Practicable solution approaches for differentiated pricing of vehicle sharing systems

Mülle , Ch is ian
A icle — Published Ve sion
P ac icable solu ion app oaches o di e en ia ed p icing
o ehicle sha ing sys ems
Cen al Eu opean Jou nal o Ope a ions Resea ch
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Mülle , Ch is ian (2024) : P ac icable solu ion app oaches o di e en ia ed
p icing o ehicle sha ing sys ems, Cen al Eu opean Jou nal o Ope a ions Resea ch, ISSN
1613-9178, Sp inge , Be lin, Heidelbe g, Vol. 33, Iss. 1, pp. 145-190,
h ps://doi.o g/10.1007/s10100-024-00915-2
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Cen al Eu opean Jou nal o Ope a ions Resea ch (2025) 33:145–190
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ORIGINAL PAPER
P ac icable solu ion app oaches o di e en ia ed p icing
o  ehicle sha ing sys ems
Ch is ianMülle 1
Accep ed: 3 Ap il 2024 / Published online: 24 May 2024
© The Au ho (s) 2024
Abs ac
Vehicle sha ing sys ems ha e become inc easingly popula . Howe e , one-way ehi-
cle sha ing sys em p o ide s ace a majo challenge. The une en dis ibu ion o
ehicles ac oss loca ions caused by he une en na u e o he demand pa e ns poses
a p oblem, since he e a e accumula ions o ehicles whe e he demand is low. This
challenge can be sol ed wi h an app op ia e p icing app oach ha c ea es incen i es
o use -based eloca ion by conside ing supply-side ne wo k e ec s. While he li -
e a u e mos ly ocuses on ip-based p icing, we we e inspi ed by he majo i y o ca
sha ing p o ide s who use o igin-based minu e p icing ha di e en ia es based on
he o igins o en als, such as Sha e Now. The e o e, we de elop wo di e en and
p ac icable solu ion app oaches o de e mine spa ially and empo ally di e en ia ed
o igin-based minu e p ices ha ake in o accoun supply-side ne wo k e ec s. The
i s solu ion app oach does no di e en ia e be ween en als and demand and calcu-
la es con inuous p ices o e e y pe iod and loca ion. The second solu ion app oach
de e mines he ehicle dis ibu ion o each pe iod and hen calcula es he op imal
p ices o each pe iod backwa ds. Ex ensi e compu a ional expe imen s show ha
ou solu ion app oaches an icipa e supply-side ne wo k e ec s and hus gene a e a
nea -op imal p o i in less compu a ional ime compa ed o mo e complex bench-
ma ks om he li e a u e. In a sensi i i y analysis we addi ionally show ha he
esul s a e obus agains s ochas ici y o demand and ha he solu ion app oaches
pe o m well o di e en p ice se s.
Keywo ds Vehicle sha ing sys ems· Ca sha ing· Di e en ia ed p icing· S a ic
p icing· O igin-based· Op imiza ion· Backwa ds solu ion app oach
Communica ed by Ul ike Leopold-Wildbu ge .
* Ch is ian Mülle
ch is ian.muelle [email p o ec ed]
1 Chai o Se ice Ope a ions, Me ca o School o Managemen , Uni e si y o Duisbu g-Essen,
Lo ha s . 65, 47057Duisbu g, Ge many
146
C.Mülle
1 In oduc ion
Vehicle sha ing sys ems (VSSs), such as ca sha ing, bike sha ing, o scoo e
sha ing, a e speci ic sha ed mobili y sys ems (Mou ad e al. 2019). They allow
use s o lexibly and spon aneously en ehicles o indi idual ips o a sho
pe iod o ime (A aç e al. 2021). In con as o o he popula sha ed mobili y sys-
ems, hese indi idual ips a e conduc ed by he use . VSSs a e one componen
o a sus ainable mobili y concep in ligh o he ongoing deba e abou he clima e
c isis (Tu an e al. 2023). Fo example, ideally, ca , bike o scoo e sha ing sys-
ems wi h widely a ailable ehicles do no only pa ially eplace p i a e ehicles
bu can also help o educe emissions, especially, i he ips a e made wi h an
elec ic ehicle.
This, in combina ion wi h a high lexibili y, a e easons why he concep o
ca sha ing has become inc easingly popula among p o ide s and cus ome s in
ecen yea s. Fo example, in he EU, he sha e o sold ca s used o new mobili y
(ca sha ing, ide hailing, ide sha ing, and obocabs) is p edic ed o ise om 2%
in 2015 up o 15% in 2025 (Des a is 2017).
VSS p o ide s ope a e in a business a ea wi h a gi en lee -size. The VSS
can be ope a ed by di e en ins i u ions, such as public municipali ies o p i a e
companies. Cus ome s use a mobile applica ion o ge he loca ion and p ice o
a ailable ehicles. The VSS p o ide canno selec cus ome s o ips (no ip
selec ion), bu i does se p ices o loca ions in di e en pe iods, which a ec s
demand. The VSS p o ide s a e in e es ed in he cen al objec i e o maximiz-
ing p o i (minimizing cos s) (Pan uso 2022). In he bes case, his objec i e
is achie ed wi h he help o a p ac icable solu ion app oach. P ac icable he e
means ha he p o ide is able o success ully execu e o implemen he solu ion
app oach in p ac ice wi hou any u he knowledge, e.g. abou addi ional p e-
p ocessing s eps.
We make a dis inc ion be ween one-way o ee- loa ing and wo-way VSS. In
one-way VSS, he cus ome can pick up he ehicle a one s a ion and d op i o
a any o he s a ion o a he same s a ion (o in ee- loa ing VSS, he cus ome
can pick up he ehicle and d op i o anywhe e in he business a ea). In wo-way
VSS, he cus ome has o d op o he ehicle a he same s a ion whe e she picked
i up. We ocus on one-way (and ee- loa ing) VSS as a lexible and con enien
al e na i e o p i a e ehicles, which enables use s o pick up and d op o ehi-
cles anywhe e wi hin he p o ide ’s business a ea. Howe e , one-way (and also
ee- loa ing) VSS p o ide s ace a main challenges: The dis ibu ion o ehicles
is une en ac oss loca ions, because o igin and des ina ion loca ions a e in lu-
enced by he cus ome ’s p e e ences. These p e e ences and hus he demand a y
du ing he day (une en na u e o he a el pa e n). This is he so-called " ide
phenomenon", which ep esen s he oscilla ion o demand in ensi y h oughou
he day (spa io- empo al demand asymme ies) (Côme 2014; Jo ge and Co eia
2013). Besides, he cu en numbe o a ailable ehicles in an loca ion a ec s he
numbe o u u e en als in ha loca ion. The e o e, i mus be aken in o accoun
ha a en al no only dec eases he cu en amoun o ehicles in a o igin loca ion
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P ac icable solu ion app oaches o di e en ia ed p icing…
and inc eases he amoun o ehicles in he des ina ion loca ion, bu also has an
impac on u u e en als in hese wo loca ions (supply-side ne wo k e ec s).
Neglec ing hese supply-side ne wo k e ec s leads o an accumula ion o absence
o ehicles a popula loca ions. As a esul , he sys em is no longe able o se e
demand and may lose cus ome s (Di Febb a o e al. 2012).
Wi h ega ds o he imbalance, a possible solu ion is eloca ion. Mo e p e-
cisely, eloca ion can be dis inguished in o ope a o -based and use -based eloca-
ion. Ope a o -based eloca ion in ol es eposi ioning o ehicles by employees,
while use -based eloca ion is pe o med by cus ome s and is incen i ized by he
p o ide . The ope a o -based eloca ion inc eases he ope a ional cos s, as p o id-
e s need addi ional s a and ex a equipmen , e.g. ope a o -based eloca ion o
bikes is handled wi h ucks (Dö e l e al. 2017). Al hough ope a o -based elo-
ca ion is one o he main cos d i e s (Jo ge and Co eia 2013) and ( ime) ine -
icien ( ehicle canno be used while i is eloca ed by s a ) (Schi e e al. 2021),
almos all VSS p o ide s use i o ob ain educed imbalance.
A mo e cos -e ec i e me hod is use -based eloca ion (B endel e al. 2016).
Ideally, he p o ide se s p ices o encou age cus ome s o d i e om a low-
demand loca ion o a high-demand loca ion (Angelopoulos e al. 2016). Fu he -
mo e, no addi ional ehicles o addi ional ips a e needed. In sho , use -based
eloca ion is mo e p e e able om an en i onmen al and economic pe spec i e
(Clemen e e al. 2017), as i is a mo e sus ainable and cos -e ec i e al e na i e o
ope a o -based eloca ion (S okkink and Ge oliminis 2021).
In addi ion, p o ide s should also conside cus ome p e e ences ega ding
p icing. Cus ome s ha e wo dis inc p e e ences:
1. Cus ome s p e e an easy and com o able booking p ocess in VSSs, i.e. booking
a ehicle wi hou much e o . They canno and/o do no wan o disclose hei
des ina ion o he du a ion o he in ended en al. The e o e, VSS p o ide s do no
know he des ina ion in ad ance. I VSS p o ide s ask cus ome s o ( u h ully)
disclose hei in ended des ina ion, his would conside ably change he cus ome s
expe ience o VSSs and hus would be unaccep able in mos p ac ical se ings.
2. Cus ome s p e e clea and anspa en p ices. This means ha cus ome s would
like o know (minu e) p ices be o e he en al s a s.
The easies way o sa is y bo h cus ome p e e ences is o se a uni p ice (min-
u e p ice is he same ega dless o loca ion and ime). Howe e , a one-size- i s-
all p ice ails o mee he VSS p o ide s challenge (une en ehicle dis ibu ion).
Ano he way o p ice he wo cus ome p e e ences is o choose disc e e p ices
om a p ede ined p ice se ha a y (only) based on whe e and when a en al
s a s (o igin-based p icing). This p ede ined p ice se has a clea numbe o p ice
poin s. The al e na i e, i.e., displaying p ices o all po en ial ips combina ions
( ip-based p icing) in ad ance o a en al, is imp ac icable in gene al, gi en ha
ee- loa ing VSS o one-way s a ion-based VSS p o ide s o en disc e ize hei
business a ea in o up o hund ed zones (Mülle e al. 2023).
148
C.Mülle
Thus, he challenge o imbalance, he objec i e o maximiza ion, and cus-
ome p e e ences can be easonably add essed wi h an app op ia e o igin-based
di e en ia ed p icing app oach wi h disc e e p ices o m a p e-de ined p ice se .
I incen i izes cus ome s o imp o e u u e ehicle dis ibu ion (use -based elo-
ca ion) by conside ing a longe ime ho izon (e.g., one day, one week), which
means ha supply-side ne wo k e ec s a e su icien ly aken in o accoun . This is
in line wi h he business decision o Sha e Now (o igin-based p ices om a p ice
se wi h h ee di e en p ice poin s).
Agains his backg ound, his pape ocuses on o igin-based p icing, whe e he
ee- loa ing o one-way VSS p o ide se s p ices depending on a en al’s ime
and o igin and om a disc e e p ice se . Since o igin-based p icing is mos com-
monly used in cu en p ac ice, he p o ide s o such VSSs do no ha e o change
hei booking p ocess by asking cus ome s o he des ina ion o en al du a ion.
This means a mo e e icien in e ac ion be ween use and p o ide and an easie
implemen a ion. To be p ecise, in his wo k, we conside he p oblem o di e en-
ia ed p icing o ee- loa ing o one-way VSS p o ide s wi h a ocus on i s p ac-
icabili y, by using di e en heu is ic solu ion app oaches.
The con ibu ions o ou wo k a e he ollowing:
• Fi s , o he bes o ou knowledge, we a e one o he i s o ocus on p ac i-
cable, o igin-based di e en ia ed p icing, which is highly ele an in p ac ice.
The p oposed p icing mechanism is anspa en o he cus ome . The p ob-
lem’s p ac ical ele ance is ensu ed by, among o he hings, a coope a ion
wi h Sha e Now, Eu ope’s la ges ca sha ing p o ide .
• Second, he solu ion app oaches we de elop a e p oblem-speci ic, easy o
implemen and new. The i s one is a simpli ied model, so ha he p oblem
can be sol ed quickly, e en o la ge ins ances. The second solu ion app oach
is a backwa ds algo i hm o de e mining he bes p ices o all loca ion-
pe iod-combina ions. Thus, he majo ad an age o bo h app oaches is ha
hey equi e no p e-p ocessing and ye p oduce he same esul s as exis ing,
mo e complex benchma ks in clea ly less ime. Thus, hese new solu ion
app oaches a e bene icial o p ac i ione s o ge a p ac icable and s aigh -
o wa d solu ion and o esea che s o benchma k o he upcoming solu ion
app oaches.
• Thi d, we gene a e a numbe o ele an manage ial insigh s based on an
ex ensi e compu a ional s udy and a sensi i i y analysis wi h di e en p ob-
lem sizes, conside ing a ious ele an pa ame e se ings and demand pa -
e ns.
The emainde o he pape is o ganized as ollows. In Sec .2, we e iew he ele-
an li e a u e, ocusing on di e en ia ed p icing p oblems using op imiza ion. In
Sec .3, we desc ibe he p oblem and p esen wo p oposed solu ion app oaches.
Sec ion4 con ains he compu a ional s udy. A e he compu a ional s udy, we
pe o m a sensi i i y analysis in Sec .5. Sec ion6 concludes he pape and gi es
an ou look on u u e esea ch. The appendix p o ides addi ional esul s o he
compu a ional expe imen s and unde lying ma hema ical model.

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P ac icable solu ion app oaches o di e en ia ed p icing…
2 Li e a u e
The li e a u e on VSS op imiza ion is qui e ex ensi e, so o gene al o e iews we
e e o he ollowing pape s:
• bike sha ing: DeMaio (2009), Fishman e al. (2013), Ricci (2015)
• ca sha ing: Jo ge and Co eia (2013), Fe e o e  al. (2015a), Fe e o e  al.
(2015b), Illgen and Höck (2019), Nansubuga and Kowalkowski (2021),
Es andabadi e al. (2022)
• VSS in gene al: Lapo e e al. (2015, 2018), A aç e al. (2021), wi h a ocus on
sus ainabili y: Tu an e al. (2023,Chap e 5.1)
In ou li e a u e e iew, we ocus on di e en ia ed p icing in VSSs in he sense ha
he p icing does no depend on he sys em’s cu en s a e (e.g. cu en ehicle dis-
ibu ion, see A aç e al. 2021). Fu he mo e, we only conside pape s ha apply
collec i e, no indi idual p icing ( a ge ed o all cus ome s, see Pan uso 2022) using
op imiza ion. We exclude pape s ha apply business ules (e.g. Ruch e al. 2014;
B endel e al. 2016; Wagne e al. 2015; Ba h e al. 2004).
In he ollowing, we in oduce dimensions o di e en ia ed p icing app oaches
(Sec .2.1). Using hese dimensions, Sec .2.2 conside s di e en ia ed p icing and
Sec .2.3 p esen s u he li e a u e ha de eloped single-pe iod solu ion app oaches
in a olling-ho izon ashion o dynamic p icing.
2.1 Dimensions o di e en ia ed p icing
We p opose wo dimensions o he cus ome pe spec i e and ou dimensions o he
p o ide pe spec i e o s uc u e he di e en solu ion app oaches (see also Table1
in Appendix 1). The ollowing dimensions desc ibe he cus ome pe spec i e:
1. Spa io- empo al p icing: O igin-based p ices depend only on he ime and loca-
ion o a en al’s s a . O he a ian s a e des ina ion-based p ices (p ices depend
on loca ion and ime o des ina ion) o ip-based p ices (p ices depend on bo h
o igin and des ina ion).
2. Numbe o possible p ices: Some p icing app oaches se only one p ice o all
loca ions and pe iods, whe eas o he s se di e en p ices selec ed om a disc e e
se o p ices (p ice lis ). S ill o he s ei he ha e a de ined uppe and lowe bound
o p ices, o he es ic ion ha p ices mus be posi i e, o no es ic ions ega d-
ing he p ices a all.
The ollowing ou dimensions cha ac e ize he p o ide pe spec i e:
1. Con ol o en als: The e a e p o ide s ha can in luence he numbe o en als
only by he p ice (p ice con ol). Howe e , he e a e also p o ide s ha can addi-
ionally ejec eques s ( ip selec ion).
150
C.Mülle
2. Objec i e: Di e en p icing app oaches aim ei he a imp o ing he dis ibu ion
(balance) o ehicles in he VSS o a inc easing p o i (o educing cos s).
3. Fo esigh : Some p icing app oaches de e mine p ices based on he cu en
pe iod wi hou conside ing he supply-side ne wo k e ec s o he nex pe iod(s)
(myopic). In con as , o he p icing app oaches addi ionally conside how p icing
decisions in he cu en pe iod a ec u u e ehicle supply, and hus u u e en als
a each loca ion in subsequen pe iods, by conside ing supply-side ne wo k e ec s
(an icipa i e).
4. Addi ional pa ame e s: Some p icing app oaches equi e some p e-p ocessing,
o ins ance, addi ional es ima ion o pa ame e s in ad ance o pe o m he p ice
de e mina ion.
2.2 Li e a u e ondi e en ia ed p icing
Mos o he published pape s conside ing di e en ia ed p icing in VSS deal wi h ip-
based p ices. The e a e solu ion app oaches ha use a luid app oxima ion o de e mine
p ices. In luid app oxima ions, he model se s he p ices so ha he en als o a s a ion
ma ch he demand, i.e. he e is no dis inc ion be ween demand and en als. This means
ha he p ice is no cons ained (e en nega i e p ices a e possible). Wase hole and Jos
(2012) p opose a luid app oxima ion o he e enue-maximizing ip-based p icing
p oblem, which is he uppe bound o he s ochas ic model i demand and supply a e
scaled o in ini y. Guo and Kang (2022) also p opose a luid model o maximize p o i ,
which conside s he p icing and e-balancing p oblem o elec ic ehicles.
Howe e , o he pape s dis inguish be ween demand and en als. Mo e p ecisely,
en als depend on supply, demand and p ices. Only posi i e p ices a e de e mined
he e. We dis inguish be ween pape s whe e he p o ide canno ejec use s and
con ols en als only by p ice (i.e. p ice con ol) and pape s whe e he p o ide can
ejec use s (i.e. ip selec ion).
Fi s , we conside pape s whe e he p o ide can (indi ec ly) ejec use s. Xu e al.
(2018) o mula e a mixed-in ege non-linea and non-con ex p og am. On his basis,
hey de elop a compu a ionally ac able mixed-in ege con ex p og am which has he
same objec i e in he op imum, and sol e he la e a bi a ily close o op imali y. Jiao
e al. (2020) in eg a e ip selec ion and p ice incen i es wi h use -based eloca ion in
one model. The e o e his model dis inguishes be ween h ee di e en ypes o demand:
(1) po en ial a el demand, (2) adop ed demand (demand a e cus ome s see he p ice),
and (3) inal se ed demand (a e ip selec ion). They conside a mixed-in ege non-
linea p og am o maximize he p o i and p opose an i e a i e algo i hm be ween he
wo decomposi ion p og amming sub-p oblems (a linea mas e sub-p oblem and a non-
linea sub-p oblem). Huang e al. (2020) compa e ope a o -based and use -based elo-
ca ion. They o mula e wo mixed-in ege non-linea p og ams o he use -based elo-
ca ion. The i s one se s ip-based p ices, whe eas he second op imizes pick-up and
d op-o ees. They sol e bo h p og ams wi h a combined olling-ho izon and i e a ed
local sea ch heu is ic. Lu e al. (2021) use ano he model o mula ion, i.e. a bi-le el
non-linea p og am in which he p o ide de e mines p o i -maximizing p ices on he
uppe le el. In his case, he p ices a e wi hin he p e iously de ined bounds. The lowe
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P ac icable solu ion app oaches o di e en ia ed p icing…
le el’s objec i e minimizes cus ome s’ o al cos by a bina y choice be ween wo modes
o anspo a ion (sha ed ehicles s. p i a e ehicles). In an in e p e a ion o a disc e e
choice model, en als a e addi ionally bounded om abo e by a logi model. The au ho s
ans o m he bi-le el p og am o a single-le el one using Ka ush–Kuhn–Tucke condi-
ions, and heu is ically sol e i wi h a gene ic algo i hm.
Second, we conside pape s whe e he p o ide s only con ol en als by p ice.
Jo ge e  al. (2015) o mula e a p o i -maximizing ip-based p icing p oblem as
mixed-in ege non-linea , non-conca e p og am, which is no ac able o eal-
wo ld-ins ances. They p opose an i e a ed local sea ch me a-heu is ic o sol e he
p og am. Ren e al. (2019) ex end he p e ious p og am o include a ehicle-g id
in e ac ion o elec ic ehicles. They use non-linea sol e s.
Huang e al. (2020) combine ac ical and ope a ional decisions o a one-way s a-
ion-based ca sha ing sys em on a mixed-in ege non-linea p og am. This p og am
op imizes p o i by conside ing lee size, p icing (bo h ac ical) and eloca ion. They
linea ize his p og am, decompose i in o wo in e dependen s ages, and de elop a
g adien sea ch me hod o sol e he wo s ages. Zhang and Kan (2018) o mula e a
non-linea , non-conca e p og am ha maximizes he p o i o an en i e planning
ho izon o a s a ion-based, one-way ca sha ing sys em by se ing ip-based p ices.
Pa icle swa m op imiza ion is used o sol e he p og am.
In con as o he p esen ed li e a u e abo e, Soppe e al. (2022) conside o igin-
based, di e en ia ed p icing o a one-way o ee- loa ing VSS. They o mula e a
p o i -maximizing, mixed-in ege linea p og am ha dis inguishes be ween en als
and demand. This p og am de e mines o igin-based p ices om a disc e e p ice se .
The en als o a loca ion a e calcula ed as a minimum o demand and supply (num-
be o idle ehicles). The demand can be in luenced only by he p ice (p ice con ol).
Since he model canno be sol ed due o i s complexi y, Soppe e al. (2022) p o-
pose a solu ion me hod using alue unc ion app oxima ion. To apply his me hod,
he p o ide has o es ima e some pa ame e s in p e-p ocessing.
We also conside a one-way o ee- loa ing VSS p o ide , who de e mines o i-
gin-based p ices om a disc e e p ice se (i.e. p ice con ol). We sugges wo solu-
ion app oaches o sol e his mixed-in ege linea p og am wi hou he need o es i-
ma e any pa ame e s in p e-p ocessing. This makes hem mo e p ac icable.
2.3 Fu he li e a u e
In addi ion, he e a e o he models o VSSs ha a e de eloped in such a way ha
hey ha e o be sol ed o each single pe iod wi h he cu en ly a ailable da a (wi h-
ou conside ing supply-side ne wo k e ec s). Al hough hey apply dynamic p icing,
hey ha e simila i ies wi h ou p oblem.
The ollowing pape s do no dis inguish be ween demand and en als. They
assume ha p ices do no a ec he demand, bu a ec he cus ome s’ des ina ions.
P omme e al. (2014) p opose a model p edic i e con ol app oach. The objec i e
o he quad a ic p og am is a weigh ed sum o he de ia ion om an op imal ehicle
dis ibu ion and he cos o incen i e paymen s. Chemla e al. (2013) use a linea
p og am o de e mine he numbe o cus ome s who change hei a el plans due
o he p ice incen i e in o de o each he gi en a ge in en o y o ehicles o
152
C.Mülle
each s a ion. Haide e al. (2018) o mula e a bi-le el p og am, whe e he uppe le el
de e mines p ices and minimizes ehicle imbalance, while he lowe le el ep esen s
he cos -minimizing ou e choice o cus ome s. The p oblem is ans o med in o a
single-le el p og am. Wang and Ma (2019) conside he objec i e o keeping he
ehicle in en o y wi hin a ce ain ange o a pe iod. Fo his pu pose, hey de ine
lowe and uppe h esholds o each s a ion. The numbe o en als om o o a s a-
ion can be a ec ed by pick-up and d op-o ees. To his end, hey o mula e a sim-
ple quad a ic p og am o calcula e such op imal dynamic ees.
O he pape s dis inguish be ween demand and en als. Pan uso (2020, 2022) o -
mula es an ex ensi e mixed-in ege wo-s age s ochas ic p og am, which maximizes
he p o i by se ing ip-based p ices and decides abou ope a o -based eloca ion.
Pan uso (2020) also p oposes a compac in ege p og amming e o mula ion and
compa es he wo o mula ions in e ms o ease o solu ion. Pan uso (2022) p oposes
an exac solu ion algo i hm o he mixed-in ege wo-s age s ochas ic p og am.
3 Solu ion app oaches o di e en ia ed p icing
In his sec ion, we i s de ine he o igin-based di e en ia ed p icing p ob-
lem in VSSs (Sec .3.1) o in oduce he p oblem, and hen desc ibe wo solu ion
app oaches o di e en ia ed p icing. One solu ion app oach is o use a simpli ied
model (Sec .3.2), he o he is o calcula e p ices backwa ds (Sec .3.3).
3.1 P oblem s a emen andno a ion
The e is a ee- loa ing VSS p o ide , which disc e izes he business a ea in Z zones
Z=1, ..., Z
) which can be ea ed as s a ions. This VSS p o ide se s di e en ia ed
o igin-based p ices om a disc e e p ice se
PM
. This means o all en als, which
ha e he same o igin, he same minu e p ice
pi,
is cha ged. The conside ed ime
ho izon is subdi ided in o T pe iods (
T=1, ..., T
). The VSS p o ide maximizes
i s p o i by se ing minu e p ices
pi,
o e e y loca ion
i∈Z
and pe iod
∈T
,
ega dless o he des ina ion. The minu e p ices a e chosen om a gi en p ice se
PM
=p
1, ...,
p
M wi h M p ice poin s.
We also ha e he ollowing assump ions conce ning demand, en al ealiza ion and
dynamics. Demand depends on he p ice. A base demand (demand a he median p ice)
di
,
j
,
o each en al combina ion
i−j
o possible o igins and des ina ions a pe iod is
gi en. We assume ha i he p ice is lowe ( han he median p ice), demand inc eases
and i he p ice is highe ( han he median p ice) demand dec eases. Thus, we scale he
base demand wi h sensi i i y ac o s
m
, which depend on he chosen p ice
pi, =pm
om a gi en p ice se
PM
=p
1, ...,
pM
wi h M p ice poin s (like in Soppe e al. 2022).
A low p ice co esponds o a high sensi i i y ac o and ice e sa (like in Özkan 2020;
Soppe e al. 2022). The ad an age o using hese p ice poin s
pm∀m=1..M
om he
p ice se s and he associa ed sensi i i y ac o s
m∀m=1..M
is ha all (e en nonlinea )
demand unc ions can be ep esen ed by hese p ice se s and sensi i i y ac o s wi hou
a ec ing he (linea ) model. The sensi i i y ac o s
m∀m=1..M
can be de e mined
159
P ac icable solu ion app oaches o di e en ia ed p icing…
• BASE deno es a benchma k using cons an uni o m p icing. He e we use he
base p ice p
i,
=p
(2)
∀i∈Z, ∈T.
• MOD48h deno es he solu ion o he model wi h a gi en ime limi o he sol e
o 48h in which all 48 pe iods a e op imized simul aneously.
• ROL-H is a basic olling-ho izon app oach and is con igu ed wi h di e en ho i-
zon leng hs H (ROL-1, ROL-4, ROL-8). No e ha his benchma k wi h
H=1
ep esen s he myopic solu ion ha only conside s one pe iod in each subs i u e
p oblem.
• ADP-H is he ADP decomposi ion solu ion app oach p esen ed in Soppe e al.
(2022), and is con igu ed wi h di e en ho izon leng hs H (ADP-1, ADP-4,
ADP-8). I uses a alue unc ion app oxima ion o app oxima e he u u e a e
he ho izon H ha is being conside ed.
Each combina ion o se ings and DSRs o ms an ins ance in ou expe imen s. We
implemen he algo i hms in Py hon 3.8 and sol e all solu ion app oaches and
benchma ks wi h Gu obi 9.1.2. In all scena ios, we se he op imali y gap o ze o and
he ime limi is se o one hou o ADP-H, ROL-H, BAW-ROL-1, BAW-MODSIM
and o 48h o MOD48h and MODSIM. We execu e he compu a ions on a wo ks a-
ion wi h an AMD Ryzen 9 3900 X 12-Co e p ocesso wi h 12 co es and 64 Giga-
by e RAM. Please no e ha we use he MILP o Soppe e al. (2022) (see Appendix
12) o e alua e he compu ed p ices o he solu ion app oaches and he benchma ks.
4.3 Resul s
In his sec ion we p esen he esul s ega ding he analyses o p o i (Sec .4.3.1),
p icing (Sec .4.3.2), en als (Sec .4.3.3) and compu a ional ime (Sec .4.3.4).
4.3.1 P o i
We begin wi h a compa ison o he di e en solu ion me hods and he benchma ks
by iden i ying he imp o emen o e BASE. The po en ial is g aphically shown
in Fig.3. I depic s he p o i ob ained wi h he di e en solu ion app oaches and
benchma ks ( o he la e conside ed ADP-H and ROL-H in dependence o he ho i-
zon leng hs H on he ho izon al axis) ela i e o he p o i wi h BASE, which he
0%-line ma ks. The p o i s ob ained by MODSIM, BAW-ROL-1, BAW-MODSIM
and MOD48h a e ho izon al lines as hey do no depend on ho izon H.
We obse e ha MOD48h yields a p o i inc ease o a leas 13.5% o e BASE.
Fo SMALL wi h
DSR =1∕3
MOD48h yields he op imal solu ion. Fo LARGE
and a highe DSR han 1/3, MOD48h does no ind any easible solu ion wi hin 48h.
The myopic solu ion ROL-1 p o ides a leas 5% mo e p o i han BASE.
The di e ence be ween ROL-1 and MOD48h (in he ins ance SMALL,
DSR =1∕3
) shows he whole e ec o conside ing supply-side ne wo k e ec s. The
exac supply-side ne wo k e ec o la ge ins ances canno be de e mined because i
is no possible o de e mine he op imal solu ion wi hin 48h. In he ins ance whe e
an op imal solu ion can s ill be de e mined wi hin 48h (SMALL,
DSR =1∕3
), he

160
C.Mülle
p o i s o MODSIM, and BAW-ROL-1, BAW-MODSIM and ADP-8 a e e y simi-
la o he p o i o he op imal solu ion (MOD48h). Fu he mo e, i can be shown
ha he p o i s o ADP-8, BAW-ROL-1 as well as BAW-MODSIM and MODSIM
a e almos he same in all o he ins ances (see Fig.1 in Appendix). BAW-ROL-1
(BAW-MODSIM) gene a es a p o i ha is a mos 0.55 (0.22) pe cen age poin s
highe o 0.16 (0.06) pe cen age poin s lowe han ADP-8. MODSIM gene a es a
p o i ha is a mos 0.19 pe cen age poin s highe o 1.26 pe cen age poin s lowe
han ADP-8. This leads o he conclusion ha hese solu ion app oaches and he
benchma ks conside he supply-side ne wo k e ec s bes .
In con as , he p o i o ROL-8 is some imes e y simila (e.g. Fig.3a) o he
p o i o MOD48h, some imes clea ly lowe (e.g. Fig.3b, c).
Thus, we can d aw he ollowing conclusions:
1. Al hough he p oposed solu ion app oaches (MODSIM, BAW-ROL-1, BAW-
MODSIM) a e a he s aigh o wa d, hey achie e equi alen esul s o mo e
complex benchma ks (MOD48h, ADP-8).
2. The compa ison o p o i o bo h new solu ion app oaches (MODSIM, BAW)
shows ha hey success ully conside supply-side ne wo k e ec s, as do he
benchma ks MOD48h and ADP-8.
4.3.2 P icing decisions
The an icipa i e solu ion app oaches MODSIM, BAW-ROL1 and BAW-MOD-
SIM conside supply-side ne wo k e ec s as he benchma ks ADP-8 and MOD48h
in con as o ROL-1 (see Sec .4.3.1). These supply-side ne wo k e ec s a e also
e lec ed in he p icing decisions.
The p icing decisions o selec ed solu ion app oaches and benchma ks
a e depic ed as p ice ables in Fig.4 (see Fig.13 in Appendix4 o all solu ion
app oaches and benchma ks) o SMALL wi h
DSR =
1
∕3
. Fo he sake o simplic-
i y, we only analyze he p icing decision o he benchma ks ROL-1 (as myopic p ic-
ing), ROL-8, MOD-48h, ADP-1 and ADP-8.
Since MOD48h (which is he op imal solu ion in his ins ance) conside s he sup-
ply-side ne wo k e ec s h oughou he day, we compa e he p ice able o he o he
solu ion app oaches and benchma ks wi h ha o MOD48h. I is ob ious ha he
p ice able o ROL-1 is e y di e en om he p ice able o MOD48h, which shows
Fig. 3 Rela i e p o i inc ease (SMALL, MEDIUM, LARGE),
DSR =
1
∕3
161
P ac icable solu ion app oaches o di e en ia ed p icing…
he s ong impac o supply-side ne wo k e ec s. The p ice ables o ROL-8, ADP-8,
MODSIM, BAW-ROL-1 and BAW-MODSIM a e e y simila o he p ice pa e n
o MOD48h. Tha indica es ha hese solu ion app oaches and benchma ks conside
supply-side ne wo k e ec s.
On an agg ega e le el, hese di e ences become also isible in compa ing he p o-
po ion o di e en p ices o he solu ion app oaches and benchma ks. In SMALL
wi h
DSR =1∕3
, o example, ROL-1 esul s in 2% low, 77% base, and 22% high
p ices (see Fig.5). P icing decisions o MOD48h consis s o 34% low, 29% base, and
37% high p ices. The p opo ions o di e en p ices o ADP-8 (43% low, 19% base,
38% high p ices), MODSIM (45% low, 16% base, 39% high p ices), BAW-ROL-1
(36% low, 28% base, 36% high p ices) and BAW-MODSIM (35% low, 25% base,
37% high p ices) a e also simila o he p opo ion o di e en p ices o MOD48h,
especially o he high p ices. This shows ha he be e supply-side ne wo k e ec s
a e cap u ed, he mo e he esul ing p icing decisions esemble he op imal p icing.
Thus, we can d aw he ollowing conclusions:
1. Supply-side ne wo k e ec s a e isible in he p ice ables and p ice p opo ions
o MOD48h.
2. MODSIM, BAW-ROL-1 and BAW-MODSIM c ea e p ice ables which a e e y
simila o hose o MOD48h and ADP-8, which indica es ha hey conside sup-
ply-side ne wo k e ec s e ec i ely.
Fig. 4 P icing di e en solu ion app oaches and benchma ks (SMALL),
DSR =
1
∕3
. G een: L = low
p ice, yellow: B = base p ice, ed: H = high p ice
Fig. 5 P ice p opo ions. G een: low p ice, yellow: base p ice, ed: high p ice
162
C.Mülle
4.3.3 Ren als
The conside a ion o supply-side ne wo k e ec s is e iden in p o i and p icing. We
also conside he cou se o en als o SMALL wi h a
DSR =1∕3
(see Fig.6). Since
p ices a ec demand and demand a ec s en als, we s udy he ex en o which sup-
ply-side ne wo k e ec s a e e iden o en als. Fo he sake o simplici y, we only
analyze he p icing decision o he benchma ks BASE, ROL-1 (as myopic p icing),
ROL-8, MOD-48h and ADP-8.
The i s hing o no ice is ha du ing he pe iods wi h low demand, BASE esul s
in ewe en als han MOD48h and du ing he peaks i gene a es mo e en als han
MOD48h. Fu he mo e, i is appa en ha he en al-cu e o ROL-1 luc ua es simi-
la o he en al-cu e o BASE, whe eas he en al-cu e o MOD48h luc ua es less
han bo h. Ano he ema kable ea u e is ha he en al cu es o MOD48h, ADP-8,
BAW-ROL-1 and BAW-MODSIM lie almos on op o each o he . This means ha
he en al cu es o ADP-8, BAW-ROL-1, and BAW-MODSIM a e hidden behind
he en al cu e o MOD48h in Fig.6. The en al cu e o MODSIM de ia es only
sligh ly om hese h ee en al cu es.
F om his we conclude ha all an icipa i e solu ion app oaches and benchma ks
(ROL-8, ADP-8, MODSIM, BAW-ROL-1, BAW-MODSIM) in his se ing conside
he supply-side ne wo k e ec s simila ly and ha apa om he simila p ices, he
en al cu es o MOD48h, ADP-8 BAW-ROL-1, BAW-MODSIM and MODSIM
a e also e y simila .
4.3.4 Compu a ional ime
An impo an aspec o he p ac icabili y o solu ion app oaches is, among o he s,
he compu a ional ime. Fo his pu pose, we compa e he compu a ional ime o
SMALL, MEDIUM and LARGE (see Fig.7 o
DSR =1∕3
).
Fi s , we conside he di e en benchma ks. MOD48h (2.5 h up o o e 48h)
always akes he longes ime o SMALL. Fo MEDIUM and LARGE, MOD48h
akes he ull gi en ime o 48h. The compu a ional imes o he olling-ho izon
and decomposi ion solu ion app oaches depend on hei ho izon. ROL-1 (6s up
o unde 1 min) akes he second lowes compu a ional ime. ROL-4 (18s up o 8
min) equi es a simila compu a ional ime. In con as , ROL-8 (20 min up o o e
12h) equi es a clea ly longe compu a ional ime han ROL-1 and ROL-4. The
Fig. 6 Ren als o e he day
(SMALL),
DSR =
1
∕3
163
P ac icable solu ion app oaches o di e en ia ed p icing…
o de o compu a ional imes o he di e en ho izons is simila o he decompo-
si ion solu ion app oaches. Mo e p ecisely, ADP-1 (4 min up o o e 12 min) and
ADP-4 (4 min up o o e 1.5 h) s ill equi e ela i ely sho compu a ional imes,
whe eas he compu a ional ime o ADP-8 (23 min up o o e 17.5 h) is clea ly
longe . Thus, when compa ing each di e en ho izon leng h o he olling-ho i-
zon app oaches wi h he decomposi ion solu ion app oaches i is ob ious ha he
decomposi ion solu ion app oaches need longe compu a ional imes. This is due
o he conside a ion o u u e s a es in he decomposi ion solu ion app oaches.
Howe e , i should also be no ed ha he benchma ks ADP-1, ADP-4 and ADP-8
equi e pa ame e es ima ion in ad ance. The addi ional compu a ional ime o
pa ame e es ima ion o all pe iods, which lies be ween less han 1h o SMALL
and oughly 2h o LARGE, hus, mus be conside ed (Soppe e al. 2022).
Second, we conside he p oposed solu ion app oaches. MODSIM (1 s up
o 6s) akes he leas compu a ional ime o SMALL, MEDIUM and LARGE.
BAW-ROL-1 (1.5 min up o 35 min) and BAW-MODSIM (1 min up o o e 3.h)
need ela i ely sho compu a ional imes.
Compa ing benchma ks wi h solu ion app oaches, ADP-8 and MOD48h,
which conside he supply-side ne wo k e ec s e ec i ely, equi e mo e compu-
a ional ime han MODSIM, BAW-ROL-1 and BAW-MODSIM. In mos cases,
ADP-4 and ROL-8 equi e longe compu a ional imes han he p oposed solu ion
app oaches. ROL-4 akes less ime han BAW-MODSIM and BAW-ROL-1, bu
mo e ime han MODSIM.
Fig.15 in Appendix 5 depic s he esul s o SMALL, MEDIUM and LARGE
wi h all DSRs. We also in es iga e he compu a ional ime in dependency o he
numbe o zones o
DSR =1∕3
(see Fig.16inAppendix 6). Fo his analysis, he
ime limi o he calcula ions was emo ed, bu a a ge op imali y gap o 2% was
added. All p oposed solu ion app oaches p o ide esul s o all conside ed num-
be o zones. This shows ha he new solu ion app oaches a e also applicable o
ealis ic zone sizes.
F om he ob ained compu a ional imes we can conclude he ollowing.
1. ROL-1 and ROL-4 equi e sho compu a ional imes bu do no conside he
supply-side ne wo k e ec s e ec i ely (see Sec .4.3.1).
2. ADP-8 and MOD48h equi e long compu a ional imes.
Fig. 7 Compu a ional ime (SMALL, MEDIUM, LARGE),
DSR =1∕3
164
C.Mülle
3. MODSIM equi es a sho compu a ional ime and and is a p e e ed op ion due
o he compa able esul s in Sec .4.3.1 wi h he bes benchma ks (MOD48h,
ADP-8).
4. BAW-ROL-1 and BAW-MODSIM need abou he same and clea ly less compu-
a ional ime han he benchma ks ha gi e simila esul s (ADP-8, MOD48h).
Thus, hey a e also p e e ed op ions.
5. MODSIM, BAW-ROL-1 and BAW-MODSIM p o ide esul s e en o ealis ic
ins ances (e.g.
Z=81
).
5 Sensi i i y analysis
In his sec ion, we pe o m a sensi i i y analysis o show how s able he solu ions o
he di e en app oaches a e. Fi s , we s udy he solu ion s abili y in a s ochas ic en i-
onmen (see Sec .5.1). Second, we analyze he e ec o di e en in e als be ween
p ices in p ice se s and di e en numbe s o p ices in p ice se s by modi ying he
disc e e p ice se (see Sec .5.2). Thi d, we apply he di e en solu ion app oaches
and benchma ks o one week (see Sec .5.3). Fou h, we s udy he impac o a s a
solu ion on esul s and compu a ional ime (see Sec .5.4).
5.1 S ochas ic demand
We analyze he obus ness o he p ices gene a ed by he di e en solu ion
app oaches and benchma ks in a s ochas ic en i onmen . Fo his pu pose, we use
he mul iplica i e s ochas ic unc ion, which gene a es a s ochas ic demand
Di,j,
(Tallu i and an Ryzin 2004):
Di,j, =di,j,
⋅
𝜉
whe e
𝜉
is a s ochas ic e o e m
which is assumed o ollow a no mal dis ibu ion
N(1, 𝜎)
. We e alua e all scena ios,
i.e., SMALL, MEDIUM and LARGE wi h all DSRs. Fo each scena io, we con-
side di e en deg ees o s ochas ici y, exp essed by di e en s anda d de ia ions
𝜎∈0, 0.1, 0.2, 0.3, 0.4
o he ac o
𝜉
. These alues a e in he ange o demand
unce ain ies we obse ed in p ac ice. Fo each o he esul ing combina ions o sce-
na io and deg ee o s ochas ici y, we d aw
S=1, 000
demand ma ices.
Fig.8 illus a es he esul s o SMALL, MEDIUM and LARGE wi h
DSR =
1
∕3
.
On he e ical axis, he mean alue o he ela i e p o i inc eases wi h espec o
BASE. On he ho izon al axis, he s anda d de ia ion
𝜎
is a ied. O e all, he solu-
ion app oaches MODSIM, BAW-ROL-1 and BAW-MODSIM a e obus o he
s ochas ici y o demand. Simila o he p o i s o he benchma ks ROL-1, ROL-8,
ADP-1 and ADP-8, he ela i e p o i s o MODSIM, BAW-ROL-1 and BAW-MOD-
SIM dec ease sligh ly wi h inc easing s ochas ici y. The o de o he di e en solu-
ion app oaches and benchma ks wi h espec o hei pe o mance does no change
in mos ins ances. MODSIM, BAW-MODSIM and BAW-ROL-1 deli e p o i s ha
a e no wo se han he benchma k ADP-8 o all scena ios and all s ochas ici ies (see
Fig.17 inAppendix 7).

165
P ac icable solu ion app oaches o di e en ia ed p icing…
5.2 Di e en p ice se s
In his sec ion, we analyze he impac o he p ice se s on he pe o mance o he
di e en solu ion app oaches and benchma ks. Fo his pu pose, we use a common
s anda d demand pa e n (SMALL,
DSR =1∕3
). Fu he mo e, in each ins ance we
use 0.30 €/min as base p ice. We conduc wo expe imen s (see Table2 in Appendix
8). Fi s , we in es iga e 20 di e en p ice se s wi h h ee p ices each. The in e als
be ween p ices a e he same wi hin a p ice se , bu di e be ween p ice se s. We use
in e als o 0.01 €/min, 0.02 €/min up o 0.2 €/min (see Table2 in Appendix 8).
Second, we in es iga e he impac o he numbe o p ices on he pe o mance. He e,
he in e als a e he same o each p ice se , bu we inc ease he numbe o p ice
poin s o i e, se en and nine. P ice sensi i i ies change in acco dance wi h p ices.
No e ha we exclude he ADP benchma ks o his analysis due o he high e o
o es ima ing he pa ame e s in p e-p ocessing. Thus, we ocus on he solu ion
app oaches MODSIM, BAW-ROL-1 and BAW-MODSIM and on he benchma ks
ROL-8 and MOD48h only.
Fig.9 illus a es he esul s o di e en in e als be ween he p ices and di e -
en numbe o p ices in he p ice se . On he e ical axis, he mean alue o he
ela i e p o i inc eases wi h espec o BASE. On he ho izon al axis, he in e al
be ween he p ices (see Fig.9a) o he numbe o p ices (see Fig.9b) in he p ice se
is a ied.
Fig. 8 S ochas ic e alua ion o solu ion app oaches and benchma ks (SMALL, MEDIUM, LARGE),
DSR =
1
∕3
Fig. 9 Sensi i i y analysis o di e en p ice se s (SMALL), DSR
=
1
∕3
166
C.Mülle
In Fig.9a he ela i e p o i o all solu ion app oaches and benchma ks, excep
o MODSIM, ends o inc ease wi h he di e ence be ween he p ices in he p ice
se up o he di e ence o 0.14 €/min, a e which i dec eases. This shows ha he
p ices should be chosen easonably. The ela i e p o i o MODSIM inc eases un il
he di e ence o 0.08 €/min. The ea e , he p o i cu e d ops sha ply. We assume
ha his is due o he di e ence be ween he disc e e p ice poin s (
p(1)
,
p(2)
,
p(3)
).
MODSIM, which con e s he con inuous p ices o he simpli ied model in o dis-
c e e p ices, calcula es la ge p ice di e ences be ween he con inuous and he dis-
c e e p ice. Thus, i he p ice di e ence be ween p ice poin s is g ea e han 0.08 €/
min, he la ge he p ice di e ence, he wo se he esul s o MODSIM. In eali y,
howe e , di e ences o 0.05 €/min and 0.06 €/min a e obse ed o Sha e Now.
In addi ion, in Fig.9b he ela i e p o i also inc eases as he numbe o p ice
poin s in he p ice se inc eases. The p o i inc ease is deg essi e. Fu he mo e,
ROL-8, MODSIM, BAW-ROL-1, BAW-MODSIM and MOD48h pe o m simila ily
well o h ee and i e p ice poin s, whe eas ROL-8, BAW-MODSIM and MOD48h
pe o m be e han MODSIM and BAW-ROL-1 o se en and nine p ice poin s.
Wi h ega ds o he wo se pe o mance o BAW-ROL-1 compa ed o BAW-
MODSIM, he i s s ep o calcula ing ehicle dis ibu ions seems o be decisi e.
Mo e p ecisely, BAW-MODSIM, which has a sui able solu ion app oach MODSIM
o de e mine he ehicle dis ibu ions, pe o ms be e han BAW-ROL-1, which has
he benchma k ROL-1 o de e mine he ehicle dis ibu ions.
F om his we can conclude ha BAW-MODSIM is he bes p ac icable solu ion
app oach. I p o ides simila esul s as MOD48h and ROL-8.
5.3 P icing o one week wi hou ope a o ‑based eloca ion
In his sec ion, we examined he de e mina ion o p ices o an en i e week wi h-
ou ope a o based eloca ion. Fo his pu pose, we assume ha each day has he
same demand pa e n. Fo his s udy, we look a he ela i e a e age p o i o a day
compa ed o he p o i o p icing wi h BASE, he cou se o en als, and he p ice
s uc u e.
5.3.1 P o i
Fi s , we conside he ela i e a e age p o i pe day o one week (see Fig.10a),
which is ela i e o he a e age p o i pe day o BASE. I is ob ious ha supply-
side ne wo k e ec s ha e o be aken in o accoun . This can be seen in he di e ence
be ween myopic p icing ROL-1 and an icipa i e p icing ROL-8, MODSIM, BAW-
ROL-1, BAW-MODSIM o MOD48h. Second, we conside he ela i e p o i pe
day o one week (see Fig.10b), which is ela i e o he p o i o BASE a he i s
day. Simila o he obse a ion abo e, he p o i o MODSIM, BAW-ROL-1, BAW-
MODSIM, ROL-8 and MOD48h is simila and highe han he p o i o ROL-1. I is
also ob ious ha he o de o he di e en solu ion app oaches and benchma ks wi h
ega d o p o i does no change o e he se en days.
167
P ac icable solu ion app oaches o di e en ia ed p icing…
In addi ion, we no e ha he p o i is highes on he i s day compa ed o he ol-
lowing days o all solu ion app oaches and benchma ks (see Fig.10b). Howe e ,
his shows ha ope a o -based eloca ion can be wo hwhile.
Conside ing compu a ional imes, MOD48h needs 336.44 h, ROL-8 needs 8.46
h, while BAW-ROL-1 needs 7.6 min, BAW-MODSIM 7 min and MODSIM 0.07
min o calcula e p ices ( o a mo e de ailed analysis see Sec .4.3.4)
5.3.2 P icing decisions
We examine he p icing decisions (see Figs.18 in Appendix 9). Excep o he i s
day, we see a epea ing p ice pa e n o each day. Mo eo e , i is ob ious ha he
myopic p icing app oach (ROL-1, Fig.18a in Appendix 9) leads o a clea ly di -
e en p ice able han he solu ion app oaches and benchma ks ha include sup-
ply-side ne wo k e ec s (ROL-8, MODSIM, BAW-ROL-1, BAW-MODSIM,
MOD48h). Howe e , he e a e also di e ences be ween hem. Fo example, ROL-8
(see Fig.18c in Appendix 9) se s high and low p ices less equen ly han MOD-
SIM, BAW-ROL-1, BAW-MODSIM and MOD48h (see Fig.18 in Appendix 9). The
di e en p ice ables can be explained by he di e en p icing decisions on he i s
day, which p obably lead o di e en ehicle dis ibu ions on he i s day and hus
di e en s able p ice pa e ns o days 2-7.
5.3.3 Ren als
We examine en als o e he se en days (see Fig.11). Again, we no e ha a e he
ini ial day, he en al cou se o e he ollowing days is iden ical. Fu he mo e, we
obse e ha BASE and he myopic p icing app oach (ROL-1) esul in numbe s
o en als ha luc ua e clea ly mo e han o he o he s. ROL-8, MODSIM, BAW-
ROL-1, BAW-MODSIM, MOD48h a e simila .
We can he e o e conclude he ollowing:
Fig. 10 P o i o a week wi hou ope a o -based eloca ion (SMALL),
DSR =1∕3
168
C.Mülle
1. Each solu ion app oach and each benchma k c ea es a (daily) egula en al pa -
e n (see Fig.11) and p icing (see Fig.18 in Appendix 9) a e he i s day. These
egula pa e ns ( o day 2 o day 7) can be iden i ied by he equal con ibu ion
ma gins (see Fig.10b).
2. The conside a ion o supply-side ne wo k e ec s in solu ion app oaches and
benchma ks is use ul and leads o clea ly highe p o i s, e en in longe pe iods.
The conside a ion o hese supply-side ne wo k e ec s can be obse ed in he
p ice ables (see Fig.18 in Appendix 9).
3. The solu ion app oaches MODSIM, BAW-ROL-1 and BAW-MODSIM p o ide
he same esul s as he benchma ks ROL-8 o MOD48h (see Fig.10a), bu need
less compu a ional ime.
5.4 Impac o as a solu ion
In Sec .4.3, we no ice ha e en o some ins ances (
Z=25
,
DSR =2∕3
and
Z=25
,
DSR =3∕3
), he e a e no esul s o MOD48h since he sol e does no ind a easi-
ble solu ion wi hin he gi en ime limi o 48h. The e o e, we wan ed o in es iga e
wo aspec s in his sec ion: Fi s , whe he a s a solu ion imp o es he solu ion qual-
i y and second, whe he and how a s a solu ion a ec s he compu a ional ime. We
use he BASE solu ion as a s a solu ion in all ins ances.
The compa ison leads o he ollowing indings: All applied solu ion app oaches
and benchma ks wi h a s a solu ion esul in no no iceable imp o emen o he
esul s ega ding p o i (see Table6 in Appendix 11). No ably, he s a solu ion has
a posi i e e ec on he compu a ional ime o mos ins ances (a educ ion o com-
pu a ional ime up o 92 min, see Table7 in Appendix 11). Howe e , a s a solu ion
can also ha e nega i e e ec s on he compu a ional ime o some ins ances. Fo
example, o MOD48h a he smalles ins ance (
Z=9
,
DSR =1∕3
), i has a clea ly
nega i e ime e ec (+ 46h).
Ne e heless, he use o a s a solu ion esul s in solu ions o MOD48h in all
ins ances wi hin he ime limi . F om his i can be concluded ha i makes sense o
use BASE as a s a solu ion. This applies in pa icula o eal ins ances.
Fig. 11 Ren als o e se en days (SMALL),
DSR =1∕3
175
P ac icable solu ion app oaches o di e en ia ed p icing…
P o i o di e en scena ios
See Fig.14.
Fig. 14 Rela i e p o i inc ease (SMALL, MEDIUM, LARGE)

176
C.Mülle
Compu a ional ime o di e en scena ios
See Fig.15.
Compu a ional ime o di e en amoun o zones
See Fig.16.
Fig. 15 Compu a ional ime (SMALL, MEDIUM, LARGE)
177
P ac icable solu ion app oaches o di e en ia ed p icing…
S ochas ic demand
As a echnical ema k, no e ha in he s ochas ic demand model, demand eali-
za ion
Dij <0
could po en ially esul in pa icula o high alues o
𝜎
(see he
co esponding discussion in Tallu i and an Ryzin (2004,Chap e 7.3.4)). We
co ec o his by se ing nega i e d aws o 0. No e ha he small posi i e bias
esul ing om his unca ion is no ele an o ou s udy, as o each deg ee o
s ochas ici y, we use he same 1000 scena ios o all app oaches we compa e
(Fig.17).
Fig. 16 Compu a ional ime
o di e en amoun o zones,
DSR =1∕3
178
C.Mülle
O e iew o e di e en p ice lis s
See Table2.
Fig. 17 P icing wi h di e en solu ion app oaches (SMALL),
DSR =1∕3
179
P ac icable solu ion app oaches o di e en ia ed p icing…
Table 2 O e iew o e di e en p ice lis s
Expe imen 1: di e en in e als Expe imen 2: di e en numbe o p ices
P ice in e al P ice se Sensi i i y ac o s Numbe
o p ices
P ice se Sensi i i y ac o s
0.01 €/min
{0.29, 0.30, 0.31}
{1.04, 1, 0.96}
3
{0.24, 0.30, 0.36}
{1.25, 1, 0.75}
0.02 €/min
{0.28, 0.30, 0.32}
{1.08, 1, 0.92}
5
{0.18, 0.24, 0.30, 0.36, 0.42}
{1.50, 1.25, 1,
0.75, 0.50}
0.03 €/min
{0.27, 0.30, 0.33}
{1.12, 1, 0.87}
7
{0.12, 0.18, 0.24,
0.30,0.36,0.42,
0.48}
{1.75, 1.50, 1.25,
1,0.75,0.50,
0.25}
0.04 €/min
{0.26, 0.30, 0.34}
{1.17, 1, 0.83}
9
{0.06, 0.12, 0.18,
0.24,0.30,0.36,
0.42, 0.48, 0.54}
{2.00, 1.75, 1.50,
1.25,1,0.75,
0.50, 0.25, 0}
0.05 €/min
{0.25, 0.30, 0.35}
{1.21, 1, 0.79}
0.06 €/min
{0.24, 0.30, 0.36}
{1.25, 1, 0.75}
0.07 €/min
{0.23, 0.30, 0.37}
{1.29, 1, 0.71}
0.08 €/min
{0.22, 0.30, 0.38}
{1.33, 1, 0.67}
0.09 €/min
{0.21, 0.30, 0.39}
{1.37, 1, 0.62}
0.10 €/min
{0.20, 0.30, 0.40}
{1.42, 1, 0.58}
0.11 €/min
{0.19, 0.30, 0.41}
{1.46, 1, 0.54}
0.12 €/min
{0.18, 0.30, 0.42}
{1.50, 1, 0.50}
0.13 €/min
{0.17, 0.30, 0.43}
{1.54, 1, 0.46}
0.14 €/min
{0.16, 0.30, 0.44}
{1.58, 1, 0.42}
0.15 €/min
{0.15, 0.30, 0.45}
{1.62, 1, 0.37}
0.16 €/min
{0.14, 0.30, 0.46}
{1.67, 1, 0.33}
0.17 €/min
{0.13, 0.30, 0.47}
{1.71, 1, 0.29}
0.18 €/min
{0.12, 0.30, 0.48}
{1.75, 1, 0.25}
0.19 €/min
{0.11, 0.30, 0.49}
{1.79, 1, 0.21}
0.20 €/min
{0.10, 0.30, 0.50}
{1.83, 1, 0.17}
180
C.Mülle
One week p icing
See Fig.18.
(A)
(B)
(a)
(b)
(c)
(a)
(b)
(c)
(d)
Fig. 18 P icing o one week wi h di e en solu ion app oaches (SMALL)
DSR =1∕3
. G een: low p ice,
yellow: base p ice, ed: high p ice

181
P ac icable solu ion app oaches o di e en ia ed p icing…
S ochas ic e alua ion
See Tables3, 4 and 5.
Table 3 Mean p o i inc ease (SMALL, MEDIUM, LARGE),
DSR =
1
∕3
Mean p o i inc ease wi h espec o BASE in %
Z=9
Z=16
Z=25
𝜎=0
𝜎=0.1
𝜎=0.2
𝜎=0.3
𝜎=0.4
𝜎=0
𝜎=0.1
𝜎=0.2
𝜎=0.3
𝜎=0.4
𝜎=0
𝜎=0.1
𝜎=0.2
𝜎=0.3
𝜎=0.4
BASE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
ROL-1 6.24 5.71 5.73 5.79 5.78 9.05 5.67 5.67 5.67 5.61 11.44 6.24 6.23 6.22 6.22
ROL-4 9.41 12.06 12.16 12.29 12.36 12.35 8.84 8.78 8.45 8.15 14.17 9.24 9.07 8.90 8.74
ROL-8 11.68 14.50 14.64 14.79 14.83 13.97 10.91 10.67 10.24 9.83 14.71 11.54 11.38 11.24 11.09
ADP-1 14.25 11.55 11.65 11.75 11.81 14.20 12.66 12.13 11.60 11.00 14.25 14.10 13.77 13.44 13.12
ADP-4 14.57 14.59 14.79 15.03 15.20 14.67 13.20 12.57 11.95 11.35 14.82 14.36 14.05 13.72 13.40
ADP-8 14.72 15.27 15.46 15.62 15.72 14.65 13.45 12.83 12.22 11.64 14.78 14.48 14.20 13.91 13.60
MODSIM 13.46 14.67 14.74 14.85 14.97 14.23 13.90 13.39 12.80 12.26 14.25 13.30 13.11 12.90 12.69
BAW-ROL-1 14.87 15.26 15.31 15.36 15.37 14.59 14.08 13.48 12.88 12.32 14.70 14.61 14.30 13.99 13.66
BAW-MODSIM 14.73 15.36 15.41 15.47 15.49 14.71 13.62 12.97 12.34 11.73 14.80 14.41 14.06 13.72 13.36
182
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Table 4 Mean p o i inc ease (SMALL, MEDIUM, LARGE),
DSR =
2
∕3
Mean p o i inc ease wi h espec o BASE in %
Z=9
Z=16
Z=25
𝜎=0
𝜎=0.1
𝜎=0.2
𝜎=0.3
𝜎=0.4
𝜎=0
𝜎=0.1
𝜎=0.2
𝜎=0.3
𝜎=0.4
𝜎=0
𝜎=0.1
𝜎=0.2
𝜎=0.3
𝜎=0.4
BASE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
ROL-1 8.14 8.26 8.29 8.35 8.28 9.53 9.31 9.13 9.04 8.99 9.05 9.04 9.05 9.05 9.07
ROL-4 12.65 12.84 12.91 12.99 12.98 12.65 12.43 12.20 12.05 11.99 12.35 12.30 12.26 12.22 12.17
ROL-8 13.96 14.09 14.16 14.22 14.14 14.64 14.37 14.15 14.02 13.97 13.97 13.93 13.82 13.73 13.63
ADP-1 13.69 13.59 13.77 14.00 14.10 14.42 13.94 13.73 13.62 13.55 14.23 14.13 14.03 13.93 13.84
ADP-4 13.87 14.00 13.98 13.99 13.90 14.79 14.44 14.15 13.96 13.85 14.59 14.57 14.46 14.35 14.24
ADP-8 13.93 14.10 14.12 14.16 14.07 14.87 14.53 14.31 14.16 14.06 14.71 14.60 14.52 14.43 14.34
MODSIM 13.38 13.73 13.74 13.80 13.77 14.12 14.55 14.63 14.61 14.60 14.20 14.29 14.30 14.25 14.18
BAW-ROL-1 13.96 13.88 13.87 13.89 13.82 14.80 14.50 14.20 13.99 13.84 14.67 14.51 14.41 14.31 14.22
BAW-MODSIM 13.94 13.94 13.94 13.97 13.90 14.79 14.62 14.33 14.12 13.99 14.65 14.64 14.53 14.43 14.33
183
P ac icable solu ion app oaches o di e en ia ed p icing…
Table 5 Mean p o i inc ease (SMALL, MEDIUM, LARGE),
DSR =
3
∕3
Mean p o i inc ease wi h espec o BASE in %
Z=9
Z=16
Z=25
𝜎=0
𝜎=0.1
𝜎=0.2
𝜎=0.3
𝜎=0.4
𝜎=0
𝜎=0.1
𝜎=0.2
𝜎=0.3
𝜎=0.4
𝜎=0
𝜎=0.1
𝜎=0.2
𝜎=0.3
𝜎=0.4
BASE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
ROL-1 10.09 10.23 10.30 10.42 10.45 11.83 11.72 11.60 11.48 11.51 11.44 11.41 11.31 11.18 11.08
ROL-4 12.64 12.78 12.74 12.73 12.64 14.36 14.31 14.08 13.64 13.30 14.17 14.09 13.93 13.73 13.55
ROL-8 13.29 13.37 13.39 13.38 13.30 15.23 15.03 14.69 14.25 13.95 14.71 14.56 14.37 14.13 13.90
ADP-1 13.26 13.08 13.04 13.03 12.95 14.81 14.87 14.64 14.34 13.98 14.25 14.16 14.02 13.87 13.75
ADP-4 13.20 13.24 13.17 13.13 13.06 15.25 15.04 14.73 14.35 14.02 14.70 14.67 14.52 14.35 14.21
ADP-8 13.31 13.38 13.40 13.39 13.30 15.26 14.95 14.63 14.26 13.81 14.80 14.66 14.50 14.34 14.18
MODSIM 12.93 13.33 13.30 13.29 13.21 14.90 14.90 14.69 14.28 13.84 14.25 14.18 14.09 13.99 13.89
BAW-ROL-1 13.31 13.14 13.06 13.01 12.94 15.25 15.12 14.78 14.40 14.04 14.82 14.60 14.42 14.18 13.95
BAW-MODSIM 13.31 13.24 13.17 13.13 13.06 15.14 15.05 14.68 14.20 13.79 14.78 14.70 14.55 14.39 14.21
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C.Mülle
Impac o as a solu ion
See Tables6 and 7.
Table 6 P o i wi h and wi hou s a solu ion
Di e ence in p o i inc ease wi h espec o BASE in %
Z=9
Z=16
Z=25
DSR =1∕3
DSR =2∕3
DSR =3∕3
DSR =1∕3
DSR =2∕3
DSR =3∕3
DSR =1∕3
DSR =2∕3
DSR =3∕3
ROL-4 0.02 0.00 0.00 0.00 0.37 0.09 −0.04 −0.15 −0.04
ROL-8 0.02 0.00 0.00 −0.06 −0.02 −0.09 −0.33 −0.01 0.04
MOD48h 0.00 0.00 0.00 0.07 0.02 0.25 0.30
ADP-1 0.00 0.00 −0.06 0.00 −0.04 0.00 0.00 0.00 0.02
ADP-4 0.00 0.00 0.00 0.00 0.04 0.14 0.00 −0.01 −0.01
ADP-8 0.00 0.00 0.00 −0.07 0.05 0.01 −0.04 0.00 −0.04
BAW-ROL-1 −0.01 0.00 0.00 0.00 0.04 0.00 0.00 0.00 0.00
BAW-MODSIM 0.00 0.00 0.00 0.00 −0.04 0.00 0.00 0.00 −0.05