A next generation approach to estimating customer demand ... · These approaches tend to utilize the availability of classes (or more generally, availability of choice sets) to isolate

Research Article

A next generation approach to estimatingcustomer demand via willingness to payReceived (in revised form): 1st January 2014

Edward KambourPROS Inc., 3100 Main Street, Suite 900, Houston, TX 77382, USA

Edward Kambour has been with PROS since September of 1998 as a member of the Science and ResearchDepartment. He currently holds the title of Principal Research Scientist. His responsibilities include sciencestatistical support across all PROS applications, research aimed at improving existing forecast methods as wellas deriving new methodologies, and science liaison with Software Development for implementation andmaintenance of the science algorithms. He earned his PhD in Statistics at Texas A&M University. He hasauthored, contributed or presented a number of leading edge Revenue Management research topics such asBayesian forecasting (configuration and practice), optimizing in the presence of uncertainty, mathematicalunconstraining, low fare demand models, the effects of competition, customer segmentation and price demandmodels. He received the Best Technical Presentation Award at the 2003 and 2008 AGIFORS Reservations andYield Management Meetings.

Correspondence: Edward Kambour, PROS Inc., 3100 Main Street, Suite 900, Houston, TX 77382, USAE-mail: [email protected]

ABSTRACT Price-driven passenger behavior has been addressed in many different ways over the lastseveral years. These approaches tend to utilize the availability of classes (or more generally, availability ofchoice sets) to isolate passenger’s buying behavior and address spiral down/fare dilution. The new approachwill instead examine only the products purchased and the prices paid, to isolate passenger priceable demandbehavior. In this article, we will cover the concepts and the underlying statistical forecast model and showexamples of the willingness to pay method in practice.Journal of Revenue and Pricing Management (2014) 13, 354–365. doi:10.1057/rpm.2014.20;published online 27 June 2014

Keywords: airline revenue management; willingness to pay; price-sensitive demand; demand curveestimation; forecasting

INTRODUCTIONOver the last 15 years or so, there have beenmany who have noted that the independentclass demand approach to revenue managementwas flawed. The concept underlying this obser-vation was that passengers were not completelyfenced into their own specific fare product, butthat instead passengers would choose among aset of currently available products. Proposalsincluded the use of sell-up probabilities orinput elasticity/Q-forecasting (Hopperstad,2000; Belobaba and Hopperstad, 2004), hybrid

forecast models (Kambour et al, 2001, Boydet al, 2004; Walczak and Kambour, 2014) andmore general choice models (Talluri and vanRyzin, 2004, Vulcano and van Ryzin, 2006 andFiig et al, 2012). The sell-up and Q-forecastingapproaches depend on expert input of theelasticity, whereas the hybrid and choiceapproaches both are designed to generatedemand forecasts. For both the hybrid andchoice models these forecasts take as input thehistorical choice sets available to prospectivepassengers, where the choice sets depend on

© 2014 Macmillan Publishers Ltd. 1476-6930 Journal of Revenue and Pricing Management Vol. 13, 5, 354–365www.palgrave-journals.com/rpm/

mailto:[email protected]

http://dx.doi.org/10.1057/rpm.2014.20

http://www.palgrave-journals.com/rpm

the historical class availability. The goal of thisresearch is to create a new forecast model that

1. directly forecasts elasticity,2. captures price-driven (priceable) demand and3. does not depend on historical availability.

We have termed this new approach the will-ingness to pay (W2P) model.

The W2P model is built to capture purepriceable demand, defined as demand that willpurchase the lowest available fare rather than byindividual fare class among classes within a givenset of classes. This can work within a fare familystructure, wherein we model the demand inde-pendently between fare families, while the faresfor the classes within the fare family are simplydifferent price points. The W2P model uses onlybookings information, the volume of bookingsand the fares that were paid. There will be anadditional assumption regarding the fare availabilitythat the carrier provides to prospective passengers,which we label the Dart Game Approach. Thefinal result of the W2P model is a demand curveforecast that combines both elasticity and volume.

W2P MODEL CONCEPTIn the W2P model, we characterize the faresthat are offered to passengers as targeted towardthe maximum fare that the passengers are will-ing to pay. We label this the Dart GameApproach. Conceptually, we model the lowestfare offered as a dart throw at a dart board where

the bulls-eye is the potential passenger’s max-imum W2P, as in Figure 1.

As there is uncertainty in the passenger’sW2P, we model the offers as randomly dis-persed around the W2P bulls-eye. For example,if there are three such historical offerings toa particular passenger, we expect them to becentered on his maximum fare, but not all ofthem right on the bulls-eye, as in Figure 2.

The prospective passenger will only bookwhen the offered fare is less than or equal to hismaximum price. In the terms of the dart game,this means that we only observe the darts thatland to the left of the bulls-eye, as in Figure 3.

In addition there are many different prospec-tive passengers, all with different (though simi-lar) maximum prices. As a result, the observed

Figure 1: Dart board – Passenger’s maximum fare is thebulls-eye.

Figure 3: Dart board – Only see darts to the left of thebulls-eye.

Figure 2: Dart board – Passenger’s maximum fare is thebulls-eye, offered fares are darts.

Estimating customer demand via willingness to pay

355© 2014 Macmillan Publishers Ltd. 1476-6930 Journal of Revenue and Pricing Management Vol. 13, 5, 354–365

purchased fares are censored (that is, we onlysee the fares that were paid) with two sourcesof randomness, the variability across passengersand the uncertainty in the individual passengermaximum price.

W2P STATISTICAL MODELFor a given offer, say offer i, label the offeredfare as Yi and the passenger’s maximum price asWi. We model the variability in the maximumprices across passengers, Wi, as normally distrib-uted with mean μ and variance σ2. We modelthe offered fare, Yi, given Wi=wi, as normallydistributed with mean wi and variance σ2.Note that we are modeling the two sources ofvariance as homogeneous, that is the error in theestimation of the W2P is set equal to the naturalvariability among passengers. Mathematically,

Wi �N μ; σ2� �

Yi jWi ¼ wi �N w; σ2� �

As pointed out in the previous section, thedata will not contain either the Yi or Wi, butinstead the observations consist of the paid fares.That is, for an observed purchase j, Xj consists ofthe offered fare, given that it was less than thepassenger’s maximum price.

Xj ¼ Yi Yi⩽j Wi

To estimate the W2P mean and variance(μ and σ2), we will create the probability dis-tribution of the observed purchased fare, that is,the X-vector. We first form the joint distribu-tion of Yi and Wi then condition on Yi<Wi.

The likelihood function for μ and σ2 isproduct of the probability distributions foracross all purchased prices. That is,

L μ; σ2� � ¼ Y

j

f xj� �

from which we can find maximum likelihoodestimators of the W2P, μ and σ2. Owing to thecomplexity of the joint distribution and, thus,the likelihood function, there are not closedform estimators. Instead, the likelihood maxi-mization necessitates an iterative method.

W2P MODEL EXAMPLETo illustrate the approach, we examine a setof actual paid fare data. The histogram belowshows the historical fares paid for a majorinternational airline for a short-haul Origin-Destination, over the period of a year, within agiven fare family. The fares in the histogramhave been listed in US dollars. The average

f wi; yið Þ ¼ f wið Þf yi wijð Þ ¼ 12πσ2

exp -wi - μð Þ22σ2

( )exp -

yi -wið Þ22σ2

( )

f xj� � ¼ f yi yi⩽j wið Þ ¼

Zwi⩾yi

12πσ2


( )exp -

yi -wið Þ22σ2

( )dwi

Zyi

Zwi⩾yi

12πσ2


( )exp -

yi -wið Þ22σ2

( )dwidyi

¼

12πσ2

Z1wi¼yi

exp -2w2

i - 2 μ + yið Þwi + μ2 + y2i2σ2

� �dwi

0:5

¼ 1πσ2

exp -yi - μð Þ24σ2

( ) Z1wi¼yi

exp -2 wi -

μ + yi2

� �22σ2

( )dwi

Kambour

356 © 2014 Macmillan Publishers Ltd. 1476-6930 Journal of Revenue and Pricing Management Vol. 13, 5, 354–365

purchased fare was approximately US$220,with considerable variation (Figure 4).

Recall that this data contains purchased fares,which are always less than or equal to eachpassenger’s W2P. We can estimate the meanand variance of the W2P distribution, asdescribed in the previous section; intuitivelyone can think of this as unconstraining thepaid fare information to get at the true W2P.Figure 5 shows the same histogram of paid faresalong with the fitted W2P distribution.

Note that the W2P fit mean has shiftedto the right, allowing for the unconstrainingof the paid fares to the max purchase price fora customer. We estimate the mean W2P atapproximately $300, in this example.

EXTENDING FROM W2P TO THEDEMAND CURVEWe can move to an estimate of the probabilitythat a given passenger will purchase at a given

0%

2%

4%

6%

8%

10%

12%

20 60 100

140

180

220

260

300

340

380

420

460

500

540

580

620

660

700

740

780

820

860

900

940

980

% B

ooki

ngs

Purchased Fare Values

Sample Mean = 220.33

Sample Standard Deviation = 128.96

Figure 4: W2P example, historical fares purchased.

0 100 200 300 400 500 600 700 800 900 1000

%B

ooki

ngs

Purchased Fare Values

W2P Mean = 307.08

W2P Standard Deviation = 108.74

Figure 5: W2P example, historical fares purchased and fitted W2P.



price by examining the survivor function for theestimated W2P distribution, fW. That is,

SW pð Þ ¼Z1p

fW wð Þdw

These probabilities can then be used tocalculate the demand elasticity, or buy-up per1-centages. This satisfies criteria 1 and 3 from theIntroduction, but does not provide a demandvolume estimate. In this section, we will showthat we can estimate the demand directly fromthe history and combine it with the elasticityfrom the survivor function to yield a forecastdemand curve.

Let D and Y a bivariate random variable, inour particular case D is the booking demandvolume at a given fare offer Y. Let the demandcurve that describes the price demand relation-ship, be a function, g. That is,

E D Y ¼ yj½ � ¼ g yð Þ

By iterated expectation, we can show thatthe marginal expectation for D is,

E D½ � ¼ EY ED Yj D½ �� ¼ EY g Yð Þ½ �

We create the Taylor expansion for g, aboutthe expected offered price, Y, and substituteinto the marginal expectation for the demandvolume,

g yð Þ ¼ g E Y½ �ð Þ

+X1i¼1

g ið Þ E Y½ �ð Þ y -E Y½ �ð Þii !

Taylor Seriesð Þ

So,

EY g Yð Þ½ � ¼ g E Y½ �ð Þ

+X1i¼1

g ið Þ E Y½ �ð ÞEY Y -E Y½ �ð Þi� �i !

Recall that based on the W2P statisticalmodel, Y is a normally distributed randomvariable.

We begin by examining the different termsin the summation, beginning with

EY Y -E Y½ �ð Þi� �As Y is a normal random variable, all of the

odd moments are zero, so the Taylor expansionreduces to

EY g Yð Þ½ � ¼g E Y½ �ð Þ

+X1i¼1ieven

g ið Þ E Y½ �ð ÞEY Y -E Y½ �ð Þi� �i !

Now, let us examine the other term in thesummation,

g ið Þ E Y½ �ð Þg represents the demand function, that is theexpected demand at a given price. We know thatit is related to the W2P survivor function, S.Specifically, we know that they have the sameshape, as S captures the elasticity in g. Thus, g is aconstant multiplied by the survivor function S,which, based on the W2P model, is the survivorfunction for a normal random variable (W ).Thus, the ith derivative of g is a constant timesthe (i-1)st derivative of a normal probabilitydensity function.

For a normal random variable, one canshow that the odd derivatives of a normal pdf,evaluated at its expectation are all zero. Themost straight-forward approach is to note therelationship between the normal probabilitydensity function and the Hermite polynomials,but this can also be shown directly via mathe-matical induction. The implication is that theeven derivatives of g are a constant multiplied bythe odd derivatives of a normal pdf evaluated atits mean, which are known to be zero. That is,

g ið Þ E Y½ �ð Þ ¼ 0; for i even

Thus, the Taylor expansion reduces furtherto

EY g Yð Þ½ � ¼ g E Y½ �ð ÞThat is, the marginal expectation for the

demand volume is equal to the demand curveevaluated at the mean price. More succinctly,

Kambour


the point (E[Y], E[D]) lies on the demandcurve, as shown in Figure 6.

Thus, if we can use the expected demandbased on the history, together with the expectedoffered price (equal to the W2P mean), alongwith the probability of purchase to yield theforecasted demand at a given price.

One additional note is that one can applymore sophisticated forecast models to yieldexpected demand and expected price, for exam-ple, time series approaches that capture season-ality and dynamic changes.

NOTES REGARDING THE W2PMODEL ASSUMPTIONSAs noted in the W2P Statistical Model section,the model is based on underlying assumptionsregarding the W2P distribution and the offeredfares. Specifically, the model assumes the W2Pdistribution and the conditional distribution ofthe offered fares are normally distributed, withhomogenous variance. The assumption of homo-geneity in variance relies on the concept that theoffered fares are as precise as the underlyingvariability in the passsengers’ W2P. The assump-tion is also made that the conditional mean of thefare offered to a given passenger lies at the W2Pfor that passenger; that is that the offers aretargeted as the passenger’s W2P. This assumptionregarding the offered fare distribution explains whythe model does not include the percentage of timethat given fares (or fare classes) are available; theconcept is that the fare availability is a product of

the prospective passengers’ W2P and is, thus,captured indirectly in the paid fare information.

The normality assumptions are important interms of the derivation of the model, as we doapply a maximum likelihood approach. It is alsoimportant to note the normality assumptionallows for the reduction of the Taylor Seriesrepresentation, as the odd moments of a normalrandom variable and the even derivatives of thenormal Survivor function are all zeros. As such,one approach to check for normality and extendthe model would be to utilize normalizingtransformations (Box and Cox, 1964; Tukey,1977, Draper and Smith, 1981).

AIRLINE CASE STUDYTo illustrate the methodology, we applied theW2P approach to an airline data set, containingbookings and paid fares. The data set in questionis for a short haul, with strong local demand.We will investigate a leisure fare family withfour classes; the four classes are genericallylabeled as Y, B, M and Q going from highestto lowest fare within the family. The fares havebeen scaled and are represented in US dollars.We applied a generalized dynamic linear model(GDLM) for the price and volume forecasts,thus the forecasts will show seasonality and willput more weight on the most recent observa-tions than earlier bookings or paid fares.

Figure 7 shows a histogram of the historicalfares paid, along with the estimated W2Pdistribution.

We can see that there is considerable varia-bility in the historical fare data, with an averagepaid fare of approximately $140. The W2P fitalso shows considerable variability, but estimatesthe mean W2P for passengers at about $164.

Figure 8 shows a high-level aggregation ofthe future forecast by price. This is aggregatedover all the flights for a year of departures. It alsoshows the current fare value for each of theclasses in the fare family.

We note that the demand tends to berelatively stable around the Q and M fares, butwe see a large drop as we move to B or Y.

Price

Demand(E[Price],E[Demand])

Figure 6: Forecast demand curve.



As noted, we applied a GDLM model forthe price and volume components. This allowsus to capture trends and seasonal patterns inboth volume and price (and thus demand leveland elasticity). Figure 9, below, shows the

historical bookings totals by departure weekand the forecasted demand at the expectedW2P for a year of future departures. Thiscorresponds to the point (E[Y ], E[D]) on thedemand curve for each future departure.

0

100000

200000

300000

400000

500000

600000

85.00 105.00 125.00 145.00 165.00 185.00 205.00 225.00

An

nu

alD

eman

d

Fare Value/Price

Demand Q M B Y

Figure 8: Aggregated demand curve.

87 91 94 97 100

103

106

109

112

115

118

121

124

127

130

134

137

140

143

146

149

152

155

158

161

164

167

170

173

177

180

183

186

189

192

195

198

201

204

207

210

213

216

220

223

226

229

232

235

238

241

Purchased Price W2P

Figure 7: W2P example, historical fares purchased and fitted W2P.

Kambour


Note that there has been a distinct shiftupward in demand, about 65 per cent of theway through the historical departures. The

forecast profile, at the expected passengerW2P, has adjusted appropriately. Further theforecast is capturing the general seasonal pattern

0

2000

4000

6000

8000

10000

12000

Wee

kly

Bo

oki

ng

s

Departure Week

History Forecast

Figure 10: Historical bookings and forecast at Q fare.

0

2000

4000

6000

8000

10000

12000

Wee

kly

Bo

oki

ng

s

Departure Week

History Forecast

Figure 9: Historical bookings and forecast at expected W2P.



inherent in the data, aside from individual spikesin demand.

To illustrate the priceable demand, we nowexamine the same time series chart, but now

with forecasts at the current fare values asso-ciated with the four classes in the fare family.These are shown in Figures 10–13. This allowsus to see how the forecasts are expected to

0

2000

4000

6000

8000

10000

12000

Wee

kly

Bo

oki

ng

s

Departure Week

History Forecast

Figure 11: Historical bookings and forecast at M fare.

0

2000

4000

6000

8000

10000

12000

Wee

kly

Bo

oki

ng

s

Departure Week

History Forecast

Figure 12: Historical bookings and forecast at B fare.

Kambour


respond to changes in the offered price, that is,class availability.

Note that while the seasonal forecast patternsare somewhat similar, they are not identical.

This is particularly evident when comparing theprofiles for classes Q or M to B or Y. Thisindicates that the demand level and the elasticitymay both be showing seasonality.

0

2000

4000

6000

8000

10000

12000

Wee

kly

Bo

oki

ng

s

Departure Week

Peak

Low

History Forecast

Figure 14: Historical bookings and forecast at Y fare.

0

2000

4000

6000

8000

10000

12000

Wee

kly

Bo

oki

ng

s

Departure Week

History Forecast

Figure 13: Historical bookings and forecast at Y fare.



To illustrate the dynamics in the elasticityforecast, we will examine the demand curve fitfor two different departure weeks. These twospecific weeks are explicitly shown in Figure 14.

Figure 15 shows the forecasted demandcurve for the obvious peak departure for class Y,as marked in above. Figure 16 shows thedemand curve for the low departure week.

0

2000

4000

6000

8000

10000

12000

85.00 105.00 125.00 145.00 165.00 185.00 205.00 225.00

Dem

and

Fare Value/PriceDemand Q M B Y

Figure 16: Forecast demand curve – Low departure week.

0

2000

4000

6000

8000

10000

12000

Dem

and

Fare Value/PriceDemand Q M B Y

85.00 105.00 125.00 145.00 165.00 185.00 205.00 225.00

Figure 15: Forecast demand curve – Peak departure week.

Kambour


The demand curve forecasts differ consider-ably, between the peak and the low departureweeks. There is much greater price impact ondemand evident in the low period.

CONCLUSIONIn this article, we have proposed a new forecastmodel that will work within a fare family and isdesigned to

1. directly forecast elasticity;2. capture price-driven (priceable) demand;

and3. not depend on historical availability.

We have put forward the underlying statisticalmodel with its assumptions and shown resultsfrom an airline case study. In the case study, wesee how the historical paid fares and bookingstranslate to forecasts of the elasticity anddemand curve. Though the mapping is notsimple, or even closed form, these initial checksdo yield an example of the how theW2P modelwould work in practice.

An important next step will be to compareand contrast the W2P forecasts with otherapproaches for estimating priceable demand,for example, the hybrid approach that utilizesfare availability as an input. Another step wouldentail a proper method to compare actuals withforecasts; this would entail either appropriatelyconstraining the W2P forecast to compare withactual booking counts or manipulating theactuals to compare the W2P forecast. There arealso plausible extensions to the forecast model,

including the application of transformationmethods, relaxations of the homogeneityassumption as well as examination of the offerdistribution, particularly allowing for aiming tothe left or right of passengers’ W2P.

REFERENCESBelobaba, P. and Hopperstad, C. (2004) Algorithms for revenue

management in unrestricted fare markets. Presented at the AGIFORSRevenue Management Conference, Auckland, New Zealand.

Box, G.E.P. and Cox, D.R. (1964) An analysis of transformations.Journal of the Royal Statistical Society, Series B 26: 211–243.

Boyd, E., Kallesen, R. and Kambour, E. (2004) Changes inpassenger purchasing behavior and their impact on revenuemanagement models. Presented at the AGIFORS RevenueManagement Conference, Auckland, New Zealand.

Draper, N.R. and Smith, H. (1981) Applied Regression Analysis.New York: Wiley.

Fiig, T., Isler, K., Hopperstad, C. and Olsen, S. (2012) Forecastingand optimization of fare families. Journal of Revenue and PricingManagement 11: 322–342.

Hopperstad, C. (2000) Modeling sell-up in PODS, enhancementsto existing sell-up algorithms. Presented at the AGIFORSReservations and Yield Management Study Group. New York.

Kambour, E., Boyd, E. and Tama, J. (2001) The impact of buy-down on sell-up, unconstraining, and spiral down. Presented atthe AGIFORS Reservations and Yield Management Conference,Bangkok, Thailand.

Talluri, K. and van Ryzin, G. (2004) Revenue managementunder a general discrete choice model of consumer behavior.Management Science 50(1): 15–33.

Tukey, J.W. (1977) Exploratory Data Analysis. Reading, MA:Addison-Wesley.

Vulcano, G. and van Ryzin, G. (2006) Choice-based revenuemanagement: An empirical study of estimation and optimiza-tion. Presented at the AGIFORS Joint Revenue Management,Distribution & Cargo Study Group Meeting, Cancun, Mexico.

Walczak, D. and Kambour, E. (2014) Revenue management forfare families with price sensitive demand. Journal of Revenue andPricing Management, doi:10.1057/rpm.20143, published online21 February 2014.



Documents

A next generation approach to estimating customer demand ... · These approaches tend to utilize the availability of classes (or more generally, availability of choice sets) to isolate