flight, so that
each offered fare class represents one product i2I. When products only differ in
their fares, customers either book the cheapest offer or they do not book at all. Let
the sales period tconsist of a set of time slices s2f1;...;Tg, so that at most one
customer considers to book per time slice. The forecast aims to predict expected
bookings ^
btisper period t, product i, and time slice s.
Let customer requests arrive according to a Poisson process with arrival rate k.
Furthermore, let each customer be willing to pay a minimum fare of f0. Let
customers’ individual willingness-to-pay wbe exponentially distributed with mean
f0
,0\1. Here, indicates the customers’ price-sensitivity. If demand parameter
Benchmarking filter-based demand estimates for... 65
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values are constant over the sales period, Xtincludes two values: demand arrival
rate ktand price-sensitivity t.
Let the fare of the cheapest currently available class, fi, be equal to or larger than
the minimum accepted fare, fif0. Then, the probability of observing a booking,
given this offer, is
PðwfiÞ¼exp fif0
f0

:ð2Þ
For each time slice sof the sales period, the expected number of bookings for the
cheapest offer iis ^
btis¼1
Tktexp fi
f0

. For all other products, the expected
number of bookings is 0. This demand behavior corresponds to the nonlinear
component of the computational study’s demand model.
3.2 Optimization and inventory
Scarce capacity limits the range of feasible inventory controls. For example, when
there are only two seats left unsold on a flight, the airline cannot accept more than
two bookings (neglecting overbooking). For simplicity, we assume a constant
capacity and express feasible inventory controls as a function of the current
accepted bookings B0. Thereby, we can derive the set of feasible controls, A, from a
function over the accepted bookings, B0!A. Then, the optimization problem is to
select the optimal availability function a
tðB0Þ. This function computes optimal
inventory controls A
tfor period t. Depending on the inventory system, A
tcan
prescribe a number of units to be sold or whether the product should be offered per
time slice of the sales period or not.
For example, in an inventory system that implements booking limits, the optimal
availability function may specify that, across the entire sales in period t, no more
than four seats should be sold in the 100 Euro fare class on the 7 a.m. flight. Using
the bookings observed for this class iup to time s,btisit states: atis¼4btis. Thus,
when two bookings are already accepted, a
tis¼42¼2. Alternatively, an
inventory system that relies on boolean controls requires a different availability
function, e.g.
a
tðbtisÞ¼ 0;if btis4;
1;otherwise:
ð3Þ
This availability function returns a 1 if class ishould be offered at time sof period t,
and a 0 otherwise. Of course, it also needs to specify whether to offer all other
products for all other time slices, which we neglect in the equation above.
Revenue is maximal when inventory controls A
tmaximize the product of
expected bookings ^
Btand fares fi. In principle, optimal controls could result from
enumerating all feasible availability functions, computing the expected revenue, and
selecting the availability function that returns revenue-optimal inventory controls.
In practice, the large number of feasible availability functions renders full
66 P. Bartke et al.
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enumeration intractable. This motivates the development of more efficient
optimization algorithms and heuristics.
Example: optimization We exemplify revenue optimization on the approach
implemented in the computational study.It optimizes inventory controls by
combining dynamic programming as described in Talluri and van Ryzin
(2005, Chapter 3) and fare transformation as described in Fiig et al. (2009). We
chose this optimization approach as it mimics the real-world system that produced
the empirical data underlying the simulation scenarios.
Fare transformation lets demand models that allow for customers choice provide input
for optimization algorithms that assume independent demand. Fiig et al. (2009)showthat
the resulting models are equivalent with respect to the optimal availability controls.
Fare transformation assumes that all products can be ordered such that all
feasible inventory controls are nested. For example, if products are ordered by their
fare, nested controls define that offering product imeans also offering product i0
with fi\fi0. This condition is frequently fulfilled in airline applications, where
booking classes are ordered by descending order of fare.
Fare transformation assigns new, so-called transformed demand and transformed
fares to each product. Per sales period tand product i, transformed demand ^
BTR
ti the
increase of expected bookings expected when offering the product. Let Pki
^
Btk be
the sum of expected bookings from offering the imost expensive products in period
t. Then, transformed demand for product iis defined as ^
BTR
ti ¼Pki
^
Btk 
Pki1
^
Btk for all but the most expensive product. For that, transformed demand
equals the expected bookings, ^
BTR
ti ¼^
Bti.
The transformed fare value is set such that multiplying this new value with the
transformed demand value from above yields the respective total increase of
expected revenue: let Pki
^
Btk fkbe the total revenue from offering the imost
expensive products in period t. For all but the most expensive product, the
transformed fare is then given as fTR
ti ¼ð
Pki
^
Btk fkÞð
Pki1
^
Btk fkÞ. The
transformed fare of the most expensive product is its actual fare, fTR
ti ¼fti.
These modified fare and demand values feed into a dynamic program for revenue
optimization following a formulation described in Talluri and van Ryzin
(2005, Chapter 2). This model assumes that at each time slices sof the sales
period, at most one customer request can arrive. Let RðsÞdescribe the expected
revenue in slice s. If a customer request arrives for product iand is accepted as a
booking, RðsÞ¼fTRti. Otherwise, RðsÞ¼0. The Bellman equation of this model is
VsðxÞ¼Vsþ1ðxÞþEmax
u20;1fðRðsÞDVsþ1ðxÞÞug;

ð4Þ
where VsðxÞis the value function with boundary conditions VTðxÞ¼08xand
Vsð0Þ¼08s. The value DVsþ1ðxÞ¼Vsþ1ðxÞVsþ1ðx1Þis the expected mar-
ginal value of capacity in time-slice s.
Talluri and van Ryzin (2005) show that the optimal policy is to accept all
requests for products with fTR
ti DVsþ1ðxÞ. The vector DVsþ1ðxÞindicates so-called
Benchmarking filter-based demand estimates for... 67
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bid prices for each time slice sof the sales period, with one bid price for each unit of
remaining capacity. The bid prices control the offered products: only products with
a transformed fare that exceeds than the current bid price are offered.
4 Filter-based demand estimation for revenue management
Table 2introduces the notation that formalizes the estimation approaches outlined
in this section. For simplicity, we assume that forecast and optimization update only
after period thas passed and before period tþ1 starts. We drop index twherever it
is not explicitly needed.
We express revenue management as the following state-space model:
Xtþ1¼Xtþet;etNð0;QÞð5Þ
BtPoiðHðAt;XtÞÞ:ð6Þ
Again, Xtare the parameter values that define demand in sales period t.Btare the
bookings observed during that period. Atare the inventory controls that were
implemented throughout t.HðAt;XtÞdescribes the demand function, which com-
bines inventory controls Atand demand parameters Xtto generate bookings.
BtPoiðHðAt;XtÞÞ denotes that the observed bookings are a Poisson-distributed
random variable with mean HðAt;XtÞ.
If HðAt;XtÞwere a linear function of Xt, a simple Kalman filter could find the
least-squares estimate of Xtgiven Bt. For example, in a simple independent demand
model, each value in Xtdescribes the expected demand for a specific product. Then,
Hcan be formulated as HiðAt;XtÞ¼Xti if product iis offered (Ati [0); otherwise
HiðAt;XtÞ¼0. His obviously linear in Xtand a simple Kalman filter can estimate
Xt.
However, in most revenue management systems, HðAt;XtÞis not linear, as
exemplified by the demand model described in Sect. 3.1. Therefore, the simple
Kalman filter is not directly applicable. This motivates us to adapt the unscented
Kalman filter and the particle filter.
4.1 Unscented Kalman filter
The general idea of the unscented Kalman filter is to represent the demand estimate
as a multivariate normal distribution. However, when H(A,X) is nonlinear, every
update distorts the demand estimate such that it no longer precisely follows a
multivariate normal distribution. For example, consider a random variable Rthat
follows a uni-variate normal distribution. Transforming Rwith some function f
yields a new random variable R0¼fðRÞ, which is no longer guaranteed to be
normal-distributed. In an extreme case, fcould map all values to a constant, such as
fðxÞ¼1.
However, the unscented Kalman filter can still approximate Xby assuming that it
follows a multivariate normal distribution. It achieves this by computing the update
68 P. Bartke et al.
123

Table 2 Notation introduced to describe estimation approaches
Kalman filter
XL,XCDemand parameters of the linear and the nonlinear part of the demand
function, respectively
LðA;XLÞLinear component of the demand function
CðA;XCÞNon-linear component of the demand function
VNð0;diagðHðA;XÞÞÞ Normal-distributed error term with mean 0 and covariance matrix
diag(H(A,X))
PtCovariance matrix of demand parameters for sales period t
U,UTAn upper triangular matrix and its transpose, respectively
rjSigma point calculated by the unscented Kalman filter, indexed by
j21;...;2jXCj
ck-parameter from Julier and Uhlmann (1997), renamed to avoid confusion
with demand volume
gjTransformation of Sigma point rjcalculated by the unscented Kalman filter
^
BC,^
BLSets of bookings expected from the nonlinear and the linear demand
component, respectively
a,bScaling parameters from Julier and Uhlmann (1997)
KKalman gain
Particle filter
Pt¼f^
Xtk;k¼1;...;NgSet of particles computed for sales period t, indexed by k
Wt¼fxtk;k¼1;...;NgSet of particle weights computed for sales period t, indexed by k
pðXjX0:t1;k;B0:tÞImportance function highlighting parts of the demand parameter space
pðBtjAt;^
XtkÞConditional probability linking observed bookings to observed
inventory controls and estimated parameters
pð^
Xtkj^
Xt1kÞConditional probability linking estimated parameters for sales
period tto those estimated for sales period t1
Maximum-likelihood estimation
p(B,A,X) Joint probability of observing bookings band availability Agiven demand
parameters X
X0:t,A0:t,B0:tTrajectories of demand parameters, inventory controls, and bookings from period 0 to t
pðX0ÞPrior probability of parameter set Xbefore any bookings are observed
pð^
Xt0j^
Xt01ÞConditional probability linking estimated parameters for sales period t0to those estimated
for period t01
pðBt0jX;At0ÞConditional probability of observing bookings Bt0given demand parameters Xand
inventory controls At0
LðXÞLikelihood function of demand parameters X
Q1Inverse of covariance matrix Q
Benchmarking filter-based demand estimates for... 69
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in a set of characteristic points in the parameter space of Xrequired to explain each
new set of observations B. Then, it uses these updated characteristic points to find
the best-fitting multivariate normal distribution for the new estimate.
In this paper, we adapt the original unscented Kalman filter to accommodate a
demand function that can be decomposed into a linear and a multiplicative non-
linear part. The result is a combination of the classical and the unscented Kalman
filter. This assumption does not limit the set of possible demand functions, since the
linear part can always be set to 0; then, the non-linear part represents the complete
demand function. We suggest using the decomposition for computational speed and
numerical stability. The linear demand component is still estimated by a regular
Kalman filter, using a least-squares estimate. Only those components of Xthat are
relevant for the non-linear component require the non-linear approximation
algorithm.
Let LðA;XLÞdenote the linear demand component and CðA;XCÞdenote the non-
linear component. To enable this formulation, we split the demand parameters into
two non-overlapping partitions X¼XL;XC
fg
. Then, we formalize the demand
function decomposition by
HðA;XÞ¼LðA;XLÞþCðA;XCÞ:ð7Þ
If no linear demand component can be isolated, we propose to set LðA;XLÞ¼0.
When there is no non-linear component, set CðA;XCÞ¼0 and apply the original
Kalman filter.
In our formulation, the unscented Kalman filter assumes additive Gaussian
observation errors. Hence, the observed bookings Bresult from summing up the
linear and the non-linear demand component and error term V:
B¼LðA;XLÞþCðA;XCÞþVVNð0;diagðHðA;XÞÞÞ:ð8Þ
Here, Vis normal-distributed with mean 0 and covariance matrix diag(H(A,X)).
The covariance matrix diagðHðA;XÞcarries H(A,X) on its diagonal, where all other
entries equal zero. Thus, the variance of observed bookings Bequals their expected
mean, as for a Poisson distribution.
When estimating demand for sales period tþ1, at every update, the unscented
Kalman filter computes ^
Xtþ1and a covariance matrix Ptþ1. Together, these define
the approximate current belief about demand. In other words, ^
Xtþ1and Ptþ1are the
parameters of the approximate posterior distribution of the true demand parameters
Xtþ1, considering all observations up to and including sales period t.
To estimate ^
Xtþ1, the unscented Kalman filter decomposes the covariance matrix
Ptinto an upper triangular matrix U, such that UUT¼Pt. From U, it computes the
set of 2jXCjþ1 sigma points rj. Here, jXCjis the number of parameters in the non-
linear demand component. Then,
r0¼^
Xt;ð9Þ
70 P. Bartke et al.
123

rj¼^
Xtþffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jXjþc
pUjj¼1;...;jXCj;ð10Þ
rj¼^
Xtffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jXjþc
pUjjXCjj¼jXCjþ1;...;2jXCj:ð11Þ
In Eqs. 9–11,Ujis the jth column of U.|X| is the overall number of demand
parameters. cis the k-parameter from Julier and Uhlmann (1997).
1
Deriving Eqs. 9–
11 from the original formulation is straight-forward, exploiting linearity and the
upper triangular form of Uwhenever possible.
The unscented Kalman filter applies the non-linear part of the demand function
C(A,X) to each sigma point rjfor observed availability A:
gj¼CðA;rjÞj¼0;...;2jXCj:ð12Þ
From this, it computes expected bookings ^
Bby summing up the expected bookings
from the non-linear demand component, ^
BC, and the expected bookings from linear
demand component, ^
BL:
^
BC¼cþjXLj
cþjXjg0þ1
2ðjXjþcÞX
2jXCj
j¼1
gjð13Þ
^
BL¼LðA;^
XLÞð14Þ
^
B¼^
BLþ^
BC:ð15Þ
Additionally, the algorithm determines the booking covariance matrix P^
BC^
BCfrom
the non-linear demand component. To this end, it applies a scaling parameter aand
sets b¼3 following (Julier and Uhlmann 2004):
P^
BC^
BC¼cþjXLj
cþjXjþð1a2þbÞ

ðg0^
BCÞðg0^
BCÞT
þ1
2ðjXjþcÞX
2jXCj
j¼1
ðgj^
BCÞðgi^
BCÞT:
ð16Þ
The unscented Kalman filter calculates the cross-covariance PX^
BCbetween demand
parameters Xand expected bookings from the non-linear component, ^
BC,as
PX^
BC¼1
2ðjXjþcÞX
2jXCj
j¼1
ðrj^
XÞðgj^
BCÞT:ð17Þ
By combining PX^
BCand the upper left jXLjjXLjblock of P,PL, it computes the
total covariance, P^
B^
B,as
1
We renamed the k-parameter from Julier and Uhlmann (1997)tocto avoid confusion with kas used to
denote demand volume in the forecast example and in the simulation study.
Benchmarking filter-based demand estimates for... 71
123

P^
B^
B¼LðA;XLÞPLLðA;XLÞTþP^
BC^
BCþdiagðHðA;ð^
XÞÞ:ð18Þ
With the left jXjjXLjblock of P,PCL, it constructs the total cross-covariance, PX^
B,
as
PX^
B¼PCL LðA;XLÞTþPX^
BC:ð19Þ
The Kalman gain K¼PX^
BP1
^
B^
Blets the values estimated for ^
B,P^
B^
Band PX^
Benter the
regular Kalman filter update and prediction functions. Up to this point, all com-
putations exclusively relied on observations made during period t. From this, the
demand estimate for period tþ1, ^
Xtþ1, is calculated based on Kand covariance Q:
^
Xtþ1¼^
XtþKðBt^
BtÞð20Þ
Ptþ1¼PtKP ^
B^
BKTþQ:ð21Þ
4.2 Particle filter
In contrast to the unscented Kalman filter, the particle filter is a more general
extension of the Kalman filter. As it does not assume the belief function to have any
parametric form, the particle filter can be adapted to any form of state-space model.
The unscented Kalman filter still assumes that the current belief function of the
demand parameters Xcan be approximated by a multivariate Normal distribution
with reasonable accuracy. In the case of highly non-linear demand models, this may
no longer be true. The particle filter remedies this limitation of the unscented
Kalman filter by dropping any assumptions on the parametric form of the belief
function. It represents the current belief function of the demand parameters as a
cloud of so called ‘‘particles’’. Each particle represents a hypothesis or point
estimate on the actual parameter values. The particle filter updates the particles’
likelihood as new observations arrive. However, large numbers of particles are
required to represent the belief function with high accuracy, especially so if Xis
high-dimensional, i.e. the vector of demand parameters has many components. The
increases the computational effort for the particle filter.
Let Ptbe the set of particles ^
Xtk;k¼f1;...;Ng.Each ^
Xtk describes estimated
demand parameter values for period t. Let Wtbe a corresponding sets of weights
xtk;k¼f1;...;Ng. Each weight describes the corresponding particle’s likelihood
of being accurate.
At the end of period t, a new set of controls Atand bookings Btwere observed.
From these, the algorithm generates new particles and evaluates their likelihoods to
compute particle weights.
Drawing new particles from a uniform distribution over the whole parameter
space would create many particles with very small likelihoods, causing excessive
computational effort. To avoid this, we apply a standard particle filter with
importance sampling as described in Doucet et al. (2000). Importance sampling puts
most particles in regions of high interest. These regions are defined by a so-called
72 P. Bartke et al.
123

importance function pðXjX0:t1;k;B0:tÞ. The importance function assigns a weight to
each particle based on the particle’s past trajectory and the booking history.
The particle filter relies on two conditional probabilities:
1. pðBtjAt;^
XtkÞ, the conditional probability of bookings Btbeing observed given
controls Atand demand estimate ^
Xtk. Here, we assume that the observed
bookings per product are Poisson-distributed. To define this probability, let ^
btis
be the expected number of bookings for product iin time slice s:
^
btis¼HðAt;^
XtkÞ. Substituting this arrival rate and the observation Btinto the
distribution function of the Poisson distribution yields
pðBtjAt;^
XtkÞ¼ Y
i2I;s2f1;...;Tg
^
bBtis
tis
Btis!erisð22Þ
2. pð^
Xtkj^
Xt1kÞ, the conditional probability of estimated demand parameters ^
Xtk
resulting from the evolution of a demand estimate ^
Xt1kfrom the previous
period t1. Hence, pð^
Xtkj^
Xt1;kÞdescribes the state evolution. Equation 5
states that this probability follows a multivariate Gaussian distribution with
mean ^
Xt1;kand covariance Q.
For each particle ^
Xtk, the particle filter algorithm iterates over the following
steps:
1. Sample ^
Xtk pðXj^
X0:t1;k;B0:tÞ.
2. Compute importance weights x0
t;kas
x0
t;k¼xt1;kpðBtjAt;^
XtkÞpð^
Xtkj^
Xt1;kÞ
pð^
Xtkj^
X0:t1;k;B0:tÞ:ð23Þ
3. Normalize importance weights xt;k¼x0
tk
Px0
tk
.
Together, weights and particles form a discrete distribution, which approximates the
actual continuous posterior distribution. The expected estimate for period tis the
mean of the particles in t,^
Xt¼1
NPxtk ^
Xtk.
Importance sampling keeps particle weights as evenly distributed as possible.
The optimal importance function pðXjX0:t1;k;B0:tÞ¼pðXjXt1;Bt;AtÞminimizes
the variance of particle weights to achieve this. In our model, this function has no
closed form. If it did have a closed form, say a multivariate Normal distribution, we
could generate a new set of particles by sampling from this distribution function
directly. Instead, we can only approximate the importance function locally around X
by a multivariate Gaussian distribution. A second-order Taylor expansion yields the
covariance R¼L
00ðXÞ1and mean m¼XþRL
0ðXÞ, where LðXÞis the log-
likelihood function: LðXÞ¼log pðXjXt1;Bt;AtÞ. Its first two derivatives are
Benchmarking filter-based demand estimates for... 73
123

LðXÞ¼const:1
2ðXXt1ÞTQ1ðXXt1Þ
þX
i;s:HðAt;XÞis[0
Btislog HðAt;XÞisHðAt;XÞis

;ð24Þ
rXLðXÞ¼Q1ðXXt1Þ
þX
i;s:HðAt;XÞis[0
rxHðAt;XÞisBtis
HðAt;XÞis
1

;ð25Þ
DX
XLðXÞ¼Q1
þX
i;s:HðAt;XÞi;s[0DX
XHðAt;XÞi;sBt;i;s
HðAt;XÞis
1

ðr
XHðAt;XÞisÞðrXHðAt;XÞisÞTBtis
H2ðAt;XÞis;
ð26Þ
where DX
Y¼ðr
XÞðrYÞT¼ðo
oX1;...;o
oXmÞTðo
oY1;...;o
oYnÞ.
Particle X, around which the log-likelihood function is approximated, should be
the mode of pðXjXt1;Bt;AtÞ. This mode can be found numerically by applying
Newton’s iterative method. Constructing this importance function is computation-
ally expensive when Xincludes many parameters. However, it enables the
estimation to focus on areas of the parameter space that agree with the current
observation.
Eventually, this filter can still degenerate. Here, degeneration means that the
resulting weights are skewed to emphasize a single particle. To overcome this,
particles have to be resampled from time to time. As recommended in Doucet et al.
(2000), we resample whenever the estimated number of effective particles N¼
1
Px2
tk
is smaller than some minimum fraction of N.
During resampling, each particle ^
Xtk is replaced by an existing particle ^
Xtk0.To
this end, we draw index k0randomly with replacement from f1;...;Ngwith
probabilities xtk. The new weights are x0
tk ¼1
N. This does not alter the particle
distribution moments’ expectation. As a result, the number of effective particles N
equals the total number of particles N.
4.3 Quantifying the uncertainty of estimates
Both unscented Kalman filter and particle filter not only compute demand estimates,
but also provide a covariance matrix that quantifies the estimate’s uncertainty.
Thereby, estimates can be ranked and compared, supporting revenue management
analysts and robust revenue management strategies.
Figure 2illustrates how uncertainty is quantified for an estimate of two demand
parameters. Here, one parameter describes the price elasticity and one describes the
volume of demand. Three panels illustrate three potential outcomes. Every panel
74 P. Bartke et al.
123

depicts the value intervals where the true parameter values are expected with
probabilities of 68% (continuous line, 1r) and 95% (dotted line, 2r).
Panel (a) shows a state of low uncertainty and no parameter correlation. In (b),
uncertainty is higher. Panel (c) represents high uncertainty and correlating
parameters. From (c), decision makers cannot tell whether a low number of
bookings was caused by low demand volume or by high price elasticity. The
deliberate creation and absorption of more, varied observations into the belief
function could transform this state into one resembling (b) or even (a).
Of course, sophisticated demand models include far more than just two
parameters. Accordingly, visualizing uncertainty information for such models
becomes difficult. However, we suggest using techniques such as principal
component analysis to aggregate parameters in meaningful ways for two- or
three-dimensional analysis. Parameter rankings and automated strategy selection
can still be conducted on the original level of dimensionality.
4.4 Classical estimation methods
This paper benchmarks the filter-based estimators’ efficiency on that of sequential
estimation and maximum-likelihood estimation. Therefore, the remainder of this
section outlines these estimation approaches.
4.4.1 Sequential estimation
Sequential estimation computes each demand parameter value independently. This
simple method is commonly implemented in older revenue management systems
relying on independent demand models. When assuming independent demand, the
bookings per product can be treated as individual time series and forecasted in
isolation. Thus, each product’s expected demand is characterized by one parameter
in X. For each, sequential estimation calculates the value that deviates least from the
current observations, keeping all other parameters constant.
Fig. 2 Estimate uncertainty
Benchmarking filter-based demand estimates for... 75
123

Consider the example demand model from Sect. 3.1 . Recall that the number of
expected bookings for the lowest available booking class iis defined as
^
btis¼1
Tktexp fi
f0
 ð27Þ
The parameters of the demand model are ktand . These two parameters are
estimated from observed bookings in booking class iby sequential estimation. This
is done by keeping one parameter constant and solving the above equation for the
other parameter. Let btisdenote the number of observed bookings in booking class i
and let ktand tdenote the current estimate of the demand parameters in period t.
The new, preliminary estimates are then defined as follows:
k0
tþ1¼T btisexp tfi
f0
 ð28Þ
0
tþ1¼log T  btisf0
ktfi
 ð29Þ
The values derived for 0
tþ1from the above equation occasionally assume extreme
levels. To reduce the impact of these outliers, we limit the values for 0
tþ1to an
interval [0.2, 4].
New estimates are computed using a linear combination of the old estimates kt
and tand the new, preliminary ones k0
tþ1and 0
tþ1with a smoothing rate of a:
ktþ1¼ktð1aÞþk0
tþ1að30Þ
tþ1¼tð1aÞþ0
tþ1að31Þ
In our simulation study, we performed preliminary experiments to determine a¼
0:2 as a good smoothing rate was set through initial testing to c¼0:2. Both the
exponential smoothing and the outlier detection with regard to tcompensate the
‘‘overshooting’’ behavior of this method to some extent.
From a theoretical perspective, sequential estimation has little appeal. It is,
however, representative of the way demand is still estimated in basic revenue
management systems. To practitioners, it might seem appealing to apply such
existing estimation algorithms to more complex demand models. After all, often
enough, simple heuristics perform adequately in practical settings. We include this
approach in the computational study to highlight the problems of applying it to
complex demand models.
4.4.2 Maximum-likelihood estimation
Let p(B,A,X) describe the joint probability of observing bookings Band
availability Agiven demand parameter values X. Maximum-likelihood estimation
seeks a trajectory X0:tthat maximizes this probability given controls A0:tand
bookings B0:tas observed up to the end of period t.
76 P. Bartke et al.
123

p(B,A,X) can be derived from the original state-space model in Eqs. 5and 6as
pðB;A;XÞ¼pðX0ÞY
t
t0¼1
pðBt0jAt0;Xt0ÞpðXt0jXt01Þ:ð32Þ
Maximizing Eq. 32 or its logarithm over Xis a very high dimensional problem. To
make it more tractable, we limit the historical data to a rolling history of obser-
vations from t0to t. For these periods, we assume Xto be constant. In other words,
we assume the same parameter values to govern the demand across all periods from
t0to t. We also assume that the initial Xt0is known and equals the estimate from the
preceding period, i.e., Xt0¼^
Xt1. Since Xt0acts as a prior in the above equation, we
chose to set Xt0equal to its most recent estimate ^
Xt1. Then, the joint probability
function is defined as
pðB;A;XÞ¼pðXj^
Xt1ÞY
t
t0¼t0þ1
pðBt0jX;At0Þ;ð33Þ
where the probability pðXj^
Xt0Þresults from the multivariate normal distribution with
mean ^
Xt0and covariance matrix Q. The conditional probability pðBtjX;AtÞis the
product of the Poisson probability distribution functions with rates
ki;s¼HðX;AtÞi;s.
This yields the log-likelihood LðXÞ:
LðXÞ¼log pðB;A;XÞ
¼1
2ðX^
Xt0ÞTQ1ðX^
Xt0Þ
þX
t
t0¼t0þ1X
i;s:HðX;At0Þis[0
Bt0islog ðHðX;At0ÞisÞHðX;At0Þis
þconst:ð34Þ
This likelihood function is maximized by calculating the root of the first derivative
via the iterative Newton method. Because LðXÞis generally not concave, a global
maximum is not guaranteed. In preliminary experiments, the Newton method
converged almost always to the global maximum. One reason for this is that the
maximum is expected to be close to the current estimate ^
Xt1, which is, therefore, an
excellent starting value.
5 Simulation study
Simulations provide a way to assess revenue management performance under
laboratory conditions. They avoid the high risk and cost of empirical tests while
providing stable and controllable conditions. This particularly applies to demand
estimation, as in real-world settings, the true demand model and its parameter
values are never perfectly known or even stable.
Benchmarking filter-based demand estimates for... 77
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This section provides numerical results from simulation scenarios that were
calibrated on empirical airline data. It compares the performance of the unscented
Kalman filter and the particle filter to sequential estimation and maximum-
likelihood estimation. We measure estimator efficiency by scaling the estimation
error on the Posterior Crame
´r–Rao bound. In addition, we compare revenue to the
potential revenue opportunity, i.e., the maximum achievable revenue per scenario.
The posterior Crame
´r–Rao Bound represents the minimally achievable estima-
tion error (Tichavsky et al. 1998). In a simulation study, the true dynamics of
demand are known, so that the exact posterior Crame
´r–Rao bound can be computed.
In a real-world setting, an assumed theoretical demand process could only
approximate actual demand and therefore the posterior Crame
´r–Rao bound could
also only be approximated.
5.1 The simulation setting
To compare estimation performance in a full airline revenue management system,
we extended the simulation system REMATE developed at Lufthansa German
Airlines (Gerlach et al. 2010). REMATE models a full airline revenue management
system with forecast, optimization, and inventory. To test and benchmark
estimation methods, we implemented them in the simulation using Java. To
provide unbiased benchmarks, we also implemented a demand model that fulfills
the estimates’ assumptions.
5.1.1 Revenue management simulation
Creating a simulation experiment first requires defining a set of flights by carrier,
origin, destination, and departure time. Next, each carrier offering flights has to be
assigned a revenue management system. Given this supply, the simulation also
requires a set of parameters to guide the generation of artificial demand.
To model a best-case scenario for benchmarking, we implemented a demand
generation process that relies on the hybrid model also underlying the demand
forecast. Thus, as outlined in Sect. 5.1.2, the artificial demand actually follows the
model that each approach attempts to estimate. Particular effort was spent on
empirically calibrating the demand parameters. This calibration relies on fare and
booking data as reported for a European network carrier during the years
2001–2012.
The simulation is stochastic, in that it randomly varies the demand model’s
parameters per simulation run following an AR(1) process as formalized by Eq. 1.
Thereby, the resulting artificial demand evolves from one run to the next,
mimicking the evolution of demand over sales periods.
Once all simulation runs have been processed, the experiment’s output, such as
estimator efficiency and revenue, are available for analysis. As supply and demand
are constant across simulation experiments, result differences are exclusively caused
by the estimation approach employed.
Demand is forecasted from historical inventory controls and bookings via each of
the benchmarked methods. Subsequently, the simulation system optimizes inventory
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controls by a combination of dynamic and linear programming as described in Sect.
3.2. This optimization approach was chosen as it closely resembles the approach
underlying the empirical data used for calibration. It returns binary availability
parameters atis, indicating the availability of class iat day sof sales period t.
In each slice of the sales period, requesting customers decide whether and what to
book. Bookings are recorded in the simulation inventory. The simulation forecast
uses these records to predict bookings for the next period.
5.1.2 Artificial demand generation
To generate artificial demand, the simulation implements a demand function
H(A,X) that models hybrid demand as described in Fiig et al. (2009). We choose
this demand model for its compatibility with the implemented optimization model
that also underlies the empirical data. It includes two components, one representing
independent demand and one representing price-dependent demand. Per product i,
the independent demand function LðA;XÞicomputes the number of bookings
expected from customers who only consider to buy product i. The dependent
demand function CðA;XÞicomputes the number of bookings expected from
customers who always book the cheapest available product.
The independent demand component LðA;XÞicomputes expected demand ^
biper
product ias follows: if iis offered at time sof period t,atis¼1, there result xi
expected bookings. Otherwise, the number of expected bookings is zero:
LðA;XÞi¼xiif atis¼1;
0 else.
ð35Þ
Dependent demand C(A,X) includes two parameters. First, a volume parameter k
represents the volume of expected demand at a reference price f0. Second, a price
elasticity parameter describes how the volume of expected demand decreases as
the price of the cheapest available product increases. Analytically, the choice of f0is
arbitrary, as the demand parameters kand can be adapted for any reference fare.
When iis the cheapest offer with fare fi, the number of requests expected for ican
be expressed as
CðA;XÞi¼kexp fi
f0
1

:ð36Þ
As the dependent demand component assumes all customers to buy the cheapest
offer i, for all other products j6¼ i,CðA;XÞj¼0.
The sum of independent demand component LðA;XÞiand price-dependent
demand component CðA;XÞidesignates the expected bookings as
HðA;XÞi¼LðA;XÞiþCðA;XÞi:ð37Þ
Airline industry expertise suggests that price elasticity varies over the sales period:
price-elastic leisure demand arrives before price-inelastic business demand.
Therefore, we model price elasticity as a degree-two Lagrange polynomial in the
Benchmarking filter-based demand estimates for... 79
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square root of the number of days before departure. This function has three
parameters: elasticity at 360 days before departure 360 , at 60 days before departure
60, and at departure 0. At time sbefore departure, price elasticity is a linear
combination of the two parameters defined for times bordering s.
5.1.3 Experimental set-up
For parsimony, we consider only a single compartment on a single flight. Thus, a
product describes a combination of fare class and flight, and customers compare
available classes, but cannot book a different flight. Consequentially, we speak of
classes rather than of products in this section.
A typical airline network contains a variety of routes: short-haul and long-haul,
dominated by business or leisure demand. Revenue management has to perform
well across all types of routes. Therefore, we deliberately replicate that variety, as
this study aims to consider the applicability of filter-based estimators in practice. To
this end, we calibrated simulation scenarios on a 12-year time-series (2001–2012) of
tariffs, fares, and monthly bookings reported for a European network carrier’s top
ten domestic, continental, and intercontinental routes. As a side-effect, this
calibration also evaluates the estimators’ sensitivity with regard to the distribution
of independent demand, which we found to be the largest difference between the
scenarios.
The number of classes offered per flight and their fares were also derived from
the empirical data set and are listed in Table 3. Capacity is set to 100 seats for
domestic and continental routes, and to 200 seats for intercontinental routes.
From one sales period to the next, simulated demand evolves according to an
AR(1) process of the form given by Eq. 1. From the empirical data, we estimate
mean values of as the starting value for the demand evolution. Furthermore, we
estimate the diagonal of Qin Eq. 1and artificially add correlation to Q. The
resulting values are listed in Table 4, where DbD indicates days before departure.
Table 3 Fares (in EUR) per
class Product Domestic Continental Intercontinental
A 269 609 1614
B 229 504 1099
C 189 429 799
D 149 348 699
E 129 290 614
F 114 213 539
G 99 181 479
H 89 149 439
I 79 119 399
J 69 101 352
K 59 87 319
L 49 49 201
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To calibrate independent demand per fare class, we differentiate classes based on
the underlying tariffs: we assume that bookings in published tariffs mirror
dependent demand, while bookings in corporate, bulk, or target group tariffs
represent independent demand. Consequentially, independent demand shares are
low (see Table 4). Table 5an uneven distribution of independent demand over
classes. This is because many corporate and other special tariffs are only assigned to
a small subset of classes.
Finally, for each type of route, we vary the arrival rate kto recreate the empirical
span of demand-to-capacity ratios and evaluate the estimators’ response to this.
Three ratios were implemented: 0.8, 1.0, and 1.2. When demand is low (0.8),
Table 4 Simulation parameterization per route type
Simulation parameter Domestic Continental Intercontinental
Capacity in offered seats 100 100 200
Number of products per route 12 12 12
Price elasticity coefficients 
360 DbD 1.16 0.622 1.227
60 DbD 0.963 0.539 0.924
0 DbD 0.204 0.157 0.182
Relative variance of change
Price elasticity diagðQÞ= 0.00933 0.00673 0.00888
Demand volume diagðQkÞ=k0.0151 0.0102 0.0159
Pair-wise price elasticity correlation
360 and 60 DbD 0.9 0.9 0.9
60 and 0 DbD 0.9 0.9 0.9
360 and 0 DbD 0.81 0.81 0.81
Independent demand share 3.5% 3.9% 7.6%
Base fare f0100 100 500
Table 5 Distribution of
independent demand over
classes per route type
Class Domestic (%) Continental (%) Intercontinental (%)
A33 2 1
B 0 14 1
C56 5 5
D0 3 1
E 6 42 2
F1 1 4
G1 4 6
H1 6 11
I 1 20 11
J0 2 10
K0 1 33
L0 0 15
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capacity is not scarce, so that revenue management is mostly based on exploiting
customers’ willingness to pay. When demand is high (1.2), capacity is scarce and
revenue management has to avoid stock-out situations. In such situations, capacity
is already sold-out when high-value demand arrives late in the sales period. In
conclusion, the study considers nine scenarios.
From a scenario’s demand parameters, the simulation creates a number of
artificial customers as drawn from a Poisson distribution according to k. Each
customer’s maximum willingness to pay is drawn from an exponential distribution.
Flags indicate whether a customer’s choice behavior is dependent or independent.
This study analyses the system’s long-term behavior from a steady state on.
Therefore, we consider a burn-in period after initializing the forecast in the first run.
We found that 100 sales periods are a sufficient burn-in period to allow the
estimation algorithms to reach a steady state. Results are then averaged over another
100 sales periods to generate narrow confidence intervals.
Testing four estimation approaches on nine demand scenarios yields 36
experiments. Each experiment includes 100 stochastically independent simulation
runs, each processing 200 consecutive sales periods. Total runtime was about 40
hours on a computer with an Intel Core i5 2.6 GHz processor and 4GB of RAM.
5.2 Numerical results
We compare approaches with regard to estimator efficiency and revenue perfor-
mance. Estimator efficiency is defined as the quotient of the posterior Crame
´r–Rao
bound and estimates’ mean squared error. The Crame
´r–Rao bound is the theoretical
minimum of estimates’ mean squared error. As such, the ratio is a unit-less number
1. Since multiple parameters are estimated for multiple products and time slices,
both the Crame
´r–Rao bound and the mean squared error are covariance matrices.
The trace of these matrices (the sum of diagonal elements) indicates estimator
efficiency. It increases as the mean squared error gets closer to the theoretical
minimum, the Crame
´r–Rao bound. Estimator efficiency equals 1 for an estimation
error that corresponds to the Crame
´r–Rao bound.
We evaluate revenue by computing the percentage gap to the revenue
opportunity. To that end, we transform the simulation demand parameters into an
artificially accurate forecast. Using this forecast in the optimization yields the
revenue opportunity.
5.2.1 Estimator efficiency
Figures 3,4and 5depict estimator efficiency on the y-axis; they indicate the
estimated mean and 95% confidence intervals. Note that as a ratio of the estimates’
mean squared error and the Crame
´r–Rao bound, estimator efficiency is a unit-less
number.
Figure 3aggregates results across all scenarios. The efficiency of sequential
estimation is close to 0, as the mean squared error exceeds the posterior Crame
´r–
Rao bound by several orders of magnitude. The particle filter’s error exceeds the
posterior Crame
´r–Rao bound by a factor of about three, leading to an efficiency of
82 P. Bartke et al.
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0.0
0.1
0.2
0.3
0.4
0.5
LSE MLE UKF PF
Estimation Method
Estimator Efficiency
Fig. 3 Estimator efficiency across scenarios; boxes represent the mean and the 95% confidence interval.
LSE sequential estimation, MLE maximum-likelihood estimation, UKF unscented Kalman filter, PF
particle filter
Low Medium High
0.0
0.2
0.4
0.6
LSE MLE UKF PF LSE MLE UKF PF LSE MLE UKF PF
Estimation Method
Estimator Efficiency
Fig. 4 Estimator efficiency by demand volume
Domestic Continental Intercont
0.0
0.2
0.4
0.6
0.8
LSE MLE UKF PF LSE MLE UKF PF LSE MLE UKF PF
Estimation Method
Estimator Efficiency
Fig. 5 Estimator efficiency by route type
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0.3. Both unscented Kalman filter and maximum-likelihood estimation perform
significantly better. The unscented Kalman filter slightly outperforms maximum-
likelihood estimation, but the difference is not statistically significant.
Figure 4aggregates results to highlight the estimators’ sensitivity to demand
volume. When demand is high, efficiency suffers across all methods. One
explanation is that high demand reduces the quality of availability information.
From high demand forecasts, optimization computes restrictive inventory controls,
i.e., high bid prices. When bid prices are high, availability may change with each
booking and every update of the bid price vector. None of the estimates account for
availability on such a fine level; all merely approximate the availability of classes
per time slice.
Figure 5aggregates scenarios by route type. For intercontinental routes, all
methods show reduced performance. Note that this effect is statistically significant
for maximum-likelihood estimation and unscented Kalman filter, but not for particle
filter. While the differences are not statistically significant, the unscented Kalman
filter outperforms other methods in this scenario.
To analyze the performance difference on intercontinental routes, we consider
their difference to domestic routes. Price elasticity coefficients and their relative
variance do not significantly differ from domestic to intercontinental routes.
However, intercontinental routes include about double the amount of independent
demand and this demand is much more concentrated in the cheaper fare classes.
Discussions with industry experts revealed that this might be due to a higher share
of code-share bookings. During demand estimation, a significant share of
independent demand in the cheapest available class adds noise: now, when
expected and actual bookings differ in the cheapest available class, this may be due
to a shift in the amount of price-sensitive demand, a shift in price-elasticity, or a
shift in the amount of independent demand.
5.2.2 Revenue
Figures 6,7and 8depict the percentage gap to the revenue opportunity on the y-
axis; they indicate the estimated mean and 95% confidence intervals. Figure 6
displays the results across scenarios, while Figs. 7and 8aggregate results by
demand volume and route type.
Across all scenarios, sequential estimation yields significantly less revenue than
all other methods. Particle filter comes in second. Maximum-likelihood estimation
and unscented Kalman filter achieve the highest revenues, with only a slight gap to
the opportunity. The revenue difference between the unscented Kalman filter and
the maximum-likelihood estimation is not statistically significant.
The different aggregation levels illustrated by Figs. 7and 8show stable qual-
itative results across all groups of scenarios. Differences in demand volume or route
type do not affect the qualitative order of revenue performance.
Clearly, estimator efficiency and revenue are closely related. An efficiency of about
0.4 suffices to let both maximum-likelihood estimation and the unscented Kalman filter
earn almost the full revenue opportunity. A 0.1 reduction in efficiency based on the
particle filter induces only a revenue loss of less than two percent points. In contrast, the
84 P. Bartke et al.
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very low efficiency of sequential estimation leads to a revenue opportunitygap of almost
15%. However, these effects are not as pronounced for intercontinental routes. This
could, again, be due to the comparatively high ratio of independent demand.
−15
−10
−5
0
LSE MLE UKF PF
Estimation Method
Revenue Loss (%)
Fig. 6 Overall revenue opportunity gap in %; boxes represent the mean and the 95% confidence interval.
LSE sequential estimation, MLE maximum-likelihood estimation, UKF unscented Kalman filter, PF
particle filter
Low Medium High
−15
−10
−5
0
LSE MLE UKF PF LSE MLE UKF PF LSE MLE UKF PF
Estimation Method
Revenue Loss (%)
Fig. 7 Revenue opportunity gap in %, by demand volume
Domestic Continental Intercont
−20
−15
−10
−5
0
LSE MLE UKF PF LSE MLE UKF PF LSE MLE UKF PF
Estimation Method
Revenue Loss (%)
Fig. 8 Revenue opportunity gap in %, by route type
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6 Conclusion
This paper presented two filter-based approaches to demand estimation for revenue
management and benchmarked them on classical approaches. We realized this by
embedding alternative demand forecasts in a fully simulated airline revenue
management system. This enabled us to measure performance not only in terms of
efficiency given the posterior Crame
´r–Rao bound, but also in terms of revenue. The
resulting simulation study implemented empirically calibrated scenarios, represent-
ing the route types observed in a typical airline network based on data provided by
an industry partner. The objective of this study was to test the estimators’
performance on a realistic range of routes.
Clearly, sequential estimation is inferior to the more sophisticated methods, both
in terms of estimator efficiency and revenue. We conclude that sequential estimation
cannot support complex demand models such as the hybrid model implemented
here. From a practitioner’s standpoint, the large revenue gap of such a popular
method is especially troublesome. For any airline network of a reasonable size, the
added cost and complexity of more sophisticated methods are clearly a good
investment.
In addition to an adapted version of the unscented Kalman filter, we proposed to
employ a particle filter. This approach appeared promising as the particle filter does
not assume a particular form of the belief function. However, depending on the
scenario, the particle filter is slightly inferior to the alternatives—increasing its
performance by increasing the number of particles would also significantly increase
the computational cost. Across all scenarios, this leads to a performance gap to the
unscented Kalman filter as well as maximum-likelihood estimation that is just
barely significant at the 95% confidence level, both in terms of estimator efficiency
and revenue.
Finally, maximum-likelihood estimation and the unscented Kalman filter perform
similarly well across all scenarios according to both metrics. The unscented Kalman
filter, however, is computationally more efficient and additionally computes the
estimated parameters’ covariance. Thus, it indicates demand uncertainty, which
could be useful for revenue management analysts: if the forecast differs strongly
from the analyst’s intuition, this measure of uncertainty could support decisions
with regard to forecast adjustments. Furthermore, it could also support methods of
robust revenue optimization relying on upper and lower bounds of demand as
suggested in Lan et al. (2008) or Perakis and Roels (2010).
Future research could consider further approaches to demand estimation for
revenue management with regard to computational effort, versatility, and measure-
ments of demand uncertainty. While we emphasized the versatility of the filter-
based approaches, the computational study is clearly limited in that it only
benchmarks approaches on a single demand model as found in practice. As more
detailed demand models require finer aggregation levels, the ratio of observations to
parameters decreases. Here, new methods to overcome the resulting problem of
small numbers are called for. Finally, we regard the potential conflict between
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robust revenue maximization and active learning approaches that sacrifice potential
revenue to increase forecast accuracy as an interesting venue for future work.
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