Cuella -Usaquén, Daniel; Ulme , Ma lin W.; An ons, Oli e ; A linghaus, Julia C.
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Dynamic mul i-pe iod ecycling collec ion ou ing wi h
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Sugges ed Ci a ion: Cuella -Usaquén, Daniel; Ulme , Ma lin W.; An ons, Oli e ; A linghaus, Julia
C. (2025) : Dynamic mul i-pe iod ecycling collec ion ou ing wi h unce ain ma e ial quali y, OR
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ORIGINAL ARTICLE
Dynamic mul i‑pe iod ecycling collec ion ou ing
wi hunce ain ma e ial quali y
DanielCuella ‑Usaquén1· Ma linW.Ulme 2 · Oli e An ons3,4·
JuliaC.A linghaus3,4
Recei ed: 16 Feb ua y 2024 / Accep ed: 6 Janua y 2025 / Published online: 8 Feb ua y 2025
© The Au ho (s) 2025
Abs ac
We conside he p oblem o collec ing and p ocessing was e ma e ial. A a p oduc-
ion acili y, a known amoun o in en o y is equi ed o p oduc ion (e.g., pape ) o
e e y pe iod. Ins ead o new ma e ial, he acili y elies on collec ed and p ocessed
was e ma e ial (e.g., pape was e). This ma e ial is collec ed om egional was e
collec ion loca ions. The amoun o was e ma e ial pe loca ion is unce ain, as is he
quali y o he collec ed was e, i.e., he esul ing in en o y when p ocessing he ma e-
ial. I he in en o y is insu icien a he end o a pe iod, cos ly new ma e ial mus be
bough . Each pe iod, decisions a e made abou how much was e ma e ial o collec
om which loca ion and how o ou e he collec ion ehicles acco dingly. Ideally,
in en o y is buil o hedge agains quali y unce ain y and o ensu e e icien ou ing
ope a ions in u u e pe iods. We p opose a s ochas ic lookahead me hod ha sam-
ples a se o scena ios and sol es a simpli ied wo-s age s ochas ic p og am in e e y
pe iod. We show he alue o ou me hod o wo case s udies, one based on eal-
wo ld da a om Sachsen-Anhal , Ge many, and one om he li e a u e wi h da a
om he Uni ed Kingdom. We u he conduc a de ailed analysis o ou me hod and
he p oblem cha ac e is ics. The esul s show ha ou me hod e ec i ely an icipa es
all sou ces o unce ain y, educing cos signi ican ly compa ed o benchma k poli-
cies. This supe io pe o mance is due o app op ia e s a e-dependen supplie selec-
ion ha conside s he pe cen age o ma e ial loss, a ailable ma e ial, and ou ing
cos o cu en and u u e pe iods.
Keywo ds Rou ing· Ci cula economy· Sequen ial decision p ocess· S ochas ic
lookahead
Ex ended au ho in o ma ion a ailable on he las page o he a icle
700
D.Cuella -Usaquén e al.
1 In oduc ion
The ci cula economy p oposes eplacing linea sou cing o p oduc ion ma e ials
ia global supply chains wi h ecycling local was e ma e ials (such as pape , plas-
ics, glass, me als, elec onic pa s, e c.). These ma e ials a e collec ed om local
was e supplie s (e.g., was e acili ies), cleaned, so ed, and hen in eg a ed in o he
p oduc ion p ocess. Ci cula economy p ojec s kick o wo ldwide, and he Eu o-
pean Union ecen ly issued a Ci cula Economy Ac ion Plan (h ps:// en i onmen .
ec. eu opa. eu/ s a egy/ ci cu la - econo my- ac ion- plan_ en). In he Ge man s a e o
Sachsen-Anhal , a la ge esea ch conso ium o Uni e si ä Magdebu g, F aunho e
Ins i u e o Fac o y Ope a ion and Au oma ion IFF Magdebu g, and Max Planck
Ins i u e o Dynamics o Complex Technical Sys ems Magdebu g wo k on imp o -
ing ope a ions in he ci cula economy in hei esea ch clus e Sma P oSys (h ps://
www. sma p osys. o gu. de).
The e a e many easons o he su ge in ci cula economy p ojec s, no only he
high cos o new ma e ial and he inc easing unce ain y in global supply chains due
o poli ical con lic s and na u al disas e s bu also he commi men o mo e en i on-
men ally sus ainable p oduc ion. Replacing classical linea sou cing o new ma e i-
als wi h ecycled local ma e ials b ings se e al new challenges in p oduc ion plan-
ning and logis ics. In his wo k, we ocus on ou o hese challenges ela ed o he
collec ion o ma e ials om he supplie s o ensu e a s eady a ailabili y o ma e ials
o p oduc ion. Fi s , in con as o la ge shipmen s o new goods, collec ing smalle
ba ches o was e ma e ials om local supplie s equi es de ailed ou ing conside a-
ions and he co esponding anspo a ion cos . Second, he a ailable quan i ies o
ma e ials a local supplie s depend on local consump ion o ecycling, making hem
unce ain. Thi d, he collec ed was e ma e ials equi e cleaning and so ing. As he
quali y o he was e ma e ials is unce ain, he inal quan i y ha can be used is also
unce ain. Fou h, in con as o ope a ions wi h less unce ain y, decisions a e in e -
connec ed and mus an icipa e po en ial u u e e en s o ensu e e ec i e u iliza ion
o esou ces, such as p ocessing and lee capaci y. These ou challenges combined
lead o a complex planning p oblem o companies elying on ci cula economy
sou cing.
The p oblem can be modeled as a s ochas ic dynamic mul i-pe iod in en o y
ou ing p oblem. I is a dynamic and mul i-pe iod p oblem because a sequence
o decisions mus be made pe iod by pe iod o e he planning ho izon. These
decisions a e connec ed as e e y decision impac s he s a es subsequen deci-
sions a e made in. I is s ochas ic because he e a e wo s ochas ic componen s:
i s , he a ailable quan i ies a each supplie in he nex pe iod(s), and second,
he ne quan i y o ma e ials ha can e en ually be used o p oduc ion. In en o y
o ne ma e ial a he p oduc ion acili y can be buil wi h a negligible holding
cos . Howe e , he quan i y o was e ma e ials ha can be p ocessed e e y pe iod
is limi ed. E e y pe iod, decisions a e made abou he supplie s o isi o col-
lec ion, he quan i y o collec om each supplie , and he ou ing o anspo a-
ion ucks o collec ions. Besides anspo a ion cos s, linea backo de sou cing
cos s can occu i he ne quan i y does no sa is y he ma e ial demand a he
701
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
p oduc ion acili y and new ma e ial has o be used ins ead. The challenge is now
o de e mine cos -e icien collec ion quan i ies and ou es, hedge agains po en-
ial backo de sou cing cos s due o insu icien ne quan i ies, and ideally, build
some in en o y o u u e pe iods.
We employ he sequen ial decision p ocess amewo k p oposed by (Powell
2021) o model he dynamic and s ochas ic componen s o ou p oblem h ough
s a es, decisions, ealiza ion o cos , obse a ion o exogenous in o ma ion, and an-
si ions o nex s a es. In ou solu ion app oach, we app oach he challenges as ol-
lows. Fi s , we gene a e a se o mul i-pe iod scena ios in e e y pe iod o cap u e he
wo sou ces o unce ain y. Each scena io con ains ealiza ions o was e quan i ies
a he supplie s in he ollowing pe iods and, o each supplie and he cu en and
u u e pe iods, he pe cen ages o collec ed quan i ies ha can be used o p oduc-
ion (i.e., he ne quan i ies). Based on he scena ios, we sol e an auxilia y model, a
wo-s age s ochas ic mixed-in ege p og am, o de e mine he supplie s o isi and
he quan i ies o collec . Ins ead o in eg a ing ou ing decisions explici ly, ou ing is
app oxima ed in he s ochas ic p og am. Once he supplie s isi and he quan i ies
o collec a e de e mined, he de ailed ou ing solu ion is de e mined ia a ou ing
heu is ic om he li e a u e.
We apply ou me hod o a eal-wo ld case in Sachsen-Anhal wi h eal geog aphi-
cal da a o p oduc ion and was e acili ies as well as eal supply da a o pape was e
ma e ial. We also es ou me hod o he was e collec ion da a p o ided by Keskin
e al. (2023). In a comp ehensi e compu a ional s udy, we de i e wo main manage-
ial insigh s. Fi s , an icipa o y decision-making educes cos s bu equi es e ec i e
managemen o dynamism and all sou ces o unce ain y h oughou he planning
ho izon. Second, supplie selec ion should balance quali y loss, supply olumes, and
ou ing e iciency, conside ing he ac o s independen ly can lead o cos o e uns.
Ou pape makes he ollowing con ibu ions:
• We in oduce a new and impo an p oblem. To he bes o ou knowledge, we
a e he i s o add ess a dynamic ou ing p oblem wi h s ochas ic yield o quali y
loss. We p esen he modeling o he p oblem as a sequen ial decision p ocess
and p o ide a comp ehensi e li e a u e su ey o he p oblem’s componen s.
• We p opose a ailo ed solu ion me hod named S ochas ic Lookahead o e Mul-
iple pe iods (STM) ha le e ages a wo-s age s ochas ic p og am and a ou -
ing heu is ic o ackle he p oblem’s main challenges: a complex decision space,
dynamic mul i-pe iod decision-making, and unce ain supply olumes and
quali y. Ou me hod ollows he gene al solu ion amewo k p oposed by Cuel-
la -Usaquén e al. (2024) bu is ailo ed o ou new p oblem, pa icula ly wi h
espec o s ochas ic quali y loss and p ocessing capaci y.
• We p esen a new se o ins ances based on eal olumes o pape collec ion in
Sachsen-Anhal , Ge many. Fu he mo e, we e alua e se e al heu is ics, discuss-
ing hei ad an ages and disad an ages using bo h eal da a and exis ing da a
om he li e a u e. The c ea ed da ase s a e a ailable online (h ps:// www. ms.
o gu. de/ Resea ch. h ml).
• We p esen a comp ehensi e analysis o he me hod and p oblem cha ac e is ics
and u he de i e aluable insigh s in o bo h he p oblem and he model.
702
D.Cuella -Usaquén e al.
The pape is ou lined as ollows. In Sec .2, he ele an li e a u e is p esen ed. The
p oblem is de ined in Sec .3. We p esen ou me hod in Sec .4, and he compu a-
ional s udy in Sec .5. The pape concludes wi h a summa y and ou look in Sec .6.
2 Li e a u e
Ou p oblem comp ises se e al componen s: in en o y ou ing, mul i-pe iod ou -
ing, aw ma e ial collec ion, and s ochas ic dynamic decision making. To he bes o
ou knowledge, he e is no wo k conside ing all ou componen s ye . Thus, in he
ollowing, we will discuss he mos ela ed wo k conside ing wo o h ee o hem.
The e is also ex ensi e wo k on he indi idual componen s, which we discuss in he
second pa o he li e a u e e iew.
2.1 Mos ela ed wo k
While ela ed p oblems ha e been explo ed in he li e a u e, hey o en lack he
dynamic aspec o e a mul i-pe iod decision ho izon. Habibi e al. (2017, 2019)
add ess decision-making challenges in hi d-pa y e e se logis ics, ocusing on in e-
g a ing End-o -Li e p oduc collec ion and disassembly p ocesses. The de e minis-
ic a ian is sol ed in Habibi e al. (2017), and he s ochas ic e sion, conside ing
unce ain ies in supply and quali y o aw ma e ials, is add essed in Habibi e al.
(2019). The au ho s p opose solu ions in he o m o Two-Phase I e a i e Heu is-
ics and a me hod based on sample a e age app oxima ion, espec i ely. In con as
o p io wo k, ou app oach in ol es add essing in en o y managemen and collec-
ion ou ing decisions on a pe iod-by-pe iod basis, an icipa ing unce ain y alues
and decisions o u u e pe iods while conside ing he ealiza ion o pe iod-speci ic
unce ain ies. Fo compa a i e pu poses, we adap hei in a-pe iod app oach in ou
benchma k policy STS. We con i m ha explici conside a ion o quali y unce ain-
ies is essen ial o single-pe iod decision-making; howe e , o ou mul i-pe iod se -
ing, a combined in a- and in e -pe iod an icipa ion is subs an ially mo e e ec i e.
Ma ko e al. (2020) add ess a mul i-pe iod solid was e collec ion p oblem using
a he e ogeneous lee o collec was e om con aine s wi h unce ain supply. They
aim o minimize cos s associa ed wi h con aine o e lows by de eloping collec-
ion ou es on a pe iod-by-pe iod basis. They p opose a me aheu is ic combined
wi h o ecas ing wi hin a olling ho izon, signi ican ly ou pe o ming de e minis ic
policies. Thei app oach inco po a es dynamic, p obabili y-based cos s o con aine
o e lows and ou e ailu es. While bo h pape s in ol e dynamic, pe iod-by-pe iod
decision-making, ou pape addi ionally ocuses on deciding he collec ion ou es
and he in en o y o ma e ial p ocessed a he depo o a oid cos o e uns due o
unsa is ied demand. Besides he unce ain olumes, ou decisions mus also an ici-
pa e he unce ain y in yield o he ecycled ma e ial.
Keskin e al. (2023) add ess a mul i-pe iod dynamic ehicle ou ing p oblem o
a was e collec ion company in he Uni ed Kingdom. They in oduce he concep o
“ ou ing”, a demand managemen echnique in which cus ome s a e encou aged o
703
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
place o de s ea lie han usual, allowing he company o combine s ops a exis ing
cus ome s wi h s ops a nea by “ ou ed” cus ome s. The supply o was e ma e ial
om cus ome s is known once he cus ome is con ac ed. A olling ho izon and
simula ion model a e used o sol e he p oblem; he au ho s p opose ules o iden-
i y p omising acili ies o call based on he expec ed supply olumes and how hey
could be in eg a ed in o he ou es. Empi ical esul s indica e subs an ial imp o e-
men s in ou ing e iciency compa ed o he no- ou ing s a egy. Unlike ou wo k, he
au ho s do no plan in en o y o e he pe iods. Following he app oach p esen ed in
Keskin e al. (2023), we es se e al supplie selec ion ules as benchma k policies.
Ou esul s indica e ha explici an icipa ion o cu en and u u e ou ing decisions
and unce ain ies is supe io o he p ac ically-inspi ed decision ules.
The conside a ion o aw ma e ial/p oduc quali y om supplie s is explo ed in
o he con ex s. The emphasis on quali y loss in supply chains o en e ol es a ound
he eshness and pe ishabili y o p oduc s (Rohme e al. 2019). Add essing a ehi-
cle ou ing p oblem, S ellingwe e al. (2021) employ a ime- and empe a u e-
dependen kine ic model o simula e he deg ada ion o p oduc s o e ime. The
quali y o p oduc s changes a e each cus ome isi , a ibu ed o luc ua ions in
he ehicle’s empe a u e du ing anspo a ion. A simila pe spec i e is p o ided
by Al a ez e al. (2022), which ackles a p oduc ion ou ing p oblem speci ically
conce ning pe ishable goods. In his con ex , he quali y o p oduc s is cha ac e ized
by a decay a e o e ime. On he o he hand, in a manu ac u ing and disassembly
con ex , Laouini e al. (2023) add ess he ma e ial yield o collec ed p oduc s, seek-
ing o sa is y demand o inished p oduc s om he collec ion o ecycled p oduc
esou ces. In con as o ou wo k, he men ioned s udies emphasize ha aw ma e-
ial quali y/yield is a ec ed o e ime and does no ha e a di ec impac on collec-
ion ou es be o e eaching he depo , as is he case wi h pe ishable p oduc s. Addi-
ionally, while unce ain y in aw ma e ial quali y loss is add essed in manu ac u ing
con ex s, ou ing decisions a e no in ol ed, and he unce ain y is no ela ed o he
quan i y o be eplenished in he p ocessing cen e .
Al a ez e al. (2021) add ess he s ochas ic in en o y ou ing p oblem (SIRP)
unde unce ain y in bo h p oduc supply and cus ome demands. The au ho s p o-
pose a heu is ic solu ion me hod based on he p og essi e hedging algo i hm, deli -
e ing high-quali y solu ions wi hin easonable unning imes o p oblems wi h a
la ge numbe o scena ios. In con as o ou wo k, he au ho s con empla e unce -
ain y in he supply o a single supplie and gene a e an icipa ion o a single pe iod
going o wa d. Ou p oposed me hodology explici ly conside s u u e unce ain ies
and ou ing o se e al supplie s.
Me hodologically, some wo ks sha e simila i ies wi h ou wo k. Elbek e al.
(2015) sol e a was e collec ion p oblem wi h unce ain y in he amoun a ailable
in he con aine s. The p oblem is modeled as a wo-s age s ochas ic p og am and
sol ed using a lookahead algo i hm. As in ou wo k, he au ho s dynamically sol e
he collec ion policy o each day o he planning ho izon bu do no conside he
p oduc quali y, in en o y, o p ocessing capaci y a he depo .
B inkmann e al. (2019, 2020) ocus on dynamic bicycle anspo a ion o main-
ain op imal in en o ies a bike sha e s a ions. The au ho s p opose a lookahead
app oach o e alua e in en o y decisions, which in ol es sampling u u e demand
704
D.Cuella -Usaquén e al.
and selec ing in en o y o minimize unme demand. The ime-in luenced looka-
head ho izon is ob ained using he alue unc ion app oxima ion (VFA). No ably,
his ho izon does no ex end o u u e ou ing decisions. In pa allel, ou esea ch
sha es simila i ies ha span bo h p oblem o mula ion and me hodological aspec s,
as we also use u u e samples o in o m in en o y decisions. Howe e , ou app oach
explici ly in eg a es ou ing and associa ed cos conside a ions in o he o ecas ing
model.
Finally, Cuella -Usaquén e al. (2024) sol e a dynamic s ochas ic mul i-pe iod
p oblem encompassing pu chasing, in en o y, and ou ing decisions o a i s -mile
ope a ion. The lookahead algo i hm p oposed samples supplie s’ pu chase p ices
as well as supply and demand a he depo . In addi ion, an adap i e algo i hm is
in oduced o cap u e he consolida ion beha io o supplie s in ou ing decisions.
Cuella -Usaquén e al. (2024) in oduce he gene al amewo k o combining wo-
s age s ochas ic p og amming wi h ou ing cos app oxima ion o dynamic decision
p oblems. Fo ou wo k, we use his amewo k o a di e en p oblem. In con as
o Cuella -Usaquén e al. (2024), we model he beha io o ma e ial quali y loss
upon a i al a he depo , conside ing p ocessing capaci y and cumula i e in en o y
a supplie s. This changes he design o he scena ios, and he wo-s age s ochas ic
p og am.
2.2 Wo k onindi idual p oblem componen s
In addi ion o he mos ele an esea ch discussed, he e is ela ed wo k in ecycling
collec ion ou ing, in en o y ou ing, and he unce ain supply and quali y loss o
aw ma e ials in manu ac u ing and supply chains. These s udies a e discussed in
de ail in he ollowing sec ions and summa ized in he subsequen able.
Table1 p esen s a summa y o he ele an li e a u e o he ields o In en o y
Rou ing P oblems (IRPs), Re e se Ma e ials Requi emen s Planning (RMRP),
Supply Chain Managemen (SCM), Vehicle Rou ing P oblems (VRPs) and Mul i-
Pe iod Rou ing (MP-R). They a e ca ego ized as ollows: P oblem ea u es indi-
ca e whe he he p oblem conside s dynamic decisions, s ochas ic pa ame e s, o
mul i-pe iod aspec s, e lec ing he impac o decisions o e a ime ho izon longe
han one pe iod; Decisions speci y whe he he p oblem in ol es ou ing, in en o y
managemen decisions, o bo h; Unce ain y highligh s whe he he wo k conside s
unce ain y in a leas one componen o he p oblem; and An icipa ion iden i ies
whe he he p oposed me hod an icipa es he impac o a decision on u u e cos s.
Fo de e minis ic p oblems, he An icipa ion column is ma ked as ’n/a,’ as an icipa-
ion is only applicable in s ochas ic p oblems.
2.2.1 Vehicle ou ing in hecon ex o ecycling collec ion
In he 1980s, he di icul ies o ou ing municipal was e disposal we e ecognized
by Ra (1983) as i s own ca ego y o Vehicle Rou ing P oblems (VRP). I s global
impac and ele ance o di e en socie al dimensions, including esou ce manage-
men , ene gy u iliza ion, ecological damages, and mone a y cos s, a e acknowledged
705
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
Table 1 Li e a u e classi ica ion, highligh ing simila i ies and di e ences o ou p oposed p oblem and he li e a u e
Pape P oblem ea u es Decisions Unce ain y An icipa ion
Dynamic S ochas ic Mul i-pe iod Rou ing In en o y Supply Quali y
IRP
Habibi e al. (2017)
✓
✓
✓
n/a
Rohme e al. (2019)
✓
✓
✓
n/a
Al a ez e al. (2022)
✓
✓
✓
n/a
Moin e al. (2011)
✓
✓
✓
n/a
Mji da e al. (2014)
✓
✓
✓
n/a
Chi saz e al. (2019)
✓
✓
✓
n/a
Be azzi e al. (2020)
✓
✓
✓
n/a
Chi saz e al. (2020)
✓
✓
✓
n/a
Chi and He (2023)
✓
✓
✓
n/a
Jieyu e al. (2024)
✓
✓
✓
n/a
Habibi e al. (2019)
✓
✓
✓
✓
✓
✓
✓
Al a ez e al. (2021)
✓
✓
✓
✓
✓
✓
B inkmann e al. (2019)
✓
✓
✓
✓
✓
✓
B inkmann e al. (2020)
✓
✓
✓
✓
✓
✓
✓
Nolz e al. (2014)
✓
✓
✓
✓
✓
✓
Mes e al. (2014)
✓
✓
✓
✓
✓
✓
✓
Ma ko e al. (2020)
✓
✓
✓
✓
✓
✓
✓
Liu e al. (2021)
✓
✓
✓
✓
✓
✓
F i i a e al. (2022)
✓
✓
✓
✓
✓
Has u k e al. (2024)
✓
✓
✓
✓
✓
✓
RMRP/SCM
Laouini e al. (2023)
✓
✓
✓
✓
✓
706
D.Cuella -Usaquén e al.
Table 1 (con inued)
Pape P oblem ea u es Decisions Unce ain y An icipa ion
Dynamic S ochas ic Mul i-pe iod Rou ing In en o y Supply Quali y
Inde u h and Langella (2006)
✓
✓
✓
✓
Rickli and Camelio (2014)
✓
✓
✓
Key anshokooh e al. (2016)
✓
✓
✓
✓
✓
Üs e and Hwang (2017)
✓
✓
✓
✓
Üs e and Memişoğlu (2018)
✓
✓
✓
✓
Liu and Zhang (2018)
✓
✓
✓
✓
✓
Memişoğlu and Üs e (2021)
✓
✓
✓
✓
✓
✓
Zhou e al. (2022)
✓
✓
✓
✓
✓
Li e al. (2023)
✓
✓
✓
✓
✓
✓
VRP
S ellingwe e al. (2021)
✓
n/a
Kim e al. (2009)
✓
n/a
De B uecke e al. (2018)
✓
n/a
Ismail and Loh (2009)
✓
✓
✓
G ule e al. (2017)
✓
✓
✓
Cook and Lod ee (2017)
✓
✓
✓
✓
Jammeli e al. (2021)
✓
✓
✓
Ky iakidis e al. (2020)
✓
✓
✓
✓
Ma ko ić e al. (2020)
✓
✓
✓
Sasha Dong e al. (2022)
✓
✓
✓
✓
MP-R
Bogh e al. (2014)
✓
✓
n/a
A che i e al. (2015)
✓
✓
n/a
713
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
EUR. The op cen e o he igu e now shows he ealiza ion o he s ochas ic
in o ma ion: he ealized loss and he ealized inc ease in supply o he nex
pe iod. As supplie s 1 and 2 a e no isi ed, he loss is no obse ed, bu only
he inc ease is shown, 20 uni s o supplie 1 and 30 uni s o supplie 2. Fo
supplie 3, he ealized loss is
30%
, highe han expec ed. The inc ease in supply
is 50. The highe - han-expec ed loss leads o an in en o y inc ease o only 70.
Gi en he demand o 80, en uni s need o be pu chased h ough he cos ly back-
o de op ion o educe he ne in en o y o ze o. Thus, a backo de cos o 1000
EUR incu s leading o a o al cos o 1195 EUR a pe iod
=1
. The nex s a e is
hen shown on he igh . The in en o y alue is
−80
, and he supply alues a e
upda ed acco ding o he ealized inc ease.
Ano he po en ial decision is p esen ed a he bo om o Fig.1. This decision
in es s ou ing cos s o inc ease he depo ’s expec ed in en o y and a oid o de s
om he backup op ion. Two ehicles a e employed. The i s one collec s all
supplies om supplie 3, simila o he i s decision. The second ehicle i s
isi s supplie 1 and collec s all 80 a ailable supply uni s, indica ed by he alue
(80) nex o he ehicle when lea ing supplie 1. Then, he ehicle isi s supplie
2 and collec s an addi ional 20 uni s, e u ning ull-back o he depo . The ou e
o he second ehicle has a du a ion o 250 minu es and a cos o 375 EUR. As
200 uni s a e collec ed o e all, he p oduc ion capaci y limi is no exceeded.
The bo om cen e shows he ealiza ion o s ochas ic in o ma ion, simila o
be o e, bu now also o he loss o supplie s 1 (
40%
) and 2 (
50%
). This col-
lec ion leads o an in en o y inc ease o
48 +10 +70 =128
and
128 −80 =48
in en o y uni s emaining a he end o he pe iod wi hou incu ing he cos o
backo de s (0 EUR), sa ing 820 EUR compa ed o he p e ious decision. In he
new s a e on he igh , he ne in en o y is he e o e
−32
.
3.3 Sequen ial decision p ocess
The p oblem a hand is a s ochas ic and dynamic decision p oblem. I is s o-
chas ic because he amoun o was e ma e ial a ailable a he supplie s is known
a he beginning o each pe iod, and he amoun o supply ha can be used o
gene a e p oduc in en o y is e ealed when i a i es a he depo . I is dynamic
because a sequence o decisions mus be made, one pe pe iod, wi h each cu -
en decision a ec ing he in en o y olumes o u u e pe iods and consequen ly
in luencing subsequen decisions.
A dynamic s ochas ic decision p oblem can be modeled as a sequen ial deci-
sion p ocess (Powell 2021), modeling he p oblem as a sequence o s a es. In
each s a e, a decision is made, and he cos is obse ed. Nex , s ochas ic in o -
ma ion is e ealed ( esul ing in u he cos ), and a ansi ion leads o he nex
s a e. In he ollowing, we de ine he s a es, he decisions, he cos unc ions, he
s ochas ic in o ma ion, and he ansi ion unc ion o ou p oblem. Fo an o e -
iew o he no a ion used, e e o Table2.
714
D.Cuella -Usaquén e al.
Table 2 O e iew o no a ion
No a ions De ini ions
Se s
MSe o supplie s
TSe o pe iods
FSe o ehicles
V
Se o e ex,
V∶= M∪{0}
A
Se o a cs,
A= {(i,j)∶i,j∈V|i≠j}
U
A se o nodes whe e
U⊂V
𝛿+(U)
Se o a cs (i,j) wi h
i∈U
and
j∈V⧵U
𝛿−(U)
Se o a cs (i,j) wi h
j∈U
and
i∈V⧵U
T′
Se o pe iods in he lookahead ho izon
Ω
Se o scena ios
Pa ame e s
d,
𝛽
Demand o new p oduc and p ocessing capaci y a he depo
QVehicle capaci y
𝜏ij
T a el ime be ween i and j o
(i,j)∈A
wi h se ice ime included
cCos o e e y ime uni a eled
Backo de cos pe uni
lmax
Maximum wo king ime pe ehicle and pe iod
𝜇
m,𝜎
m
Mean and s anda d de ia ion o p obabili y dis ibu ion o supply inc ease alue
o supplie
m∈M
𝜇𝜙
m
,𝜎
𝜙
m
Mean and s anda d de ia ion o p obabili y dis ibu ion o quali y loss pe cen age
alue o supplie
m∈M
m
′
𝜔,𝜙m
′
𝜔
Realiza ion o inc ease in supply and quali y loss pe cen age o supplie
m∈M
a pe iod
�∈T�
in scena io
𝜔∈Ω
I
Ne in en o y a he depo in pe iod
∈T
qm
Amoun o supply a ailable a supplie
m∈M
a pe iod
∈T
hSize o he lookahead ho izon
𝜏 m
Di ec ip cos om he depo o e e y supplie
m∈M
𝛾
Discoun pa ame e o ou ing cos es ima ion
𝜃s ock
Addi ional pe cen age collec ed o e expec ed supply o sa e y s ock
Va iables
zm
Amoun o supply o collec om supplie
m∈M
by ehicle
∈F
a pe iod
∈T
xij
Equal o 1 i he a c
(i
,
j
)∈
A
is ac i a ed in he ou e o ehicle
∈F
a pe iod
a pe iod
∈T
, 0 o he wise
em
Equal o 1 i supplie
m∈M
is isi ed by ehicle
∈F
a pe iod
∈T
0 o he wise
z′
m
′
𝜔
Amoun o supply o be collec ed om supplie
m∈M
in pe iod
�∈T�
in he
scena io
𝜔∈Ω
e′
m
′
𝜔
Equal o 1 i supplie
m∈M
is isi ed a pe iod
�∈T�
in he scena io
𝜔∈Ω
I′
′
𝜔
In en o y le el o new p oduc a he end o pe iod
�∈T�
in he scena io
𝜔∈Ω
y
′
𝜔
Backo de amoun a iable a pe iod
�∈T�
in scena io
𝜔∈Ω
km
′
𝜔
In en o y le el o was e ma e ial om supplie
m∈M
a he end o pe iod
�∈T�
in he scena io
𝜔∈Ω
715
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
3.3.1 Global no a ion
We deno e he se o supplie s as
m∈M
. We de ine he collec ion ne wo k as a
comple e, di ec ed g aph
G=(V,A)
. Le
V∶= M∪{0}
be he se o e ices, whe e
0 ep esen s he depo . Le
A
be he se o a cs whe e
A= {(i,j)∶i,j∈V|i≠j}
.
Fo he sub- ou elimina ion cons ain s, gi en a se
U⊂V
, we de ine
𝛿+(U)
as he
se o a cs (i,j) wi h
i∈U
and
j∈V⧵U
, and
𝛿−(U)
as he se o a cs (i,j) wi h
j∈U
and
i∈V⧵U
.
The pe iods a e deno ed as
∈T
wi h
T={1, 2, …,|T|
}. In each pe iod, he
demand is he same, deno ed as d. Fo each supplie , he inc ease in supply pe
pe iod and he quali y loss ollow known p obabili y dis ibu ions wi h mean and
s anda d de ia ion (
𝜇
m,𝜎
m)
o supply inc ease alue and
(
𝜇
𝜙
m
,𝜎
𝜙
m)
o quali y loss
pe cen age.
We assume a su icien ly la ge a ailable se o ehicles F (e.g.,
|F|=|M|
). Vehi-
cles ha e a maximum load capaci y Q, and a maximum wo king ime pe ehicle
and pe iod
lmax
. The a el ime be ween i and j o
(i,j)∈V
is deno ed by
𝜏ij
, and
o e e y ime uni a eled, he e is a cos o c. The se ice ime o load he was e
ma e ial on he ehicles is he same o all supplie s. We include he se ice imes
in he a el imes
𝜏ij
, leading o asymme ic a el imes om/ o he depo . A he
end o he collec ion, when he ehicles a i e a he depo , he sum o he quan i ies
collec ed canno exceed he p ocessing capaci y de ined as a ac o o he demand,
𝛽×d
, wi h
𝛽
being he p ocessing capaci y scaling pa ame e . The backo de cos
pe uni is de ined as .
3.3.2 S a e
A decision is made in e e y pe iod
∈T
. The s a e comp ises all in o ma ion a ail-
able o make a decision. We deno e he s a e in pe iod
∈T
as
S
. Fo ou p oblem,
he s a e
S
consis s o wo componen s, one ela ed o he depo and one ela ed o
he supplie s: The i s is he ne in en o y a he depo in pe iod , deno ed by
I
.
The second is he amoun o supply a ailable a supplie
m∈M
a pe iod , deno ed
by
qm
.
We no e ha he ne in en o y al eady cap u es he demand o he day and, he e-
o e, can be nega i e. S a e
S
can be summa ized as
S =(I ,q )
, whe e
I
is a scala
and
q
is a |M|-dimensional ec o . A he beginning o he p ocess, no in en o y is
a ailable a he depo ,
I0=−d
. The ini ial amoun o supply o supplie
m∈M
is
deno ed
qm0
.
3.3.3 Decision
We deno e a decision a pe iod
∈T
as
a
. A decision
a =(z ,x )
has wo compo-
nen s ha e lec collec ion and ou ing pa s. The collec ion pa is modeled ia deci-
sion ma ix
z =(zm )m∈M, ∈F
. I de e mines he amoun o was e ma e ial o collec
om each supplie m by each ehicle . The second pa o he decision is he de ini-
ion o collec ion ou es, modeled ia
x =(xij )i,j∈M,i≠j, ∈F
. The a iable
xij ∈{0, 1}
716
D.Cuella -Usaquén e al.
indica es i he a c om supplie i o supplie j is ac i a ed in he ou e o ehicle a
pe iod . In he ollowing, we summa ize he decision space using a mixed-in ege o -
mula ion. Fo he o mula ion, we use he auxilia y a iable
em
, which akes he alue
o one i supplie
m∈M
is isi ed by ehicle
∈F
a pe iod and 0 o he wise.
A decision
a =(z ,x )
a pe iod
∈T
is easible i he cons ain s(1–10) a e sa -
is ied. Equa ion (1) ensu es ha he p ocessing capaci y
𝛽×d
in he depo is no
exceeded. Equa ion(2) ensu es ha he collec ion should no exceed he a ailable sup-
ply o was e ma e ial o he supplie s and ha a ehicle can only collec supply i a
supplie is isi ed. Equa ion(3) ensu es ha he ehicle capaci y Q is no exceeded i
one o mo e supplie s a e isi ed. Eq.(4) imposes non-spli isi cons ain s ha ensu e
ha each supplie is isi ed by a mos one ehicle. Eq.(5) imposes ha , o each is-
i ed supplie , exac ly one a c mus en e and lea e he ela i e node. Equa ion(6) and
Eq.(7) a e he sub- ou elimina ion cons ain s and maximum a el ime cons ain s
(Mane ba and Mansini 2016). Finally, Eqs.(8–10) de ine he domain o he a iables.
(1)
∑
∈F
∑
m∈M
zm ≤𝛽
d
(2)
zm ≤qm em ,∀m∈M,∀m∈F
(3)
∑
m∈M
z
m
≤Q,∀ ∈F
(4)
∑
∈F
e
m
≤1, ∀m∈M
(5)
∑
(i,j)∈𝛿
−
({b})
x
ij
=∑
(i,j)∈𝛿
+
({b})
x
ij
=e
b
,∀b∈M,∀ ∈F
(6)
∑
(
i
,
j
)∈𝛿
+
()
x
ij
≥e
b
,∀
⊆
M,∀b∈,∀ ∈F
(7)
∑
(i,j)∈
𝜏ij
x
ij
≤lmax,∀ ∈F
(8)
zm
≥
0, ∀m∈M,∀ ∈F
(9)
em ∈{0, 1},∀m∈M,∀ ∈F
(10)
xij ∈{0, 1},∀(i,j)∈,∀ ∈F
717
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
The ou ing cos associa ed wi h decision
a
in s a e
S
is de e mined by mul iplying
he du a ion o each ou e by he cos pe uni o ime. The ou ing cos
C (S ,a )
can
be o mally de ined as:
3.3.4 S ochas ic in o ma ion and ansi ion unc ion
A e a decision
a
is aken in s a e
S
, he s a e is ans e ed o pos -decision s a e
Sa
=(I ,qa
,z )
. The collec ion decision,
z
, induces changes in he amoun o supply
a ailable a he supplie s. The inal in en o y o supply a he supplie s a pe iod is
he di e ence be ween he amoun o supply on hand and he amoun collec ed:
The exogenous in o ma ion
𝜔 +1=(z𝜔
+1, 𝜔
+1)
e eals he in en o y ha can be gen-
e a ed om he supply collec ed and he addi ional supply a ailable a he supplie s.
The ealiza ion o gene a ed in en o y,
z𝜔
+1
, impac s bo h he sa is ac ion o demand
and he inal in en o y a he depo . We de ine
I𝜔
+1
as he di e ence be ween he
e ealed amoun o ma e ial ha can be used and he ne in en o y le el a pe iod :
In he case ha he in en o y a he depo is no enough o mee he demand in pe iod
, a backo de cos is incu ed o he pu chase o ma e ial o mee he demand. The
ealized backo de cos is de ined as:
A e applying he ansi ion unc ion
T(
S
a
,𝜔
+1)
, a new s a e
S +1=(I +1,q +1)
is
eached. Fi s ly, he ne in en o y
I +1
is upda ed, conside ing he ealiza ion o he
inal in en o y le el o he new p oduc a he depo and he demand o pe iod
+1
.
Secondly, he supply o was e ma e ial a ailable
qm +1
is upda ed based on he inal
in en o y a each supplie and he newly gene a ed amoun . We de ine his upda e as
ollows:
(11)
C
(S ,a )=c⋅
∑
∈F
∑
(i,j)∈A
𝜏ijxij
.
(12)
q
a
m =qm −
∑
∈F
zm ,∀m∈M
.
(13)
I𝜔
+1=I +z𝜔
+1.
(14)
Ce(S ,a ,𝜔
+
1)=
⋅
max(0, −I𝜔
+1).
(15)
I +1=max(0, I𝜔
+1)−d
(16)
qm +1=
q
a
m +
𝜔
m +1
,
∀
m
∈
M
.
718
D.Cuella -Usaquén e al.
3.3.5 Policy
A solu ion o a sequen ial decision p ocess is a policy
𝜋
. A policy assigns a deci-
sion
a =A𝜋(S )
o e e y s a e
S
. The o e all se o policies is de ined as
Π
. An op i-
mal solu ion
𝜋∗∈Π
minimizes he expec ed cos ha is composed o he ou ing
cos and he backo de cos :
s a ing om s a e
S0
. Te m
Ce(S ,A𝜋(S ))
indica es he expec ed backo de cos .
4 Me hod
E en he small example in Sec .3 al eady highligh s he challenges in decision-mak-
ing o his p oblem. Fi s , unce ain y mani es s no only in he supply pe pe iod
bu also in he loss, i.e., he pe cen age o collec ed ma e ial ha canno be used.
Second, a s a e’s decision comp ises a complex in en o y ou ing p oblem. Thus,
solu ion me hodology mus accoun o he wo sou ces o unce ain y wi hin he
pe iod and o he ollowing pe iods and mus be able o ackle he complex decision
space ho oughly.
To his end, we p opose a s ochas ic lookahead me hod based on he gene al
amewo k o Cuella -Usaquén e al. (2024). Ou me hod is deno ed STochas ic
lookahead o e Mul iple pe iods (STM). In he ollowing, we gi e an o e iew o
he unc ionali y o ou me hod. Fo de ails and a pseudo-code ep esen a ion, we
e e o AppendixA.1.
The idea o STM is o cap u e cu en and u u e s ochas ici y by sol ing a s o-
chas ic lookahead model based on a se o sampled mul i-pe iod scena ios. Each
scena io comp ises he ealiza ion o loss in he cu en pe iod and he ealiza ions
o supply and loss in u u e pe iods. Then, a decision is aken ha minimizes he
a e age cos while conside ing all scena ios gene a ed. Ma hema ically, he s ochas-
ic lookahead can be modeled as a wo-s age s ochas ic p og am. Sol ing he ull
s ochas ic p og am explici ly is compu a ionally in ac able since ou ing decisions
in he cu en and u u e pe iods mus be conside ed. Ins ead, we assume di ec ips
bu app oxima e ou ing consolida ion ia a discoun pa ame e . Cuella -Usaquén
e al. (2024) ha e shown ha his p ocedu e wo ks e y well o ou ing p oblems
wi h a limi ed numbe o s ops pe ehicle due o capaci y cons ain s, as gi en o
ou p oblem. A e he collec ion decisions a e de e mined, he de ailed ou ing is
done ia a heu is ic. Fig.2 p esen s he sequence o ob aining a decision using STM,
ollowing he amewo k in oduced in Cuella -Usaquén e al. (2024). In a s a e, he
s a e in o ma ion is obse ed and a se o mul i-pe iod scena ios a e gene a ed. Wi h
he scena ios and he ou ing discoun ac o
𝛾
, a educed wo-s age s ochas ic p o-
g am is c ea ed and sol ed (S ep I). The solu ion p esc ibes he selec ion o sup-
plie s o be isi ed and he quan i ies o be collec ed. Based on his in o ma ion,
(17)
𝜋
∗=a gmin
𝜋∈Π
𝔼
[∑
∈T
(C(S ,A𝜋(S )) + Ce(S ,A𝜋(S ))
|
S0)
],
719
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
he explici collec ion ou es a e de e mined (S ep II). S eps I and II o Fig.2 a e
illus a ed in mo e de ail in Fig.3 o he s a e p e iously in oduced in he example
(Sec .3.2). He e, a ime , we assume a o wa d pe iod ho izon o h ee pe iods and
wo scena ios, one a he op and one a he bo om (
𝜔1,𝜔2)
. In he i s s ep, shown
on he le , STM sol es a wo-s age s ochas ic p og am wi h one join decision o
ime ( i s -s age decisions) and indi idual decisions in he scena ios o imes
+1
,
+2
,
+3
(second-s age decisions). All decisions assume di ec ips and a e e alu-
a ed ia discoun ed di ec ip cos s. In his example, o pe iod , wo supplie s (and
hei espec i e collec ion olumes) a e selec ed. Then, in he second s ep, shown on
he igh , he decision a ime is ans e ed o a ou ing decision. In he ollowing,
we i s p esen he s ochas ic lookahead model and hen he ou ing heu is ic.
4.1 The s ochas ic lookahead model
In each s a e
S
du ing pe iod , he s ochas ic lookahead model sol es an auxilia y
model (Fleckens ein e al. 2023), a wo-s age s ochas ic p og am o de e mine he
amoun o supply o collec , in en o y le els, and he supplie s o isi om he depo .
This lookahead model sol es he p oblem o e a sho e planning ho izon by combin-
ing an app oxima ion o u u e in o ma ion wi h an app oxima ion o u u e decisions.
Fig. 2 Flowcha o decision making in a gi en s a e using STM (adap ed om Cuella -Usaquén e al.
2024)
Fig. 3 Illus a ion o he STM-policy wi h simpli ied wo-s age s ochas ic p og am on he le and esul -
ing ou ing decision on he igh
720
D.Cuella -Usaquén e al.
The e o e, in he lookahead model, a o wa d pe iod ho izon
T′
is composed o he
subsequen pe iods:
, +1, …, +h
. No ably, decisions ela ed o
′>
a e employed
o e alua e he quali y o decisions o be made in pe iod . Fo each pe iod
�∈T�
, a
se o scena ios, i.e., sample pa hs
𝜔∈Ω
, is cons uc ed. Each sample pa h de e mines
he ealized amoun o supply (
m
′
𝜔
) and he quali y loss pe cen age (
𝜙m
′
𝜔
) o e e y
pe iod
�∈T�
and e e y supplie
m∈M
.
The wo-s age s ochas ic p og am is p esen ed in Eqs.(18–29). De i ed om he
model de ined in Sec .3.3, his s ochas ic p og am models mul iple scena ios and ime
pe iods. The s ochas ic p og am simpli ies he ou ing decision by assuming discoun ed
di ec ips. To his end, i in oduces wo addi ional pa ame e s,
𝜏 m
and
𝛾
. The pa am-
e e
𝜏 m
is he di ec ip cos om he depo o e e y supplie m i we collec some hing
om he supplie , wi h
𝜏 m∶= 𝜏0m+𝜏m0,∀m∈M
. The pa ame e
𝛾
se es as a dis-
coun pa ame e , es ima ing he ou ing cos , as di ec ips may o e es ima e he ac ual
ou ing cos . I akes alues in he ange [0,1]. A weigh close o ze o indica es consoli-
da ion po en ial and a weigh close o one indica es ha he supplie is usually isi ed
by di ec a el. The pa ame e is uned ia enume a ion.
The objec i e unc ion (18) minimizes he expec ed o al cos o collec ion ou ing
and backo de cos . Equa ion(19) gua an ees he in en o y low and ensu es demand
sa is ac ion a he depo , conside ing po en ial u u e alues o he quali y loss pe cen -
age. Equa ion(20) ensu es supplie s’ low o in en o y and new supply. Equa ion(21)
and Eq.(22) ensu e no exceeding he capaci y o he ehicles, along wi h he con-
s ain s o no spli isi s and no exceeding he p ocessing capaci y o he depo . Non-
an icipa i i y cons ain s, ep esen ed by Eq.(23) and Eq.(24), ensu e ha he i s -
pe iod decisions on he ho izon
T′
, co esponding o s a e
S
, emain consis en ac oss
sample pa hs. Finally, Eqs.(25–29) de ine he a iable domains.
Compa ed wi h he model p esen ed in Sec .3.3, he wo-s age s ochas ic p og am
does no ake in o accoun he ehicle-speci ic ou ing decision,
xij
being eplaced
by supplie selec ion decisions
e′
m
′
𝜔
. A backo de amoun a iable
y
′
𝜔
is added. The
cons ain s (1–4) a e adjus ed o cons ain s (20–22); and he cons ain s (5–7) a e
emo ed. The s ochas ic p og am can be sol ed using a s anda d sol e se ing a maxi-
mum gap o
10%
o he ins ances conside ed in his pape . Le
e
and
z
ep esen he
compu a ional esul s o he decision a iables
e′
and
z′
. The alues o
e
and
z
o he
pe iod se e as inpu in o ma ion o he ou ing heu is ic.
s. .
(18)
min ∑
𝜔∈Ω
1
|
Ω
|(∑
�∈T�
c
∑
m∈M
𝛾 𝜏 me�
m �𝜔+ ⋅y �𝜔
)
(19)
I
�
�𝜔=I�
�−1𝜔+
∑
m∈M
z�
m �𝜔(1−𝜙m �𝜔)−d+y �𝜔,∀ �∈T�,∀𝜔
∈Ω
(20)
km
�
𝜔
=k
m
�
−1𝜔
+
m
�
𝜔
−z
�
m
�
𝜔
,∀m∈M,∀
�
∈T
�
,∀𝜔
∈Ω
721
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
4.2 Rou ing heu is ic
A e sol ing he wo-s age s ochas ic p og am, which e u ns he alues o he sup-
ply collec ed (
z
) and supplie selec ion (
e
) a a s a e
S
, he ou ing heu is ic p o-
ceeds o de e mine he ou ing decision,
xij
, by sol ing a Dis ance Cons ained
Capaci a ed Vehicle Rou ing P oblem (DCVRP). The concep ual p ocess is ou lined
below, wi h u he de ails a ailable in Cuella -Usaquén e al. (2024). Fi s , a gian
ou is c ea ed using he nea es neighbo algo i hm based on he selec ed suppli-
e s, ollowing he app oach ou lined in Cuella -Usaquén e al. (2021). Subsequen ly,
an augmen ed g aph is cons uc ed using he gian ou and he supply collec ed
(
z
). The spli p ocedu e om P ins (2004) ex ac s he pool o ou es om he aug-
men ed g aph, espec ing ehicle capaci ies and maximum a el ime cons ain s.
The cons uc ion o he augmen ed g aph ollows a Di ec ed Acyclic G aph (DAG)
s uc u e. Subsequen ly, a single-sou ce sho es pa h p oblem is sol ed o ind he
se o ou es ha minimizes a el ime (Co men e al. 2022). This in ol es using a
opological o de ing o e ices, esul ing in a complexi y o O(|A|). The esul ing
ou es a e hen implemen ed in he decision a iable
xij
, whe e each ou e co e-
sponds o a ehicle assignmen .
(21)
z�
m
�
𝜔
≤Qe
�
m
�
𝜔
,∀m∈M,∀
�
∈T
�
,∀𝜔
∈Ω
(22)
∑
m∈M
z�
m �𝜔≤𝛽d,∀ �∈T�
,∀𝜔
∈Ω
(23)
z
′
m 𝜔
=
∑
𝜔
′
∈Ω
z′
m 𝜔′
|
Ω
|
,∀m∈M,∀𝜔
∈Ω
(24)
e
′
m 𝜔
=
∑
𝜔
′
∈Ω
e′
m 𝜔′
|
Ω
|
,∀m∈M,∀𝜔
∈Ω
(25)
e′
m ′𝜔∈{0, 1},∀m∈M,∀ ′∈T′,∀𝜔∈Ω
(26)
z′
m
′
𝜔≥0, ∀m∈M,∀ ′∈T′,∀𝜔∈Ω
(27)
I′
′𝜔≥0, ∀ ′∈T′,∀𝜔∈Ω
(28)
km ′𝜔≥0, ∀m∈M,∀ ′∈T′,∀𝜔∈Ω
(29)
y ′𝜔≥0, ∀ ′∈T′,∀𝜔∈Ω
722
D.Cuella -Usaquén e al.
5 Compu a ional s udy
In his sec ion, we p esen ou compu a ional s udy. We i s desc ibe he es
ins ances and he benchma k policies. Nex , we p o ide he implemen a ion de ails
and pa ame e uning. We hen analyze he alue o single-pe iod and mul i-pe iod
an icipa ion o ou me hod. Finally, we in es iga e he decision-making o ou
me hod in de ail.
5.1 Ins ances
We p esen wo main ins ance se ings o geog aphy, supply and demand. One se -
ing is based on he ecycling collec ion da a om he Ge man s a e o Sachsen-
Anhal , and he second one is adap ed om he Uni ed Kingdom (UK) was e col-
lec ion da a p esen ed in Keskin e al. (2023). The ins ances di e in he numbe o
supplie s and hei geog aphical dis ibu ion.
• Sachsen-Anhal : The da a comp ises loca ions and supply da a o nine pape col-
lec ion acili ies, as well as a pape p ocessing acili y sp ead o e he en i e y
o Sachsen-Anhal . The a el dis ances be ween he loca ions a e calcula ed ia
Google Maps on ee oads and mul iplied by a ac o o 1.3 o mimic po en ial
a ic. We ha e access o he mon hly supply da a in ons o each supplie om
Janua y 2021 o Decembe 2022. Based on he da a, we calcula e he expec ed
supply pe supplie and pe iod as he a e age weekly supply. We assume he sup-
ply ollows a no mal dis ibu ion and i s he mean and s anda d de ia ion o he
da a acco dingly. The expec ed alues ange in he in e al
𝜇∈[0.3, 7.2]
ons
and he s anda d de ia ion in he in e al
𝜎∈[0.07, 0.9]
.
• UK-ins ances om Keskin e al. (2023): The da a comp ises a el imes and sup-
ply da a in li e s om a was e collec ion company ope a ing in a smalle egion
o he Uni ed Kingdom. The a el ime is ob ained using he coo dina es o he
cus ome s (supplie s). The supply dis ibu ion is calcula ed om a eal-wo ld
da ase ha co e s h ee mon hs o was e collec ions o wo d i e s ope a ing
om one depo . Fo hese ins ances, we ocused on he cus ome s wi h a highe
was e olume. To his end, we so ed he supplie s by expec ed alue in descend-
ing o de and selec ed he i s 30 cus ome s. The supply ollows a no mal dis i-
bu ion wi h
𝜇
anging in in e al [40.23,278.64] li e s o he di e en supplie s
and he s anda d de ia ion
𝜎∈[50.04, 512.96]
. E en hough nega i e alues a e
highly unlikely, we unca e he supply dis ibu ions a 0 li e s (and
2
⋅
𝜇
li e s o
ensu e symme y).
Fo bo h da a se s and ou main expe imen s, we assume a ime ho izon o 20 pe iods
(e.g., e lec ing weekly collec ions). We se he demand pe pe iod o
50%
pe cen o
he expec ed supply pe pe iod. We u he se he p ocessing capaci y as wo imes
he demand, and he e is no ini ial in en o y o new p oduc a he depo . In p ac ice,
he a e age quali y loss o pape is a ound
20%
(AF&PA 2024). Consequen ly, he
729
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
he supplie s wi h highe supply olumes is pos poned o he second pe iod. He e,
ou me hod an icipa es he supply inc ease om pe iod one o pe iod wo. Ins ead
o sending a ehicle o he supplie s in he i s pe iod and e u ning i o he depo
hal -emp y, i sa is ies he demand o nea by supplie s in he i s pe iod. In he sec-
ond pe iod, supplie s 3 and 47 a e isi ed indi idually, each consuming an en i e
ehicle capaci y. As be o e, we can also obse e a ime-consis en pa e n o many
supplie s, e.g., o supplie s 13, 31, 46, and 45.
Fig. 7 Supplie selec ion pe cen age o e he planning ho izon
730
D.Cuella -Usaquén e al.
5.5.2 P oblem dimensions
Nex , we analyze he impac o changes in loss ola ili y, demand unce ain y and
p ocessing capaci y. Fo he ollowing analysis, we inc ease he numbe o ins ance
ealiza ions o 100 o smoo he alues.
Loss ola ili y
One impo an ea u e conside ed in he p oblem is unce ain y in quali y loss.
To in es iga e he impac o loss unce ain y u he , we a y he ola ili y. Besides
ou o iginal a ia ion wi h a coe icien o a ia ion (COV) o 0.1, COVs o 0, 0.2,
and 0.5 a e es ed o he Sachsen-Anhal case and policies Myopic, STS, and STM.
We u he apply a a ian o STM whe e loss is in eg a ed in o he scena ios ia
expec ed alues only, policy STM(q). The esul s a e shown in Fig.8. The x-axis
depic s he COV, and he y-axis ep esen s he objec i e alues o he di e en poli-
cies. We obse e ha he Myopic policy pe o ms inc easingly wo se wi h inc eas-
ing loss ola ili y. This esul can be expec ed since his policy only ope a es on
expec ed alues. In con as , policy STS explici ly conside s loss ola ili y in he sce-
na ios. I s pe o mance does no only s ay cons an bu e en inc eases wi h inc eas-
ing ola ili y. The eason is again ha wi h di e en loss alues in he scena ios,
STS builds in en o y. We u he obse e ha policy STM is no a ec ed much by
inc easing COVs. Thus, e en when ola ili y is e y unce ain, ou p oposed policy
p o es o be qui e e ec i e. Finally, we obse e ha he di e ence be ween STM and
STM(q) is a he small, abou
3%
o he case o 0.5 COV.
Demand unce ain y
In ou main expe imen s, we assume ha he demand pe pe iod is s a ic and
de e minis ic. We now analyze he impac o elaxing his assump ion. In he new
Fig. 8 Changing he coe icien o a ia ions o he loss dis ibu ion
731
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
expe imen s, we assume ha he demand in a pe iod only e eals a e all collec-
ions we e made (In p ac ice, he demand in a pe iod e lec s he amoun o p oduc
needed o p oduc ion in he beginning o he nex pe iod. The e o e, i migh only
become known a he end o he cu en pe iod). In ou new expe imen s, demand
pe pe iod ollows a no mal dis ibu ion. The expec ed alue is he same as he ixed
demand in ou main expe imen s,
50%
o he expec ed supply. We es coe icien o
a ia ions (COVs) o 0, 0.1, 0.2, 0.3, 0.4, and 0.5, whe e a COV o 0 ep esen s he
o iginal a ia ion. We apply he STM, STS, and Myopic policies o his new se o
ins ances. In he gene a ion o scena ios, we now also sample demand.
The esul s o he Sachsen-Anhal case a e p esen ed in Fig.9. The x-axis ep-
esen s he COV alues, and he y-axis shows he objec i e unc ion o he policies.
Unce ain y in demand signi ican ly a ec s he pe o mance o he Myopic policy,
whose objec i e is o mee demand a i s expec ed alue wi hou accumula ing
in en o ies, leading o high backo de cos s when demand exceeds he a e age. In
con as , he STS and STM policies demons a e a he s able pe o mances as he
COV inc eases, indica ing ha unce ain demand can be cap u ed well ia he sce-
na ios. As expec ed, STM achie es supe io esul s, because i cap u es u u e pe i-
ods as well.
P ocessing capaci y
In ou main expe imen s, we assume a p ocessing capaci y o
𝛽=2
imes he
daily demand. Since he (expec ed) expec ed loss o e all supplie s is 0.2 in ou
ins ances, his means ha in en o y o abou 1.6 imes he daily demand can be
p oduced pe pe iod. We now analyze he impac o p ocessing capaci y by es -
ing
𝛽=1, 1.5, 2, 2.5, 3
imes he daily demand. We apply policies STM, STS, and
Myopic o he se o new ins ances.
The esul s o Sachsen-Anhal a e shown in Fig.10. The x-axis shows he
maximum p ocessing capaci y. The y-axis shows he ela i e di e ence o STM
in ou main se ing wi h a capaci y o
𝛽=2
. We obse e ha wi h mo e capaci y,
imp o emen s a e ma ginal. Thus, o ou main se ing, p oduc ion capaci y is no
a bo leneck. Once capaci y dec eases, we see an inc ease in cos . The easons
o he cos a e h ee old. Fi s , he likelihood o cos ly backo de s inc eases wi h
loss unce ain y and educed capaci y. Second, he policies may decide o a el
longe ou es o collec a supply o highe quali y, i.e., he smalle expec ed loss.
Fig. 9 Changing he coe icien
o a ia ions o he demand
dis ibu ion
732
D.Cuella -Usaquén e al.
Thi d, wi h limi ed capaci y, in en o y canno be buil o hedge agains supply
unce ain y in u u e pe iods.
We now in es iga e he impac o he capaci y cons ain on decision-making
in mo e de ail. To his end, we plo he a e age p ocessing capaci y u iliza ion
o e he ime ho izon. The esul s a e depic ed in Fig.11. The x-axis shows he
pe iod, and he y-axis shows he capaci y u iliza ion in ons (We ecall ha he
ne in en o y demand is 10 ons pe day). We make wo main obse a ions. Fi s ,
wi h a dec easing capaci y limi , he usage becomes mo e le el. This beha io can
be expec ed since, wi h limi ed p ocessing capaci y, in en o y canno be buil ,
and e e y day, (nea ly) all capaci y is used o a oid cos ly backo de s. Howe e ,
we also obse e some pa e ns in cases wi h high capaci y limi s. Gi en ha he
alues a e calcula ed o e 100 uns, he di e ences canno be explained by noise
only. Fo example, we obse e, ha o highe capaci ies
𝛽=2
and
𝛽=2.5
, in
he i s , ou h and se en h pe iod, signi ican ly mo e p ocessing capaci y is used
while in o he pe iods (e.g., wo, i e, and eigh ), less supply is p ocessed. In he
i s pe iod, no ini ial in en o y is gi en, and he policies decide o collec signi i-
can ly mo e han is needed o build an in en o y o u u e pe iods. Consequen ly,
Fig. 10 Pe o mance o chang-
ing p ocessing capaci y pa am-
e e
𝛽
(baseline
𝛽
=
2
)
Fig. 11 P ocessing capaci y u iliza ion o e ime o di e en capaci y limi s
𝛽
733
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
he policy is lexible in he second pe iod and may decide o sa e ou ing cos s by
collec ing less. The same epea s o e ime, e.g., wi h pe iods 4 and 5 o 16 and
17. These esul s indica e ha he e migh be alue in ha ing mo e lexibili y con-
ce ning he p ocessing capaci y, e.g., by scheduling wo king shi s dynamically.
6 Conclusion andou look
In his pape , we ha e p esen ed a dynamic mul i-pe iod ecycling collec ion p ob-
lem wi h unce ain y in he a ailable u u e supply and he supply’s quali y. We
ha e add essed his challenge using a ailo ed s ochas ic lookahead me hod, which
employs a wo-s age s ochas ic p og am o make decisions on supplie selec ion and
quan i ies o be collec ed, while ou ing is sol ed heu is ically in a second s ep. We
ha e used an app oxima e ou ing cos wi h a pa ame e ized discoun ac o o make
he s ochas ic p og am ac able and inco po a e ou ing decisions, u u e scena ios,
and pe iods. Compu a ional esul s, using ins ances om a eal collec ion ope a ion
in Ge many and ins ances om he li e a u e, illus a e ha ou me hod e ec i ely
an icipa es sou ces o unce ain y and op imizes he use o p ocessing capaci y a
he depo and a ailable supply om supplie s.
The e is a a ie y o u u e esea ch oppo uni ies. In ou wo k, we assumed ha
each supplie ’s quali y loss p obabili y dis ibu ion is known and s a ic. Fu u e wo k
may elax his assump ion. Fo example, quali y loss dis ibu ions may be unknown
ini ially and could be app oxima ed based on obse ed quali ies ia online lea ning.
Fu u e wo k may ace a challenging ade-o be ween e ec i e ou ing (exploi a-
ion) and e ec i e lea ning (explo a ion). In addi ion, we ha e assumed ixed se -
ice imes o collec ma e ial om he supplie s. Add essing s ochas ic and quan-
i y-dependen se ice imes would be an in e es ing a enue o u he s udy. This
would in oduce new challenges in collec ion decisions wi h espec o cos and
olumes. Ano he in e es ing a enue is a close conside a ion o he p ocessing o
he supply. Fo example, u u e wo k may add ano he in en o y o non-p ocessed
supply ins ead o assuming only he p ocessed in en o y. Fu he mo e, in ou wo k,
we assumed p ocessing capaci y is a cons ain . We u he ha e shown ha he
capaci y is no consumed equally pe pe iod bu a dis inc peak pe iods. Based on
his insigh , u u e esea ch may add lexibili y o p ocessing capaci y, e.g., allow-
ing he p ocessing o mo e ma e ial in one pe iod bu less in a subsequen pe iod.
Fu he mo e, p ocessing capaci y may be s ochas ic because o unce ain p ocessing
ime o a ying ene gy cos s pe pe iod. Fu he , we obse ed consis ency pa e ns
in he pe iods speci ic supplie s we e isi ed. Such consis ency may allow o be e
planning and coope a ion wi h he was e collec ion acili ies. Fu u e esea ch may
in es iga e how consis ency may be in eg a ed in o decision-making explici ly and
o wha cos .
Finally, ou li e a u e su ey e ealed ha wo k on ou ing p oblems wi h quali y
o yield unce ain y is somewha limi ed. Howe e , such unce ain y is p esen in
a ious p oblems, no only in ecycling bu also in collec ing a ming p oduc s o
e e se logis ics.
734
D.Cuella -Usaquén e al.
Appendix A
In he Appendix, we desc ibe he logic o he dynamic componen s, he e ela ion o
s ochas ici y, and he decision-making p ocess o ou me hod o e he planning ho i-
zon. We hen p esen de ails on he ixed ou e policy and he esul s o addi ional
expe imen s.
A.1 Decision making on heplanning ho izon
Algo i hm1 p esen s he pe iod-by-pe iod low o in o ma ion and decision making
using he STM-policy. The algo i hm ecei es wo ypes o inpu . Fi s , i p ocesses
p oblem in o ma ion such as he planning ho izon (T), supplie de ails (M), beha io
o he unce ain y sou ces (
𝜇
,
𝜎
,
𝜇𝜙
,
𝜎𝜙
), a el ime and ehicle cha ac e is ics
(
𝜏,Q,lmax
), and cos in o ma ion (c, ). Second, i uses he pa ame e s o he p o-
posed s ochas ic lookahead me hod such as he size o he lookahead ho izon (h),
he numbe o sample pa hs wi hin he lookahead (
|Ω|
), and he discoun pa ame e
o ou ing app oxima ion (
𝛾
). The algo i hm e u ns he cumula i e cos o e he
en i e planning ho izon (
To alCos
).
Algo i hm1 s a s by ini ializing he ne in en o y a he depo (
I1
) and supply a
he supplie s (
q1
) o he i s s a e and se s he o al cos o he planning ho izon o
ze o, his alue is s o ed in he a iable
To alCos
(lines 1 - 4). In e e y pe iod (lines
5 - 17), he unc ion SampledMul ipe iodScena ios (line 6) cons uc s he supply
and pe cen age loss ealiza ions o be e alua ed wi hin he s ochas ic p og am. This
unc ion akes as inpu he in o ma ion om each unce ain y sou ce (
𝜇
,
𝜎
,
𝜇𝜙
,
𝜎𝜙
)
and gene a es a numbe
|Ω|
o sample pa hs associa ed wi h he o wa d pe iod ho i-
zon o size h espec i e o he cu en pe iod . Wi h he s a e in o ma ion (
S
), he
gene a ed scena ios (
Ω
), he discoun pa ame e (
𝛾
), and addi ional p oblem in o -
ma ion, he wo-s age s ochas ic p og am is sol ed by calling he unc ion S ochas i-
cLookahead (line 7). The solu ion o he s ochas ic p og am e u ns he selec ed sup-
plie s (
e
) and he quan i ies o be collec ed a each o hem (
z
). This in o ma ion,
along wi h he a el ime (
𝜏
) and he cos pe minu e a elled (c), is gi en o he
Rou ingHeu is ic unc ion (line 8), which e u ns he explici collec ion ou es (
x
).
Wi h he ou es and quan i ies o be collec ed decided, he decision
a
o he pe iod
is comple ed (line 9). Then, he ou ing cos (
C
) is calcula ed by conside ing he
a el ime o he ou es and he cos pe minu e (line 10). Ha ing he ac ion
a
o
he s a e
S
, he pos -decision s a e
Sa
is gene a ed (line 11). Then, he unce ain y
ela ed o he in en o y ha can be used om he ma e ial collec ed (
z𝜔
+1
) and he
addi ional supply a ailable om supplie s (
𝜔
+1
) is e ealed (line 12), and he ne
new in en o y (
I𝜔
+1
) is calcula ed based based on ma e ial gene a ed (line 13). I he
ne in en o y is nega i e, he cos o backo de s (
Ce
) is incu ed o mee demand
(line 14). Finally, he cos s o ou ing and backo de s o he cu en pe iod a e accu-
mula ed wi hin he a iable
To alCos
(line 15), and he new s a e
S +1
is calcula ed
using he ansi ion unc ion (line 16). This p ocedu e con inues un il he e a e no
mo e pe iods in he planning ho izon.
735
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
Algo i hm1 Decision-making o he STM-policy o e he planning ho izon
A.2 Solu ion app oach based on ixed collec ion ou es
This sec ion p esen s he Fixed-policy, which u ilizes ixed, pe iod-dependen col-
lec ion ou es based on his o ical ou es obse ed. We i s explain how we simula e
he his o ical ou es and hen desc ibe how he policy cons uc s a solu ion. Nex ,
we de ail how we de e mine he numbe o ou es pe pe iod.
We c ea e 100 uning ins ances o iden i y p omising ou es ollowing he se -
ing desc ibed in Sec .5.1. Each ins ance is sol ed using he STM-policy. We ex ac
he di e en ou es o each pe iod and coun he numbe o imes each ou e was
used. Rou e symme y is excluded in his coun ; o example, ou e
0−1−2−0
is conside ed equi alen o ou e
0−2−1−0
. The ou es o each pe iod a e hen
so ed in descending o de based on hei equency o use and subsequen ou es
a e emo ed in case hey con ain al eady co e ed supplie s o ensu e no epea ed
supplie s.
S a ing wi h he ou e wi h he highes equency o use in each pe iod, he pol-
icy collec s ma e ial by ollowing he o de o he supplie s on he ou e, ensu ing
he ehicle capaci y is no exceeded. I no all supplie s on a ou e a e selec ed due
o capaci y cons ain s, he ou ing and i s cos a e e-adjus ed based on he selec ed
736
D.Cuella -Usaquén e al.
supplie s. This p ocess is epea ed wi h he nex ou e un il he p ocessing capaci y
a he depo is ull o all a ailable ou es o he pe iod ha e been used.
The numbe o ou es pe pe iod should be enough o a oid high back-
o de cos s bu no so la ge as o gene a e cos o e uns o unnecessa y col-
lec ions. The maximum numbe o ou es pe pe iod
∈T
is de ined as
B
=min(
⌈
⋅(1+𝜃Fixed)
⌉
,
|
K
|)
whe e
is he a e age numbe o ou es used in
pe iod o e he 100 uning ins ances,
𝜃Fixed
is a con ol pa ame e ha allows o
a ying he lee size, and
K
is he o de ed ou es a ailable o he pe iod .
The pa ame e
𝜃Fixed
is uned ia enume a ion. We es ed
𝜃Fixed
alues in
{0.0, 0.1, 0.25, 0.5, 1.0, 1.5, 2.0}
. The alues ha minimize he expec ed objec i e
unc ion we e selec ed using 30 uning ins ance ealiza ions. Fo he Sachsen-Anhal
case,
𝜃Fixed
was ixed a 0; o he UK-ins ance case, i was ixed a 1.5.
A.3 Di e ence be weenSTS andmyopic
Figu e12 illus a es he di e ences in decision-making be ween he STS and Myopic
policies du ing he i s pe iod o he planning ho izon, s a ing wi h an iden ical
ini ial s a e o bo h policies. The igu e shows he di e ence in ou ing cos , supply
collec ed, esul ing in en o y, and backo de s. The alues a e calcula ed as he di -
e ence be ween he a e age alue o STS and he a e age alue o Myopic.
We obse e ha he ou ing cos is he same o bo h policies. Howe e , he STS
policy collec s a la ge amoun han he Myopic policy (0.5 ons mo e on a e age).
This addi ional amoun esul s om he di e en loss scena ios, leading o highe
in en o y and ewe backo de s.
A.4 Mul i‑pe iod policy decision making
In his sec ion, we del e in o he in a-pe iod beha io o policies in mo e de ail.
Table3 p o ides insigh s in o a ious indica o s o each policy. A he op a e he
indica o s o he Sachsen-Anhal case, and a he bo om a e he indica o s o he
Fig. 12 Compa ison o decisions made by STS and Myopic policies in he single pe iod e sion
737
Dynamic mul i‑pe iod ecycling collec ion ou ing wi h…
UK-ins ances case. The Rou ing indica o e e s o he a e age accumula ed ou ing
cos o e he planning ho izon. Then, we p esen he a e age numbe o supplie s
isi ed pe pe iod. The Backo de s indica o e lec s he a e age accumula ed back-
o de cos o e he planning ho izon. Backo de pe iodici y indica es he a e age
numbe o pe iods un il he backo de cos is incu ed again. Finally, he Loss indi-
ca o shows he a e age loss incu ed by each policy.
We obse ed ha he supe io pe o mance o STM is due o a balanced app oach
ac oss all indica o s. Single- ocus policies nega i ely impac o he indica o s, a ec -
ing o e all pe o mance, pa icula ly he pe cen age o loss and he numbe o
supplie s isi ed (Quali y+M, Volume+M, Dis ance+M). The Fixed-policy o e s
ad an ages by cap u ing supplie isi beha io bu lacks lexibili y in some pe i-
ods, leading o inc eased ou ing and backo de cos s. Al hough he policies o he
Myopic+M and STS+M do no isi many supplie s, hey incu highe ou ing cos s,
indica ing spo adic isi s o unsui able supplie s. The STM-policy has he ewes
supplie isi s bu main ains s able pe iodici y and backo de cos s. These indings
sugges ha an icipa ing po en ial supplie e en s allows o be e supplie selec-
ion, e ec i ely balancing a ailable ma e ial and losses.
A.5 Vehicle pa ame e s
In he ollowing, we analyze he pe o mance o he policies in case ehicle pa ame-
e s change. Fi s , we in es iga e he impac o he ehicle capaci y by es capaci ies
o 5 ons (small uck) and 20 ons ( uck wi h aile ). We compa e he changes o
he policies wi h espec o STM and ou basis Sachsen-Anhal se ing wi h 10 ons
capaci y. The esul s a e shown in Fig.13. We obse e ha ou policy becomes e en
mo e e ec i e i capaci y is inc eased. No ably, o he 20 ons ins ances, policy
Myopic achie es simila esul s as ou policy wi h 10 ons capaci y. In he case o
e y limi ed uck capaci ies o 5 ons, we obse e a subs an ial inc ease in cos .
Thus, ha ing su icien ly la ge ucks is e y impo an o he company.
Table 3 Analysis o mul i-pe iod policy indica o s
Indica o s Quali y+M Volume+M Dis ance+M Fixed Myopic+M STS+M STM
Sachsen-Anhal
Rou ing (€) 29045 17255 17670 15349 15795 12806 10635
Supplie s isi ed 4.5 2.0 2.8 1.9 2.0 1.5 1.3
Backo de s (€) 0.3 0.0 51.7 45.0 0.0 0.0 10.9
Backo de pe iodici y 7.0 – 9.2 20.0 – – 11.5
Loss (%) 17.9 22.6 24.8 24.3 24.6 21.8 23.3
UK-ins ances
Rou ing (€) 19902 8177 20527 15380 6841 6908 5861
Supplie s isi ed 8.6 2.4 9.3 4.6 3.1 3.2 2.7
Backo de s (€) 1.3 28.2 5.7 636.1 7.8 60.4 22.8
Backo de pe iodici y 11.0 12.5 2.5 10.7 7.0 11.3 14.8
Loss (%) 14.7 19.5 20.5 19.5 19.7 19.4 19.3
738
D.Cuella -Usaquén e al.
Nex , we analyze he ou ing cos . In ou main expe imen s, we assume a cos
o 1.5 Eu os pe minu e o a el. Now, we a y he cos om 0.5 o 1.0 up o 3
Eu os pe minu e. The esul s a e shown in Fig.14, again ela i e o he base se ing.
Rega dless o he ou ing cos , STM pe o ms supe io . The policy is pa icula ly
e ec i e when he ou ing cos is small.
A.6 Backo de cos
In his sec ion, we show he esul s o a ying backo de cos in Fig.15. We un he
h ee policies o backo de cos s o 50, 100, 250, 500, 1000, and 2000 pe on. We
obse e ha he esul s emain compa ably s able. Ideally, backo de cos is a oided.
Only o cases wi h (un ealis ically) cheap backo de cos o 50, all policies swi ch
om collec ion ou ing o backo de .
Fig. 13 Changing o ehicle
capaci y
Fig. 14 Changing o ou ing
cos
