Lee, Chungjae; Dalmeije , Ke in; Van Hen en yck, Pascal; Zhang, Peibo
A icle
Op imizing Au onomous T ans e Hub Ne wo ks:
Quan i ying he po en ial impac o sel -d i ing ucks
EURO Jou nal on T anspo a ion and Logis ics (EJTL)
P o ided in Coope a ion wi h:
Associa ion o Eu opean Ope a ional Resea ch Socie ies (EURO), F ibou g
Sugges ed Ci a ion: Lee, Chungjae; Dalmeije , Ke in; Van Hen en yck, Pascal; Zhang, Peibo (2024) :
Op imizing Au onomous T ans e Hub Ne wo ks: Quan i ying he po en ial impac o sel -d i ing
ucks, EURO Jou nal on T anspo a ion and Logis ics (EJTL), ISSN 2192-4384, Else ie , Ams e dam,
Vol. 13, Iss. 1, pp. 1-15,
h ps://doi.o g/10.1016/j.ej l.2024.100141
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EURO Jou nal on T anspo a ion and Logis ics
jou nal homepage: www.else ie .com/loca e/ej l
Op imizing Au onomous T ans e Hub Ne wo ks: Quan i ying he po en ial
impac o sel -d i ing ucks
Chungjae Lee a, Ke in Dalmeije a,∗, Pascal Van Hen en ycka, Peibo Zhang b,a
aH. Mil on S ewa School o Indus ial and Sys ems Enginee ing, Geo gia Ins i u e o Technology, Uni ed S a es o Ame ica
bGoizue a Business School, Emo y Uni e si y, Uni ed S a es o Ame ica
ARTICLE INFO
Keywo ds:
Au onomous T ans e Hub Ne wo ks
Au onomous ucking
Load planning
Mixed-in ege linea p og amming
Case s udy
ABSTRACT
Au onomous ucks a e expec ed o undamen ally ans o m he eigh anspo a ion indus y. In pa icula ,
Au onomous T ans e Hub Ne wo ks (ATHNs), which combine au onomous ucks on middle miles wi h human-
d i en ucks on he i s and las miles, a e seen as he mos likely deploymen pa hway o his echnology.
This pape p esen s a amewo k o op imize ATHN ope a ions and e alua e he bene i s o au onomous
ucking. By exploi ing he p oblem s uc u e, his pape in oduces a low-based op imiza ion model o his
pu pose ha can be sol ed by blackbox sol e s in a ma e o hou s. The esul ing amewo k is easy o
apply and enables he da a-d i en analysis o la ge-scale sys ems. The powe o his app oach is demons a ed
on a sys em ha spans all o he Uni ed S a es o e a ou -week ho izon. The case s udy quan i ies he
po en ial impac o au onomous ucking and shows ha ATHNs can ha e signi ican bene i s o e adi ional
anspo a ion ne wo ks.
1. In oduc ion
Sel -d i ing ucks a e expec ed o undamen ally ans o m he
eigh anspo a ion indus y. Mo gan S anley es ima es he po en-
ial sa ings om sel -d i ing ucks a $168 billion annually o he
Uni ed S a es alone (G eene,2013). Addi ionally, au onomous ans-
po a ion may imp o e on- oad sa e y, and educe emissions and a ic
conges ion (Sho and Mu ay,2016;Slowik and Sha pe,2018).
SAE In e na ional de ines di e en le els o d i ing au oma ion,
anging om L0 o L5, co esponding o no-d i ing au oma ion o ull-
d i ing au oma ion (SAE In e na ional,2018). The cu en ocus is on
L4 echnology (high au oma ion), which aims a deli e ing au oma ed
ucks ha can d i e wi hou any human in e en ion in speci ic do-
mains, e.g., on highways. The ucking indus y is ac i ely in ol ed in
making L4 ehicles a eali y. Daimle T ucks, one o he leading hea y-
du y uck manu ac u e s in No h Ame ica, acqui ed a majo i y s ake
in sel -d i ing uck de elope To c Robo ics, which laid ou a oadmap
o launch au onomous ucks in 2027 (T anspo Topics,2023). Au-
onomous ucking company TuSimple has ecen ly comple ed he i s
d i e less es s on Chinese public oads (TechC unch,2023). In he US,
Au o a Inno a ion eamed up wi h FedEx o haul eigh be ween Fo
Wo h and El Paso, Texas, and he company epo s ha 60,000 miles
ha e been comple ed wi hou inciden s (FedEx,2022). These a e jus
some o he companies in ol ed in au onomous ucking, and o he s
∗Co espondence o: 755 Fe s D NW, A lan a, GA 30318, Uni ed S a es o Ame ica.
E-mail add ess: [email p o ec ed] (K. Dalmeije ).
include Emba k, Ga ik, Kodiak, and Plus (Flee Owne ,2021;Fo bes,
2021;F eigh Wa es,2021).
A s udy by Viscelli desc ibes di e en scena ios o he adop ion o
au onomous ucks by he indus y (Viscelli,2018). The mos likely
scena io, acco ding o some o he majo playe s, is he ans e hub
business model (Viscelli,2018;Be ge ,2018;Shahandash e al.,2019).
Joanna Bu le , head o Daimle ’s global au onomous echnology g oup,
o example, s a ed ha ‘‘We a e s aying lase ocused on U.S. hub- o-
hub, on-highway’’ (T anspo Topics,2023). An Au onomous T ans e
Hub Ne wo k (ATHN) makes use o au onomous uck po s, o ans e
hubs, o hand o aile s be ween human-d i en ucks and d i e less
au onomous ucks. Au onomous ucks hen ca y ou he anspo a-
ion be ween he hubs, while egula ucks se e he i s and las
miles (see Fig. 1). O de s a e spli in o a i s -mile leg, an au onomous
leg, and a las -mile leg, each o which se ed by a di e en ehicle.
A human-d i en uck picks up he eigh a he cus ome loca ion,
and d ops i o a a nea by ans e hub. A d i e less sel -d i ing uck
mo es he aile o a ans e hub close o he des ina ion, and ano he
human-d i en uck pe o ms he las leg.
The ATHN applies au oma ion whe e i coun s: Mono onous high-
way d i ing is au oma ed, while mo e complex local d i ing and cus-
ome con ac is le o humans. Global consul ancy i m Be ge es i-
ma es ope a ional cos sa ings be ween 22% and 40% in he ans e
h ps://doi.o g/10.1016/j.ej l.2024.100141
Recei ed 12 Feb ua y 2024; Recei ed in e ised o m 15 July 2024; Accep ed 5 Augus 2024
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100141
A ailable online 6 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/ ).
C. Lee e al.
Fig. 1. Example o an au onomous ans e hub ne wo k.
hub model, based on cos es ima es o h ee example ips (Be ge ,
2018). A ecen whi e pape published by Ryde Sys em, Inc. and he
Socially Awa e Mobili y Lab s udies whe he hese sa ings can be a -
ained o ac ual ope a ions and ealis ic o de s in Full T uckload (FTL)
shipping (Ryde Sys em and Socially Awa e Mobili y Lab,2021). I
models ATHN ope a ions as a scheduling p oblem and uses a Cons ain
P og amming (CP) model o minimize emp y miles and p oduce sa ings
om 27% o 40% on a case s udy in he Sou heas o he Uni ed S a es.
The cu en pape is an ex ension o he Ryde whi e pape ha
subs an ially imp o es, simpli ies, and gene alizes he me hodology. I
is he culmina ion o wo yea s o esea ch in o he co e compu a ional
di icul y o op imizing ATHN ope a ions, epo ed in he Ryde whi e
pape and in echnical epo s by he au ho s (Dalmeije and Van Hen-
en yck,2021;Lee e al.,2022). The CP model p esen ed in Dalmeije
and Van Hen en yck (2021) p oduces solu ions ha ou pe o m he
cu en ope a ions, bu ha do no p o ide a bound on op imali y. Lee
e al. (2022) in oduces a Column Gene a ion (CG) app oach and a
bespoke Ne wo k Flow (NF) model. I is shown ha he CP solu ion can
be mo e han 10% om op imal, and ha he NF model can quickly
p oduce solu ions wi hin 1% om op imali y. These ea lie indings
mo i a e he low-based op imiza ion model in his pape ha exploi s
he p oblem s uc u e and is sol ed o op imali y by blackbox sol e s in
a ma e o hou s. The esul ing amewo k is easy o apply and enables
he da a-d i en analysis o la ge-scale sys ems. I has also enabled a
ollow-up s udy on he ole o hub capaci ies in ATHNs (Lee e al.,
2023).
The powe o he new me hodology is demons a ed on an FTL
sys em ha spans all o he Uni ed S a es o e a ou -week ho izon,
expanding bo h he egion and ime ho izon used in ea lie epo s.
The case s udy quan i ies he po en ial impac o sel -d i ing ucks and
shows ha ATHNs yield signi ican bene i o e adi ional anspo a-
ion ne wo ks. The main con ibu ions o his wo k can be summa ized
as ollows:
1. The pape p o ides a high-le el amewo k o op imize ATHN
ope a ions.
2. The pape demons a es ha his enables he s udy o la ge-scale
sys ems, equi ing only a blackbox sol e .
3. The pape uses ealis ic o de da a o quan i y he po en ial
impac o FTL au onomous ucking in he US on a na ional scale.
The emainde o his pape is o ganized as ollows. Sec ion 2
p esen s an o e iew o he li e a u e. Sec ion 3p o ides he p oblem
desc ip ion and Sec ion 4discusses he me hodology o op imizing
ATHNs. This me hodology is applied o a case s udy in he US ha is
in oduced in Sec ion 5. The baseline esul s and he analysis o he
po en ial impac o au onomous ucking a e p esen ed in Sec ion 6
and a de ailed sensi i i y analysis is p o ided by Sec ion 7. Finally,
Sec ion 8p o ides he conclusions.
2. Li e a u e e iew
As au onomous echnology ad ances, mo e pape s a e s udying he
e ec o au onomous ehicles on anspo a ion sys ems. Flämig (2016)
p o ides an o e iew o he di e en ways ha au onomous ehicles
can be used bo h on public in as uc u e and on p i a e p ope y
(e.g., wa ehouses o company g ounds). In he u ban anspo a ion
se ing, de Almeida Co eia and an A em (2016) s udies he e ec o
au onomous ehicles on a ic delays and pa king demand in a ci y.
The au ho s use con ex op imiza ion o de e mine a ic assignmen s
and a mixed in ege nonlinea o mula ion o assign au onomous ehi-
cles o households. A case s udy o he ci y o Del , The Ne he lands,
demons a es a posi i e impac on he oad ne wo k.
In he eigh anspo a ion con ex , ou ing and scheduling p ob-
lems wi h au onomous ucks ha e gained a en ion e y ecen ly. Chen
e al. (2021) conside s scheduling a pla oon o au onomous ucks o
educe ai esis ance when a eling be ween wo seapo e minals in
Singapo e. The au ho s p esen a mixed in ege second-o de -cone o -
mula ion ha is sol ed wi h a column-gene a ion based heu is ic. In he
a ea o se ice ne wo k design, Sche e al. (2018) p oposes a p oblem
whe e a human-d i en uck leads a pla oon o au onomous ehicles in
he i s ie o ci y logis ics. An a c-based mixed in ege p og amming
model on a ime-space ne wo k is p esen ed, bu empi ical obse a ions
show ha only small p oblem ins ances a e ac able. Sche e al.
(2020) ex ends his wo k by in oducing a dynamic disc e iza ion
disco e y app oach ha ou pe o ms a comme cial sol e , and also
p esen a heu is ic o quickly gene a e solu ions.
In he Less-Than-T uckload (LTL) con ex , Al Hajj Hassan e al.
(2022) s udies he daily load planning p oblem unde di e en le els
o au oma ion. The pape ocuses on modi ying a gi en base plan o
deal wi h dynamic load eques s and o he aspec s ha a e impo an
du ing ope a ions, including d i e egula ions whe e d i e s a e in-
ol ed. The au ho s p esen a column-gene a ion based heu is ic o
sol e indus y-based ins ances wi h up o 20 hubs and 1500 loads o e
a one-week ho izon.
In e ms o he p oblem s uc u e, op imizing ull uckload ATHN
ope a ions can be seen as a Pickup and Deli e y P oblem wi h Time
Windows (PDPTW), whe e ucks pick up and d op o loads wi hin
he ime windows p esc ibed by he cus ome s. The book To h and
Vigo (2014) p o ides a su ey o his ehicle ou ing p oblem and
o he a ian s. Howe e , ins ead o ou ing, his pape will exploi he
p oblem s uc u e and ake he pe spec i e o scheduling a sequence o
asks (combined pickups and deli e ies), which is closely ela ed o he
Vehicle Scheduling P oblem wi h Time Windows (VSPTW, Des osie s
e al. (1995)). These p oblems a e well s udied, and se e al exac and
heu is ic solu ion me hods exis . Fo example, F eling e al. (2001)
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100141
2
C. Lee e al.
p esen s a solu ion me hod based on he p imal–dual algo i hm ame-
wo k o VSPTW wi h a single depo . Ribei o and Soumis (1994)
p oposes a column-gene a ion app oach o he VPSTW wi h mul iple
depo s, and Hadja e al. (2006) p esen s a b anch-and-cu algo i hm
o he same p oblem. S einzen e al. (2010) conside s sol ing he ime-
ex ended a ian o he VSPTW wi h mul iple depo s using a heu is ic
based on he b anch-and-p ice amewo k. Campbell and Sa elsbe gh
(2004) p esen s inse ion heu is ics o ehicle ou ing and scheduling
p oblems.
The Vehicle Rou ing P oblem wi h Full T uckloads (VRPFL, A una-
pu am e al.,2003) is he speci ic a ian ha pe haps mos s uc u ally
esembles he ATHN p oblem. Simila o he cu en pape , he VRPFL
asks o minimum-cos uck ou es o se e a se o loads ha a e
speci ied by an o igin, des ina ion, and a pickup ime window. A una-
pu am e al. (2003) p oposes a b anch-and-p ice amewo k as he
solu ion app oach. The au ho s assume ha each o de consumes he
ull capaci y o he uck, and he same assump ion is made o op-
imizing ATHN ope a ions, which e lec s ha au onomous ucks a e
expec ed o be mos ly used o long-haul ips. A c ucial echnical
di e ence be ween (A unapu am e al.,2003) and he cu en pape is
ha au onomous ucks a e assumed o be comple ely in e changeable.
This will allow o a low-based op imiza ion model ha is amenable
o blackbox sol ing.
This pape in oduces a high-le el amewo k o op imize ATHN
ope a ions. The goal o his amewo k is o p o ide a p ac ical way o
s udy la ge-scale au onomous FTL sys ems and o quan i y he po en ial
impac o au onomous ucking. P e ious wo ks o en ely on ad anced
op imiza ion echniques such as cu ing planes o column gene a ion,
o p o ide me hods ha do no scale o indus y-sized p oblems. Fo
example, he la ges p oblem conside ed by A unapu am e al. (2003)
in ol es only 5 hubs and 160 loads. In con as , his pape exploi s he
p oblem s uc u e o p o ide a model ha is blackbox sol able on a
la ge scale (up o 200 hubs and 6000+ loads o e a ou -week ho izon).
Ano he bene i o he high-le el amewo k is ha i can be used o
gene a e a base plan ha o ms he basis o he ope a ional decisions,
e.g., as s udied by Al Hajj Hassan e al. (2022) o LTL ucking.
3. P oblem desc ip ion
This sec ion in oduces he p oblem o op imizing ATHN ope a ions,
while he solu ion me hodology is p esen ed in Sec ion 4.Table 1
summa izes he nomencla u e o he p oblem desc ip ion. The goal is
o se e a se o 𝑛 ull uckloads 𝐿a minimum cos wi h a combina ion
o deli e ies h ough he au onomous ne wo k and di ec deli e ies
wi h egula ucks. Each load 𝑙∈𝐿is iden i ied by an o igin loca ion
𝑜(𝑙), a des ina ion loca ion 𝑑(𝑙), and a planned depa u e ime, o elease
ime, 𝑟(𝑙). The au onomous ne wo k is based on a se o ans e hubs
𝑉𝐻. E e y load 𝑙∈𝐿is associa ed wi h an o igin hub ℎ+
𝑙∈𝑉𝐻nea
he o igin 𝑜(𝑙)and a des ina ion hub ℎ−
𝑙∈𝑉𝐻nea he des ina ion 𝑑(𝑙).
Solu ion. A solu ion consis s o h ee ypes o decisions ha a e made
join ly. Fi s , i is de e mined how each load 𝑙∈𝐿is se ed. I is
assumed ha he e a e exac ly wo op ions:
•Au onomous: The load ollows he pa h 𝑜(𝑙)→ℎ+
𝑙→ℎ−
𝑙→𝑑(𝑙).
The i s and las legs a e pe o med by a egula uck, while he
connec ion be ween he hubs is se ed by an au onomous uck.
•Di ec : The load ollows he pa h 𝑜(𝑙)→𝑑(𝑙)→𝑜(𝑙). Bo h legs
a e se ed by a single egula uck ha e u ns emp y. No e ha
he case s udy will conside challenging o de s ha ac ually incu
such an emp y e u n in p ac ice.
Second, he au onomous legs (ℎ+
𝑙→ℎ−
𝑙) o he loads ha a e se ed
au onomously a e combined in o ou es o a mos 𝐾≥0au onomous
ucks. No e ha hese ou es may include emp y eloca ions om ℎ+
𝑙 o
ℎ−
𝑙′be ween loads 𝑙and 𝑙′. I is assumed ha su icien egula ucks a e
a ailable o pe o m he adi ional legs. The co esponding cos s will
be cap u ed in he objec i e unc ion, bu he egula uck ou es a e
no modeled explici ly. This is mo i a ed by he ac ha , in p ac ice,
he i s - and las -mile p oblems a e no e y cons ained. Thi d, i
is decided a which ime each load is picked up. I is assumed ha
e e y load 𝑙∈𝐿admi s a lexibili y o 𝛥≥0a ound he planned
depa u e ime 𝑟(𝑙), leading o a ime window o [𝑟(𝑙) − 𝛥, 𝑟(𝑙) + 𝛥] o
pickup. This ime window is ansla ed o ℎ+
𝑙,ℎ−
𝑙, and 𝑑(𝑙)acco ding
o he a el imes o main ain his lexibili y h oughou . The a el
imes include ime o loading and unloading he au onomous uck,
which is assumed o be 𝑆≥0. A solu ion is easible i each load
is se ed au onomously o di ec ly, all implied au onomous legs a e
co e ed by au onomous uck ou es, and he au onomous uck ou es
a e easible wi h espec o ime. No e ha i is always easible o
eplica e he cu en si ua ion by se ing all loads di ec ly and no using
any au onomous ucks.
Loca ion g aph. Be o e de ining he objec i e, i is con enien o de ine
aloca ion g aph. The loca ion g aph models all ele an loca ions and
po en ial connec ions in he ATHN. Le he loca ion g aph be deno ed
by he di ec ed g aph 𝐺= (𝑉 , 𝐴). Ve ex se 𝑉con ains a e ex
o e e y hub loca ion, and wo e ices o e e y load 𝑙∈𝐿 ha
co espond o he o igin 𝑜(𝑙)and he des ina ion 𝑑(𝑙), espec i ely. A cs
𝑎∈𝐴a e de ined om e e y o igin o he hubs ( adi ional i s mile),
be ween all he hubs (au onomous middle mile), om he hubs o e e y
des ina ion ( adi ional las mile), and be ween o igin and des ina ion
di ec ly ( adi ional di ec deli e y and emp y e u n). No e ha he
a cs be ween he hubs o m a comple e g aph. E e y a c 𝑎∈𝐴is
associa ed wi h a dis ance 𝑐𝑎≥0and a a el ime 𝜏𝑎>0ob ained
om OpenS ee Map (2021). Fo con enience, he cos and a el ime
om 𝑖∈𝑉 o i sel a e de ined as 0.
Objec i e. The objec i e is o se e all loads a minimum cos . While
any non-nega i e a c-addi i e cos s uc u e is suppo ed, his pape
will de ine cos as he o al dis ance in adi ional mileage equi alen .
Au onomous ucks incu a cos o (1 − 𝛼)𝑐𝑎 o e e y a c 𝑎∈𝐴on hei
ou es, including a cs ha ep esen emp y eloca ions. The pa ame e
𝛼∈ [0,1] discoun s he au onomous dis ance o co ec o educed
labo cos . A di ec ip o load 𝑙∈𝐿has a cos equal o i s dis ance o
𝑐𝑜(𝑙)𝑑(𝑙)+𝑐𝑑(𝑙)𝑜(𝑙). No e ha he discoun does no apply o egula ucks.
Finally, each i s /las -mile a c 𝑎∈𝐴is assigned a cos o 1
1−𝛽𝑐𝑎. The
pa ame e 𝛽∈ [0,1) ep esen s he i s /las -mile ine iciency, which
assumes ha a ac ion 𝛽o he i s /las -mile ou e mileage would be
emp y. The ac o 1
1−𝛽inc eases he cos o he i s /las -mile a cs o
compensa e o he ac ha hese ou es a e no modeled explici ly.
The o al objec i e is he sum o he abo e componen s and can be
in e p e ed as he o al dis ance measu ed in equi alen adi ional
mileage.
4. Me hodology
This sec ion in oduces he me hodology ha enables a la ge-scale
da a-d i en s udy o quan i y he impac o sel -d i ing ucks. The
nomencla u e o his sec ion is summa ized by Table 2. P ac ical
assump ions and p ep ocessing s eps lead o a model ha is easy o
implemen , can immedia ely be sol ed by blackbox sol e s, and is
highly ex ensible. The sec ion ends by p o iding a p ac ical guide
o enable egional and empo al analysis in his amewo k, which
equi es only mino modi ica ions o he inpu and he model.
Task g aph. The op imiza ion model conside s he p oblem o op i-
mizing ATHN ope a ions om he pe spec i e o scheduling asks o
au onomous ucks. Simila ans o ma ions a e common in he a c-
ou ing li e a u e (e.g., see Black e al. (2013)). One ask 𝑡∈𝑇is
c ea ed o e e y load 𝑙∈𝐿. I an au onomous uck pe o ms a ask, i
means ha he co esponding load is se ed h ough he au onomous
ne wo k, and his uck se es he middle mile. I a ask is no pe -
o med by any au onomous uck, his means ha he co esponding
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100141
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C. Lee e al.
Table 1
Nomencla u e p oblem desc ip ion.
Symbol De ini ion
Se s and g aphs
𝐿Se o loads, each load 𝑙∈𝐿consis s o an o igin 𝑜(𝑙), a des ina ion 𝑑(𝑙)and a elease ime 𝑟(𝑙).
𝑉𝐻Se o au onomous ans e hub loca ions, 𝑉𝐻⊆ 𝑉 .
𝐺= (𝑉 , 𝐴), loca ion g aph ha models loca ions and connec ions in he ATHN.
𝑉Se o loca ions.
𝐴Se o loca ion a cs, each a c (𝑖, 𝑗) ∈ 𝐴co esponds o a el om loca ion 𝑖∈𝑉 o loca ion 𝑗∈𝑉.
Pa ame e s
𝑛Numbe o loads, 𝑛=|𝐿|.
ℎ+
𝑙O igin hub o load 𝑙∈𝐿,ℎ+
𝑙∈𝑉𝐻.
ℎ−
𝑙Des ina ion hub o load 𝑙∈𝐿,ℎ−
𝑙∈𝑉𝐻.
𝐾Maximum numbe o au onomous ucks.
𝛥Flexibili y a ound he planned depa u e ime (depa up o 𝛥ea lie o la e han planned).
𝑆Au onomous uck loading/unloading ime.
𝑐𝑎Dis ance o a el loca ion a c 𝑎∈𝐴,𝑐𝑎≥0.
𝜏𝑎Time o a el loca ion a c 𝑎∈𝐴,𝜏𝑎>0.
𝛼Discoun ac o o au onomous mileage, 𝛼∈ [0,1]
𝛽Fi s /las -mile ine iciency, 𝛽∈ [0,1)
Table 2
Nomencla u e me hodology.
Symbol De ini ion
Se s and g aphs
𝑇Se o asks, each ask 𝑡∈𝑇co esponds one- o-one o a load 𝑙(𝑡) ∈ 𝐿, and ep esen s se ing his load on he
ATHN wi h pickup ime 𝑝(𝑡)a i s o igin hub.
𝐺= (
𝑉 ,
𝐴), ask g aph ha models he sequence o asks.
𝑉Se o e ices {0,…, 𝑛 + 1} wi h sou ce 0, sink 𝑛+ 1, and asks 1,…, 𝑛.
𝐴Se o ask a cs, each a c (𝑡, 𝑡′) ∈
𝐴indica es ha e ex 𝑡∈
𝑉is ollowed immedia ely by e ex 𝑡′∈
𝑉.
Pa ame e s
𝜏𝑎Du a ion o ask a c (𝑡, 𝑡′) ∈
𝐴,𝜏𝑎>0, which consis s o loading a uck o ask 𝑡, d i ing be ween hubs, unloading,
and eloca ing o he o igin hub o ask 𝑡′.
𝐶𝑡Baseline cos o se ing load 𝑙(𝑡)di ec ly wi h a egula uck.
𝑐𝑎Cos o ask a c 𝑎∈
𝐴, which is he di e ence be ween se ing load 𝑙(𝑡)compa ed o he baseline.
𝑀𝑡𝑡′=𝑝(𝑡) − 𝑝(𝑡′)+2𝛥+𝜏𝑡𝑡′, o 𝑡, 𝑡′∈𝑇, su icien ly la ge big-M o Cons ain s (2e).
Va iables
𝑥𝑡Con inuous a iable ha indica es he s a ime o ask 𝑡∈𝑇.
𝑦𝑎Bina y a iable ha akes alue one i 𝑎∈
𝐴is selec ed (i.e., he co esponding asks a e pe o med sequen ially by
he same ehicle), and ze o o he wise.
load is se ed di ec ly by a egula uck. No e ha while pe o ming
asks is op ional, all loads a e se ed in he end: pe o ming a ask only
indica es ha he ask is se ed au onomously. App op ia e bene i s
will be assigned o pe o ming asks o ma ch he cos s uc u e in
Sec ion 3.
To cap u e his pe spec i e, he loca ion g aph is ans o med in o
a di ec ed ask g aph
𝐺= (
𝑉 ,
𝐴)in which he nodes a e asks and he
a cs indica e he sequence o asks pe o med by he same au onomous
uck. The se
𝑉includes a sou ce node 0whe e each sequence s a s,
and a sink node 𝑛+ 1 whe e i ends. As he nodes now ep esen asks
ins ead o loca ions, he numbe o nodes in he ask g aph is ypically
la ge han he numbe o nodes in he loca ion g aph. A cs a e de ined
om he sou ce o he asks, be ween he asks (bi-di ec ional), and
om he asks o he sink. Fig. 2 p o ides an illus a i e example. The
loca ion g aph shows ou hubs and wo loads 𝑙1and 𝑙2associa ed wi h
asks 𝑡1and 𝑡2, espec i ely. Visi ing node 𝑡1means ha an au onomous
uck loads a o igin hub ℎ+
𝑙1, d i es o des ina ion hub ℎ−
𝑙1, and unloads
he e. I also implies ha he i s and las miles a e pe o med by
egula ucks (no pic u ed). A e ha , he au onomous uck may
ei he pe o m ano he ask 𝑡2∈𝑇, which i s equi es a eloca ion
om ℎ−
𝑙1 o ℎ+
𝑙2, o i may end i s sequence. Loads o which he
co esponding ask is no co e ed a e se ed by a di ec ip wi h a
egula uck (no pic u ed). This means ha all ope a ions in he ATHN
a e cap u ed in he ask g aph by a se o pa hs om he sou ce o he
sink, whe e each pa h co esponds o an au onomous uck.
Rou es and cos s. Op imizing ATHN ope a ions now amoun s o choos-
ing a se o easible au onomous uck ou es ha minimize he o al
cos . A ou e is de ined as a simple pa h in he ask g aph om sou ce
o sink, oge he wi h a s a ing ime o e e y ask. A cs be ween asks
𝑡1, 𝑡2∈𝑇model he passage o ime be ween picking up loads 𝑙1and
𝑙2, espec i ely. Tha is, he du a ion is de ined as 𝜏𝑡1𝑡2=𝑆+𝜏ℎ+
𝑙1ℎ−
𝑙1
+
𝑆+𝜏ℎ−
𝑙1ℎ+
𝑙2
>0, which sums he ime o loading, pe o ming he middle
mile o load 𝑙1, unloading, and eloca ing o he s a ing poin o load
𝑙2. Task 𝑡1mus s a in he co ec ime window, which is ob ained
by shi ing he o iginal ime window o load 𝑙1by he ime i akes o
pe o m he i s mile. This ime window is gi en by [𝑝(𝑡1)−𝛥, 𝑝(𝑡1)+𝛥],
whe e 𝑝(𝑡1) = 𝑟(𝑙1) + 𝑡𝑜(𝑙1)ℎ+
𝑙1
.
No co e ing ask 𝑡1∈𝑇is associa ed wi h a cons an baseline
cos o
𝐶𝑡1 o pe o ming a di ec ip. This alue is gi en by
𝐶𝑡1=
𝑐𝑜(𝑙1)𝑑(𝑙1)+𝑐𝑑(𝑙1)𝑜(𝑙1). I ask 𝑡1is pe o med, he cos on he ou going a c
eplaces he baseline cos wi h he app op ia e cos s o se ing he load
au onomously. Mo e p ecisely, i ask 𝑡1appea s in a sequence ollowed
by ask 𝑡2∈𝑇, he cos 𝑐𝑡1𝑡2o a c (𝑡1, 𝑡2) ∈ 𝐴is de ined as ollows:
𝑐𝑡1𝑡2=1
1 − 𝛽𝑐𝑜(𝑙1)ℎ+
𝑙1
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟
i s mile
+ (1 − 𝛼)𝑐ℎ+
𝑙1ℎ−
𝑙1
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟
middle mile
+1
1 − 𝛽𝑐ℎ−
𝑙1𝑑(𝑙1)
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟
las mile
+ (1 − 𝛼)𝑐ℎ−
𝑙1ℎ+
𝑙2
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟
eloca ion
−
𝐶𝑡1
⏟⏟⏟
di ec
.
(1)
Along he same lines, sou ce a cs 𝑎∈
𝐴ha e cos 𝑐𝑎= 0 and sink
a cs omi he eloca ion e m. No e ha 𝑐𝑎<0when an au onomous
deli e y is p e e ed o e a di ec deli e y, which encou ages he ask
o be pe o med.
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100141
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C. Lee e al.
Fig. 2. Cons uc ing he ask g aph om he loca ion g aph.
Op imiza ion model. The op imiza ion p oblem can now be s a ed as
ollows:
min ∑
𝑡∈𝑇
𝐶𝑡+∑
𝑎∈
𝐴
𝑐𝑎𝑦𝑎,(2a)
s. .∑
𝑎∈𝛿+
𝑡
𝑦𝑎≤1 ∀𝑡∈𝑇 , (2b)
∑
𝑎∈𝛿+
𝑡
𝑦𝑎=∑
𝑎∈𝛿−
𝑡
𝑦𝑎∀𝑡∈𝑇 , (2c)
∑
𝑎∈𝛿+
0
𝑦𝑎≤𝐾, (2d)
𝑥𝑡′≥𝑥𝑡+𝜏𝑡𝑡′−𝑀𝑡𝑡′(1 − 𝑦𝑎),∀𝑡, 𝑡′∈𝑇 , (𝑡, 𝑡′) ∈
𝐴(2e)
𝑥𝑡∈ [𝑝(𝑡) − 𝛥, 𝑝(𝑡) + 𝛥] ∀𝑡∈𝑇 , (2 )
𝑦𝑎∈B∀𝑎∈
𝐴. (2g)
Le 𝑥𝑡∈ [𝑝(𝑡)−𝛥, 𝑝(𝑡)+𝛥]be he s a ime o ask 𝑡∈𝑇. The a iable
𝑦𝑎∈Bis he low on a c 𝑎∈
𝐴, i.e., i akes alue one i he asks a e
pe o med sequen ially by he same ehicle and ze o o he wise. Fo
con enience, le 𝛿+
𝑣and 𝛿−
𝑣deno e he ou -a cs and in-a cs o 𝑣∈
𝑉,
espec i ely. P oblem (2) hen models he op imiza ion o ATHN op-
e a ions. Objec i e (2a) minimizes he sys em cos as discussed abo e.
Cons ain s (2b) equi e ha each ask is pe o med a mos once, and
Cons ain s (2c) ensu e low conse a ion. The numbe o ehicles is
limi ed by Cons ain (2d). Cons ain s (2e) a e Mille , Tucke , and
Zemlin (1960) cons ain s ha model he passage o ime and elimina e
cycles. I is s aigh o wa d o show ha he cons an s
𝑀𝑡𝑡′=𝜏𝑡𝑡′−(𝑝(𝑡′) − 𝛥)
⏟⏞⏞⏞⏟⏞⏞⏞⏟
lowe bound on 𝑥𝑡′
+(𝑝(𝑡) + 𝛥)
⏟⏞⏞⏟⏞⏞⏟
uppe bound on 𝑥𝑡
(3)
a e su icien ly la ge o make he cons ain inac i e when 𝑦𝑎= 0.
Finally, Eqs. (2 )–(2g) de ine he a iables. Fo a consis en analysis,
he solu ion is pos p ocessed o shi he s a imes o as ea ly in ime
as possible.
4.1. Accele a ion echniques
The size o P oblem (2) can be educed signi ican ly by ecognizing
ha many a cs (𝑡, 𝑡′) ∈
𝐴a e ei he i ially ime- easible because ask
𝑡′is planned much la e han ask 𝑡, o i ially ime-in easible because
ask 𝑡′is planned much ea lie han ask 𝑡. These obse a ions a e
o malized in he ollowing p oposi ion.
P oposi ion 1 (P ep ocessing Rules).Le 𝑎𝑡=𝑝(𝑡) − 𝛥and 𝑏𝑡=𝑝(𝑡) + 𝛥
be he ea lies and la es possible s a ime o ask 𝑡∈𝑇, espec i ely. The
ollowing p ep ocessing ules a e alid o a c (𝑡, 𝑡′) ∈
𝐴be ween wo asks
𝑡, 𝑡′∈𝑇.
1. A c is always ime easible: 𝑏𝑡+𝜏𝑡𝑡′≤𝑎𝑡′⇒ emo e Cons ain (2e)
o a c (𝑡, 𝑡′).
2. A c is ne e ime easible: 𝑎𝑡+𝜏𝑡𝑡′> 𝑏𝑡′⇒ emo e a c (𝑡, 𝑡′) om
𝐴.
P oo . By de ini ion, 𝑥𝑡is only allowed o ake alues in 𝑥𝑡∈ [𝑎𝑡, 𝑏𝑡].
The condi ion in he i s ule implies 𝑥𝑡+𝜏𝑡𝑡′≤𝑏𝑡+𝜏𝑡𝑡′≤𝑎𝑡′≤𝑥𝑡′⇔
𝑥𝑡′≥𝑥𝑡+𝜏𝑡𝑡′ o all easible alues o 𝑥𝑡and 𝑥𝑡′. I ollows ha he ime
cons ain is edundan and can be emo ed. Simila ly, he condi ion
in he second ule implies 𝑥𝑡+𝜏𝑡𝑡′≥𝑎𝑡+𝜏𝑡𝑡′> 𝑏𝑡′≥𝑥′
𝑡⇔𝑥′
𝑡< 𝑥𝑡+𝜏𝑡𝑡′.
I ollows ha 𝑦𝑡𝑡′= 1 would iola e Cons ain (2e). As such, 𝑦𝑡𝑡′may
be se o ze o, which is achie ed by simply emo ing he a c. Hence,
hese p ep ocessing ules a e alid. □
Bo h p ep ocessing ules elimina e ime cons ain s, which can
make hem inc edibly powe ul. I all ime Cons ain s (2e) a e elim-
ina ed, hen he 𝑥- a iables (2 ) a e au oma ically sa is ied, and he
emainde o P oblem (2) can be seen as a minimum-cos ne wo k
low p oblem. The only cons ain s ha a e no in s anda d o m a e
Cons ain s (2b) and (2d), bu hey ake he o m o node capaci ies ha
can be handled h ough node spli ing (Ahuja e al.,1993). I is well-
known ha he min-cos low p oblem exhibi s he in eg ali y p ope y,
and can be sol ed in polynomial ime by linea p og amming. As he
lexibili y 𝛥dec eases, he p ep ocessing ules become mo e e ec i e,
and P oblem (2) ge s close o a minimum-cos ne wo k low p oblem.
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100141
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C. Lee e al.
Fig. 3. A c il e ing o empo al decomposi ion (
𝐴in ed and
𝑉in blue).
In ac , his si ua ion is eached o he no- lexibili y 𝛥= 0 case, when
e e y a c (𝑡, 𝑡′) ∈
𝐴ei he sa is ies Rule 1 o Rule 2 and all ime
cons ain s a e elimina ed. In o mally, i is easie o op imize ATHN
ope a ions when he e is less lexibili y, o he poin whe e i becomes
p o ably easy wi hou lexibili y.
MIP s a . In addi ion o p ep ocessing, his pape will y o imp o e
he op imiza ion p ocess by p o iding he sol e wi h an ini ial easible
solu ion. This solu ion is known as a MIP s a and p o ides an uppe
bound ha can assis he b anch-and-bound p ocess. Rega dless o he
lexibili y 𝛥, a easible solu ion can be calcula ed e icien ly by sol ing
he case when lexibili y is se o ze o. The calcula ed solu ion is hen
used as a s a ing poin o he ac ual p oblem. To a oid he o e head
o building an addi ional model, he sol e is p o ided a pa ial MIP
s a o only 𝑥𝑡=𝑝(𝑡) o all 𝑡∈𝑇, which is su icien o ind he same
solu ion.
The ac ha 𝛥= 0 is easy o sol e and gua an ees a alid
uppe bound is speci ically because he ucks a e au onomous. The
main echnical di e ence is ha au onomous ucks a e comple ely
in e changable, while human-d i en ucks need o be dis inguished
o ensu e ha d i e s e u n o hei speci ic s a ing poin (A una-
pu am e al.,2003) o ha hey do no exceed he maximum d i ing
ime (G onal e al.,2003). Ne wo k low elaxa ions ha agg ega ed
d i e s ha e been used o de i e lowe bounds (G onal e al.,2003),
bu i is no ob ious how o ans o m he ou come in o a easible
solu ion. Fo au onomous ucks hese human ac o s do no apply,
which enables he amewo k in his pape .
4.2. Regional and empo al decomposi ion
The amewo k in his pape is easily ex ended o pe o m egional
and empo al decomposi ion. This equi es only mino modi ica ions o
he inpu and o he model.
Regional decomposi ion. The goal o he egional decomposi ion is o
compa e he global op imiza ion o ATHN ope a ions o a si ua ion in
which each egion (e.g., he Sou h o he US) has dedica ed au onomous
ucks ha only pick up loads ha s a in ha egion. In his case,
ucks can se e loads wi hin hei egion and loads ha a e mo ing
ou o he egion. Howe e , a e lea ing he egion o d op o a load,
he uck has o e u n be o e i can pe o m ano he ask. The model
will be modi ied o join ly op imize how ucks a e assigned o egions
and how o ope a e he ATHN unde hese es ic ions. Pe o ming an
analysis in his se ing helps answe ques ions abou he scale a which
au onomous ucks a e e ec i e and whe e hey should be deployed.
Regional decomposi ion can easily be pe o med by il e ing a cs
om he ask g aph. Fi s assign e e y ask 𝑡∈𝑇 o a egion based
on he loca ion o he o igin hub ℎ+
𝑙(𝑡). Nex , emo e all a cs (𝑡, 𝑡′) ∈
𝐴
be ween asks 𝑡, 𝑡′∈𝑇i he egions a e di e en . I ollows ha when
a low eaches a ask ha s a s om a speci ic egion, he e is no
way o each asks ha s a om a di e en egion, as in ended. The
amoun o low om he sou ce o each egion ep esen s he amoun
o ucks ha a e assigned o ha egion. The uck assignmen s and
ope a ions a e hen join ly op imized by sol ing his il e ed ins ance
o P oblem (2).
Tempo al decomposi ion. The goal o he empo al decomposi ion is o
plan ATHN ope a ions on a olling ho izon (e.g., one week a a ime),
a he han o a ull pe iod a once (e.g., ou weeks). Op imizing o e
a sho e ho izon equi es less in o ma ion and is easie compu a ion-
ally. Howe e , he model does no explici ly ebalance ucks a he
end o he pe iod. This means ha op imizing myopically may lea e he
ucks ill-posi ioned o he nex pe iod. The empo al decomposi ion
can be used o explo e hese ade-o s.
Implemen ing a olling ho izon o P oblem (2) is ela i ely s aigh -
o wa d. A p ac ical way o do so is by eusing he exis ing s uc u es
and modi ying he model as li le as possible. Fig. 3 p o ides an
illus a i e example. Fi s , build he ask g aph o he ull pe iod.
Fo a gi en ho izon, iden i y he a cs
𝐴o he pa ial ou es c ea ed
p e iously (wi hou sink a cs). Fix hese a cs 𝑎∈
𝐴 o 𝑦𝑎= 1 in he
op imiza ion model o s ay consis en wi h he pas . To plan o he
cu en ho izon, only he ollowing nodes
𝑉a e ele an : he sou ce,
he sink, he cu en ou e endpoin s, and he asks ha s a du ing he
ho izon. Now il e he ask g aph o only keep he a cs in
𝐴and he
a cs in he subg aph induced by
𝑉. This makes i impossible o plan
ou side o he ho izon. The model is sol ed and he s eps a e epea ed
un il he ull pe iod is planned.
5. Case s udy
To quan i y he impac o au onomous ucking on a ealis ic ans-
po a ion ne wo k, a case s udy is p esen ed o he dedica ed ans-
po a ion business o Ryde Sys em, Inc. (Ryde ). Ryde is one o he
la ges anspo a ion and logis ics companies in No h Ame ica, and
p o ides lee managemen , supply chain, and dedica ed anspo a ion
se ices o o e 50,000 cus ome s.
Da a. Ryde has p o ided a da ase ha is ep esen a i e o i s dedi-
ca ed anspo a ion business in he US, educing he scope o o de s
ha a e s ong candida es o au oma ion. The case s udy ocuses
on he challenging o de s ha cu en ly consis o a single deli e y
ollowed by an emp y e u n ip. These o de s a e highly ine icien
and con ibu e signi ican ly o he o e all emp y mileage, such ha
ATHN can po en ially ha e a big impac . The challenging o de s also
allow o a clean compa ison o he cu en si ua ion: These a e o de s
o which Ryde was unable o ind a backhaul, and e u ning emp y
a e deli e y is how hey would be se ed in p ac ice. The challenging
o de s a e con e ed in o loads wi h an o igin, des ina ion, and planned
depa u e ime. The ATHN ope a ions a e op imized o loads ha s a
du ing he i s ou weeks o Oc obe 2019. This co esponds o 6842
loads, wi h an a e age dis ance o 390 km (242 mi).
Ne wo k design. To design an e ec i e ne wo k, i is impo an o selec
hubs ha a e 1. close o equen ly used o igins and des ina ions o
minimize he i s /las mile, and 2. easily accessible om he highway
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100141
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C. Lee e al.
Fig. 4. ATHN design o 100 hubs (Ci cle a ea p opo ional o numbe o assigned loads).
o enable au oma ion be ween he hubs. This is achie ed by i s using
he K-means algo i hm in Sciki -lea n (Ped egosa e al.,2011) o clus e
he o igins and des ina ions in o he desi ed numbe o hubs, based on
da a om Oc obe o Decembe 2019. Nex , he cen oids a e mapped
on o he closes US uck s op ob ained om he U.S. Depa men o
T anspo a ion (2019), acco ding o ha e sine dis ance. The o igin and
des ina ion hubs ℎ+
𝑙≠ℎ−
𝑙 o load 𝑙∈𝐿a e chosen o minimize he
alue o 𝑐𝑜(𝑙)ℎ+
𝑙
+(1−𝛾)𝑐ℎ+
𝑙ℎ−
𝑙
+𝑐ℎ−
𝑙𝑑(𝑙), whe e 𝛾∈ [0,1] is a discoun ac o
o au onomous ucks. Fo 𝛾= 0 his minimizes he o al dis ance,
o 𝛾= 1 his minimizes he i s /las -mile dis ance, and 𝛾∈ (0,1)
minimizes a combina ion o he wo. By aking bo h he o igin and he
des ina ion in o accoun , he ule allows o assigning hubs in he igh
di ec ion ha a e no necessa ily he closes . Fig. 4 isualizes he 100-
hub design o 𝛾= 40%, in which he a ea o each hub is p opo ional
o he numbe o loads ha a e assigned o i . I can be seen ha
many loads a e concen a ed in he Sou h (pu ple) and in he No heas
( ed).
Expe imen al se ings. Table 3 p o ides an o e iew o he baseline
pa ame e alues used in he case s udy. Va ious sensi i i y analyses
will be pe o med o obse e how hese pa ame e s a ec he sys em.
The baseline includes wo au onomous discoun ac o s: 𝛼= 25% and
𝛼= 40%. This esul s in a conse a i e es ima e o he bene i s o
au onomous ucking, which is p edic ed o be 29% o 45% cheape
pe mile (Engholm e al.,2020). All expe imen s use a consis en hub-
assignmen ule wi h au onomous discoun ac o 𝛾= 40% o ensu e
ha he esul s a e compa able. A highe alue o 𝛾 ends owa ds load
pa hs ha include mo e au onomous mileage. The baseline also uses
mul iple alues o 𝐾 o obse e he impac o inc easing a ailabili y
o au onomous ucks, wi h he alue 𝐾= 100 as he s anda d.
These ins ances a e sol ed sequen ially by inc easing he igh -hand
side o Cons ain (2d) and eop imizing. All s eps in Sec ion 4a e
implemen ed in Py hon 3.9 and he ATHN ope a ions a e op imized
wi h Gu obi 9.5.2. Gu obi is gi en h ee hou s o sol ing ime o each
ins ance, unless s a ed o he wise. Each expe imen is un on a Linux
machine wi h dual In el Xeon Gold 6226 CPUs on he PACE Phoenix
clus e (PACE,2017), using a single node wi h 24 co es and 192 GB
o RAM. I memo y is insu icien , he expe imen is epea ed on a
Table 3
Baseline pa ame e alues o he case s udy.
Pa ame e Value
𝑛6842 loads
|𝑉𝐻|100 ans e hubs
𝐾∈ {0,50,100,150,200,250} au onomous ucks
𝛥1 h pickup- ime lexibili y
𝑆30 min au onomous uck loading/unloading ime
𝛼∈ {25%,40%} discoun o au onomous mileage
𝛽25% i s /las -mile ine iciency
𝛾40% discoun o au onomous mileage du ing hub-assignmen
P ep ocessing Applied
MIP s a Disabled
Sol e ime limi 3 h
machine wi h 384 GB o RAM. No e ha i high-memo y machines
a e no a ailable, memo y could also be aded o compu ing ime by
educing he numbe o pa allel h eads.
6. Baseline esul s
This sec ion discusses he baseline esul s o he case s udy. I
i s p esen s compu a ional esul s o demons a e ha he p esen ed
amewo k can handle la ge-scale sys ems. Nex , i analyzes he impac
o au onomous ucking o he case s udy.
6.1. Compu a ional esul s
Table 4 p esen s he compu a ional esul s o he baseline in-
s ances. The ins ances di e by he discoun o au onomous mileage 𝛼
and he numbe o ehicles 𝐾. As desc ibed in he p e ious sec ion, he
ins ances o di e en 𝐾a e un sequen ially and euse he same model,
as would be done in p ac ice o s udy he sys em. The ‘LP Relaxa ion’
columns p esen he ime o sol e he linea p og amming elaxa ion
(be o e cu s) and he co esponding oo gap. The ‘B anch and Bound’
columns summa ize he ull b anch-and-bound p ocess, epo ing he
numbe o nodes in he ee, he solu ion ime (10,800 i he ime limi
o h ee hou s is eached), and he inal gap. The able omi s he ime
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100141
7
C. Lee e al.
Table 4
Baseline compu a ion s a is ics.
Pa ame e s LP elaxa ion B anch and bound
𝛼 𝐾 Seconds Gap % Nodes Seconds Gap (%)
25%
0 0 00 0 0
50 218 19.50 1 10,800 0.03
100 170 6.44 1 4,133 0
150 183 1.81 1 3,261 0
200 181 0.70 1 2,754 0
250 177 0.40 1 2,850 0
40%
0 0 00 0 0
50 268 25.10 1 10,800 0.18
100 206 10.40 3382 6,430 0
150 232 3.20 1 2,972 0
200 207 1.00 1 3,149 0
250 189 0.74 1 2,026 0
o building he model, which was less han six minu es, and he ime
o p esol e, which ook less han ou minu es in all cases.
Despi e he ac ha each model has o e 22M bina y a iables and
close o 300k cons ain s, Gu obi is able o ind op imal solu ions in
mos cases. Fo 𝐾= 0, he p oblem is i ial and is sol ed immedia ely
in p esol e. The cases wi h ewe ehicles a e challenging o he
sol e , p esumably because he asks a e packed mo e densely in o he
schedule, as will be discussed in Sec ion 7.2. Only he 𝐾= 50 cases
whe e no sol ed o op imali y, and emain a 0.03% and 0.18% gap.
Bo h o 𝛼= 25% and 𝛼= 40% he sol e ends o keep adding cu ing
planes a he han b anch. This s a egy is success ul o sol e all o he
ins ances o op imali y wi hin he ime limi . The only ins ance ha
s ands ou is 𝛼= 40% and 𝐾= 100, which explo es 3382 nodes in he
b anch-and-bound ee. Fo his ins ance, he log shows ha a gap o
0.01% is ound a e 4845 s. When he gap is s ill a 0.01% a 5921 s,
he sol e decides o s a b anching o close he gap. This beha io
can likely be explained by symme y in he solu ion space, e.g., i wo
ehicles swap hal o hei asks, he solu ion is likely o be o simila
quali y. The esul is ha good solu ions a e ound quickly, bu i akes
a subs an ial numbe o cu s o b anches o ind he op imum. O e all,
he compu a ions o he baseline ins ances show ha he p oposed
me hodology can ind op imal o close o op imal solu ions in a sho
amoun o ime compa ed o he planning ho izon.
Using MIP s a s. Enabling MIP s a o ces he sol e o cons uc an
ini ial easible solu ion be o e s a ing he sea ch (Sec ion 4). Fig. 5
p o ides an example o he e ec o enabling MIP s a o he 𝛼= 25%
baseline wi h 𝐾= 100 ucks. No e ha hese es s a e un indepen-
den ly wi hou eusing he model o 𝐾= 50 as in he expe imen s
abo e. Wi hou any guidance, Gu obi akes 520 s o epo he i s
op imali y gap o 33%. A 741 s, a signi ican ly be e solu ion o 1.25%
gap is ound, and he p oblem is sol ed o op imali y in unde an hou .
When MIP s a is enabled, i akes mo e ime o he sea ch p ope o
s a , and he i s gap is epo ed a 1139 s. Howe e , spending ime
o cons uc an ini ial easible solu ion immedia ely leads o a gap o
only 1.19% because o he imp o ed uppe bound. The ull p oblem is
sol ed in unde one hou and 15 min.
Two obse a ions a e made o he case s udy. Fi s , using a MIP
s a does no seem o imp o e solu ion ime, bu i does c ea e a
mo e p edic able esul . Especially i he ins ance canno be sol ed o
op imali y, he MIP s a is mo e likely o p oduce a easonable solu ion
be o e he ime limi . Second, he small ini ial gap o he MIP s a
sugges s ha he ini ial solu ion is al eady o high quali y. Recall ha
his ze o- lexibili y 𝛥= 0 case can be seen as a min-cos low p oblem,
which gi es p ac i ione s he possibili y o a oid comme cial so wa e
and ins ead plan ATHN ope a ions wi h highly-e icien open sou ce
sol e s such as he LEMON g aph lib a y (Dezső e al.,2011). As mos
Fig. 5. Op imali y gap o e ime o 𝛼= 25% and 100 ucks.
ins ances can be sol ed o op imali y, MIP s a s will only be enabled
o he di icul la ge- lexibili y ins ances in Sec ion 7.1.
6.2. Impac o au onomous ucking
Fig. 6 p esen s he impac o au onomous ucking o he baseline,
whe e au onomous mileage is discoun ed by ei he 𝛼= 25% o 𝛼=
40%. I is clea ha in oducing au onomous ucks leads o subs an ial
bene i s. Fig. 6(a) shows ha he i s 50 ucks al eady lowe he
ope a ional cos o he sys em (including i s /las miles) by he equi -
alen o mo e han one million adi ional kilome e s. E.g., a $1.25/km
(≈$2/mile) o adi ional ucks, his co esponds o a alue o abou
$1.3M pe ou weeks o $16.9M pe yea . The pe cen age sa ings o
he o e all sys em a e p o ided in Fig. 6(c). These sa ings ange om
20% o 50 ucks in he mo e expensi e scena io o 37% o 250 ucks
when au onomous ucking is less expensi e. I is in e es ing o obse e
ha adding ehicles clea ly sa is ies he law o diminishing e u ns.
As mo e ehicles a e added, mo e loads a e se ed au onomously
(Fig. 6(b)) and mo e sa ings a e ob ained (Fig. 6(c)), bu he bene i s
le el ou a abou 100 ucks o he Ryde case s udy. No e ha his
is a ela i ely small numbe o ucks compa ed o he 6842 loads,
which e lec s he ac ha au onomous ucks can ope a e a ound he
clock. Fig. 6(d) looks a he sa ings pe cen age only o loads ha a e
se ed au onomously. I can be seen ha , on a e age, loads ha a e
se ed au onomously sa e be ween 31% and 42% in cos s compa ed o
adi ional anspo a ion.
Table 5 and Fig. 7 di e deepe in o he esul s o 𝛼= 25% and
100 ucks speci ically. The able shows ha he o al dis ance d i en
in he ATHN (including emp y miles) is 13.0% lowe han o he
cu en sys em. These sa ings a e due o he lexibili y o au onomous
ucks ha can ope a e h oughou he nigh and ne e need o e u n
home. This allows o only 29% emp y miles on he au onomous middle
mile, which is a subs an ial imp o emen o e he a e o 50% in he
cu en sys em. When labo cos educ ion is aken in o accoun , he
sa ings inc ease o 25.2%. The uck schedule in Fig. 7 also shows
ha eloca ions a e small compa ed o he wo k pe o med: E e y ow
ep esen s a single uck, whe e blue ba s co espond o pe o ming
asks, and he ed ba s co espond o d i ing emp y. The schedule
is ela i ely igh , excep o he ou ‘gaps’ du ing he weekends, in
which no many loads a e planned. This is an a i ac o he da a ha
esul s om planning a ound people, Sec ion 7will explo e he alue o
inc easing lexibili y and allowing au onomous ucks o pick up loads
on any day.
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100141
8
C. Lee e al.
Decla a ion o AI and AI-assis ed echnologies in he w i ing p o-
cess
Du ing he p epa a ion o his wo k he au ho s used Cha GPT o
imp o e eadabili y. A e using his se ice, he au ho s e iewed and
edi ed he con en as needed and ake ull esponsibili y o he con en
o he publica ion.
Acknowledgmen s
This esea ch was pa ly unded h ough a gi om Ryde and
pa ly suppo ed by he NSF AI Ins i u e o Ad ances in Op imiza ion
(Awa d 2112533). Special hanks o he Ryde eam o hei in aluable
suppo , expe ise, and insigh s.
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