Mac ina, Giusy; A che i, Claudia; Gue ie o, F ancesca
A icle
Bundles gene a ion and p icing in c owdshipping
EURO Jou nal on T anspo a ion and Logis ics (EJTL)
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gene a ion and p icing in c owdshipping, EURO Jou nal on T anspo a ion and Logis ics (EJTL), ISSN
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Bundles gene a ion and p icing in c owdshipping
Giusy Mac ina a,∗, Claudia A che i b, F ancesca Gue ie o a
aDepa men o Mechanical Enginee ing Uni e si y o Calab ia, I aly
bDepa men o In o ma ion Sys ems, Decision Sciences and S a is ics, ESSEC Business School in Pa is, F ance
ARTICLE INFO
Keywo ds:
C owdshipping
Bundles gene a ion
P icing
Auc ions
ABSTRACT
C owdshipping is a new deli e y pa adigm ha exploi s he capaci y o o dina y people who o e hei own
ehicles and ee ime o pe o m deli e ies agains compensa ion. In his wo k, we conside a pee - o-pee
logis ic pla o m whe e a company ecei es o de s om i s cus ome s and assigns hem o occasional d i e s
(ODs), o c owdshippe s, who pe o m he deli e y ope a ions. We i s in es iga e he p oblem o deciding how
he o de s should be pa i ioned in o bundles, whe e a bundle is a se o o de s assigned o he same OD. Then,
we ocus on he p oblem o de e mining he compensa ion associa ed wi h each bundle, wi h he pu pose o
minimizing he o al deli e y cos s. The p icing scheme is based on he assump ion ha each OD is associa ed
wi h a willingness- o-se e unc ion, which is modeled as a andom a iable ha gi es he p obabili y ha he
OD accep s o deli e he bundle gi en he compensa ion alue. This andom a iable cap u es he es ima ion
o he willingness- o-se e unc ion ha he company has elabo a ed, o example on he basis o his o ical
da a. I he compensa ion o e ed by he company is g ea e han o equal o he willingness- o-se e alue, he
OD pe o ms he deli e y, o he wise she/he e uses. In case no OD is a ailable o deli e a bundle, hen all
packages in he bundle a e o e ed o a hi d-pa y deli e y company. We simula e wo auc ion sys ems o
he assignmen o bundles o ODs: a s a ic and a dynamic auc ion. In exhaus i e simula ion es s, we compa e
di e en p icing schemes as well as he wo auc ion sys ems, and ou line se e al manage ial insigh s.
1. In oduc ion
C owdshipping is an inno a i e deli e y s a egy in which occa-
sional d i e s (‘c owdshippe s’) o e hei se ice o deli e y compa-
nies, ypically using an online pla o m (Mckinnon,2016;Punel and
S a hopoulos,2017;Buldeo Rai e al.,2018;Alnagga e al.,2021).
The pla o m, also named ‘‘pee - o-pee pla o m’’, ypically ope a es
as ollows: he company uploads deli e y eques s, speci ying he a ea,
expec ed ea nings, and ask du a ion. Occasional d i e s (ODs) b owse
hese oppo uni ies, selec ing hose ha align wi h hei con enience,
and hen communica e hei a ailabili y o pe o m he se ice. The
pla o m subsequen ly assigns he deli e y ask o an OD, who deli e s
he pa cels and ecei es paymen upon success ul comple ion. Among
he mos success ul c owdshipping pla o ms we men ion Amazon Flex,
he c owdshipping pla o m o Amazon, My Ways o DHL, Roadie o
UPS, and Pos ma e o Ube .
This pa adigm is gaining success in he las -mile deli e y con ex .
The main eason is ha hi ing an OD is much ‘easie ’ han hi ing
a egula d i e (RD), i.e., a d i e wi h a pe manen ull con ac .
Indeed, an OD wo ks empo a ily o he company (and he/she is ypi-
cally engaged jus o he ime slo associa ed wi h he deli e y se ice
he/she accep s o pe o m) and ecei es compensa ion o he se ice
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (G. Mac ina), [email p o ec ed] (C. A che i), [email p o ec ed] (F. Gue ie o).
p o ided. RDs ins ead ha e a long- e m con ac and wo k ull- ime o
he company (and hus ecei e a ull- ime sala y). Thus, c owdshipping
o e s a g ea oppo uni y o deal wi h peak o demands by empo a ily
hi ing ODs, a oiding o inc ease he ‘ egula deli e y capaci y’ o he
company (associa ed wi h RDs) which migh hen become useless when
he peak is o . In mos cases, hi d-pa y companies a e engaged in case
o missing capaci y (nei he ODs no RDs a e a ailable o ul ill ce ain
eques s).
Implemen ing a c owdshipping sys em in ol es ca e ul conside a-
ion o nume ous ac o s. In pa icula , a c i ical key poin is es ab-
lishing app op ia e compensa ion o each deli e y o e ed o ODs. As
s a ed in Buldeo Rai e al. (2018), emune a ion is he mos in luen ial
ac o in luencing willingness o wo k as a po en ial c owdshippe
(45.36%). The e o e, de ining he policy o assigning eques s o ODs,
bo h in e ms o numbe o pa cels and compensa ion, is c ucial, and
his is he ocus o ou wo k.
In his pape , we conside a c owdshipping pla o m ha mimics
eal applica ions, like Amazon Flex, whe e cus ome eques s a e ag-
g ega ed in o so-called ‘‘blocks’’, which a e hen o e ed o he d i e s.
Blocks include he da e, ype, expec ed ea nings, s a ime, and es-
ima ed du a ion. We i s add ess he p oblem o bundles (blocks)
h ps://doi.o g/10.1016/j.ej l.2024.100142
Recei ed 18 Ap il 2023; Recei ed in e ised o m 24 June 2024; Accep ed 5 Augus 2024
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100142
A ailable online 8 Augus 2024
2192-4376/© 2024 The Au ho (s). Published by Else ie B.V. on behal o Associa ion o Eu opean Ope a ional Resea ch Socie ies (EURO). This is an open access
a icle unde he CC BY-NC-ND license ( h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ ).
G. Mac ina e al.
gene a ion, p oposing a heu is ic app oach o gene a e hem. Then, we
p opose a p icing scheme o de e mine he compensa ion associa ed
wi h each bundle. Finally, we de elop wo auc ion s a egies, a s a ic
and a dynamic auc ion, whe e bundles a e o e ed o ODs and we
simula e he co esponding accep ance.
The main ocus o ou s udy is o in es iga e he impac o p icing
and auc ion s a egies in a c owdshipping sys em and o de i e insigh s
ha migh be use ul o applica ions. Indeed, we ocus on decisions
ela ed o how bundles should be gene a ed, wha should be he
co esponding compensa ion, and how o adjus he compensa ion in
case o a dynamic auc ion.
The con ibu ions o he pape can be summa ized as ollows:
•We s udy a c owdshipping sys em in which a company ecei es
o de s om cus ome s, which a e hen o e ed o ODs o deli e y.
The company has o decide how o cons uc bundles o o de s
o he ODs and he compensa ion associa ed wi h each bundle.
I no OD accep s o deli e a bundle, all pa cels in he bundle a e
assigned o a hi d-pa y deli e y company, incu ing a highe
cos . We assume ha ODs accep o se e a bundle acco ding
o a willingness- o-se e unc ion, which is modeled as a andom
a iable gi ing he p obabili y o accep ance wi h espec o he
compensa ion o e ed.
•We p opose a solu ion app oach in which, in he i s phase, we
de e mine he bundles o be o e ed o he ODs. The co espond-
ing compensa ion o each bundle is hen de e mined in a second
phase.
•We de elop a bundle gene a ion p ocedu e based on a g eedy
algo i hm, ha akes in o accoun bo h he empo al and spa-
ial p oximi y o eques s. The compensa ion is hen de e mined
acco ding o a bundle p icing scheme, based on he willingness- o-
se e unc ion.
•We simula e wo auc ion sys ems o he assignmen o bundles
o ODs. In he i s sys em, which is called ‘‘s a ic’’, bundles
a e o e ed wi h a compensa ion ha is de e mined as desc ibed
abo e. I a bundle is no accep ed, i is o e ed o he hi d
company a a highe p ice. In he second auc ion sys em, called
‘‘dynamic’’, he company can eac o a ‘‘ ejec ion’’ by inc easing
he compensa ion associa ed wi h a bundle and epea ing he
auc ion.
•We pe o m ex ensi e simula ions on syn he ic ins ances wi h
100 and 1000 eques s. We compa e he s a ic and he dynamic
auc ions, as well as wo p icing s a egies: he one p e iously
men ioned and ano he whe e compensa ion is equal o he pu e
anspo cos o se ing he eques s belonging o he bundle,
es ima ed based on he dis ance a eled. We also un simula ions
on an ins ance gene a ed on Rio de Janei o’s eal oad ne wo k,
conside ing a case s udy o a la ge company ha ope a es in las -
mile deli e y. We p o ide manage ial insigh s, by compa ing he
p icing s a egies and he auc ion sys ems.
The es o he pape is o ganized as ollows: in Sec ion 2we
p esen a b ie li e a u e e iew on c owdshipping. In Sec ions 3and
4we desc ibe he se ing o he p oblem and he solu ion app oach,
espec i ely. In Sec ion 5we desc ibe he auc ion sys ems. Sec ion 6is
de o ed o compu a ional esul s and we inally ou line he conclusions
and u u e esea ch pe spec i es in Sec ion 7.
2. Li e a u e e iew
C owdshipping gained success no only in p ac ical applica ions,
bu also in he scien i ic communi y, as academics ha e s a ed o
s udy he p oblem om se e al pe spec i es. Le e al. (2019) e iewed
cu en p ac ices, academic esea ch, and se e al case s udies ang-
ing om supply and demand managemen o ope a ions scheduling,
while A che i and Be azzi (2021) ocused on ou ing p oblems wi h
c owdshipping. Pou ahmani and Jalle (2021) p o ided an o e iew
o he ope a ional cha ac e is ics o c owdshippping pla o ms and a
comp ehensi e e iew o he li e a u e, highligh ing he challenges and
he esea ch oppo uni ies.
In he ollowing we e iew some o he ecen con ibu ions in
c owdshipping, ocusing on deli e y applica ions. In pa icula , we
conside he wo ks add essing he h ee main p oblems a ising when
implemen ing a c owdshipping (o pee - o-pee logis ics) pla o m:
ma ching, ou ing, and p icing p oblems.
2.1. Ma ching in c owdshipping
In gene al, pee - o-pee logis ics pla o ms use wo di e en ap-
p oaches o ma ch supplie s (i.e., d i e s) and eques s (i.e., cus ome s):
cen alized o decen alized (see, Mo idi and Pazou (2019)). In he
cen alized app oach, decisions a e aken by he company managing
he pla o m. The pla o m p oposes a eques o a supplie and he
supplie may accep o ejec he p oposed eques . In he decen alized
app oach, he supplie s choose which eques s hey would like o se e.
Mo idi and Pazou (2019) p oposed a hie a chical app oach o he
solu ion o he ma ching p oblem in pee - o-pee logis ics pla o m.
The pla o m i s de e mines which eques s should be ecommended
o each supplie , conside ing he es ima ion o he supplie ’s u ili y
associa ed wi h each eques . This es ima ion is based on his o ical da a
collec ed by he pla o m. Then, he supplie s may selec which eques s
o ul ill. Each eques can be o e ed o mo e han one supplie , and
each supplie can accep o ejec he o e ed eques /s. The au ho s
p opose a bi-le el app oach o modeling he p oblem. Ho ne e al.
(2021) ex ended he wo k o Mo idi and Pazou (2019) by conside ing
a s ochas ic d i e beha io and eques s size la ge han one. Aus-
seil e al. (2022) s udied a dynamic ma ching, whe e he p oblem
is modeled as a sequen ial decision p ocess, wi h s ochas ic supply
and demand. They p oposed a mul iple scena io app oach, whe e he
solu ions o each scena io a e combined using a consensus unc ion o
de i e he inal decision. Li e al. (2019) s udied a decen alized sys em
and p oposed a mul i-agen ein o cemen lea ning app oach.
2.2. Rou ing in c owdshipping
In his sec ion we e iew he li e a u e on con ibu ions whe e
he aim is o de e mine he ou es o se e he cus ome s, ei he
pe o med by ODs o by egula d i e s. No decision is aken abou
he compensa ion o be gi en o ODs, which is ins ead ixed, ei he
as a cons an amoun o as depending on he dis ance a eled. The
con ibu ions whe e he compensa ion is pa o he decision p ocess
a e e iewed in he nex sec ion.
The li e a u e ela ed o c owdshipping in deli e y ope a ions is
qui e ecen . Following he classi ica ion p oposed by Boysen e al.
(2022), we conside h ee c owdshipping pa adigms: cus ome -based,
d i e -based, and employee-based.
The cus ome -based sys em mimics he Walma concep , s udied
by A che i e al. (2016), who in oduced he ehicle ou ing wi h ODs
(VRPOD). In his sys em, in-s o e cus ome s may help he company
in deli e ing pa cels o online cus ome s. Se e al au ho s ex ended
he p oblem p esen ed in A che i e al. (2016), conside ing di e en
ea u es (see, e.g., Mac ina e al. (2017), and Dahle e al. (2019)). Dahle
e al. (2017) and Skålnes e al. (2020) conside ed an unce ain ODs’
a ailabili y. Daya ian and Sa elsbe gh (2020) p oposed a dynamic and
s ochas ic ou ing p oblem in which no only ODs a ailabili y, bu also
cus ome s’ o de s a i e o e ime.
The d i e -based c owdshipping sys em is based on he Amazon Flex
concep , s udied by Mac ina e al. (2020). In his case, ODs a e no in-
s o e cus ome s, hence, hey do no s a hei ou e om he depo
(whe e pa cels a e picked up) bu om di e en o igins. Beh end e al.
(2019) s udied a sys em in which he supply and eques o i ems,
as well as c owdshippe a ailabili y, a e all announced on he same
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100142
2
G. Mac ina e al.
pla o m. Hence, hey join ly add essed he assignmen o supplies o
eques s and he ou ing o c owdshippe s. Se e al au ho s s udied
di e en ex ensions o his p oblem conside ing s ochas ic/dynamic
in o ma ion (see. e.g., A che i e al. (2021), Yıldız (2021), To es e al.
(2022b,a), and Di Puglia Pugliese e al. (2023)).
The las pa adigm, named employee-based c owdshipping, was p o-
posed and s udied o he i s ime by Boysen e al. (2022). In his case,
he employees d i e hei p i a e ca s a e wo k o docking places o
picking up some shipmen s, hen o he cus ome s o deli e y.
Focusing on he e alua ion o he compensa ion o he ODs, i can
be no ed ha in all he con ibu ions men ioned abo e he compensa-
ion scheme o he ODs is based on he de ou o on a ixed amoun
pe pa cel. A ew o he con ibu ions ha e, howe e , s udied di e en
compensa ion schemes. Sampaio e al. (2020) p oposed a d i e -based
VRPOD wi h pickup and deli e y, ime windows and ans e s. Since
he ime slo in which a d i e is a ailable could be sho e han he
ime equi ed o he deli e ies, he au ho s p opose a mo e lexible
se ice, in which a pa cel ha mus be deli e ed can be anspo ed by
mo e han one d i e om i s o igin o i s des ina ion. The compen-
sa ion o he ODs is based on he ime spen o he deli e ies. Dai
and Liu (2020) p oposed a wo k o ce capaci y planning and alloca-
ion model o logis ics sys ems whe e di e en ypes o wo k o ces
a e a ailable: in-house d i e s, ull- ime c owdsou ced d i e s, and
pa - ime c owdsou ced d i e s, which di e in hei cha ac e is ics
(e.g., compensa ion schemes and posi ion). They conside ed a deli e y
cos o he ull- ime c owdsou ced d i e s composed o wo elemen s:
he ixed labo cos o he ime pe iods hey wo k and he numbe o
pa cels assigned. The compensa ion scheme o he pa - ime c owd-
sou ced d i e s depends on he numbe o o de s assigned and no on
he de ou .
2.3. P icing in c owdshipping
Among he ac o s ha in luence he c owdshippe s beha io , he
deli e y compensa ion is a c i ical one. Some s udies p oposed di e en
me hods o de e mine he mos a ac i e p icing s a egy (see Le e al.
(2019) o a e iew). Among he s udies on ou ing wi h c owdship-
ping, only a ew con ibu ions ocused on he p ice de ini ion; indeed,
as men ioned abo e, he majo i y o he con ibu ions ocusing on he
ou ing p oblems wi h c owdshipping p oposed a compensa ion based
on he dis ance a eled.
Few s udies conside ed he p oblem o combining he decisions
abou ma ching, p icing, and ou ing. Recen ly, T iki (2021) in oduced
he combina o ial auc ion echnique wi hin he VRPOD. In his ame-
wo k, he ODs a e he bidde s ha can submi any combina ion o
di e en bids and hey p opose he co esponding p ice. The lowe he
submi ed p ice, he highe he p obabili y o ob ain i . The au ho
p oposed a ma hema ical o mula ion as well as wo heu is ic ap-
p oaches. In pa icula , a decomposi ion-based and a cos -compa ison
app oach a e p oposed, in which he winning bid is selec ed based on
he compa ison be ween he bid’s p ice and he cos o se ing he
same se o cus ome s by he company’s ehicles. The compu a ional
s udy highligh ed he bene i s, in e ms o cos educ ion, o using
combina o ial auc ions o assign he pa cels o he ODs.
Mancini and Gans e e (2022) p oposed a VRPOD wi h bundles.
In hei amewo k, he company i s ocuses on he gene a ion o
he bundles, and hen bundles a e o e ed o he ODs. In pa icula ,
bundles a e gene a ed ollowing wo s a egies: a adi ional clus e ing
app oach and a echnique based on he gene a ion o co ido s. ODs
make a bid o he bundles hey conside as a ac i e. The bid depends
on he OD de ou , lexibili y (maximum accep able de ou ), and will-
ingness o wo k (a pa ame e ha can assume h ee alues acco ding o
h ee le els o willingness: high, medium, and low). Bundles and bids
a e inpu s o a ma hema ical model ha assigns hem o he ODs and
c ea es also he ou es o he egula d i e s.
Table 1
Li e a u e e iew: p icing in c owdshipping.
Re e ence Rou ing ODs Bundles
compensa ion
T iki (2021) Yes Bids-based: andom bids Random
Mancini and Gans e e (2022) Yes Bids-based: ODs des ina ion, Clus e ing
lexibili y, willingness
Le e al. (2021) No De ou No
This wo k No Tou , willingness Rou e-based
Le e al. (2021) p oposed a amewo k ha in eg a es he ma ch-
ing o eques s and o e s, oge he wi h p icing and compensa ion
schemes. They i s gene a ed all he admissible ma ches cou ie s-
eques s, hen apply he p icing s a egies o de e mine he compen-
sa ion. The au ho s e alua ed ou p icing and compensa ion schemes
based on la e sus indi idual scheme se ings. In he la se ing, he
p ice and compensa ion a e he same o all eques s and deli e y
ips, while in he indi idual one, hey a e applied o each eques and
deli e y ip, sepa a ely.
In he cu en pape , we ocus on he bundling and he p icing
p oblems. Simila ly o T iki (2021), we p opose an auc ion s a egy
o assign pa cels o ODs. Howe e , ou amewo k di e s signi ican ly
om he one p oposed in T iki (2021). Indeed, in con as o he
la e one, whe e bidde s p opose and submi bids, in ou app oach
he pla o m cons uc s he bundles and de e mines he compensa ions.
Bundles a e hen o e ed o ODs, who decide whe he o accep hem
o no . Ou wo k also sha es some simila i ies wi h Le e al. (2021),
bu he app oach is di e en , as hey do no conside an auc ion
sys em. Finally, we di e also wi h espec o he wo k o Mancini
and Gans e e (2022). Fi s , we conside a pla o m whe e deli e ies
a e pe o med h ough c owdshipping only. In Mancini and Gans e e
(2022) he e a e bo h egula and ODs. They i s ocus on he bundles
gene a ion and assignmen , and hen he emaining cus ome s a e
se ed by he company’s d i e s. Second, he de ini ion o he com-
pensa ion o he bundles is also di e en . In Mancini and Gans e e
(2022) ODs make bids based on de ou , lexibili y, and a willingness o
wo k, which is ep esen ed as a pa ame e . In ou wo k, we assume
o ha e no in o ma ion abou ODs des ina ion, hence, he company
e alua es a s a ing compensa ion based on he ou e ha has o be
a eled and on he willingness- o-se e unc ion. Ou willingness- o-se e
unc ion, di e en ly om Mancini and Gans e e (2022), is ep esen ed
as a andom a iable depending on he compensa ion, and no as a pa-
ame e mul iplying he de ou . In addi ion, in Mancini and Gans e e
(2022) he au ho s p opose a ma hema ical model, in which bo h he
bundles and he bids o e ed by he ODs a e inpu da a. Then, he
ma hema ical model inds he bes assignmen . Ins ead, in ou wo k we
p opose a di e en s a egy, whe e bundles a e assigned o ODs h ough
an auc ion. A simila s a egy is p oposed in Gans e e and Ha l (2018)
and Gans e e e al. (2020), whe e a bundle gene a ion p oblem in a
collabo a i e anspo a ion sys em is s udied. Howe e , he p oblem
se ing is comple ely di e en gi en ha i a ises in a collabo a ion
amewo k whe e ca ie s can exchange anspo a ion eques s among
each o he , and decide which eques s should be inse ed in he auc ion
pool. Then, he auc ionee gene a es he bundles o eques s and o e s
hem o he ca ie s. Ca ie s gi e hei bids o he o e ed bundles,
while he auc ionee alloca es bundles o ca ie s based on hei bids,
hence he p o i s a e dis ibu ed among he ca ie s.
In Table 1 we summa ize he main ea u es o he con ibu ions
e ised in his sec ion and compa e hem wi h he ea u es o he sys em
s udied in he cu en pape .
3. P oblem desc ip ion and o mula ion
In his pape , we conside a pee - o-pee logis ic pla o m. Fig. 1
p o ides a g aphical illus a ion o how he pla o m wo ks. Speci i-
cally, he pla o m ecei es he eques s om he cus ome s. Reques s
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100142
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Fig. 1. Rep esen a ion o he pee - o-pee logis ic pla o m.
a e hen g ouped in o bundles and a compensa ion is associa ed wi h
each bundle. A e wa d, bundles a e o e ed o ODs h ough an auc ion
sys em. I a bundle is accep ed by an OD, he OD ecei es all in o ma-
ion needed o pe o m he deli e y, namely he se o cus ome s wi h
hei loca ions, he ou e, and he compensa ion. I no OD accep s o
deli e a bundle, he deli e y is pe o med by a hi d-pa y deli e y
company.
In pa icula , we suppose ha cus ome s place hei eques s he day
be o e he planning. Cus ome s may choose he ime window in which
hey a e a ailable o ecei e hei pa cel, i.e., mo ning o a e noon,
and mus indica e he deli e y loca ion. Once he pla o m collec s all
cus ome s’ o de s, he planning phase s a s. The company aces wo
p ima y decisions: i s , i has o de e mine how he eques s should be
clus e ed in bundles. Second, i has o decide wha is he compensa ion
associa ed wi h each bundle. Once bundles and compensa ions a e se ,
he co esponding in o ma ion is pos ed on he pla o m. We suppose
an in ini e popula ion o ODs is a ailable. We a e awa e ha his is
a s ong assump ion, which limi s he applicabili y o he esul s o
ou s udy. Howe e , ou esul s can s ill se e as a basis o scena ios
whe e his assump ion is emo ed. Each OD accesses he pla o m
and gi es he a ailabili y o deli e ing a bundle a he co esponding
compensa ion. Each bundle is assigned o he i s OD who accep s o
deli e i h ough an auc ion sys em. The compensa ion is paid o he
deli e y o all pa cels in he bundle. I some bundles a e no accep ed
by any ODs, hen he co esponding pa cels a e deli e ed by a mo e
expensi e hi d-pa y deli e y company. All deli e y ou es s a om
a common depo , whe e pa cels a e picked up.
We assume ha he p obabili y ha an OD accep s o deli e pa cels
depends on he compensa ion o e ed. Each OD accep s o deli e a
bundle acco ding o he /his willingness- o-se e unc ion, which co e-
sponds o a andom a iable measu ing he p obabili y o accep ance on
he basis o he compensa ion. In addi ion, we assume ha ODs a e no
willing o make long de ou s o se e he pa cels included in a bundle.
Thus we limi he leng h o he ou e associa ed wi h each bundle o
a maximum alue
𝑇. The objec i e o he company is o de ine he
bundles o eques s, and he co esponding compensa ion, in such a
way as o minimize he expec ed o al cos , which is gi en by he sum
o expec ed compensa ion o ODs plus he expec ed cos s associa ed
wi h he hi d-pa y deli e y se ice.
The p oblem can be o mula ed as a non-linea s ochas ic p og am-
ming p oblem as ollows. Le 𝑃be he se o pa cels o be se ed and
𝐵be he se o all bundles ha could be o e ed o ODs ( hus, bundles
associa ed wi h a ou e whose leng h does no exceed
𝑇). Each bundle
is associa ed wi h a bina y a iable 𝑦𝑏which is equal o 1 in case he
bundle is o e ed o ODs, 0 o he wise. Also, pa ame e 𝑎𝑝𝑏 is equal o 1
in case pa cel 𝑝is included in bundle 𝑏. Each bundle is also associa ed
wi h a second decision a iable 𝑐𝑏co esponding o he compensa ion
o e ed. The p oblem can be modeled as:
𝑚𝑖𝑛 ∑
𝑏∈𝐵
E(𝑐𝑏)𝑦𝑏(1a)
𝑠.𝑡.
∑
𝑏∈𝐵
𝑎𝑝𝑏𝑦𝑏= 1, 𝑝 ∈𝑃(1b)
𝑐𝑏≥0, 𝑏 ∈𝐵(1c)
𝑦𝑏∈ {0,1}, 𝑏 ∈𝐵(1d)
whe e E(𝑐𝑏)in (1a) deno es he expec ed cos o bundle 𝑏when com-
pensa ion 𝑐𝑏is o e ed, gi en by he sum o 𝑐𝑏 imes he p obabili y
o accep ance and he cos o he hi d-pa y deli e y se ice imes
he p obabili y o non-accep ance. Cons ain s (1b) gua an ee ha each
pa cel is included in one bundle only. Cons ain s (1c) and (1d) de ine
he domain o he a iables.
The d awback o o mula ion (1) is ha he se 𝐵is ypically
composed o exponen ially many a iables, namely, one a iable o
each easible bundle. Thus, a b anch-and-p ice app oach is needed,
whe e he elaxa ion o each node o he b anch-and-bound ee is
sol ed h ough column gene a ion.
Gi en he gene al de ini ion o he p oblem desc ibed abo e, we
now p esen he speci ic assump ions we made in ou s udy, based on
which we p opose he solu ion me hodology illus a ed in Sec ion 4.
1. The p obabili y ha an OD accep s o se e a bundle is inc eas-
ing wi h espec o he compensa ion o e ed.
2. ODs a e iden ical, i.e., hey ha e he same willingness- o-se e
unc ion. Thus, we a e no in e es ed in knowing which OD
accep s he bundle, bu simply in knowing whe he he e is a
leas one OD who accep s.
3. The e is an in ini e popula ion o ODs, so he accep ance o
a bundle has no impac on he p obabili y o accep ing he
emaining bundles. As a consequence, ODs can be agg ega ed
o de e mine he dis ibu ion unc ion measu ing he p obabili y
ha a leas one OD will accep he bundle. No e ha , despi e
seeming simplis ic, his assump ion is in line wi h cu en ends
in c owdshipping, whe e he explosion o c owdsou ced d i e s
a ailabili y is expe ienced (see h ps://www.cnbc.com/2020/
02/09/amazon- lex-d i e s-use-bo s- o-ge -mo e-wo k.h ml).
This phenomenon also leads o a homogeneous beha io ela ed
o he e o o acqui ing as many deli e ies se ices as pos-
sible, which co obo a es assump ion 2 on op, ela ed o he
homogenei y o ODs.
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As a consequence o he abo e assump ions, and in pa icula o
assump ion 3, each bundle can be conside ed sepa a ely, as he p ice
and he accep ance o a gi en bundle ha e no impac on he emaining
bundles. This means ha objec i e unc ion (1a) is sepa able by bun-
dles. Then, he expec ed cos associa ed wi h o e ing compensa ion 𝑐𝑏
o bundle 𝑏is gi en by:
𝑐𝑏𝑝𝑟𝑏+𝐶𝑏(1 − 𝑝𝑟𝑏)(2)
whe e 𝑝𝑟𝑏is he p obabili y ha an OD accep s o se e bundle 𝑏
wi h compensa ion 𝑐𝑏(gi en by he willingness- o-se e unc ion) and
𝐶𝑏is he cos (de e minis ic) o assigning all pa cels in he bundle o a
hi d-pa y deli e y company.
In he ollowing we p esen he solu ion app oach we p opose o
sol e he p oblem unde he assump ions men ioned abo e. I co e-
sponds o sol ing sequen ially wo phases aimed a de e mining:
1. How should he pa cels be clus e ed in o bundles?
2. Wha should be he compensa ion o e ed o each bundle?
4. Solu ion app oach
The solu ion app oach is composed o wo main phases:
1. Bundles gene a ion.
2. Bundles p icing.
The wo phases a e desc ibed in Sec ions 4.1 and 4.2, espec i ely.
Bundles a e cons uc ed on he basis o spa io/ empo al p oximi y, o
make hem a ac i e o ODs. Then, he compensa ion associa ed wi h
each o hem is calcula ed and inally, hey a e o e ed o ODs, h ough
an auc ion sys em.
4.1. Bundles gene a ion
In his sec ion, we p esen he p ocedu e ha cons uc s he bundles
o be o e ed o he ODs.
Gi en he assump ion men ioned in Sec ion 3, i.e., ha ODs a e
a ailable o a maximum ime
𝑇, hen bundles a e cons uc ed in such
a way ha he ou e o se e all pa cels in he bundle is no longe han
𝑇.
To cons uc he ou es, we use a g eedy algo i hm ha wo ks as
ollows. S a ing om he depo 𝑠, i i s adds he a hes cus ome
(see, e.g., Ca ic and Gold (2008), B äysy and Gend eau (2001) and
B äysy (2003)). Then, du ing each subsequen i e a ion, i adds o he
ou e he i s easible (in e ms o ime windows) and nea es cus ome ,
un il he maximum ou ing ime
𝑇is eached. The p ocedu e is i e a ed
un il all cus ome s ha e been p ocessed, i.e., hey ha e been inse ed
in a bundle. Algo i hm 1p o ides a ske ch o he app oach, whe e 𝑃is
he se o cus ome s (o pa cels). The cus ome s belonging o he same
ou e compose a bundle.
4.2. Bundles p icing
In his sec ion, we desc ibe how we de e mine he compensa ion
associa ed wi h each bundle. A bundle is composed o a se o pa cels
ha should be deli e ed by an OD.
The p ice associa ed wi h a bundle a ies wi hin a ange de ined by
a minimum and a maximum compensa ion o e ed o ODs o deli e
he pa cels included. To ix his ange, we conside he cos o he ou e
isi ing all cus ome s in he bundle, s a ing om he depo . Then,
we de ine a compensa ion ac o ha mul iplies his cos o ind he
minimum and he maximum alue o he ange. The compensa ion
o e ed is hen de e mined wi hin his ange by aking in o accoun he
OD willingness- o-se e unc ion.
We de ine as 𝑑𝑏 he cos o he ou e associa ed wi h bundle 𝑏,
calcula ed as desc ibed abo e (by he g eedy algo i hm). Le 𝑐′
𝑏be he
Algo i hm 1: G eedy [𝑃,𝑠,
𝑇]
1𝑠←depo ;
2while P is no emp y do
3ini ialize a new ou e 𝑟;
4inse 𝑠in 𝑟;
5 ind he a hes cus ome 𝑖∈𝑃 om 𝑠;
6inse 𝑖in 𝑟;
7 emo e 𝑖 om 𝑃;
8while he s opping c i e ion (
𝑇) is no me do
9 ind he nea es cus ome 𝑖′∈𝑃 o 𝑖;
10 inse 𝑖′in 𝑟;
11 emo e 𝑖′ om 𝑃;
12 𝑖←𝑖′;
13 inse he ou e 𝑟in he bundle se ;
maximum compensa ion ha is o e ed o an OD o se e pa cels in
bundle 𝑏and 𝐶𝑏be he p ice paid o he hi d-pa y deli e y company
o deli e all pa cels in 𝑏. We need o de e mine, o each bundle 𝑏∈𝐵,
he compensa ion o o e o he ODs such as o minimize he expec ed
o al cos , which includes he compensa ion paid o ODs in case o
accep ance and he p ice paid o he hi d-pa y deli e y company in
case o non-accep ance.
The expec ed cos is calcula ed wi h espec o he ODs willingness-
o-se e unc ion. In p ac ice, he company does no know wha com-
pensa ion an OD will accep o deli e ing bundle 𝑖(i.e., all pa cels in
he bundle). Thus, his willingness- o-se e unc ion is ep esen ed as a
andom a iable ha gi es he p obabili y o accep ance wi h espec
o he compensa ion o e ed. This unc ion can be es ima ed h ough
he analysis o his o ical da a. No e ha , because o he assump ion
abo e, all ODs a e associa ed wi h he same willingness- o-se e unc ion.
I he compensa ion o e ed by he company is g ea e han o equal
o he willingness- o-se e alue, hen he OD will accep o se e he
bundle, o he wise, she/he will no . We assume ha he willingness- o-
se e unc ion ollows a Gaussian dis ibu ion (𝜇𝑏, 𝜎𝑏)(no e ha he
ollowing model emains alid o any log-conca e unc ion), whe e 𝜇𝑏
is he expec ed alue o he willingness- o-se e and 𝜎𝑏is he s anda d
de ia ion. In he ollowing, we deno e as 𝑓(𝑏) he willingness- o-se e
unc ion associa ed wi h bundle 𝑏and 𝐹(𝑏) he co esponding cumula-
i e unc ion (co esponding o 𝑝𝑟𝑏in (2)). A minimum compensa ion
𝛽𝑏is de ined, i.e., he compensa ion o e ed o ODs canno be lowe
han 𝛽𝑏. We ix 𝛽𝑏=𝑑𝑏. This is easonable as he compensa ion
should a leas co e he cos o he co esponding ou e. The maximum
compensa ion 𝑐′
𝑏is ins ead ixed o 𝛽𝑏+𝛥𝑏, whe e 𝛥𝑏is he maximum
p ice, in addi ion o ou e cos , ha he company is willing o pay
o ODs’ compensa ion. I no OD accep s he bundle, he p ice 𝑝𝑏=
(𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏is paid o he hi d-pa y deli e y company, wi h 𝜁𝑏>1.
The ollowing analysis is based on de e mining an analy ical o -
mula o he op imal compensa ion 𝜂𝑏 ha should be o e ed o each
bundle 𝑏, i.e., he compensa ion ha minimizes he expec ed cos . We
no e ha a simila app oach has been used in Yildiz and Sa elsbe gh
(2019). Howe e , he se ing o he p oblem in he la e pape is
di e en . Speci ically, in Yildiz and Sa elsbe gh (2019), he au ho s
conside a se ing whe e he compensa ion is ixed a a la a e and
ODs a e associa ed wi h a willingness- o-wai unc ion, i.e., a unc ion
ha de e mines how long hey a e willing o wai be o e being o e ed
a se ice (and hus ob aining he la a e compensa ion). The au ho s
hen make a heo e ical s udy o de e mine he bes adius o se ice
assignmen , aking in o accoun he willingness- o-wai , and hey de i e
an analy ical o mula, as we do in he ollowing o de e mining he
op imal compensa ion.
Le us call 𝜂𝑏 he compensa ion o e ed o se e bundle 𝑏. Gi en he
assump ions abo e, he expec ed cos o he company associa ed wi h
bundle 𝑏is:
E(𝜂𝑏) = 𝜂𝑏𝐹(𝜂𝑏) + (1 − 𝐹(𝜂𝑏))((𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏).(3)
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Thus, he company should aim o de e mine:
min
𝜂𝑏
{E(𝜂𝑏)|𝛽𝑏≤𝜂𝑏≤𝛽𝑏+𝛥𝑏}.(4)
No e ha in (3), he i s e m, 𝜂𝑏𝐹(𝜂𝑏), is mono onically inc easing
in 𝜂𝑏while he second e m, (1 − 𝐹(𝜂𝑏))((𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏), is mono onically
dec easing. Thus, we migh expec some ‘ egula i y’ in he shape o
unc ion (3). This is indeed con i med in he ollowing p oposi ion.
P oposi ion 1. In he in e al 𝛽𝑏≤𝜂𝑏≤𝛽𝑏+𝛥𝑏, unc ion (3) is i s
con ex and hen conca e. Thus, i changes con exi y jus once.
P oo . The i s de i a i e o (3) is
E′(𝜂𝑏) = 𝜂𝑏𝑓(𝜂𝑏) + 𝐹(𝜂𝑏) − ((𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏)𝑓(𝜂𝑏)
=𝐹(𝜂𝑏) + 𝑓(𝜂𝑏)(𝜂𝑏− (𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏)
and he second de i a i e is
E′′(𝜂𝑏)=2𝑓(𝜂𝑏) + 𝑓′(𝜂𝑏)(𝜂𝑏− (𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏)
=𝑓(𝜂𝑏)(2 − 𝜂𝑏−𝜇𝑏
𝜎𝑏
(𝜂𝑏− (𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏)).(5)
As 𝑓(𝜂𝑏)>0 o each 𝜂𝑏, he sign o E′′(𝜂𝑏)depends on he
second e m o he p oduc only. Now, 2 − 𝜂𝑏−𝜇𝑏
𝜎𝑏
(𝜂𝑏− (𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏)
is quad a ic in 𝜂𝑏and i s posi i e o −𝜇𝑏−(𝛽𝑏+𝛥𝑏)∗𝜁𝑏−√(𝜇𝑏+(𝛽𝑏+𝛥𝑏)∗𝜁𝑏)2+8𝜎
2
≤𝜂𝑏≤−𝜇𝑏−(𝛽𝑏+𝛥𝑏)∗𝜁𝑏+√(𝜇𝑏+(𝛽𝑏+𝛥𝑏)∗𝜁𝑏)2+8𝜎
2. Howe e ,
−𝜇𝑏−(𝛽𝑏+𝛥𝑏)∗𝜁𝑏−√(𝜇𝑏+(𝛽𝑏+𝛥𝑏)∗𝜁𝑏)2+8𝜎
2< 𝛽𝑏. Thus, in he in e al conside ed,
E(𝜂𝑏)is con ex in [𝛽𝑏,min{𝛽𝑏+𝛥𝑏,−𝜇𝑏−(𝛽𝑏+𝛥𝑏)∗𝜁𝑏+√(−𝜇𝑏−(𝛽𝑏+𝛥𝑏)∗𝜁𝑏)2+8𝜎
2}]
and conca e in [min{𝛽𝑏+𝛥𝑏,−𝜇𝑏−(𝛽𝑏+𝛥𝑏)∗𝜁𝑏+√(−𝜇𝑏−(𝛽𝑏+𝛥𝑏)∗𝜁𝑏)2+8𝜎
2}, 𝛽𝑏+
𝛥𝑏].□
This helps in sea ching o he alue o 𝜂𝑏p o iding he minimum
expec ed cos in (4). Indeed, due o P oposi ion 1, his alue is ei he a
in e al limi s 𝛽𝑏and 𝛽𝑏+𝛥𝑏o a i s unique local minimum in be ween.
In o de o de e mine he alue 𝜂𝑏minimizing he expec ed cos (4),
we p oceed h ough a bisec ion me hod wo king as ollows. We s a
by ixing he alue o 𝜂𝑏co esponding o 𝜇𝑏. We calcula e he co -
esponding alue o he cumula i e dis ibu ion unc ion 𝐹(𝜂𝑏). Then,
we sea ch o he minimum expec ed alue o (4) by mo ing on he
le and on he igh on he cumula i e unc ion wi h a s ep equal o 𝜔.
Speci ically, we calcula e he cos (4) associa ed wi h he co esponding
alue o he cumula i e unc ion. We e mina e once we de e mine a
local minimum, which, hanks o P oposi ion 1, app oxima es he alue
o he minimum o he unc ion (4). A g aphical example is gi en in
Fig. 2 whe e he cumula i e dis ibu ion unc ion 𝐹(𝜂𝑏)is ep esen ed.
To ind he minimum expec ed alue o (4), we i s s a by ixing
𝜂𝑏=𝜇𝑏. We hen mo e on he le and he igh on he 𝐹(𝜂𝑏) unc ion
by using a s ep equal o 𝜔. We ind he associa ed p ices, i.e., 𝜂𝑙
𝑏and
𝜂𝑟
𝑏. Then, we calcula e he alue o 𝐸(𝜂𝑙
𝑏)and 𝐸(𝜂𝑟
𝑏), espec i ely. In he
example, we assume ha 𝐸(𝜂𝑙
𝑏)> 𝐸(𝜇𝑏)hence, we s op he sea ch on
he le side. On he igh side ins ead, we assume o ha e 𝐸(𝜂𝑟
𝑏)< 𝐸(𝜇𝑏)
hence, we con inue he sea ch on 𝐹(𝜂𝑏)wi h a s ep equal o 2𝜔. We
calcula e he co esponding p ice 𝜂2𝑟
𝑏and hen 𝐸(𝜂2𝑟
𝑏), we obse e ha
𝐸(𝜂2𝑟
𝑏)> 𝐸(𝜂𝑟
𝑏), hence he sea ch ends and he minimum alue is se o
𝜂𝑟
𝑏.
5. Auc ion sys ems
Once all bundles a e p iced acco ding o he p ocedu e desc ibed
abo e, hey a e o e ed o ODs h ough he dedica ed pla o m. In
p ac ice, he company o e s he bundles on he pla o m, accessed
by all ODs, whe e each bundle is associa ed wi h he co esponding
compensa ion. When he bundles a e a ailable on he pla o m, he
auc ions s a s and bundles a e o e ed o ODs. All ODs ha e access
o all bundles, meaning ha hey can see hei composi ion and com-
pensa ion. Then, o each bundle, he i s OD who accep s i ge s he
co esponding compensa ion (once he se ice is pe o med). When
he auc ion ends, he company de e mines which bundles ha e been
accep ed by ODs. The emaining will be se ed by he hi d-pa y
deli e y company. Gi en ha each bundle is independen o he o he s
(due o he assump ions desc ibed abo e), we now desc ibe he auc ion
sys em o each bundle indi idually.
We p opose wo auc ion mechanisms. The i s one is s a ic, i.e., he
p ice is de ined ini ially and he auc ion is composed o a single ound.
This means ha he sys em wo ks exac ly as desc ibed abo e: he
auc ion s a s, he bundles a e o e ed wi h he co esponding compen-
sa ion, and when he auc ion e mina es he company de e mines which
ones ha e been accep ed. Fo he accep ed bundles, he co esponding
compensa ion is paid o he ODs who accep ed hem once he se ice
is comple ed. Fo he non-accep ed bundles, he co esponding pa cels
a e se ed by he hi d-pa y deli e y company.
The second auc ion is a dynamic auc ion, speci ically, an ascending-
p ice auc ion. In his case, he auc ion is composed o mul iple uns
whe e, in each un, all bundles ha ha e no been accep ed in p e ious
uns ( hus, he en i e se o bundles in he i s un) a e o e ed.
Then, he accep ed bundles a e assigned o he co esponding ODs and
emo ed om he auc ion. Fo he emaining bundles, he compensa-
ion is inc eased and hey a e o e ed again on he nex un. When
he auc ion eaches he las un, all pa cels associa ed wi h bundles
ha ha e no been accep ed a e assigned o he hi d-pa y deli e y
company.
S a ic auc ion. The alue o 𝜂𝑏 ha minimizes he expec ed cos is
e alua ed using he bisec ion me hod desc ibed in Sec ion 4.2. The
s a ic auc ion is a single ound o bidding a his alue o compensa ion.
Dynamic auc ion. Le us assume ha he company opens he bundles’
auc ion on day 𝑑, say om 6 o 8 pm, o he bundles ha should be
se ed on day 𝑑+ 1. The company s a s wi h an ini ial es ima ion o
he ODs willingness- o-se e unc ion as desc ibed abo e and de e mines
he ini ial compensa ion h ough he bisec ion me hod, as abo e in he
s a ic auc ion. The idea o he dynamic auc ion is o ha e mul iple
uns o o e s so ha , in case he bundle is no accep ed in a un,
he company can eac acco dingly o he nex un, o example by
inc easing he compensa ion o e ed. The auc ion s a s in a i s un
by o e ing he same compensa ion de e mined in he s a ic auc ion
(which can be seen as a dynamic auc ion wi h one un only). The
auc ion epea s a e e y uns on he bundles ha we e no assigned
in he o me uns.
The e migh be wo easons why he bundle 𝑏has no ye been
accep ed (by any OD) a ime 𝜏𝑡, whe e 𝑡is he 𝑡 h un:
1. he compensa ion 𝜂𝑏is oo low;
2. he dis ibu ion unc ion 𝑓(𝑏)has been badly es ima ed.
As we assume o ha e no upda ed in o ma ion o co ec a bad
es ima ion o he unc ion 𝑓(𝑏), we ocus on a s a egy based on
upda ing he alue o 𝜂𝑏. Speci ically, le 𝜂𝑡
𝑏be he compensa ion
o e ed a un 𝑡. I he bundle 𝑏is no assigned a 𝑡 < 𝐾, hen he
co esponding compensa ion in he ollowing un 𝜂𝑡+1
𝑏is inc eased by
a ce ain pe cen age. In case 𝜂𝑡+1
𝑏≥𝛽𝑏+𝛥𝑏, hen i s alue is se o
𝜂𝑡+1
𝑏=𝛽𝑏+𝛥𝑏. This p ocedu e is epea ed un il ei he he bundle is
assigned o he auc ion e mina es. I he auc ion e mina es and bundle
𝑏is no assigned, he pa cels included a e assigned o he hi d-pa y
deli e y company a a cos equal o (𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏.
6. Compu a ional esul s
In his sec ion, we discuss he simula ion expe imen s we pe o med
in o de o e alua e he p icing s a egy and he auc ion mechanisms
p esen ed abo e. The main scope o his compu a ional s udy is o
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Fig. 2. Example o he bisec ion me hod used o ind he minimum alue o Eq. (4).
compa e he ollowing ou s a egies, in which we conside di e en
app oaches o calcula e 𝜂, i.e., he compensa ion o e ed o he OD, and
he wo auc ion sys ems p esen ed in Sec ion 5:
1. BaseP iceS a : he alue o 𝜂is equal o 𝛽, ha is he minimum
compensa ion ha can be o e ed, h ough a s a ic auc ion.
2. Op P iceS a : he alue o 𝜂is de e mined acco ding o (4), ha
is he compensa ion ha minimizes he expec ed cos , h ough a
s a ic auc ion.
3. BaseP iceDyn: he alue o 𝜂is equal o 𝛽 h ough a dynamic
auc ion.
4. Op P iceDyn: he alue o 𝜂is de e mined acco ding o (4)
h ough a dynamic auc ion.
The es o he sec ion is o ganized as ollows. We i s desc ibe in
Sec ion 6.1 he se ings o he simula ion en i onmen . Then, we ocus
on he analysis o he s a ic and dynamic auc ions in Sec ions 6.2 and
6.3, espec i ely. The wo auc ion sys ems a e compa ed in Sec ion 6.4.
In Sec ion 6.5 we p esen a case s udy based on a eal ne wo k.
6.1. Simula ion en i onmen and pa ame e s se ing
We gene a ed ou ins ances s a ing om Solomon’s benchma k
se o VRPTW ins ances (see Solomon (1987)). We conside ed wo
ins ances, namely R101 and RC101, wi h 100 cus ome s which a e an-
domly dis ibu ed in R101 and pa ially clus e ed in RC101. This allows
us o e i y i and how he esul s a e a ec ed by cus ome s’ loca ions.
Then, we gene a ed wo addi ional ins ances wi h 1000 cus ome s, by
adding 900 andom cus ome s o R101 and RC101, espec i ely, in he
same geome ic a ea, ob aining mo e dense ins ances, e e ed o as
‘‘R1001’’ and ‘‘RC1001’’. We andomly gene a ed he ime windows
o he addi ional cus ome s in R1001 and RC1001, conside ing he
ime equi ed o each he cus ome om he depo , hus gua an eeing
easibili y. We conside ed a planning ho izon [0, 𝑇𝑚𝑎𝑥], co esponding o
a wo king shi . Then, we di ided i in o 𝑚iden ical slo s o du a ion
equal o 𝑇𝑚𝑎𝑥∕𝑚. We se 𝑚= 4 in he expe imen s. This se ing i s wi h
he case whe e he ho izon ep esen s ei he he mo ning (i.e., 8:00
am–12:00 am) o he a e noon (i.e., 2:00 pm–6:00 pm) deli e y shi ,
in which each slo is one hou long. In he conside ed ins ances, 𝑇𝑚𝑎𝑥
is equal o 230 and 240 o R101 and RC101, espec i ely. The alues
o 𝑇𝑚𝑎𝑥 we e kep unchanged o he gene a ed ins ances R1001 and
RC1001. Each OD is a ailable o wo consecu i e ime slo s, i.e.,
𝑇=
𝑇𝑚𝑎𝑥∕2. This is in line wi h p ac ical applica ions o c owdshipping
whe e d i e s a e a ailable o sho ime slo s and hus can deli e
a educed numbe o pa cels.
P icing pa ame e s. We ixed he alue o he minimum compensa ion
o each bundle 𝛽𝑏equal o he cos o he associa ed g eedy easible
ou e. The maximum cos 𝛽𝑏+𝛥𝑏is ixed o 3 ∗ 𝛽𝑏. The p ice 𝐶𝑏paid o
he hi d-pa y deli e y company is se o (𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏, whe e 𝜁𝑏= 2.
We ca ied ou a i s se o simula ions in which he expec ed alue 𝜇𝑏
o 𝑓(𝑖)is cen e ed in he in e al associa ed wi h [𝛽𝑏, 𝛽𝑏+𝛥𝑏]. Then, we
conside ed h ee con idence in e als co esponding o he p obabili ies
associa ed wi h 𝛽𝑏and 𝛽𝑏+𝛥𝑏, espec i ely. In pa icula , he h ee
conside ed in e als a e: [30% − 70%],[15% − 85%] and [5% − 95%]. The
idea is o assess he e ec o a di e en a iance o he willingness- o-
se e unc ion 𝑓(𝑏): he i s in e al co esponds o a highe a iance
while he las one is associa ed wi h he smalles a iance. In he es
o he pape we will indica e hese h ee con idence in e als as 𝜎𝑙𝑜𝑤,
𝜎𝑚𝑒𝑑 and 𝜎ℎ𝑖𝑔ℎ, espec i ely. No e ha in he i s se o simula ions he
alue o 𝜇𝑏is equal o ((𝛽𝑏+𝛥) + 𝛽𝑏)∕2, as i is in he cen e o he
in e al [𝛽𝑏, 𝛽𝑏+𝛥𝑏]. We named his alue o 𝜇𝑏as 𝜇𝑚𝑒𝑑 .
Fig. 3 ep esen s he shape o he dis ibu ion when a ying he
a iance 𝜎𝑏. Looking a he igu e we may obse e ha he highe he
a iance, he b oade and lowe he peak. Hence, a highe a iance
indica es ha he poin s a e mo e sp ead ou om he mean, while
when i is lowe hey a e close o i . Focusing on ou se ing, ha ing
a highe a iance co esponds o a highe a iabili y in he ODs’
beha io , whe eas a lowe a iance indica es a mo e homogeneous
beha io .
We also conduc ed wo addi ional se s o simula ions whe e we
a ied he alue o 𝜇𝑏. Speci ically, we se :
1. 𝜇𝑏=𝜇𝑚𝑒𝑑 + (20%𝜇𝑚𝑒𝑑 ): he alue o 𝜇𝑏is shi ed on he igh . We
deno ed i as 𝜇ℎ𝑖𝑔ℎ;
2. 𝜇𝑏=𝜇𝑚𝑒𝑑 − (20%𝜇𝑚𝑒𝑑 ): he alue o 𝜇𝑏is shi ed on he le . We
deno ed i as 𝜇𝑙𝑜𝑤.
As shown in Fig. 4, which ep esen s he Gaussian dis ibu ion
cu e when a ying he alue o 𝜇𝑏, shi ing 𝜇𝑏 o he le (𝜇𝑙𝑜𝑤)
and o he igh (𝜇ℎ𝑖𝑔ℎ) allows o ep esen di e en beha io s o
he ODs’ willingness- o-se e. Due o i s symme ical and bell-shaped
na u e, he p obabili y o a Gaussian dis ibu ion has a maximum a
i s mean. Hence, when shi ing 𝜇𝑏 o 𝜇𝑙𝑜𝑤, he p obabili y ha an OD
accep s o deli e a bundle a a lowe compensa ion is highe ; on he
con a y, when shi ing o 𝜇ℎ𝑖𝑔ℎ, he ODs accep a deli e y when he
compensa ion o e ed is highe .
No e ha , o hese wo addi ional se s o simula ions, we kep
he same h ee alues o he a iance o 𝑓(𝑏). As a consequence, he
con idence in e als ha e he same wid h as abo e bu a e associa ed
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Fig. 3. Rep esen a ion o he cu es associa ed wi h he ODs willingness- o-se e unc ion when a ying he a iance 𝜎𝑏. In pa icula , 𝜎𝑙𝑜𝑤 < 𝜎𝑚𝑒𝑑 < 𝜎ℎ𝑖𝑔ℎ.
Fig. 4. Rep esen a ion o he cu es associa ed wi h he ODs willingness- o-se e unc ion when shi ing 𝜇𝑏 o he le and igh .
wi h di e en lowe and uppe limi s. Fo he sake o comple eness,
we epo a g aphical ep esen a ion o he nine se ings we ha e
conside ed, depic ed in Fig. 5, ob ained by combining he di e en
alues o 𝜇𝑏and 𝜎𝑏.
As s a ed a he beginning o his sec ion, we compa e wo p icing
s a egies ha di e on how he alue o he compensa ion 𝜂𝑏is se .
In he i s one, we se 𝜂𝑏=𝛽𝑏, while in he second 𝜂𝑏co esponds
o he alue ha minimizes he expec ed cos (4) and is calcula ed as
desc ibed in Sec ion 5. Thus, we compa e a s a egy in which we educe
he compensa ion alue o he minimum (𝜂𝑏=𝛽𝑏) bu we inc ease he
p obabili y o no assigning he bundle, wi h he s a egy in which we
minimize he expec ed cos h ough (4). In he la e case, he alue o
pa ame e 𝜔used in he bisec ion me hod is ixed a 0.005. No e ha
we made p elimina y es s by using a s ep equal o 0.05. The esul s
we e simila bu less accu a e. By se ing he s ep equal o 0.005 he
simula ion is s ill as (always uns in milliseconds), so we decided o
keep he la e o gain accu acy.
As o he dynamic auc ion, he s a ing alue o he compensa ion
𝜂𝑏is de e mined wi h ei he o he wo p icing s a egies men ioned
abo e ( hus gi ing ise o wo dynamic p icing auc ions).
S a ic auc ion. We simula ed 5000 ealiza ions o he willingness- o-se e
alue o each bundle, acco ding o he Gaussian dis ibu ion. I he
ealiza ion alue is equal o o lowe han he alue o 𝜂𝑏, he bundle is
assigned o he OD, o he wise, he bundle is no assigned and i s cos
is equal o (𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏.
Dynamic auc ion. The se ing is he same as o he s a ic auc ion. In
addi ion, in he dynamic auc ion, up o 8 uns a e pe o med. Thus, in
each un, ei he he bundle is assigned, o he compensa ion is upda ed
acco ding o he ule: 𝜂𝑡
𝑏=𝜂0
𝑏∗ (1 + 0.05𝑡), whe e 𝜂0
𝑏is he ini ial
compensa ion. Thus, he compensa ion is inc eased by 5% o he ini ial
compensa ion in each un. I a e 8 uns he bundle is s ill unassigned,
hen he cos (𝛽𝑏+𝛥𝑏) ∗ 𝜁𝑏is paid.
Fo ease o eading, in he ollowing we emo e index 𝑏 om all
no a ions as we p esen agg ega ed esul s o e all bundles.
In he nex sec ions, we analyze he esul s ob ained by he s a ic
and dynamic auc ions, espec i ely. In bo h cases, bundles a e gen-
e a ed acco ding o he app oach desc ibed in Sec ion 4.1. In he
ollowing, we concen a e on he KPI o bo h auc ions. Some s a is ics
abou he gene a ed bundles can be ound in Appendix A.1, whe eas
Table 2 epo s he pa ame e se ings.
6.2. S a ic auc ion analysis
In his sec ion we p esen he esul s associa ed wi h s a ic auc ion
sys em, conside ing he wo p icing s a egies, namely, BaseP iceS a
and Op P iceS a .
We ocus he analysis on he ins ances wi h 1000 eques s only,
as he esul s o ins ances wi h 100 eques s a e simila . We i s ly
analyze he pe cen age o unassigned bundles, on he basis o he
alues o 𝜇and 𝜎, whose end, o bo h s a egies, is depic ed in
Fig. 6. Focusing on Fig. 6(a), which depic s he esul s ob ained using
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Table 8
Compa ison be ween s a ic and dynamic auc ions o he Loggy-n601-k42 ins ance.
𝜇 𝜎 𝐺𝑎𝑝𝐴𝑢𝑐𝑡𝑖𝑜𝑛𝐶𝑜𝑠𝑡 (%)
𝐵𝑎𝑠𝑒𝑃 𝑟𝑖𝑐𝑒𝑆𝑡𝑎𝑡 s. 𝐵𝑎𝑠𝑒𝑃 𝑟𝑖𝑐𝑒𝐷𝑦𝑛 𝑂𝑝𝑡𝑃 𝑟𝑖𝑐𝑒𝑆𝑡𝑎𝑡 s. 𝑂𝑝𝑡𝑃 𝑟𝑖𝑐𝑒𝐷𝑦𝑛
𝜎ℎ𝑖𝑔ℎ 277.7% 69.2%
𝜇𝑙𝑜𝑤 𝜎𝑚𝑒𝑑 282.4% 48.9%
𝜎𝑙𝑜𝑤 238.5% 102.1%
𝜎ℎ𝑖𝑔ℎ 273.6% 62.9%
𝜇𝑚𝑒𝑑 𝜎𝑚𝑒𝑑 143.4% 42.5%
𝜎𝑙𝑜𝑤 38.5% 78.0%
𝜎ℎ𝑖𝑔ℎ 96.1% 53.1%
𝜇ℎ𝑖𝑔ℎ 𝜎𝑚𝑒𝑑 5.3% 37.0%
𝜎𝑙𝑜𝑤 0.0% 60.8%
compa ison be ween 𝑂𝑝𝑡𝑃 𝑟𝑖𝑐𝑒𝑆𝑡𝑎𝑡 and 𝑂𝑝𝑡𝑃 𝑟𝑖𝑐𝑒𝐷𝑦𝑛, whose gap alues
a e epo ed in he second column, we may obse e ha he dynamic
auc ion ou pe o ms he s a ic one in e ms o o al cos . The a e age
cos gap is on a e age equal o 62%.
To conclude, compa ing he s a ic and he dynamic auc ions, i is
clea ha he dynamic one ou pe o ms he s a ic one. The dynamic
auc ion allows o be e exploi he ODs se ice, assigning a la ge
numbe o bundles while minimizing he o e all cos s.
7. Conclusions
We p esen ed a amewo k o handle he p oblem o managing
cus ome eques s in a c owdshipping pla o m. In pa icula , we ad-
d essed he p icing p oblem, o maximize he numbe o assigned
pa cels and minimize he cos s. A e gene a ing he bundles o be
assigned o occasional d i e s, using a g eedy algo i hm, we de eloped
wo p icing s a egies and wo ypes o auc ions: s a ic and dynamic.
In he s a ic auc ion he p ice is ixed and he e is a single un. The
dynamic auc ion is ins ead an ascending-auc ion wi h mul iple uns.
As o p icing, we compa ed wo s a egies: in he i s one he p ice is
calcula ed by conside ing a base cos , ha is an es ima ion o he cos o
he ou e associa ed wi h he bundle, while in he second one, he p ice
is e alua ed conside ing he willingness- o-se e unc ion associa ed wi h
occasional d i e s. We ca ied ou a compu a ional s udy on ins ances
wi h 100 and 1000 cus ome s as well as on an ins ance de i ed om
eal da a. The esul s sugges ed ha he dynamic auc ion is mo e
e icien han he s a ic one in educing he auc ion cos and he numbe
o unassigned bundles. Focusing on he p icing s a egies, hey beha e
di e en ly in s a ic and dynamic auc ions. In he s a ic auc ion, he
second s a egy ou pe o ms he i s one in e ms o bo h educing
cos s and unassigned eques s. In he dynamic one, ins ead, e en i he
cos using he i s p icing s a egy is in se e al cases lowe han he
second one, he pe cen age o unassigned bundles is much highe . The
second s a egy assigns all he eques s in almos all cases. Hence, he
decision make has o handle he ade-o be ween se ice quali y and
cos , by selec ing he mos app op ia e p icing mechanism. As a inal
obse a ion, we ema k ha he p icing s a egy is obus wi h espec
o he ins ance size as he es s p o ided simila esul s on ins ances
wi h 100 and 1000 cus ome s as well as on he eal case ins ance wi h
600 cus ome s.
As a u u e ex ension, we aim a conside ing a combined ma ching,
p icing and ou ing p oblem, including also mo e complex eal-li e
ea u es such as dynamic a i al o eques s, ODs’ beha io , a ic con-
ges ion, delay in deli e y. Ano he in e es ing aspec is he co ela ion
be ween bundles gene a ion and expec ed cos . The e o e, a mo e so-
phis ica ed app oach in eg a ing bundles’ gene a ion and p icing could
be conside ed as u u e wo k. In addi ion, mo e sophis ica ed p icing
echniques a e wo h being in es iga ed, in pa icula dynamic p icing
s a egies o e ec i ely calcula e and adjus p ices in eal- ime.
CRediT au ho ship con ibu ion s a emen
Giusy Mac ina: Concep ualiza ion, Me hodology, So wa e, Vali-
da ion, W i ing – o iginal d a , W i ing – e iew & edi ing. Clau-
dia A che i: Concep ualiza ion, Me hodology, Supe ision, W i ing
– o iginal d a , W i ing – e iew & edi ing. F ancesca Gue ie o:
Concep ualiza ion, Me hodology, Supe ision, W i ing – o iginal d a ,
W i ing – e iew & edi ing.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Acknowledgmen s
Thanks a e due o he wo anonymous e iewe s whose use ul
commen s helped imp o ing a p e ious e sion o he pape .
Appendix A. Solomon’s based ins ances compu a ional s udy
A.1. Bundles analysis
Bundles a e cons uc ed using he g eedy algo i hm desc ibed in
Sec ion 4.1 (i.e., Algo i hm 1). The esul s o ins ances R101, RC101,
R1001, and RC1001 a e gi en in Table A.1. In pa icula , o each
ins ance, we epo in he column #𝑏𝑢𝑛𝑑𝑙𝑒𝑠 he numbe o bundles
gene a ed, in #𝑚𝑒𝑎𝑛 he a e age numbe o cus ome s (o pa cels) in
each bundle, while he columns #𝑚𝑎𝑥 and #𝑚𝑖𝑛 gi e he size o he
la ge and smalle bundles, espec i ely. Finally, in he las column, we
epo he a e age cos o he bundles, i.e., he a e age ou ing cos .
We gene a ed 16 bundles o R101, 17 o R101, 71 o R1001 and
82 o RC1001. Focusing on he numbe o cus ome s in each bundle,
o he ins ance R101 (RC101) he bundles con ain 6 (5) cus ome s on
a e age, he smalles bundle con ains 1 (2) cus ome s, while he la ges
one 11 (10) cus ome s. Conside ing ins ances wi h 1000 cus ome s, he
numbe o pa cels in each bundle in R1001 (RC1001) a ies on a e age
om a minimum o 2 (1) o a maximum o 27 (36), while he a e age
numbe o pa cels in each bundle is 14 (12). We poin ou ha we
a e no conside ing an uppe limi on he numbe o cus ome s in each
bundle, bu he e is a limi on he du a ion o he wo king shi . As in
common p ac ice, bundles wi h a la ge numbe o cus ome s migh be
ela ed o cases whe e all cus ome s belong o he same neighbo hood
o building.
A.2. Co ela ion be ween 𝜂,𝜎and 𝜇in s a ic auc ion
Focusing on he esul s ob ained using he S a ic Auc ion on
Solomon’s based ins ances, we obse e ha he alue o 𝜂when using
he Op P iceS a depends on he willingness- o-se e unc ion 𝑓, while
his is no he case wi h he BaseP iceS a . Thus, we i s show he
e ec o he expec ed alue (𝜇) and he a iance (𝜎) o he willingness-
o-se e unc ion 𝑓on he alue o 𝜂 o he Op P iceS a .Tables A.2
and A.3 epo , o each alue o 𝜇and 𝜎, he alue o he cumula i e
p obabili y associa ed wi h 𝜂de e mined acco ding o he Op P iceS a ,
and he a e age alue o 𝜂o e all solu ions, o ins ances R and RC,
espec i ely.
When analyzing he alues in Tables A.2 and A.3, we i s no ice
ha he cumula i e p obabili y, i.e., he p obabili y ha a bundle will
be accep ed a he o e ed p ice 𝜂, does no change when compa ing
ins ances wi h 100 and 1000 cus ome s. Mo ing he alue o 𝜇 o 𝜇𝑙𝑜𝑤
o 𝜇ℎ𝑖𝑔ℎ has a sligh e ec on he cumula i e p obabili y associa ed
wi h 𝜂. Ins ead, he a iance 𝜎has a s ong impac and he highes
alue o he cumula i e p obabili y is associa ed wi h he mid alue
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100142
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G. Mac ina e al.
Table A.1
Bundles gene a ed o ins ances R101, RC101, R1001 and RC1001.
R101 RC101
# bundles # mean # max # min a g cos # bundles # mean # max # min a g cos
16 6 11 1 89.53 17 5 10 2 97.15
R1001 RC1001
# bundles # mean # max # min a g cos # bundles # mean # max # min a g cos
71 14 27 2 88.86 82 12 36 1 90.21
Table A.2
Cumula i e p obabili y and a e age alue o 𝜂 o R ins ances using Op P iceS a .
𝜎Cumula i e p obabili y A g 𝜂 o Op P iceS a
𝜇𝑙𝑜𝑤 𝜇𝑚𝑒𝑑 𝜇ℎ𝑖𝑔ℎ 𝜇𝑙𝑜𝑤 𝜇𝑚𝑒𝑑 𝜇ℎ𝑖𝑔ℎ
100 R cus ome ins ances
𝜎ℎ𝑖𝑔ℎ 0.55 0.58 0.63 198.82 214.48 236.13
𝜎𝑚𝑒𝑑 0.62 0.70 0.77 205.38 223.44 241.51
𝜎𝑙𝑜𝑤 0.35 0.58 0.78 157.62 189.33 221.04
1000 R cus ome ins ances
𝜎ℎ𝑖𝑔ℎ 0.55 0.58 0.63 197.32 212.87 234.35
𝜎𝑚𝑒𝑑 0.62 0.70 0.77 203.83 221.76 239.69
𝜎𝑙𝑜𝑤 0.35 0.57 0.78 156.43 187.90 219.37
Table A.3
Cumula i e p obabili y and a e age alue o 𝜂 o RC ins ances using Op P iceS a .
𝜎Cumula i e p obabili y A g 𝜂 o Op P iceS a
𝜇𝑙𝑜𝑤 𝜇𝑚𝑒𝑑 𝜇ℎ𝑖𝑔ℎ 𝜇𝑙𝑜𝑤 𝜇𝑚𝑒𝑑 𝜇ℎ𝑖𝑔ℎ
100 RC cus ome ins ances
𝜎ℎ𝑖𝑔ℎ 0.55 0.58 0.63 215.73 232.73 256.22
𝜎𝑚𝑒𝑑 0.62 0.70 0.77 222.85 242.45 262.05
𝜎𝑙𝑜𝑤 0.35 0.58 0.78 171.03 205.43 239.84
1000 RC cus ome ins ances
𝜎ℎ𝑖𝑔ℎ 0.55 0.58 0.63 200.33 216.11 237.92
𝜎𝑚𝑒𝑑 0.62 0.70 0.77 206.93 225.13 243.33
𝜎𝑙𝑜𝑤 0.35 0.57 0.78 158.81 190.76 222.71
o he a iance. In o de o explain his beha io , we ha e o conside
he link be ween he cumula i e p obabili y and he alue o 𝜂. When
mo ing om a la ge o a medium alue o he a iance, wi h a
sligh inc ease in he alue o he o e ed compensa ion 𝜂(3%), he
cumula i e p obabili y inc eases by a ound 18%. Thus, he inc ease in
he alue o 𝜂is compensa ed by a mo e han p opo ional inc ease
o bundles ha a e assigned (and, hus, o which he company a oids
paying he high cos (𝛽+𝛥) ∗ 𝜁). When mo ing ins ead om 𝜎𝑚𝑒𝑑 o
𝜎𝑙𝑜𝑤, he cumula i e p obabili y dec eases by a ound 14% eaching a
simila alue o he one ela ed o he high a iance scena io. Howe e ,
he alue o 𝜂is on a e age 19% smalle han he one associa ed wi h
he medium a iance scena io. Thus, he smalle numbe o assigned
bundles is compensa ed by a lowe compensa ion o e ed o ODs.
Appendix B. Real ne wo k case s udy compu a ional s udy
B.1. Bundles analysis
The esul s ela ed o he analysis o he bundles a e epo ed in
Table B.4. In pa icula , column #𝑏𝑢𝑛𝑑𝑙𝑒𝑠 epo s he numbe o bundles
gene a ed, in #𝑚𝑒𝑎𝑛 we ha e he a e age numbe o cus ome s in he
bundles, while columns #𝑚𝑎𝑥 and #𝑚𝑖𝑛 epo he size o he la ge and
smalle bundles, espec i ely, and inally, in he las column, we epo
he a e age cos o he bundles, i.e., he a e age ou ing cos . The o al
numbe o gene a ed bundles is 69. The smalles bundle con ains one
cus ome , while he la ges one 36. On a e age, each bundle con ains
8 cus ome s. As in he Solomon’s based ins ances, we did no conside
Table B.4
Bundles gene a ed o he Loggi-n601-k42 ins ance.
Loggi-n601-k42
# bundles # mean # max # min a g cos
69 8 36 1 3523.82
an uppe limi o he numbe o cus ome s in each bundle, bu a limi
on he empo al ou e leng h.
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