Heuristic solutions for nonlinear dynamic pricing in the presence of multiunit demand and two-dimensional customer heterogeneity

Schu , Rou en
A icle — Published Ve sion
Heu is ic solu ions o nonlinea dynamic p icing in he
p esence o mul iuni demand and wo-dimensional
cus ome he e ogenei y
OR Spec um
Sugges ed Ci a ion: Schu , Rou en (2025) : Heu is ic solu ions o nonlinea dynamic p icing in he
p esence o mul iuni demand and wo-dimensional cus ome he e ogenei y, OR Spec um, ISSN
1436-6304, Sp inge Be lin Heidelbe g, Be lin/Heidelbe g, Vol. 47, Iss. 4, pp. 1181-1215,
h ps://doi.o g/10.1007/s00291-025-00820-3
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ORIGINAL ARTICLE
Heu is ic solu ions o nonlinea dynamic p icing
in hep esence o mul iuni demand and wo‑dimensional
cus ome he e ogenei y
Rou enSchu 1
Recei ed: 27 Feb ua y 2024 / Accep ed: 30 Ap il 2025 / Published online: 26 May 2025
© The Au ho (s) 2025
Abs ac
In his pape , we in oduce a nonlinea dynamic p icing model in he p esence o
mul iuni demand, enabling i ms o quo e sepa a e p ices o each ba ch size. This
app oach di e ges om adi ional models by accoun ing o wo-dimensional cus-
ome he e ogenei y in p oduc a ac ion and ba ch size p e e ence, each modeled
by sepa a e andom a iables in he calcula ion o cus ome s’ willingness- o-pay.
The unde lying cus ome choice model esul s in a complex o mula ion o pu chase
p obabili ies, necessi a ing conside able e o o e inemen s o de i e a manage-
able exp ession. We p esen op imali y condi ions o he s a e-wise op imiza ion
p oblem and in oduce a modi ied o mula ion wi h educed complexi y ha se es
as an uppe bound. We also p o e ha unde speci ic condi ions, he op imal solu-
ion o he modi ied model is op imal o he o iginal p oblem. In ou nume ical
s udy, hese condi ions we e consis en ly me , o e ing a p ac ical al e na i e o
de e mining op imal p ices. To add ess he compu a ional challenges o sol ing
he p oblem o op imali y, we de elop h ee e icien heu is ics wi h signi ican ly
educed un imes. Benchma king hese heu is ics agains he op imal solu ion and
o he mechanisms demons a es hei nea -op imal pe o mance. We also e alua e
he e enue po en ial o nonlinea , piecewise linea , and linea p icing schemes, p o-
iding i ms wi h ools o weigh e enue maximiza ion agains p icing simplici y o
in o m s a egic decisions. No ably, ou analysis highligh s he s ong pe o mance
o piecewise linea p icing, o e ing a p ac ical and easy- o-communica e al e na i e
o ull nonlinea p icing while achie ing ema kably high e enues.
Keywo ds Re enue managemen · Dynamic p icing· Nonlinea p icing· Mul iuni
demand· Cus ome choice
* Rou en Schu
ou [email p o ec ed]
1 Chai o P oduc ion & Logis ics Planning, Me ca o School o Managemen , Uni e si y
o Duisbu g-Essen, Lo ha s aße 65, 47057Duisbu g, Ge many
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R.Schu
1 In oduc ion
Nonlinea p icing—such as olume discoun s o special o e s like "buy 3, pay
2"—we e and s ill a e a commonly applied p icing s a egy in he e e -e ol ing
ield o e ail. This app oach is used bo h o indi idual p oduc s (e.g., cans o
soda) and ac oss p oduc lines (e.g., shi s o di e en sizes, colo s, o designs).
Enabled by ad ancemen s in digi al echnologies (e.g., e-comme ce and digi al
p ice ags), businesses can now adjus p ices in eal- ime. This allows o swi
esponses o a ying ma ke demands and in en o y le els. Howe e , mos adi-
ional dynamic p icing models ha e a c i ical limi a ion: hey usually assume ha
cus ome s pu chase only a single uni a a ime. This assump ion o e looks he
complexi ies and oppo uni ies p esen ed by mul iuni demand, which a e p e a-
len in sec o s anging om g oce ies o clo hing.
Add essing his gap, ou pape p esen s an app oach ha combines nonlinea
and dynamic p icing o op imally quo e p ices o e e y possible ba ch size o a
p oduc (o a ian wi hin a p oduc line). This amewo k is designed o maxi-
mize e enue in scena ios whe e he selling ho izon is ini e, he p oduc in en-
o y is sca ce, and cus ome s exhibi mul iuni demand. In ou con ex , mul iu-
ni demand does no necessa ily imply ha cus ome s a e p ede e mined on he
numbe o uni s hey will pu chase. Ins ead, i e e s o se ings whe e cus ome s
conside a ious quan i ies based on p icing schemes, p omo ions, o he needs o
mul iple end-use s (e.g., amily membe s, colleagues, o iends).
Fo ins ance, conside a supe ma ke dynamically adjus ing ba ch p ices o
pe ishable p oduc s, such as bake y i ems o esh p oduce. Depending on eal-
ized demand and emaining in en o y, e aile s dynamically aise ba ch p ices o
capi alize on s ong demand o lowe hem o encou age bulk pu chases as p od-
uc s app oach expi a ion. Al hough his s a egy na u ally educes was e and mi i-
ga es ea ly s ockou s, i s p ima y goal emains e enue maximiza ion gi en lim-
i ed s ock and a ini e selling ho izon. Simila dynamic nonlinea p icing schemes
apply na u ally in a ious o he sec o s, whe e he p oduc loses i s alue a a spe-
ci ic poin in ime due o ac o s such as pe ishabili y (as in ood o ho el s ays)
o he necessi y o clea aluable shel space o new in en o y (as in ashion,
seasonal goods, and consume elec onics).
Cen al o ou app oach is ou cus ome choice model ha ecognizes he wo-
dimensional he e ogenei y among cus ome s, bo h in hei a ac ion o he p od-
uc and in hei inclina ion o consume. We employ a u ili y maximiza ion ame-
wo k, whe e cus ome u ili y is modeled as a unc ion o ba ch size and he wo
cus ome -speci ic a ibu es: p oduc a ac ion and consump ion. These a ibu es
a e cus ome s’ p i a e in o ma ion and he e o e unknown o he i m. Howe e ,
we assume he i m has knowledge o he dis ibu ion o hese a ibu es ac oss
he cus ome popula ion. This allows us o ea hese a ibu es as andom a i-
ables, o ming he basis o ou andom u ili y amewo k.
Wi h hese assump ions, we align closely wi h a ecen publica ion (see Schu
2024), whe e bo h s udies aim o maximize expec ed e enues h ough dynamic
nonlinea p icing and employ he same o mula ion o cus ome s’ u ili y based
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Heu is ic solu ions o nonlinea dynamic p icing in he…
on wo cus ome -speci ic a ibu es. Acco dingly, bo h publica ions a e oo ed
in he same p oblem and use iden ical ma hema ical o mula ions o he choice
model and he gene al dynamic p og am. Howe e , unlike his publica ion, we
do no make he addi ional assump ion ha he i m can iden i y cus ome s’ p i-
a e in o ma ion—speci ically, hei a ac ion o he p oduc , hei consump ion
indica o , o bo h. This assump ion in Schu (2024) simpli ies he op imiza ion
p oblem by u ilizing cus ome s’ known a ibu es. In con as , ou model does no
ely on obse ing p i a e in o ma ion, esul ing in a mo e complex op imiza ion
p oblem. This ex ends he amewo k o se ings whe e cus ome iden i ica ion is
no possible. In his way, ou wo k gene alizes he amewo k o Schu (2024).
While echniques simila o hose used in Schu (2024), speci ically ac ion space
educ ion and uppe bound in es iga ion, can be adap ed o educe complexi y
in ou se ing, we mus adop a undamen ally di e en s a egy ha ocuses on
de eloping e ec i e heu is ics.
By no elying on p i a e in o ma ion, ou app oach emains also applicable
in scena ios whe e i ms mus quo e p ices be o e any in e ac ion wi h cus om-
e s occu s, as is ypically he case in s o es using digi al p ice ags. This lexibili y
makes ou model pa icula ly aluable o scena ios whe e ad ance p icing deci-
sions a e necessa y, o whe e he i m lacks access o cus ome s’ p i a e in o ma ion.
1.1 Resul s
Ou esea ch achie es h ee in e connec ed goals: de i ing he op imal solu ion,
de eloping e ec i e heu is ics, and e alua ing he ade-o s be ween nonlinea p ic-
ing and simple p icing schemes.
Fi s , we es ablish op imali y condi ions ha enable he compu a ion o he op i-
mal solu ion. While he op imal solu ion p o ides an impo an benchma k o e al-
ua ing he e ec i eness o ou heu is ics, i s compu a ion emains imp ac ical o
eal-wo ld applica ions due o i s subs an ial un imes, equi ing nea ly i e days in
ou la ges se ing.
Second, o o e come hese compu a ional challenges, we p opose h ee heu is ics
ha esul in well-pe o ming solu ions while signi ican ly educing un imes. Fo
example, one o hese heu is ics compu es a policy in jus 2.5min wi h an op imal-
i y gap below 0.4% in ou la ges se ing, making i nea ly 3000 imes as e han
compu ing he op imal solu ion. Simila ly, he o he heu is ics also pe o m well,
o e ing nea -op imal solu ions wi h un imes sui able o p ac ical applica ion.
Finally, using he op imal and heu is ic solu ions as benchma ks, we e alua e
he e enue po en ial o di e en p icing schemes—nonlinea , piecewise linea , and
linea p icing. Ou nume ical s udy demons a es ha nonlinea p icing consis -
en ly achie es he highes e enue. Howe e , simple p icing schemes, pa icula ly
piecewise linea p icing, can pe o m e y s ongly. This obse a ion is suppo ed
by an analysis o he s a egies employed by ou bes -pe o ming heu is ics, which
e eal wo key pa e ns. Fi s , he p icing schemes exhibi an almos piecewise linea
s uc u e. This s uc u e explains he s ong pe o mance o piecewise linea p ic-
ing in ou s udy and sugges s ha i ms could adop his simple p icing scheme o
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R.Schu
educe communica ion complexi y. By quo ing jus wo p ices—one o he i s uni
and ano he o addi ional uni s— i ms can achie e high e enues while main ain-
ing ease o implemen a ion. Second, in scena ios wi h high ini ial in en o y (
C
T≥2
)
p ices o smalle ba ches dec ease g adually o e ime. This g adual adjus men
implies ha less equen p ice changes can s ill deli e s ong pe o mance, mak-
ing such s a egies iable o i ms cons ained by echnical limi a ions, cus ome
p e e ences, o s a egic conside a ions. Con e sely, in low-s ock scena ios (
T
C≥2
),
he e enue ad an age o nonlinea p icing dec eases, and s anda d dynamic p icing
app oaches become mo e e ec i e.
1.2 Con ibu ion andou line
We con ibu e o he spa se li e a u e on mul iuni dynamic p icing by add essing
he challenging op imiza ion p oblem o dynamically quo ing ba ch p ices while
conside ing wo-dimensional cus ome he e ogenei y, ep esen ed by wo andom
a iables in he cus ome choice model. Ou main con ibu ions a e as ollows:
C1. Op imali y condi ions: We de i e op imali y condi ions by educing he com-
plexi y o he cus ome choice model. No ably, we p o e ha he op imali y
condi ion holds whe he he op imal solu ion is loca ed in he in e io o on he
bounda y o he ac ion space, allowing us o ocus on his condi ion wi hou
needing o sepa a ely in es iga e bounda y cases.
C2. P oblem modi ica ion and uppe bound: We in oduce a modi ied e sion o
he o iginal p oblem ha can be sol ed mo e e icien ly and se es as an uppe
bound. We u he p o e ha unde speci ic condi ions, he solu ion o he modi-
ied p oblem is op imal o he o iginal p oblem.
C3. De elopmen o heu is ics: We de elop h ee e icien heu is ics ha p oduce
well-pe o ming p icing policies ac oss a ious p oblem sizes. These heu is ics
achie e subs an ially sho e un imes compa ed o inding he op imal solu ion,
enabling hei use in p ac ical applica ions.
C4. Nume ical s udy: Ou nume ical s udy demons a es he e ec i eness and e i-
ciency o all h ee heu is ics. The s udy also highligh s he e enue po en ial
o linea , piecewise linea , and nonlinea p icing schemes, enabling i ms o
balance e enue maximiza ion wi h p icing simplici y.
C5. Insigh s in o p icing schemes: Ou s udy p o ides insigh s in o he s a egy o
well-pe o ming p icing schemes. No ably, ewe p ice changes and simple
p icing schemes, such as piecewise linea p icing, may s ill pe o m e y well.
In Sec .2, we e iew he ele an li e a u e, posi ioning ou esea ch wi hin he
con ex o exis ing s udies and iden i ying gaps ha ou wo k add esses. Sec ion3
in oduces he cus ome choice model, emphasizing he wo-dimensional app oach,
and p esen s he op imiza ion model.
In Sec .4, we educe he complexi y o he p obabili y unc ion, de i e op imali y
condi ions, and in oduce he modi ied p oblem ha se es as an uppe bound. This
sec ion add esses he i s wo con ibu ions, C1 and C2. I also achie es ou i s

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Heu is ic solu ions o nonlinea dynamic p icing in he…
goal (ou lined in Sec .1.1) by enabling he compu a ion o he op imal solu ion and
builds he ounda ion o e alua ing di e en p icing schemes, con ibu ing o ou
hi d goal.
Sec ion5 discusses he heu is ics we de eloped, add essing he hi d con ibu-
ion, C3. By p o iding e icien and p ac ical al e na i es o compu ing he op imal
solu ion, his sec ion achie es ou second goal om Sec .1.1.
Finally, Sec . 6 p esen s he nume ical s udy and i s indings. This sec ion
add esses he las wo con ibu ions, C4 and C5, by demons a ing he p ac ical
applicabili y o he p oposed heu is ics and gaining insigh s in o hei s a egies. I
also ul ills ou hi d goal by e alua ing he e enue po en ial o nonlinea p icing
and simple p icing schemes.
2 Li e a u e e iew
In his s udy, we b idge he domains o dynamic p icing and nonlinea p icing, in e-
g a ing he s eng hs o bo h. In Sec . 2.1, we p o ide a e iew o wo ks wi hin
hese wo dis inc ye in e connec ed domains. This ounda ional o e iew se s he
s age o a deepe explo a ion in o he specialized segmen s o mul iuni and mul i-
p oduc dynamic p icing, as well as bundle p icing, in Sec . 2.2. Mul iuni dynamic
p icing, which also co e s nonlinea p icing, ep esen s a ela i ely new ield o
esea ch wi h spa se li e a u e. Ou esea ch con ibu es o his niche, acknowledg-
ing he c i ical ole o nonlinea p icing in ca e ing o mul iuni demands. Addi ion-
ally, we examine mul ip oduc dynamic p icing and bundle p icing, domains aligned
wi h ou esea ch due o hei conside a ion o cus ome choice, whe e cus ome s
selec om mul iple op ions. Howe e , all hese s udies di e om ou app oach in
a key aspec : hey do no accoun o wo-dimensional cus ome he e ogenei y. The
assump ion o one-dimensional he e ogenei y implies ha all cus ome s sha e he
same ela i e u ili y when compa ing wo op ions, which is an un ealis ic assump-
ion in mul iuni demand scena ios. To unde line his gap, we inco po a e se e al
s udies ha , ega dless o hei speci ic ield o applica ion, add ess simila choice
beha io s. We compa e hese app oaches wi h ou own in Sec . 2.3.
2.1 Founda ions o nonlinea p icing anddynamic p icing
Nonlinea p icing is a widesp ead s a egy ac oss a ious sec o s, including el-
ecommunica ions, anspo a ion, ene gy, supply chains, and e ail. Consequen ly,
he e is a ich and di e se li e a u e on he subjec . Wilson (1993) (Pa I) p o ides
a comp ehensi e o e iew o he applica ion ields, economic p inciples, and ma -
ke ing insigh s ela ed o nonlinea p icing. While much o he exis ing li e a u e
ocuses on s a ic p icing models, a small subse o esea che s has u ned hei a en-
ion o dynamic en i onmen s, which align mo e closely wi h he hema ic ocus o
ou s udy (e.g., Dheba and O en 1986, and B aden and O en 1994).
1186
R.Schu
The concep ual ounda ions o dynamic p icing ace back o seminal s udies on
in e empo al p ice disc imina ion conduc ed 30–40yea s ago, wi h no able con i-
bu ions by S okey (1979), Landsbe ge and Meilijson (1985), and Wilson (1988).
A signi ican miles one was achie ed by Gallego and an Ryzin (1994), who we e
he i s o explo e op imal dynamic p icing o a single p oduc unde s ochas ic
demand o e a ini e selling ho izon. This pionee ing wo k led o a as amoun o
ollow-up esea ch, which was e iewed and summa ized by many au ho s, includ-
ing Bi an and Calden ey (2003), Chiang e al. (2007), and, wi h a special ocus,
Gönsch e al. (2013) and den Boe (2015), as well as in ex books such as Tallu i
and an Ryzin (2004) (Chap e 5).
2.2 Mul iuni dynamic p icing, mul ip oduc dynamic p icing, andbundle p icing
While dynamic p icing has been ex ensi ely explo ed, he speci ic a ea o mul i-
uni dynamic p icing is s ill eme ging. Elmagh aby e  al. (2008) in oduce ma k-
down p icing mechanisms wi hin a mul iuni demand amewo k, assuming com-
ple e in o ma ion abou cus ome s and hei willingness- o-pay. Le in e al. (2014)
expanded he discussion o a dynamic p icing model cha ac e ized by s ochas ic cus-
ome demand o ba ches. In hei amewo k, cus ome s eques a speci ic ba ch
size and he selle ’s subsequen p icing esponse in luences whe he he eques ed
ba ch size is pu chased. Howe e , unlike hei app oach, ou esea ch in oduces a
mo e lexible decision-making p ocess in which cus ome s can e iew all a ailable
p ices p io o eques ing he pu chase quan i y. This ad ancemen no only gi es
cus ome s g ea e lexibili y bu also p o ides i ms wi h a mechanism o s a egi-
cally s ee cus ome s’ buying decisions and pu chase olumes.
Gallego e  al. (2020) in es iga e h ee dynamic p icing s a egies: nonlinea ,
linea , and block p icing. In hei p oposed choice model, cus ome s seek o maxi-
mize hei u ili y and a e cha ac e ized by a single andom a iable, ep esen ing
one-dimensional cus ome he e ogenei y. The au ho s de elop op imali y condi-
ions and show s uc u al p ope ies. Howe e , ou esea ch di e ges undamen ally
by modeling cus ome beha io wi h wo independen a iables, he eby o e ing a
mo e g anula ep esen a ion o cus ome decision-making p ocesses and allowing
o wo-dimensional he e ogenei y. This wo-dimensional app oach ep esen s a sig-
ni ican depa u e om he con en ional one-dimensional models ( e e o Sec .2.3
o a b oade o e iew o choice models applied in simila demand se ings). Schu
(2024) explo es a scena io simila o ou s bu di e ges in assuming i ms’ access o
some o all p i a e in o ma ion abou a i ing cus ome s. This assump ion pa es he
way o pe sonalized p icing s a egies and he e alua ion o he s a egic alue o
cus ome in o ma ion. Con a y o his, ou esea ch ope a es unde he p emise o
p i a e in o ma ion, enabling he uni e sal applicabili y o ou p icing model wi h-
ou elying on he a ailabili y o de ailed cus ome insigh s.
Mul iuni dynamic p icing can be compa ed o he be e -explo ed ield o mul i-
p oduc dynamic p icing by de ining ba ches o a single p oduc as se e al di e en
“p oduc s”. Se e al a icles (see, e.g., Zhang and Coope 2009, Dong e al. 2009,
and Akçay e  al. 2010, o , o a e iew, Chen and Chen 2015) ha e in es iga ed
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Heu is ic solu ions o nonlinea dynamic p icing in he…
dynamic p icing o subs i u es. Howe e , s udies ocusing on ho izon ally di e en i-
a ed p oduc s a e less ele an o ou se ing, as hey lack a common o de in p oduc
alua ions, which iola es a co e assump ion in mul iuni demand se ing: cus ome s
do no pay mo e o ewe uni s. In his con ex , gene al s udies like Magla as and
Meissne (2006) and esea ch on e ically di e en ia ed p oduc s, such as Akçay
e al. (2010) and Liu and Zhang (2013), a e mo e closely aligned wi h ou se ing.
While Magla as and Meissne (2006) do no p o ide a speci ic willingness- o-pay
unc ion, hey demons a e he asymp o ic op imali y o a solu ion o a de e minis ic
luid app oxima ion. Akçay e al. (2010) and Liu and Zhang (2013) ely on one-
dimensional cus ome he e ogenei y ega ding quali y, which, in a mul iuni se ing,
would imply ha ma ginal willingness- o-pay o addi ional uni s dec eases a a
ixed a e ac oss all cus ome s.
A special case o mul ip oduc p icing is ound in he bundle p icing li e a u e,
whe e mul iple i ems a e combined in o a bundle. Bundle-size p icing in ol es he
i m quo ing a p ice o each bundle-size, allowing cus ome s o choose which i ems
o include in hei bundle. In his con ex , i ems can be iewed as single uni s o
he same p oduc , and bundles as ba ches. Bundle p icing aligns well wi h mul iuni
demand, as cus ome u ili y ypically inc eases wi h bundle-size, e lec ing one o
he co e assump ions o mul iuni demand. Howe e , ela ed li e a u e, such as Hon-
hon and Pan (2017) and Song and Xue (2021), o en assumes one-dimensional cus-
ome he e ogenei y. O he s udies, such as Abdallah e al. (2021), E l e al. (2020),
and Chen e al. (2024), do no ensu e dec easing ma ginal u ili ies wi h inc easing
bundle size, ano he key assump ion in mul iuni demand.
2.3 Li e a u e wi hsimila cus ome choice conside a ions
Li e a u e gene ally ag ees on he undamen al concep ha cus ome s e alua e all
a ailable op ions by assigning a mone a y alue, o en e e ed o as willingness- o-
pay, o each op ion and compa ing hese alues o quo ed p ices. The op ion wi h he
highes u ili y, i.e. he di e ence be ween willingness- o-pay and p ice, is chosen.
Since willingness- o-pay is p i a e in o ma ion known only o he cus ome , a ious
models ha e been in oduced, ypically accoun ing o cus ome he e ogenei y using
one o wo p ominen app oaches.
The i s app oach models willingness- o-pay using a ious cus ome segmen -
speci ic nominal pa ame e s, whe e he e ogenei y wi hin a segmen is cap u ed
h ough independen ly and iden ically dis ibu ed e o e ms. Hanemann (1984)
o mula es a willingness- o-pay unc ion ha is linea in quan i y, leading o con-
s an ma ginal u ili ies. Subsequen s udies, such as Allenby and Rossi (1991), Chi-
ang (1991), Bell e al. (1999), and Nai e  al. (2005), p opose non-cons an ma -
ginal u ili ies o be e e lec dec easing cus ome alua ions o addi ional quan i y.
Quad a ic o mula ions o hose non-cons an ma ginal u ili ies can be ound in
Lamb ech e al. (2007) and Iyenga e al. (2008). Iyenga and Jedidi (2012) in o-
duce a gene aliza ion ha encompasses bo h quad a ic and powe u ili y unc ions
as special cases. These s udies o en employ mul inomial logi models, assuming
Gumbel-dis ibu ed e o e ms. While his model has ce ain ad an ages, i p esen s
1188
R.Schu
a signi ican limi a ion: i canno en o ce a common o de o alua ions, leading o
ins ances whe e cus ome s may be willing o pay mo e o ewe uni s. Simila ly, in
mul ip oduc p icing, his model is less sui ed o e ically di e en ia ed p oduc s
( e e o Song and Xue 2021).
The second app oach o en elies on a single andom a iable o ep esen cus-
ome s’ p e e ences, as opposed o he mul iple pa ame e s combined wi h andom
e o e ms used in he i s app oach. This one-dimensional he e ogenei y is applied
by Mussa and Rosen (1978), whe e willingness- o-pay inc eases linea ly in quan-
i y. Fu he applica ions o his app oach, wi h non-cons an ma ginal u ili ies, can
be seen in B aden and O en (1994), Sunda a ajan (2004), Banciu e al. (2010), and
Biyalogo sky and Koenigsbe g (2014). Roche and S ole (2002) inco po a e wo-
dimensional he e ogenei y, whe e willingness- o-pay is modeled as linea in quan-
i y, weigh ed by a andom a iable ep esen ing cus ome ype, along wi h an addi-
i e andom a iable ha accoun s o b and p e e ences.
We ollow he second app oach o model cus ome choice, add essing wo-
dimensional he e ogenei y h ough wo andom a iables, bu no in he linea o m
p oposed by Roche and S ole (2002). Ins ead, ou willingness- o-pay unc ion is
based on he gene alized o mula ion o Iyenga and Jedidi (2012), inco po a ing
non-linea i y in quan i y and esul ing in non-cons an ma ginal u ili ies.
2.4 Summa y
Ou li e a u e e iew highligh s a gene al gap in esea ch on nonlinea dynamic p ic-
ing, wi h only a ew excep ions, which di e om ou wo k in hei speci ic assump-
ions. B oadening he scope, i becomes appa en ha e en in ela ed ields such as
e ically di e en ia ed mul ip oduc p icing and bundle p icing, wo-dimensional
cus ome he e ogenei y—allowing o cus ome -speci ic, non- ixed a es o ma ginal
u ili y dec ease—is no conside ed. In his ega d, ou app oach could be o in e es
o hese ields as well.
3 P oblem de ini ion
We i s p esen he andom u ili y amewo k ha de ines ou cus ome choice
model in Sec .2.3. Then, we in oduce he dynamic p og amming o mula ion ha
desc ibes ou nonlinea dynamic p icing p oblem.
3.1 Cus ome choice model
In ou se ing, cus ome s exhibi mul iuni demand bu a e no p ede e mined in
he numbe o uni s hey conside pu chasing. Ins ead, hey e alua e all a ailable
op ions, i.e., possible ba ch sizes, anging om a ba ch size o ze o o he en i e
emaining s ock
c
. This e alua ion esul s in a pe sonal mone a y alua ion o each
op ion, which is known as willingness- o-pay and deno ed by
Xj
o
j
uni s. I is
1195
Heu is ic solu ions o nonlinea dynamic p icing in he…
iden i ying hese speci ic in e als, we know p ecisely which cu e de ines he uppe
bound o ealiza ions
w
a any gi en ealiza ion
l
.
Lemma1 Fo e e y
i≠k
wi h
i,k>j
and
i− j≠0≠ k− j
, he e is a mos one
l∈(0,1)
whe e
i
−
j
∑
i−1
m=j
lm
=
k
−
j
∑
k−1
m=j
l
m
. Fo e e y
i≠k
wi h
i,k<j
and
i− j≠0≠ k− j
,
he e is a mos one
l∈[0,1]
whe e
j
−
k
∑
j−1
m=k
lm
=
j
−
i
∑
j−1
m=i
l
m
.
P oo : See Supplemen S.1.
Lemma 1, in conjunc ion wi h he obse a ion ha he equa ion
k
−
j
∑
k−1
m
=
j
lm
=
1
is sa -
is ied by a mos one
l∈[0,1]
, implies ha each cu e can only se e as he mini-
mum o maximum exclusi ely wi hin a speci ic in e al. A some poin , i will in e -
sec wi h ano he cu e and will consis en ly emain ei he abo e o below he
in e sec ing cu e. Consequen ly, o he a ious se s o in e es – namely
Λj(
)
o
ins ances whe e
1≥
min
j+1≤k≤c
�
k− j
∑
k−1
m=jlm
�
,
Λmin
ij
(
)
o cases whe e
i>j
, and
Λmax
ji (
)
o
cases whe e
j>i
—we de ine hei co esponding in e als as
[
lj( ),lj( )
]
,
[
lmin
ij ( ),l
min
ij ( )
]
, and
[
lmax
ji ( ),l
max
ji ( )
]
, espec i ely. I is impo an o no e ha some o
hese in e als may be emp y, and hus, bounda y alues ha e o be selec ed
acco dingly.
Rema k 1 By de ini ion,
𝑚𝑎𝑥
i
>
j{
l
min
ij ( )
}
=𝑚𝑎𝑥
i
<
j{
l
max
ji ( )
}
. We deno e his la ges
bounda y as
lend
j( )=𝑚𝑎𝑥
i
>
j{
l
min
ij ( )
}.
Mo eo e , i holds ha :
⋃
i>j
�
lmin
ij ( ),l
min
ij ( )
�
=
�
lj( ),l
end
j( )
�
⋃
i<j
�
lmax
ji ( ),l
max
ji ( )
�
=
�
lj( ),l
end
j( )
�
Cu en ly, ou model equi es ex ensi e no a ion o accu a ely ep esen he
p obabili y unc ion. Ne e heless, he in oduc ion o he subsequen lemma will
s eamline he no a ion equi ed, he eby enhancing he b e i y and cla i y o ou
p esen a ion.
Lemma2 Fo e e y
i>j
i holds ha
[
lmin
ij ( ),l
min
ij ( )
]
=
[
lmax
ij ( ),l
max
ij ( )
]
.
P oo : See Supplemen S.3.
This lemma no only simpli ies ou no a ion bu also ca ies ano he implica ion:
o
i>j
, he in e al
[
lmin
ij ( ),l
min
ij ( )
]
de ines he egion o ealiza ion
l
whe e he
cu e
i
−
j
∑
i−1
m
=
j
lm ma ks he uppe bound o all ealiza ions
w
ha speci y, in combina-

1196
R.Schu
ion wi h
l
, all cus ome s who choose o pu chase
j
uni s. F om Lemma 2, we know
ha
[
lmin
ij ( ),l
min
ij ( )
]
=
[
lmax
ij ( ),l
max
ij ( )
]
, which implies ha
Λmin
ij
( )=Λ
max
ij
( )
.
By
de ini ion o
Λmax
ij (
)
, he same cu e (
i
−
j
∑
i−1
m=j
l
m
) also se es as he lowe bound o all
ealiza ions
w
ha , in combina ion wi h
l
, ep esen all cus ome s who p e e pu -
chasing
i
uni s. Consequen ly, his cu e ma ks he bounda y be ween wo decision
egions ( e e o Fig.1): one associa ed wi h cus ome s op ing o pu chase
j
uni s
and he o he wi h hose p e e ing
i
uni s.
We can now sho en ou no a ion o
lij
( )=l
min
ij
( )=l
max
ij
(
)
and
l
ij( )=l
min
ij
( )=l
max
ij
(
)
. The p obabili y unc ion can be w i en as:
Wi h his e inemen o he p obabili y unc ion, we can be e explo e how p ices
and hei a ia ions impac selling p obabili y. This is c ucial o de e mining op i-
mal p ices in he s a e-wise op imiza ion (7). Howe e , some complexi y emains
due o he implici de ini ion o he bounds o he in eg als. To add ess his, i is
easonable o gain deepe insigh s in o hese bounds. One way o achie e his is by
educing he ac ion space, which will esul in a closed- o m exp ession o some o
he bounds.
4.2 Ac ion space educ ion
Upon close examina ion o ou choice model, i becomes e iden ha we only need
o conside a speci ic subse o p ices o maximize expec ed e enue. Consequen ly,
we aim o e ine he de ini ion o he ac ion space
Rc
by excluding p ices ha do no
lead o a unique ou come in e ms o selling p obabili ies and expec ed e enues.
By doing so, we simpli y ou op imiza ion p oblem wi hou he isk o excluding a
po en ially unique op imal solu ion.
The a gumen a ion o deeming ce ain p ices as i ele an is as ollows: Main-
aining mul iple di e en alues o p ice
j
ha e ec i ely nulli y demand o
j
uni s
(i.e.,
pj(
)=0
) is unnecessa y. I su ices o ha e a leas one alue o
j
(depending
on
1
,…, j−
1
, j+
1
,…, c
) o p ese e he op ion o p icing ou
j
uni s. I ele an
p ices can be iden i ied by any o he ollowing ou c i e ia.
1. Exclusion o highe p ices o smalle ba ches: We exclude any p ice
j
wi h
j> j+1
because cus ome s almos su ely ha e a highe willingness- o-pay o
(8)
pj( )=
lj( )
∫
lj( )
𝜆(l)dl +
c
�
i=j+1
lij( )
∫
lij( )
F𝜔� i− j
∑i−1
m=jlm� 𝜆(l)dl
−
j−1
�
i=0
lji( )
∫
l
ji
( )
F𝜔� j− i
∑
j−1
m=ilm� 𝜆(l)dl o j=1, …,c
.
1197
Heu is ic solu ions o nonlinea dynamic p icing in he…
j+1
uni s han o
j
uni s. The e o e, se ing
j= j+1
is su icien o elimina e
demand o
j
uni s.
2. Exclusion o p ices exceeding ba ch size h eshold: Any p ice
j
exceeding
j
is
i ele an , as he maximal willingness- o-pay o
j
uni s is
j
.
3. Exclusion o p ices wi h excessi e ma gins: We exclude any p ice
j
o which
j
− j−
1
>1
, since he maximal ma ginal u ili y o he
j
- h uni is one.
4. Exclusion o p ices wi h excessi e compa a i e ma gins: We omi p ices
j
ha
sa is y he condi ion
(
j
−
j−
1
)
1
j−1>
(
j+
1−
j)
1
j
. This inequali y implies ha a
cus ome wi h a posi i e ma ginal u ili y o pu chasing he
j
- h uni has almos
su ely also a posi i e ma ginal u ili y o pu chasing he
(j+1)
- h uni .
The ollowing lemma con i ms ha educing he ac ion space based on hese ou
c i e ia only elimina es p ices ha do no esul in unique ou comes in e ms o sell-
ing p obabili ies and e enues.
Lemma3 The sea ch o he op imal p ice ec o can be es ic ed o he se .
P oo : See Supplemen S.4.
In Sec .3.1, we ha e seen ha
lk
and
lk
play a c ucial ole in calcula ing selling
p obabili ies. Wi h he ac ion space educ ion, we a e now able o shed mo e ligh on
he de ini ion o hese pa ame e s.
Lemma4 I holds ha
l1(
)=0
,
lk
( )=
(
k
−
k−1)
1
k−1=l
k−1
(
)
,
2≤k≤c
,
lc
( )=
1
o all
∈Rc
.
P oo : See Supplemen S.5.
Wi h Lemma 4, we ob ain a closed- o m exp ession o
lk(
)
and
lk(
)
(
1≤k≤c
),
which simpli ies ou p obabili y unc ion. Addi ionally, he p oo o his lemma
shows ha
lk
,
k−
1
(
)=lk(
)
, ex ending he closed- o m exp ession o hese speci ic
bounds. While he implici de ini ions o o he bounds, such as
lkj(
)
and
lkj
(
)
, gen-
e ally canno be eplaced by explici de ini ions, we can s ill explo e how hese
bounds change wi h small a ia ions in p ices, laying he g oundwo k o calcula ing
pa ial de i a i es and inding op imal solu ions.
These bounds co espond o he in e als
Λmin
kj
(
)
and
Λmax
kj (
)
o
k>j
and a e
de ined as he in e sec ion poin s o wo cu es, p o ided hese in e als a e non-
emp y ( o an illus a i e e e ence, see Fig. 2). By iden i ying hese bounds as
R
c=
{
∈ℝc∶0≤ 1≤⋯≤ c≤c, j≤j∀j, j− j−1≤1
o j≥2, and
(
j− j−1
)
1
j−1≤
(
j+1− j
)
1
j o 2 ≤j≤c−1
}.
1198
R.Schu
in e sec ion poin s, we es ablish ha o e e y uppe bound
lkj
(
)
(wi h a single
excep ion as no ed in Rema k 1), he e exis s a co esponding lowe bound
lij(
)
such
ha
l
kj( )=l
ij
(
)
and
k
−
j
∑
k−1
m=j
�
lkj( )
�
m=
i
−
j
∑
i−1
m=j
�
lkj( )
�
m.
We now wan o discuss he impac o a small p ice a ia ion, speci ically chang-
ing
m
while keeping o he ba ch p ices cons an . Fo
m∈{j,k,i}
, his would ob i-
ously shi he loca ion whe e bo h cu es in e sec . Howe e , as long as he p ice
a ia ion is su icien ly small, i would no a ec which in e als a e adjacen and
connec ed a he in e sec ion poin , i.e. he equa ion
l
kj( )=l
ij
(
)
would s ill hold,
albei wi h a di e en alue. Fo
m∉{j,k,i}
, a su icien ly small p ice a ia ion
would nei he change he ma ching o
l
kj(
)
and
lij(
)
no a ec he alue o hese wo
bounds.
Howe e , he e a e a e ins ances o ambigui y ha we ha e no explici ly
add essed. Speci ically, when he p ice ec o esul s in h ee cu es in e sec ing a
a single poin , any wo o hese h ee cu es can be used o de e mine he in e sec-
ion poin . In his scena io, one o he co esponding in e als consis s o exac ly
one elemen , which is he in e sec ion poin . This ambigui y esol es wi h e en a bi-
a ily small changes in one o he p ices
m
associa ed wi h hese cu es, as he
h ee cu es no longe in e sec a he same spo , causing he one-elemen in e al o
ei he become emp y o expand o co e an in ini e numbe o elemen s. In he i s
case, we would use he o he wo cu es o de e mine he in e sec ion poin . In he
second case, we would de e mine wo in e sec ion poin s using he combina ions o
he cu e belonging o he one-elemen in e al wi h bo h o he cu es. While hese
cases o ambigui y a e no explici ly co e ed in he emainde o his sec ion, please
no e ha hey can be easily esol ed.
The ollowing ema k summa izes he obse a ions abo e and will come in handy
in he de elopmen o op imali y condi ions.
Rema k 2 I holds ha :
• Fo e e y
l
kj( )
≠
l
end
j
(
)
he e is
l
ij
(
)
such ha
l
kj( )=l
ij
(
)
and
k
−
j
∑
k−1
m=j
�
lkj( )
�
m=
i
−
j
∑
i−1
m=j
�
lkj( )
�m
. Mo eo e , 𝜕
𝜕
m
lkj( )=𝜕
𝜕
m
lij(
)
o all
m
and
𝜕
𝜕
m
lkj( )=
𝜕
𝜕
m
lij( )=
0
o
m∉{j,k,i}
.
•
lj+
1
(
)=lj+
1,
j(
)
wi h
j+1
−
j
(
lj+1( )
)
j=
1
. Mo eo e ,
𝜕
𝜕
m
lj+1( )=
𝜕
𝜕
m
lj+1,j(
)
o all
m
and
𝜕
𝜕 m
lj+1( )=
𝜕
𝜕
m
lj+1,j( )=
0
o
m∉{j,j+1}
.
•
lj(
)=lj,j−1(
)
wi h
j
−
j−1
(
lj( )
)
j−1=
1
. Mo eo e ,
𝜕
𝜕 m
lj( )=
𝜕
𝜕
m
lj,j−1(
)
o all
m
and
𝜕
𝜕
m
lj( )=
𝜕
𝜕
m
lj,j−1( )=
0
o
m∉{j−1, j}
.
1199
Heu is ic solu ions o nonlinea dynamic p icing in he…
4.3 Op imali y condi ions
Wi h he p e ious sec ion, we ga he ed enough in o ma ion ega ding he p obabil-
i y unc ion o ad ance o one o ou main goal: inding he op imal solu ion o (7).
Be o e we engage in he pa ial di e en ia ion o he objec i e unc ion, we i s
wan o elabo a e mo e on he pa ial di e en ia ion o p obabili y unc ion (8). The
calcula ion o
𝜕
𝜕
i
pj(
)
a ies a li le depending on he ollowing h ee cases:
i>j
,
i<j
, and
i=j
.
Lemma5 I holds:
(1) Fo
i>j
, 𝜕
𝜕
i
pj( )=
lij( )
∫
l
ij
( )
1
∑
i−1
m=jlm 𝜔
�
i− j
∑
i−1
m=jlm
�
𝜆(l)
dl
.
(2) Fo
i<j
, 𝜕
𝜕
i
pj( )=
lji( )
∫
l
ji
( )
1
∑
j−1
m=ilm 𝜔
�
j− i
∑
j−1
m=ilm
�
𝜆(l)
dl
.
(3) Fo
i=j
,
𝜕
𝜕 i
pj( )=−
∑
c
k=i+1
lki( )
∫
l
ki
( )
1
∑
k−1
m=i
lm 𝜔
�
k− i
∑
k−1
m=i
lm
�
𝜆(l)dl −
∑
i−1
k=0
lik( )
∫
l
ik
( )
1
∑
i−1
m=k
lm 𝜔
�
i− k
∑
i−1
m=k
lm
�
𝜆(l)
dl
.
P oo : See Supplemen S.6.
Wi h addi ional knowledge abou he p obabili y unc ion, we now can u n ou
ocus on he i s -o de condi ion. The e o e, we calcula e he pa ial de i a i es o
he objec i e unc ion o (7):
𝜕
𝜕
i
�c
�
j=1
pj( )⋅� j−Δ
jV −1(c)�
�
=pi( )+
c
�
j=1�𝜕
𝜕 i
pj( )�⋅� j−Δ
jV −1(c)�
=pi( )+
i−1
�
j=1

lij( )
∫
l
−ij
( )
1
∑i−1
m=jlm 𝜔� i− j
∑i−1
m=jlm� 𝜆(l)dl ⋅� j−Δ
jV −1(c)
�
−
c
�
k=i+1

lki( )
∫
l
−ki
( )
1
∑k−1
m=ilm 𝜔� k− i
∑k−1
m=ilm� 𝜆(l)dl ⋅� i−Δ
iV −1(c)�
−
i−1
�
k=0

lik( )
∫
l
−ik
( )
1
∑i−1
m=klm 𝜔� i− k
∑i−1
m=klm� 𝜆(l)dl ⋅� i−Δ
iV −1(c)�
+
c
�
j=i+1

lji( )
∫
l
−
ji
( )
1
∑
j−1
m=i
lm 𝜔� j− i
∑
j−1
m=i
lm� 𝜆(l)dl ⋅
�
j−Δ
jV −1(c)
�
.
1200
R.Schu
While he i s -o de condi ion is a necessa y condi ion o local maxima, he
global maximum o e a closed and bounded space does no gene ally need o sa -
is y his condi ion. Howe e , he ollowing p oposi ions s a es ha , ega dless o
whe he he op imal solu ion lies on he bounda y o he ac ion space
Rc
o wi hin
i s in e io , i always sa is ies he i s -o de condi ion.
P oposi ion1 The op imal solu ion o (7) mee s o e e y ba ch size
i
he i s -o de
condi ion:
P oo : See Supplemen S.7.
Rema k 3 Fo
𝜔∼U[0, 1]
he i s -o de condi ion simpli ies o
𝜕
𝜕
i
�
∑
c
j=1pj( )⋅
�
j−Δ
jV −1(c)
�
�=2pi( )−
�
F𝜆
�
li+1( )
�
−F𝜆
�
li( )
��
−∑c
k=i+1
lki( )
∫
l
ki
( )
ΔkV −1(c)−Δ
iV −1(c)
∑k−1
m=i
lm 𝜆(l)dl +∑i−1
k=0
lik( )
∫
l
ik
( )
ΔiV −1(c)−Δ
kV −1(c)
∑i−1
m=k
lm 𝜆(l)
dl
.
Mo eo e , i holds ha
∑
c
i=1pi( ∗)=1
2−
∑
c
k=1
lk0( ∗)
∫
l
k0
( ∗)
ΔkV −1(c)
∑
k−1
m=0lm 𝜆(l)
dl
. Thus, he o e all
selling p obabili y is less han o equal o
0.5
and dec easing wi h oppo uni y cos s.
In he inal pe iod (
=1
), he e a e no oppo uni y cos s, so P oposi ion 1 applies
wi h
ΔjV −1(c)=0
o all
j
in hese s a es. To conclude his sec ion, we ocus on a
scena io wi hou oppo uni y cos s, whe e bo h andom a iables
𝜆
and
𝜔
a e uni-
o mly dis ibu ed. Unde hese condi ions, inding he op imal solu ion in he inal
s a es u he simpli ies. Mo eo e , he op imal solu ion does no lie on he bounda y
o ou ac ion space, implying ha none o he co esponding selling p obabili ies a e
educed o ze o.
P oposi ion2 Le
𝜔,𝜆∼U[0, 1]
. I
ΔkV −1(c)=0
o e e y
k≤c,
he op imal solu-
ion o (7) is an in e io poin o
Rc
and ul ills pi( )=
1
2(
l
i+
1( )−l
i
( )
)
o e e y
i≤c
.
P oo : See Supplemen S.8.
Al hough we ha e o mula ed he op imali y condi ions, inding he solu ion ha
ul ills hese equa ions wi hin each s a e is a di icul ask. We a e acing a sys em
wi h
c
nonlinea equa ions ha a e addi ionally plagued by in eg als and implici ly
de ined a iables. Many o hese challenges a ise om he analy ical in ac abili y
o he choice model. To add ess his, we explo e a modi ied op imiza ion p oblem in
he ollowing sec ion. This al e na i e o mula ion is simple o sol e and p o ides
𝜕
𝜕
i
(
c
∑
j=
1
pj( )⋅
(
j−Δ
jV −1(c)
))
=
0.

1201
Heu is ic solu ions o nonlinea dynamic p icing in he…
an uppe bound o he o iginal p oblem. No ably, unde speci ic condi ions, he
op imal solu ion o he modi ied o mula ion also cons i u es he op imal solu ion o
ou o iginal p oblem.
4.4 Modi ied o mula ion
In his sec ion, we p opose a modi ica ion o ou choice model, which leads o a
simpli ied p oblem ha ac s as an uppe bound o ou o iginal p oblem. The main
aspec o his o mula ion, howe e , is he po en ial o i s op imal solu ion o also
be he op imal solu ion o he o iginal p oblem. Speci ically, we de elop easy-
o- e i y condi ions o his solu ion o ensu e i s op imali y wi h espec o he
o iginal p oblem.
The modi ica ion o he cus ome choice model in ol es educing he compe i-
ion be ween di e en op ions. In pa icula , ins ead o calcula ing he p obabili y
ha
U
k=max
j=
0,
…
,
c{
Uj
}
, we now calcula e he p obabili y ha
U
k=max
j=
0,
k−
1,
k
,
k+
1
{
Uj
}
.
The eby, we educe compe i ion be ween a ailable op ions and, hus, only com-
pa e ou ins ead o
c+1
op ions. We deno e demand by his modi ied choice
model as
pm
j(
)
and w i e pm
j( )=ℙ
(
Uj=max
k=0,j−1,j,j+1
{
Uk
})≥
pj(
)
. A s a e-wise
op imiza ion wi h
pm
j(
)
he e o e se es as an uppe bound o (7).
Technically, h ough he modi ica ion, he choice model is no longe a choice
model. By summing up all p obabili ies, we ge a alue g ea e han o equal o
1
,
i.e.
∑C
j=0
pm
j
( )
≥∑C
j=0
pj( )=
1
, which does no sa is y one o he main p ope ies o
choice models.
Simila conside a ions as in Sec .4.3 lead o he ollowing ema k.
Rema k 4 When
pm
j(
)
is used ins ead o
pj(
)
, P oposi ions 1 and 2 s ill hold.
This modi ica ion signi ican ly educes complexi y in ou choice model (see Sup-
plemen S.9 o an exempla y calcula ion o
pm
j(
)
o a scena io whe e
𝜔,𝜆∼U[0, 1]
), hus imp o ing he de e mina ion o he op imal solu ion
m
o he
modi ied o mula ion compa ed o inding he op imal solu ion o he o iginal p ob-
lem. U ilizing he ac ha he modi ied model cons i u es an uppe bound o he
o iginal model, i su ices o e i y ha i s op imal solu ion
m
is easible o he
o iginal model and esul s in he same objec i e alue. To acili a e his e i ica ion,
we p opose an app oach ha a oids he exhaus i e checking whe he
pm
j(
m)=p
j
(
m)
o e e y
j
. Ins ead, we in oduce a se o s aigh o wa d and e i iable condi ions,
designed o simpli y he p ocess o p o ing
m
’s op imali y o he o iginal model.
P oposi ion 3 I
ul ills
lend
j
( )
≤
l
end
j+
1(
)
o e e y
j
, wi h
1≤j<c
, hen
pm
j(
)=p
j
(
)
o e e y
j
.
P oo : See Supplemen S.10.
1202
R.Schu
Rema k 5 The condi ions o P oposi ion3 a e sa is ied by linea p icing schemes.
Addi ionally, piecewise linea p icing schemes, whe e a dis inc p ice is se o he
i s uni and a cons an ma ginal p ice applies o any addi ional uni , also sa is y
he condi ions o P oposi ion3.
In ou nume ical s udy, we calcula ed he op imal solu ion o he modi ied model
o e e y combina ion o
T≤40
and
C≤120
, wi h
𝜔,𝜆∼U[0, 1]
. The condi ions
o P oposi ion 3 we e always sa is ied, showing ha , in hese ins ances, he op i-
mal solu ion ob ained om he modi ied model is also op imal o he o iginal p ob-
lem. Howe e , ou nume ical s udy e eals ha sol ing each s a e o he modi ied
model leads o long compu a ion imes o medium-sized p oblems (see Sec .6.2).
The e o e, we aim o add ess hese d awbacks by de eloping e icien and e ec i e
heu is ics.
5 Heu is ic design
Dynamic p icing decisions o en need o be made online and in eal- ime. Gi en he
long compu a ion imes equi ed o ind he op imal solu ion, we de eloped h ee
heu is ics o ackle he dynamic op imiza ion p oblem (5). These heu is ics aim o
compu e well-pe o ming policies e icien ly, wi h sho un imes, ensu ing p ac ical
applicabili y in dynamic nonlinea p icing scena ios.
In his sec ion, we in oduce h ee heu is ics o add ess he dynamic op imiza ion
p oblem (5). Two o hese app oaches, p esen ed in Sec .5.1, build on he esul s
o Schu (2024) and use he op imal solu ion in a se ing whe e he i m has access
o cus ome s’ p i a e in o ma ion, i.e., hei base willingness- o-pay and hei con-
sump ion indica o , espec i ely. The hi d app oach, p esen ed in Sec .5.2, can be
desc ibed as a decomposi ion in uni s. The eby, we allow cus ome s o buy he
j
- h
uni o he p oduc wi hou buying he uni s
1, 2, …,j−1
. E en hough his does
no e lec eali y, i cons i u es an easy o sol e op imiza ion p oblem, enabling us
o de ise ba ch p icing s a egies o he o iginal p oblem based on he solu ions
ob ained.
5.1 App oaches 1 and2: expec ed op imal ba ch p ices
We can e icien ly compu e ealiza ion-dependen op imal ba ch p ices
j (c|w)
and
j (c|l)
o ealiza ion
w
and
l
, espec i ely. Technically, hese ealiza ion-dependen
ba ch p ices a e hemsel es andom a iables, aising he ollowing idea: By calcu-
la ing he expec ed alue o hese ealiza ion-dependen op imal ba ch p ices, we
cons uc a policy o op imiza ion p oblem (5).
Bo h app oaches ollow he same idea, di e ing only in he de e mina ion o he
ealiza ion-dependen op imal ba ch p ices:
j (c|w)
o App oach 1 and
j (c|l)
o
App oach 2. Beyond his dis inc ion, bo h app oaches ollow he same subsequen
s eps. Consequen ly, we will explain he emaining s eps wi hou dis inguishing
1203
Heu is ic solu ions o nonlinea dynamic p icing in he…
be ween bo h app oaches and w i e
j (c|x)
ins ead o
j (c|w)
and
j (c|l)
o deno e
he ealiza ion-dependen op imal ba ch p ices.
This amewo k is applicable ac oss a wide spec um o dis ibu ion unc ions,
including, bu no limi ed o, uni o m, iangula , no mal, exponen ial, Weibull,
Gumbel, and gamma dis ibu ions, along wi h hei unca ed e sions, albei wi h
some cons ain s on pa ame e selec ions. No ably, o scena ios whe e
𝜔∼U[0, 1]
,
App oach 2 p o ides a closed- o m exp ession o he ealiza ion-dependen op imal
ba ch p ices:
j (c
�
l)=1
2
�∑
j−1
k=0lk+Δ
jVE
−1(c)
�
, wi h
j
≤
N (c
|
l)=max
j=
1,
…
,
c{
j∶Δ
1V
E
−1(c−j+1)<l
j−1}
.
Building on hese ealiza ions-dependen op imal ba ch p ices, we compu e
expec ed op imal ba ch p ices:
Mo e p ecisely, his o mula ion esul s in condi ional expec ed op imal ba ch
p ices, whe e we only ake ealiza ions o
𝜆
and
𝜔
in o accoun ha lead o pos-
sible economic sales, i.e.
j+
1,
(c
|
⋅)−
j
(c
|
⋅)
≥
Δ1V
E
−1
(c+1−j
)
. O he ealiza ions
a e economically i ele an and can dis o esul s, gi en he lack o a clea p icing
s a egy in hese cases. Thus, hese e en s whe e we e ain om selling a e no used
o compu e ou policy.
Excep o he scena io whe e we ha e a closed- o m exp ession o
j (c|l)
, we
eso o nume ically calcula ing
j (c|x)
o se e al ealiza ions
x
o accu a ely
de i e
E
j
(c
)
(in Sec .6, we use a sample size o
100
; o an analysis o he adeo
be ween accu acy and un imes, e e o Supplemen S.11). Finally, we calcula e he
expec ed e enue- o-go de i ed by expec ed op imal ba ch p ices
E
j
(c
)
:
wi h he same bounda y condi ions as he o iginal p oblem (5).
Rema k 6 By using subop imal ba ch p ices, we ge a lowe bound o op imiza ion
p oblem (5). Thus, i holds ha
V
(c)
≥
V
E
(c
)
o e e y
( ,c)
.
We sum up he i s wo heu is ics ha a e based upon he idea o expec ed op i-
mal ba ch p ices by he ollowing pseudo code:
(9)
E
j (c)=
1
∫
0
j (c|x)⋅1{j<N (c|x)} (x)dx
1
∫
0
1{j<N (c
|
x)} (x)dx
o j=1, …,c
.
(10)
V
E
(c)=
c
∑
j=1
pj
(
E
(c)
)
⋅
(
E
j (c)+VE
−1(c−j)
)
+
(
1−
c
∑
j=1
pj
(
E
(c)
))
⋅VE
−1(c)
,
1204
R.Schu
5.2 App oach 3: decomposi ion inuni s
Ou nex algo i hm employs a decomposi ion s a egy. The basic idea is ha cus om-
e s ha e he lexibili y o buy he
j
- h uni o he p oduc e en hough hey migh no
buy uni s
1
o
j−1
. As e e y uni o he p oduc is he same, he e is no dis inc ion
be ween he
1
s ,
2
nd o
j
- h uni o he han he numbe cus ome s ha e al eady in
hei baske . Thus, his decomposi ion is me ely heo e ical wi hou ha ing immedi-
a e p ac ical applicabili y. Howe e , i esul s in a g ea ly simpli ied op imiza ion
p oblem. A hypo he ical cus ome now aces
c
dis inc bina y decisions ins ead
o one decision wi h
c+1
op ions. This, in u n, enables us o sol e
c
dis inc and
a he simple independen op imiza ion p oblems ins ead o one complex p oblem.
Wi h his simpli ica ion, we can de i e ba ch p ices ha can o m a policy o add ess
op imiza ion p oblem (5).
To conside his decomposi ion, we mus change he cus ome choice model.
Cus ome s s ill s i e o maximize hei u ili y. Bu , ins ead o pu chasing
j
uni s i
and only i
U
j=max
j=0,…,c{
Uj
}
wi h
U0=0
deno ing he no-pu chase op ion, hey
decide o e e y single uni whe he hey wan o pu chase i o no . This decision is
based upon whe he he addi ional willingness- o-pay o he
j
- h uni is a leas as
high as he addi ional p ice he cus ome has o pay, i.e.
Xj
−X
j−
1=𝜔⋅(𝜆)
j−1≥
j
−
j−1
. I he cus ome s decide o pu chase he
j
- h uni ,
hey mus pay
j
− j−1
. Fo example, o gi en ba ch p ices, a cus ome migh only
be willing o pu chase he second and ou h uni due o he willingness- o-pay
cu e. In his case, he cus ome pays
(
2− 1
)
+
(
4− 3
)
o ge
2
uni s o he p od-
uc in o al.
The decomposi ion app oach educes he complexi y o he choice model. The
model i sel becomes easie as he decision be ween se e al op ions is b oken down
o se e al bina y independen decisions. This me hod a oids he need o de e mine
a single p ice ec o ha encompasses all ba ch p ices and o p edic he cus ome ’s
1211
Heu is ic solu ions o nonlinea dynamic p icing in he…
To summa ize, mechanism D closely aligns wi h he op imal mechanism, mi -
o ing mean e enues and mean pu chases o e ime. In con as , mechanism
E(𝜔)
adop s a mo e agg essi e p icing s a egy, selling mo e uni s han D and he op imal
policy o e he en i e selling ho izon, seemingly by o e ing lowe p ices.
This pa e n is also e iden in Fig.6, whe e we ha e depic ed he e olu ion o
ba ch p ices o he scena io whe e no cus ome makes a pu chase. Speci ically, he
igu e shows ba ch p ices in he s a es
(c, )
o
c=20
and
=10, 9, …,1
. In he
igu e, he lowes cu e co esponds o he p ice o a ba ch o size one, he sec-
ond-lowes line ep esen s he p ice o a ba ch o size wo, and so on. O e he
en i e selling ho izon, mechanism D quo es highe p ices han mechanism
E(𝜔)
.
Addi ionally, p ices om bo h mechanisms a e dec easing o e ime. Despi e he
dis inc app oaches used o calcula e
D
(c
)
and
E(𝜔)
(c
)
, he esul ing cu es sha e a
simila s uc u e, indica ing an unde lying s uc u e well-pe o ming policies ha e in
common.
Ba ch p ices o small numbe s o uni s a e i ually linea in ba ch size (apa
om he i s uni , he second, hi d, e c. uni s cos nea ly he same). Fo la ge num-
be s o uni s, ba ch p ices a e con exly inc easing in ba ch size. Wi h a conca ely
inc easing willingness- o-pay cu e, hese p icing schemes au oma ically p e en
selling a la ge ba ch o e en he whole s ock (
C=20
) o only one cus ome . Finally,
i is no able ha p ices o small ba ches me ely change o e ime whe eas p ices o
big ba ches no iceably dec ease.
In Fig.5, we ha e seen ha bo h heu is ics esul in selling p ocesses whe e a
cus ome pu chases app oxima ely 1.5 uni s on a e age. The e o e, we also wan
o examine he e olu ion o ba ch p ices in a scena io whe e al e na ely wo uni s
and one uni a e sold, s a ing wi h a pu chase o 2 uni s in
=10
. As a esul , we
analyze he s a es
(c, )=(20,10)
,
(18,9)
,
(17,8)
,…,
(6,1)
. As he i m canno o e
ba ches ha a e no co e ed by capaci y any longe , mos o he cu es a e e mi-
na ed du ing he selling ho izon.
The pa e n he cu es d aw looks nea ly he same o bo h heu is ics. Ba ch
p ices ob ained by
E(𝜔)
a e lowe han hose ob ained by D. The gaps be ween ba ch
p ices a e nea ly same-sized o smalle ba ches (again, s a ing wi h he wo-uni
Fig. 5 Mean e enues (le ) and mean pu chases ( igh ) a e e y poin in ime,
=10, 9, …,1
wi h
C=20
, and
𝜔,𝜆∼U[0, 1]

1212
R.Schu
ba ch) and a e inc easing o bigge ba ches. A e selling, ba ch p ices o bigge
ba ches inc ease. This e ec is mo e p onounced a e selling wo uni s in compa i-
son o selling one uni . I is a well-obse ed p icing beha io in (s anda d) dynamic
p icing ha p ices inc ease a e a sale ook place. Howe e , his only holds pa -
ially in ou mul iuni se ing as p ices o small ba ches usually dec ease slowly and
s eady o e he selling ho izon.
In conclusion, we ha e obse ed wo no able e ec s in Figs.6 and 7. Fi s , p ices
o small ba ches ( ela i e o he emaining s ock) dec ease o e ime, ega dless o
whe he a sale akes place o no . This holds o he discussed se ings wi h a ea-
sonably la ge s ock, i.e.,
C
T
=
2
. In scena ios wi h a small s ock, i.e.,
T
C
=
2
, p ices
o small ba ches do no always dec ease om one pe iod o he nex ; o main ain
b e i y, we ha e excluded he co esponding igu es. Second, p ices o small and
medium ba ches inc ease app oxima ely linea ly wi h ba ch size, s a ing om he
wo-uni ba ch. This linea i y explains he s ong pe o mance o a piecewise linea
p icing scheme obse ed in ou nume ical s udy. Such a p icing scheme is easy o
communica e, as he i m can quo e wo p ices—one o he i s uni and one o
each addi ional uni —ins ead o a long lis o p ices o e e y possible ba ch size.
Fig. 6 E olu ion o ba ch p ices wi hou a pu chase o D (le ) and
E(
𝜔
)
( igh ) o e
=10,9, …,1
wi h
C=20
, and
𝜔,𝜆∼U[0, 1]
Fig. 7 E olu ion o ba ch p ices wi h pu chases a e e y pe iod o D (le ) and
E(𝜔)
( igh ) o e
=10,9, …,1
wi h
C=20
, and
𝜔,𝜆∼U[0, 1]
1213
Heu is ic solu ions o nonlinea dynamic p icing in he…
7 Conclusion
In his s udy, we in oduced a nonlinea dynamic p icing model, based on a cus-
ome choice model ha cap u es wo-dimensional cus ome he e ogenei y in e ms
o p oduc a ac ion and consump ion inclina ion. Despi e he complexi y o he
esul ing p obabili y unc ion, we we e able o simpli y i by le e aging s uc u al
p ope ies and emo ing i ele an ba ch p ices om he ac ion space. We hen p e-
sen ed op imali y condi ions o he s a e-wise op imiza ion model. Addi ionally,
we in oduced a modi ied model ha , unde ce ain condi ions, yields an op imal
solu ion ha is also op imal o he o iginal p oblem. Howe e , inding he op imal
solu ion o he s a e-wise op imiza ion model emains compu a ionally challenging.
To add ess hese di icul ies, we de eloped h ee no el heu is ics: one ha uses a
decomposi ion app oach and wo ha calcula e expec ed op imal p ices.
In ou nume ical s udy, wo heu is ics pe o med excep ionally well, wi h an op i-
mali y gap o less han
0.4%
and
0.6%
, while he hi d heu is ic was only sligh ly
behind. Mo eo e , hey signi ican ly ou pe o med o he mechanisms. The esul s
also highligh he e enue po en ial o nonlinea p icing, allowing i ms o compa e
s aigh o wa d p icing schemes, such as linea and piecewise linea p icing, wi h
nonlinea p icing. We u he analyzed bo h bes pe o ming heu is ics and ound
se e al in e es ing cha ac e is ics o well-pe o ming p icing policies: In cases wi h
easonably la ge s ocks (in ou se ing wi h, e.g.,
C
T≥2
), ba ch p ices o small
ba ches a e slowly dec easing o e ime. This is pa icula ly in e es ing as i is an
indica o ha changing p ices a a lowe a e (no a e e e y cus ome ) migh s ill
pe o m well in he p esence o mul iuni demand. This makes he ob ained poli-
cies also applicable in se ings whe e he i m canno sus ain equen changes in
ba ch p ices due o, e.g., echnical easons, cus ome s’ eluc ance, o s a egic con-
side a ions. On he o he hand, in scena ios wi h a limi ed s ock (in ou se ing wi h,
T
C≥2
), he impo ance o nonlinea p icing is declining, whe eas a ypical (s and-
a d) dynamic p icing s uc u e becomes mo e and mo e ele an .
Ano he inding is ha ba ch p ices a e nea ly linea o low- and medium-sized
ba ches s a ing wi h he wo-uni ba ch. This explains he s ong pe o mance o
a piecewise linea p icing scheme in ou nume ical s udy, which has he bene i o
being easy- o-communica e. Ins ead o displaying a long lis con aining p ices o
e e y possible ba ch size, he i m could quo e wo p ices—one o he i s uni
and one o addi ional uni s. Compu ing he op imal piecewise linea p icing scheme
e icien ly emains a challenge a his s age.
Supplemen a y In o ma ion The online e sion con ains supplemen a y ma e ial a ailable a h ps:// doi.
o g/ 10. 1007/ s00291- 025- 00820-3.
Acknowledgemen s The au ho would like o hank he anonymous e e ees o hei aluable sugges-
ions and eedback which con ibu ed o an imp o ed quali y o he esul s o he pape .
Funding Open Access unding enabled and o ganized by P ojek DEAL.
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