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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.

123

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.

Benchmarking ﬁlter-based demand estimates for... 59

123

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 t1 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 t1. 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

AtSet 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

etNð0;QÞMultivariate Gaussian random variable with zero mean and covariance matrix Q, used

to describe demand evolution from period tto period tþ1

Benchmarking ﬁlter-based demand estimates for... 63

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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.

123

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

Xtþ1¼XtþetetNð0;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

Xtþet0.

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

123

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, fif0. Then, the probability of observing a booking,

given this offer, is

PðwfiÞ¼exp fif0

f0

:ð2Þ

For each time slice sof the sales period, the expected number of bookings for the

cheapest offer iis ^

btis¼1

Tktexp 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: atis¼4btis. Thus,

when two bookings are already accepted, a

tis¼42¼2. Alternatively, an

inventory system that relies on boolean controls requires a different availability

function, e.g.

a

tðbtisÞ¼ 0;if btis4;

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.

123

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 Pki

^

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 ¼Pki

^

Btk

Pki1

^

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 Pki

^

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

Pki

^

Btk fkÞð

Pki1

^

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;1fðRðsÞDVsþ1ðxÞÞug;

ð4Þ

where VsðxÞis the value function with boundary conditions VTðxÞ¼08xand

Vsð0Þ¼08s. The value DVsþ1ðxÞ¼Vsþ1ðxÞVsþ1ðx1Þis 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

123

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;etNð0;QÞð5Þ

BtPoiðHðAt;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.

BtPoiðHðAt;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

VNð0;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:t1;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^

Xt1kÞConditional probability linking estimated parameters for sales

period tto those estimated for sales period t1

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^

Xt01ÞConditional probability linking estimated parameters for sales period t0to those estimated

for period t01

pðBt0jX;At0ÞConditional probability of observing bookings Bt0given demand parameters Xand

inventory controls At0

LðXÞLikelihood function of demand parameters X

Q1Inverse of covariance matrix Q

Benchmarking ﬁlter-based demand estimates for... 69

123

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;XCÞþVVNð0;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^

Xtﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jXjþc

pUjjXCjj¼jXCjþ1;...;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

cþjXjg0þ1

2ðjXjþcÞX

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

cþjXjþð1a2þbÞ

ðg0^

BCÞðg0^

BCÞT

þ1

2ðjXjþcÞX

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

2ðjXjþcÞX

2jXCj

j¼1

ðrj^

XÞðgj^

BCÞT:ð17Þ

By combining PX^

BCand the upper left jXLjjXLjblock 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;XLÞPLLðA;XLÞTþP^

BC^

BCþdiagðHðA;ð^

XÞÞ:ð18Þ

With the left jXjjXLjblock of P,PCL, it constructs the total cross-covariance, PX^

B,

as

PX^

B¼PCL LðA;XLÞTþPX^

BC:ð19Þ

The Kalman gain K¼PX^

BP1

^

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^

XtþKðBt^

BtÞð20Þ

Ptþ1¼PtKP ^

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:t1;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!erisð22Þ

2. pð^

Xtkj^

Xt1kÞ, the conditional probability of estimated demand parameters ^

Xtk

resulting from the evolution of a demand estimate ^

Xt1kfrom the previous

period t1. Hence, pð^

Xtkj^

Xt1;kÞdescribes the state evolution. Equation 5

states that this probability follows a multivariate Gaussian distribution with

mean ^

Xt1;kand covariance Q.

For each particle ^

Xtk, the particle ﬁlter algorithm iterates over the following

steps:

1. Sample ^

Xtk pðXj^

X0:t1;k;B0:tÞ.

2. Compute importance weights x0

t;kas

x0

t;k¼xt1;kpðBtjAt;^

XtkÞpð^

Xtkj^

Xt1;kÞ

pð^

Xtkj^

X0:t1;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:t1;k;B0:tÞ¼pðXjXt1;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 R¼L

00ðXÞ1and mean m¼XþRL

0ðXÞ, where LðXÞis the log-

likelihood function: LðXÞ¼log pðXjXt1;Bt;AtÞ. Its ﬁrst two derivatives are

Benchmarking ﬁlter-based demand estimates for... 73

123

LðXÞ¼const:1

2ðXXt1ÞTQ1ðXXt1Þ

þX

i;s:HðAt;XÞis[0

Btislog HðAt;XÞisHðAt;XÞis

;ð24Þ

rXLðXÞ¼Q1ðXXt1Þ

þX

i;s:HðAt;XÞis[0

rxHðAt;XÞisBtis

HðAt;XÞis

1

;ð25Þ

DX

XLðXÞ¼Q1

þX

i;s:HðAt;XÞi;s[0DX

XHðAt;XÞi;sBt;i;s

HðAt;XÞis

1

ðr

XHðAt;XÞisÞðrXHðAt;XÞisÞTBtis

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 pðXjXt1;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 N¼

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

Tktexp 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 btisexp tfi

f0

ð28Þ

0

tþ1¼log T btisf0

ktfi

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:

ktþ1¼ktð1aÞþk0

tþ1að30Þ

tþ1¼tð1aÞþ0

tþ1að31Þ

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;Xt0ÞpðXt0jXt01Þ: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^

Xt1. Since Xt0acts as a prior in the above equation, we

chose to set Xt0equal to its most recent estimate ^

Xt1. Then, the joint probability

function is deﬁned as

pðB;A;XÞ¼pðXj^

Xt1ÞY

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

¼1

2ðX^

Xt0ÞTQ1ðX^

Xt0Þ

þX

t

t0¼t0þ1X

i;s:HðX;At0Þis[0

Bt0islog ðHðX;At0ÞisÞHðX;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 ^

Xt1, 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

123

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

78 P. Bartke et al.

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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;XÞi¼kexp 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

123

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

80 P. Bartke et al.

123

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 diagðQÞ= 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

Benchmarking ﬁlter-based demand estimates for... 81

123

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.

123

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

Benchmarking ﬁlter-based demand estimates for... 83

123

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

Benchmarking ﬁlter-based demand estimates for... 85

123

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

86 P. Bartke et al.

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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.

References

Araman VF, Caldentey R (2011) Revenue management with incomplete demand information.

Encyclopedia of Operations Research Wiley

Azadeh SS, Marcotte P, Savard G (2014) A taxonomy of demand uncensoring methods in revenue

management. J Revenue Pricing Manag 13(6):440–456

Ball MO, Queyranne M (2009) Toward robust revenue management: competitive analysis of online

booking. Oper Res 57(4):950–963

Besbes O, Zeevi A (2009) Dynamic pricing without knowing the demand function: risk bounds and near-

optimal algorithms. Oper Res 57(6):1407–1420

Carvalho A, Puterman M (2015) Dynamic optimization and learning: how should a manager set prices

when the demand function is unknown? Instituto de Pesquisa Econo

ˆmica Aplicada (Ipea)

Chung B, Li J, Yao T, Kwon C, Friesz T (2012) Demand learning and dynamic pricing under competition

in a state-space framework. Eng Manag IEEE Trans 59(2):240–249

Doucet A, Godsill S, Andrieu C (2000) On sequential Monte Carlo sampling methods for Bayesian

ﬁltering. Stat Comput 10(3):197–208

Farias VF, Jagabathula S, Shah D (2013) A nonparametric approach to modeling choice with limited data.

Manag Sci 59(2):305–322

Fiig T, Isler K, Hopperstad C, Belobaba P (2009) Optimization of mixed fare structures: theory and

applications. J Revenue Pricing Manag 9(1):152–170

Gerlach M, Cleophas C, Frank M (2010) Introducing REMATE: revenue management simulation in

practice. In: AGIFORS working group revenue management and cargo, New York

Gordon N, Salmond D, Smith A (1993) Novel approach to nonlinear/non-Gaussian Bayesian state

estimation. Radar Signal Process IEE Proc 140:107–113

Haensel A, Koole G (2011) Estimating unconstrained demand rate functions using customer choice sets.

J Revenue Pricing Manag 10(5):438–454

Julier S, Uhlmann J (2004) Unscented ﬁltering and nonlinear estimation. Proc IEEE 92(3):401–422

Julier S, Uhlmann J (1997) a new extension of the kalman ﬁlter to nonlinear systems. In: Signal

processing, sensor fusion, and target recognition VI; Proceedings of the conference, pp 182–193

Kalman R (1960) A new approach to linear ﬁltering and prediction problems. Trans ASME J Basic Eng

82(Series D):35–45

Keskin NB, Zeevi A (2014) Dynamic pricing with an unknown demand model: asymptotically optimal

semi-myopic policies. Oper Res 62(5):1142–1167

Kitagawa G (1996) Monte Carlo ﬁlter and smoother for non-gaussian nonlinear state space models.

J Comput Graph Stat 5(1):1–25

Kwon C, Friesz T, Mookherjee R, Yao T, Feng B (2009) Non-cooperative competition among revenue

maximizing service providers with demand learning. Eur J Oper Res 197(3):981–996

Lan Y, Gao H, Ball MO, Karaesmen I (2008) Revenue management with limited demand information.

Manag Sci 54(9):1594–1609

Li J, Yao T, Gao H (2009) A revenue maximizing strategy based on bayesian analysis of demand

dynamics. In: SIAM Proceedings: mathematics for industry, society for industrial and applied

mathematics, San Francisco, pp 174–181

Lobo M, Boyd S (2003) Pricing and learning with uncertain demand. Fuqua School of Business, Duke

University (Preprint)

Mukhopadhyay S, Samaddar S, Colville G (2007) Improving revenue management decision making for

airlines by evaluating analyst-adjusted passenger demand forecasts. Decis Sci 38(2):309–327

Mu

¨ller P (1991) Monte Carlo integration in general dynamic models. In: Statistical multiple integration:

Proceedings of an AMS-IMS-SIAM joint research conference, pp 145–162

Perakis G, Roels G (2010) Robust controls for network revenue management. Manuf Serv Oper Manag

12(1):56–76

Scholz FW (2004) Maximum likelihood estimation. Wiley. doi:10.1002/0471667196.ess1571.pub2

Benchmarking ﬁlter-based demand estimates for... 87

123

Stefanescu C (2009) Multivariate customer demand: modeling and estimation from censored sales.

Available at SSRN 1334353

Talluri K, van Ryzin GJ (2005) The theory and practice of revenue management. Springer, New York

Tichavsky P, Muravchik C, Nehorai A (1998) Posterior Crame

´r-Rao bounds for discrete-time nonlinear

ﬁltering. IEEE Trans Signal Process 46(5):1386–1396

van Ryzin GJ, McGill J (2000) Revenue management without forecasting or optimization: an adaptive

algorithm for determining airline seat protection levels. Manag Sci 46(6):760–775

van Ryzin GJ, Vulcano G (2015) A market discovery algorithm to estimate a general class of

nonparametric choice models. Manag Sci 61(2):281–300

Vulcano G, van Ryzin GJ, Ratliff R (2012) Estimating primary demand for substitutable products from

sales transaction data. Oper Res 60(2):313–334

Weatherford L (2016) The history of unconstraining models in revenue management. J Revenue Pricing

Manag 15(3):222–228

88 P. Bartke et al.

123