Guillen Reyes, Fe nando O.; Gend eau, Michel; Po in, Jean-Y es
A icle
A me aheu is ic o a ime-dependen ehicle ou ing
p oblem wi h ime windows, wo ehicle lee s and
synch oniza ion on a oad ne wo k
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: Guillen Reyes, Fe nando O.; Gend eau, Michel; Po in, Jean-Y es (2024) : A
me aheu is ic o a ime-dependen ehicle ou ing p oblem wi h ime windows, wo ehicle lee s
and synch oniza ion on a oad ne wo k, 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. 1-20,
h ps://doi.o g/10.1016/j.ej l.2024.100143
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/325215
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
Con en s lis s a ailable a ScienceDi ec
EURO Jou nal on T anspo a ion and Logis ics
jou nal homepage: www.else ie .com/loca e/ej l
A me aheu is ic o a ime-dependen ehicle ou ing p oblem wi h ime
windows, wo ehicle lee s and synch oniza ion on a oad ne wo k
Fe nando O. Guillen Reyesa,c, Michel Gend eau b,c, Jean-Y es Po in a,c,∗
aDépa emen d’in o ma ique e de eche che opé a ionnelle, Uni e si é de Mon éal, Mon éal, Canada
bDépa emen de ma héma iques e de génie indus iel, Poly echnique Mon éal, Mon éal, Canada
cCen e In e uni e si ai e de Reche che su les Réseaux d’En ep ise, la Logis ique e le T anspo (CIRRELT), Mon éal, Canada
ARTICLE INFO
Keywo ds:
Vehicle ou ing p oblem
Road ne wo k
Time-dependen
Time windows
T ans e poin s
Synch oniza ion
Me aheu is ic
Slack induc ion by s ing emo als
ABSTRACT
In his wo k, we ex end he ime-dependen ehicle ou ing p oblem wi h ime windows on a oad ne wo k
by conside ing wo ypes o ehicles, la ge and small, o se e cus ome s. Mo i a ed om ci y logis ics
applica ions, la ge ehicles a e o bidden om he down own a ea. Acco dingly, goods mus be ans e ed
om la ge o small ehicles o se e down own cus ome s. This leads o synch oniza ion issues a ans e
poin s, which a e special loca ions wi hou s o age capaci y. The p oblem is no a pu e wo-echelon ehicle
ou ing p oblem, since cus ome s ou side o he down own a ea can be se ed di ec ly by la ge ehicles.
The p oblem is u he compounded by he p esence o ime-dependen a el imes ha a e de ined on
he a cs o he oad ne wo k and a e used o model conges ion pe iods. To sol e his di icul p oblem, we
p opose an adap a ion o he Slack Induc ion by S ing Remo als me aheu is ic, which is s a e-o - he-a o
he classical capaci a ed ehicle ou ing p oblem. Compu a ional esul s on a se o es ins ances wi h di e en
cha ac e is ics empi ically demons a e he op imiza ion capabili ies o his new me aheu is ic on a p oblem
which is much mo e complica ed han he capaci a ed ehicle ou ing p oblem.
1. In oduc ion
Al hough he ehicle ou ing p oblem (VRP) has been widely s ud-
ied o a long ime, ime-dependen a ian s ha e spu ed he in e es
o esea che s only ecen ly. Time-dependency is an impo an issue,
since he ime o a el om one poin o ano he in a ne wo k o en
depends on he depa u e ime (c. ., ush hou s). Fu he mo e, no
only does he ime o a el along a pa h be ween wo cus ome s may
change depending on he depa u e ime, bu e en he bes pa h o use
may also change. Thus, ecen s udies ha e exploi ed he addi ional
in o ma ion a ailable in a oad ne wo k o accoun o mul iple possible
pa hs be ween wo cus ome s, which is o en e e ed o as he ime-
dependen ehicle ou ing p oblem wi h ime windows on a oad
ne wo k (TDVRPTWRN). In his pape , we conside an ex ension o his
p oblem whe e bo h la ge (black) and small (g een) deli e y ehicles
a e in ol ed and whe e some pa s o he oad ne wo k a e o bidden
o one ype o ehicles o he o he . Fo example, he down own a ea
is no accessible o la ge ehicles, whe eas a eas a om down own
a e no accessible o small ehicles (e.g., bicycles). Since he goods o
be deli e ed a e ini ially loaded in la ge ehicles, a cus ome loca ed
in an a ea no accessible o hem can only be se ed h ough a ans e
o i s demand om a la ge o a small ehicle. This ans e akes place
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (F.O.G. Reyes), [email p o ec ed] (M. Gend eau), [email p o ec ed] (J.-Y. Po in).
a special loca ions wi h no s o age capaci y, known as ans e poin s
(TPs). This also leads o synch oniza ion issues be ween he wo ypes
o ehicles a ans e poin s. In he ollowing, his di icul deli e y
p oblem will be e e ed o as he TDVRPTWRN wi h ans e poin s o
TDVRPTWTPRN.
Ou p oblem needs o be dis inguished om p oblems wi h in e -
media e acili ies, wi h o wi hou s o age capaci y, since he e is no
acili y as such o ans e goods. I mus also be dis inguished om
wo- o mul i-echelon VRPs whe e ehicles a e o ganized in o a s ic
hie a chical s uc u e o deli e goods o cus ome s. In ou p oblem,
black ehicles can e y well se e cus ome s di ec ly, as long as hey do
no belong o o bidden a eas. Ou con ibu ion lies in he adap a ion
o a s a e-o - he-a me aheu is ic o he capaci a ed VRP (CVRP) o a
much mo e di icul p oblem ha in ol es wo ypes o ehicles, h ee
ypes o cus ome s, ime-dependen a el imes and synch oniza ion
be ween he wo ypes o ehicles o ans e goods a ans e poin s.
As a as we know, his p oblem has ne e been add essed in he
li e a u e.
In he ollowing, Sec ion 2 i s e iews p oblems ela ed o ou s,
namely VRPs on oad ne wo ks, ime-dependen VRPs and VRPs wi h
in e media e acili ies. Sec ion 3 hen p ecisely desc ibes ou p oblem.
h ps://doi.o g/10.1016/j.ej l.2024.100143
Recei ed 11 Ma ch 2024; Recei ed in e ised o m 30 Augus 2024; Accep ed 13 Sep embe 2024
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
A ailable online 18 Sep embe 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/ ).
F.O.G. Reyes e al.
The o iginal implemen a ion o he me aheu is ic Slack Induc ion by
S ing Remo als (SISR) o sol ing he CVRP is desc ibed in Sec ion 4.
Then, Sec ion 5in oduces ime issues ha a ise in he TDVRPTWTPRN,
in pa icula calcula ion o ime bounds o check inse ion easibili y in
cons an ime and synch oniza ion be ween black and g een ehicles a
ans e poin s. Speci ic modi ica ions o he o iginal SISR implemen-
a ion ha a e equi ed o add ess he TDVRPTWTPRN a e p esen ed
in Sec ion 6. Then, compu a ional esul s ob ained on es ins ances
de i ed om a benchma k o he TDVRPTWRN a e epo ed. Finally,
a conclusion ollows.
2. Li e a u e e iew
The main ea u es o ou p oblem a e (1) he conside a ion o a
ull oad ne wo k, (2) ime-dependen a el imes and (3) in e media e
poin s o ans e goods om one ype o ehicle o ano he . P oblems
wi h hese cha ac e is ics a e b ie ly e iewed in he ollowing.
2.1. VRPs on oad ne wo ks
In many VRPs, i is implici ly assumed ha he bes pa h be ween
wo cus ome s (o cus ome and depo ) can be uniquely iden i ied in
he unde lying oad ne wo k. Then, a so-called cus ome -based g aph
is cons uc ed, wi h nodes ha co espond o he cus ome s plus he
depo and wi h an a c be ween each pai o nodes ha s ands o he
co esponding bes pa h. Howe e , i is o en he case ha a single bes
pa h canno be iden i ied a p io i, o example when mul iple a ibu es
(dis ance, ime, cos ) a e associa ed wi h each oad segmen in he oad
ne wo k. Tha is, he bes pa h is no necessa ily he same depending
on he a ibu e conside ed. Fu he mo e, ade-o s be ween di e en
a ibu es a e disca ded i a single pa h is conside ed. Acco dingly,
wo king wi h a cus ome -based g aph educes he solu ion space and
may lead o an o e es ima ion o he op imum cos . To add ess his
issue, wo app oaches a e epo ed in he li e a u e: ep esen ing he
oad ne wo k as a mul ig aph o wo king di ec ly wi h he ull oad
ne wo k.
In mul ig aphs, cus ome -based g aphs a e ex ended by in oducing
pa allel a cs be ween each pai o nodes, whe e each a c s ands o a
di e en pa h ha is wo h conside ing in he unde lying oad ne wo k.
To he bes o ou knowledge, he i s use o a mul ig aph o a
mul i-a ibu e ehicle ou ing p oblem, namely a dial-a- ide p oblem,
is epo ed in Ga aix e al. (2010). The au ho s in Ben Ticha e al.
(2018), who su eyed a numbe o pape s based on mul ig aphs (up o
2018), indica e ha he la e p o ide a e age bene i s be ween 5% and
15% when compa ed o solu ions ob ained on cus ome -based g aphs.
One di icul y wi h mul ig aphs comes om hei cons uc ion, since
he se o pa allel a cs be ween wo nodes can be la ge and may be
di icul o ob ain (mul ic i e ia sho es pa h p oblems mus be sol ed
when mo e han one a ibu e is associa ed wi h each oad segmen ).
One may se le o only a subse o all possible pa allel a cs, bu a
he expense o a educed solu ion space. On he o he hand, wo king
wi h he ull oad ne wo k p ese es he en i e solu ion space. A ew
s udies compa e he use o mul ig aphs e sus oad-ne wo k g aphs
wi h somewha di e en obse a ions. In Ben Ticha e al. (2017,2019),
he au ho s empi ically demons a e he bene i s o using a mul ig aph
ep esen a ion e sus a cus ome -based g aph o a b anch-and-p ice
algo i hm and an adap i e la ge neighbo hood sea ch (ALNS) applied
o a bi-objec i e VRP wi h ime windows (VRPTW) ha accoun s o
bo h a el ime and a el cos . Howe e , hey no e he conside able
compu a ion imes needed o compu e he mul ig aph. In he same
con ex , he au ho s in Le ch o d e al. (2014) p opose o wo k di ec ly
wi h he oad ne wo k since he p icing p oblem in he b anch-and-
p ice algo i hm can be sol ed mo e quickly, while he cons uc ion o
a mul ig aph is a oided. Also, wo king wi h a oad ne wo k appea s
o be mo e na u al and s aigh o wa d. These obse a ions mus be
con as ed wi h hose in Ben Ticha e al. (2019), whe e a compa ison
be ween a b anch-and-p ice algo i hm applied o a oad ne wo k and
a mul ig aph ep esen a ion o a bi-objec i e VRPTW ( a el ime,
a el cos ) shows ha bo h app oaches a e compe i i e, wi h a sligh
ad an age o mul ig aphs on he mo e ealis ic ins ances. The eade
in e es ed in hose issues, as well as some addi ional ones, conce ning
VRPs de ined on mul ig aphs and oad ne wo ks, is e e ed o he
su ey in Ben Ticha e al. (2018).
In he la e su ey, he au ho s also no e ha iden i ying he bes
pa h be ween wo cus ome s in a oad ne wo k is no ha simple, e en
i a single a ibu e is conside ed. This is he case in pa icula o
ime-dependen a el imes o a el cos s, as discussed in he nex
sec ion.
2.2. Time-dependen VRPs
In TDVRPTWs, he a el ime along an a c depends on he de-
pa u e ime om he o igin node. When he objec i e unc ion is
no ela ed o ime, like he classical minimiza ion o o al a eled
dis ance, he sho es pa h in he oad ne wo k be ween each pai o
cus ome s (o cus ome and depo ) can be compu ed o cons uc a
cus ome -based g aph, gi en ha he sho es pa h does no change
o e ime. The a ia ions in a el imes a e hen accoun ed o along
hese sho es pa hs. This is he app oach used in he i s s udies abou
TDVRPTWs.
To he bes o ou knowledge, he i s wo k ha add essed ime-
dependency (al hough wi hou ime windows a cus ome s) using a
cus ome -based g aph is ound in Beasley (1981). In his wo k, he ime
ho izon is di ided in o pe iods wi h a di e en a el ime ma ix o
each pe iod. This is equi alen o de ining a s ep unc ion o model
he a el ime a di e en pe iods be ween any gi en pai o nodes. A
simila app oach is p oposed in Maland aki and Daskin (1992) o he
TDVRPTW. Since he a el ime is cons an wi hin a pe iod, bu may
ab up ly change om one pe iod o he nex , he wo p e ious models
do no sa is y he Fi s -In-Fi s -Ou (FIFO) p ope y, whe e i is equi ed
ha a ehicle a eling ea lie on an a c mus a i e a he des ina ion
node ea lie han any o he ehicle a eling la e on he same a c. A
di e en model is p oposed in Hill and Ben on (1992), whe e each node
is assigned a speed a a gi en pe iod o ime, which can be in e p e ed
as he a e age speed a ound he node. Then, he a el ime on a gi en
a c be ween wo nodes is based on he a e age speed a ound hese
wo nodes. Once again, since he a el ime on a gi en a c is cons an
wi hin a pe iod, he FIFO p ope y is no sa is ied. A model ha sa is ies
he FIFO p ope y was inally p oposed in Ichoua e al. (2003). He e,
he au ho s use a s ep unc ion o model speed ( a he han a el ime)
a di e en ime pe iods. Tha is, he speed is cons an wi hin a gi en
ime pe iod, bu may change om one pe iod o he nex . The a el
ime along an a c is hen compu ed by aking in o accoun he new
speed when he ime bounda y be ween wo pe iods is c ossed. Thus,
e e y ehicle ha a els along he same a c wi hin he same pe iod has
he same speed and he speed o e e y ehicle changes simila ly when
he bounda y be ween wo gi en pe iods is c ossed. This way o model
ime-dependency has been la gely adop ed in he ollowing yea s o
sol e TDVRPTWs using exac me hods and me aheu is ics (Dona i e al.,
2008;Dabia e al.,2013;Pan e al.,2021a,b). In a ew cases, con inuous
unc ions wi h special cha ac e is ics o sa is y he FIFO p ope y ha e
also been used o model ime-dependen a el imes (Haghani and
Jung,2005;Balsei o e al.,2011). Fo a de ailed li e a u e e iew on
ime-dependen VRPs using cus ome -based g aphs (up o 2015), he
eade is e e ed o Gend eau e al. (2015).
When mo e ealis ic objec i e unc ions based on a el imes o
a el cos s a e conside ed, a new di icul y occu s ha p e en s he use
o cus ome -based g aphs. Tha is, no only does he a el ime change
along a gi en pa h in he oad ne wo k depending on he depa u e
ime om he o igin node, bu e en he bes ( as es o leas -cos )
pa h may change. This is accoun ed o by using ei he a mul i-g aph
ep esen a ion o he ull oad ne wo k. In he case o mul i-g aphs,
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
2
F.O.G. Reyes e al.
pa allel a cs be ween any gi en pai o cus ome s s and o di e en
bes pa hs in he unde lying oad ne wo k depending on he depa u e
ime om he o igin cus ome node (Ben Ticha e al.,2017,2019). In
mo e ecen wo ks, he au ho s end o use he ull oad ne wo k o
handle hese mul iple al e na i e pa hs (Ben Ticha e al.,2019,2021;
Gmi a e al.,2021), o easons al eady men ioned in Sec ion 2.1.
2.3. VRPs wi h in e media e acili ies
Due o he p esence o ans e poin s in ou p oblem, we p o ide
he e an o e iew o he li e a u e on VRPs wi h in e media e acili ies,
which a e e e ed o as sa elli es, hubs, ansshipmen poin s o c oss-
docks. They all ep esen in e media e poin s whe e goods can be
ans e ed while hey mo e om hei o igin o hei des ina ion.
In Spe anza e al. (2016), he au ho s p o ide a su ey abou in e -
media e acili ies in eigh anspo a ion, while a su ey dedica ed o
c oss-docking is ound in Van Belle e al. (2012). In Spe anza e al.
(2016), he p oblems a e di ided in o wo classes, ha is, wo-echelon
VRPs (2E-VRPs) and pickup and deli e y p oblems wi h c oss-docks
(PDPCDs). Wi h ega d o 2E-VRPs, he in e media e acili ies a e called
sa elli es and ha e some s o age capaci y. A he i s -le el o echelon,
ehicles ca y goods om a depo o sa elli es, while a he second-le el
goods a e anspo ed by o he ehicles om sa elli es o cus ome s.
Typically, a s ic hie a chy is obse ed, ha is, di ec deli e ies om
he depo o cus ome s is o bidden. In his su ey, no wo k equi es
synch oniza ion be ween ehicles a sa elli es. In he case o he su -
eyed PDPCDs, howe e , c oss-docks ha e no o li le capaci y and
synch oniza ion is equi ed. Two wo ks a e wo h men ioning, since
he e is no eal in e media e acili y, only ans e poin s (like in ou
wo k). In Bou os e al. (2011), ans e s can ake place a a bi a y
loca ions and he ehicle ha a i es i s a he ans e poin wai s
as long as necessa y o ans e goods o he o he ehicle, al hough a
wai ing penal y is incu ed. In Minic and Lapo e (2006), ansshipmen
poin s a e p ede e mined and synch oniza ion is achie ed by se ing a
ime window a hese ans e poin s.
In C ainic e al. (2009), a 2E-VRP is p oposed in he con ex o
ci y logis ics, whe e i is called a wo- ie ci y logis ics sys em. Ci y
Dis ibu ion Cen e s (CDCs), loca ed a he ou ski s o he ci y, o m
he i s ie o he sys em whe e eigh is so ed and consolida ed.
The second ie o he sys em is made o sa elli es loca ed close o o
wi hin he ci y-cen e a ea. Di e en ehicle lee s a e used o anspo
eigh om CDCs o sa elli es and om sa elli es o cus ome s. In
pa icula , ehicles o he second ie mus be adap ed o u iliza-
ion in dense ci y zones. Since i is assumed ha sa elli es ope a e
acco ding o a c oss-dock ansshipmen ope a ional model, ehicle
synch oniza ion is equi ed. Tha is, ehicles o he i s and second
ie mus mee a sa elli es a a gi en ime, wi h e y sho wai ing
ime pe mi ed. This wo k p oposes only a modeling amewo k and
no algo i hmic solu ion is de eloped. Di e en exac and heu is ic
algo i hms we e la e p oposed in C ainic e al. (2011) and Pe boli
e al. (2010,2009) o sol e a ian s o he ini ial model (e.g., no
synch oniza ion, s o age capaci y a sa elli es). A ecen wo k in Jia
e al. (2022) add esses a wo-commodi y 2E-VRP wi h synch oniza ion
a sa elli es. Two ypes o ehicles a e conside ed a he i s le el,
one o each commodi y, as opposed o he second le el whe e only
one ype o ehicles is conside ed. Synch oniza ion is only es ablished
be ween he wo ypes o i s -le el ehicles, which ha e o mee a
sa elli es o a o e iciency a he second le el. The p oblem is sol ed
wi h ALNS whe e, a each i e a ion, des oy and a epai ope a o s a e
applied o he second-le el ou s, and a econs uc ion p ocedu e is
hen applied o he i s -le el ou s in case o in easibili y, ollowed by
an imp o emen p ocedu e. The 2E-VRP epo ed in Ande luh e al.
(2017) is pa icula ly in e es ing because is sha es simila i ies wi h
ou p oblem. In his wo k, he i s -le el ehicles a e called ans and
he second-le el ehicles a e called bicycles. Simila ly, he e a e wo
classes o cus ome s depending on hei loca ion: cus ome s loca ed a
he ci y cen e a e called bike-cus ome s, while cus ome s ou side o
he cen e a e called an-cus ome s. T ans e s ake place a sa elli es,
wi h no s o age capaci y. These sa elli es a e loca ed a he bounda y
o he ci y cen e (a an can c oss he ci y cen e , bu a penal y is
incu ed). In he p oposed heu is ic me hodology based on GRASP and
pa h- elinking, he second-le el ou s a e cons uc ed be o e he i s
le el ou s. In his way, in o ma ion abou he a i al imes o bikes a
sa elli es can be used o cons uc he i s -le el ou s and accoun o
synch oniza ion. The au ho s in G angie e al. (2016) add ess a simila
p oblem called he wo-echelon mul i- ip VRP wi h sa elli e synch o-
niza ion. The p oposed me hodology also cons uc s second-le el ou s
be o e i s -le el ou s o p oduce he ini ial solu ion. Then, an ALNS
is applied wi h ea u es aimed a imp o ing synch oniza ion, like a
des oy ope a o ha emo es ips wi h he wo s synch oniza ion.
In Ka le e al. (2017), he au ho s desc ibe a c owdsou ced sys em o
u ban pa cel deli e ies, whe e uck-ca ie s isi in e media e acili ies
called elay poin s and whe e pa cels a e ans e ed o pedes ians o
cyclis s ha a e close o he end cus ome s. Howe e , i no pedes ian
o cyclis is a ailable, he uck can pe o m he deli e ies i sel . In
he p oposed sys em, he deli e y asks, as well as candida e elay
poin s, a e b oadcas on-line. Then, pedes ians and cyclis s bid o
hese deli e y asks. Thus, a bid selec ion p oblem mus be sol ed in
addi ion o he ou ing p oblem. This is done in bo h cases wi h a abu
sea ch. Ano he simila c owdsou ced sys em is desc ibed In Sampaio
e al. (2020), whe e goods can be d opped a ans e poin s o be
picked up la e by o he ehicles (i.e., ans e poin s ha e s o age
capaci y).
3. P oblem de ini ion
This pape add esses he ime-dependen ehicle ou ing p ob-
lem wi h ime windows and ans e poin s on a oad ne wo k o
TDVRPTWTPRN. As p e iously men ioned, wo ypes o ehicles wi h
di e en capaci ies a e conside ed: black (la ge) and g een (small)
ehicles. The wo se s o ehicles a e deno ed 𝐾𝐵and 𝐾𝐺, espec i ely.
The e a e also h ee ypes o cus ome s: black cus ome s ha can be
se ed by black ehicles only; g een cus ome s ha can be se ed by
g een ehicles only; and neu al cus ome s ha can be se ed by bo h
ypes o ehicles.
A oad ne wo k in his con ex is a di ec ed g aph 𝐺= (𝑉 , 𝐴), whe e
𝑉is he se o nodes o ca dinali y 𝑛and 𝐴 he se o a cs o oad
segmen s. The se o nodes is hen pa i ioned as ollow:
-𝐷= {𝑑𝑏, 𝑑𝑔}is he se o depo s wi h 𝑑𝑏 he depo o black
ehicles and 𝑑𝑔 he depo o g een ehicles;
-𝐶𝐵is he se o black cus ome s o ca dinali y 𝑛𝐵;
-𝐶𝐺is he se o g een cus ome s o ca dinali y 𝑛𝐺;
-𝐶𝐸is he se o neu al cus ome s o ca dinali y 𝑛𝐸;
-TP is he se o ans e poin s o ca dinali y 𝑛TP;
-RJ is he se o oad junc ions (i.e., any node ha is no a depo ,
a cus ome o a ans e poin ).
The TDVRPTWTPRN can be cha ac e ized as ollow:
•Each cus ome 𝑖has a demand (load) 𝑑𝑖and a se ice (dwell) ime
𝑠𝑡𝑖;
•Each cus ome 𝑖has a ime window [𝛼𝑖, 𝛽𝑖] o cons ain he se ice
s a ime. I a ehicle a i es a cus ome 𝑖be o e 𝛼𝑖, hen i mus
wai un il 𝛼𝑖 o s a he se ice. On he o he hand, a ehicle
canno a i e a e 𝛽𝑖;
•The demand o all cus ome s is assumed o be loaded in o black
ehicles a he s a ;
•Each black ehicle pe o ms a single ou e ha s a s and ends a
he black depo ; each g een ehicle pe o ms a single ou e ha
s a s and ends a he g een depo ;
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
3
F.O.G. Reyes e al.
•The black and g een depo s ha e a ime window [0, 𝑇 ], whe e 𝑇
is he end o he ime ho izon; all ehicles mus be back a hei
depo be o e o a ime 𝑇;
•A g een ehicle can se e only one cus ome a a ime. Thus, a e
depa ing om he g een depo , a g een ehicle isi s epea edly
a ans e poin , o ge a load, ollowed by he co esponding
cus ome o deli e ha load. A he end, he ehicle e u ns o
he g een depo .
•A black ehicle has a capaci y 𝑄𝑏 ha allows i o se e many
cus ome s and o ca y loads ha will be ans e ed o g een
ehicles;
•Black and g een ehicles ha e di e en speeds. Thus, each a c
(𝑖, 𝑗) ∈ 𝐴is associa ed wi h wo ime-dependen a el speed unc-
ions 𝑣𝑏
𝑖,𝑗 (𝑡)and 𝑣𝑔
𝑖,𝑗 (𝑡) o black and g een ehicles, espec i ely.
•T ans e poin s a e ixed loca ions wi hou s o age capaci y,
whe e loads a e ans e ed om a black ehicle o one o mo e
g een ehicles (we assume, wi hou loss o gene ali y, ha he
ans e ime is null). Thus, synch oniza ion be ween he wo
ypes o ehicles is equi ed a ans e poin s, which may lead o
wai ing ime o a black ehicle (i one o mo e g een ehicles ha
mus ecei e loads om he black ehicle ha e no ye a i ed a
he ans e poin ) o g een ehicles (i he black ehicle has no
ye a i ed a he ans e poin ). A black ehicle can isi he
same ans e poin mul iple imes along i s ou e; he same is
ue o g een ehicles. Each isi o ans e poin p ∈TP in a
ou e is ep esen ed by a copy which is unambiguously deno ed
𝑡𝑝𝑘
𝑗, whe e 𝑘is a ehicle and 𝑗is he copy (o isi ) index. Tha
is, copy 𝑡𝑝𝑘
𝑗co esponds o he 𝑗 h isi o ans e poin p in he
ou e o ehicle 𝑘;
•Each black cus ome is se ed di ec ly and exac ly once by a black
ehicle; each g een cus ome is se ed exac ly once by a g een
ehicle a e i s demand has been ans e ed om a black ehicle
a a ans e poin ; each neu al cus ome is se ed exac ly once,
ei he di ec ly by a black ehicle o by a g een ehicle a e i s
demand has been ans e ed om a black ehicle a a ans e
poin ;
•The objec i e is o de e mine ou es o minimal o al du a ion
such ha all cus ome s a e se ed and all cons ain s a e sa is ied.
Fig. 1 shows a ypical solu ion, wi h one black ou e s a ing om
he black depo (squa e). This ou e is iden i ied by a cs (1) o (10).
A he i s ans e poin 𝑡𝑝1( iangle), he e is a connec ion wi h he
g een ou e wi h a cs iden i ied wi h b oken lines. This ou e s a s a
he g een depo (squa e), ge s a load om he black ehicle a 𝑡𝑝1,
deli e s he load o neu al cus ome 𝑛𝑐1, ge s ano he load om he
same black ehicle a he second ans e poin 𝑡𝑝2, deli e s he load
o g een cus ome 𝑔𝑐1and e u ns o he g een depo . The a cs o he
second g een ou e a e iden i ied wi h do ed lines. This small ou e
s a s a he g een depo , ge s a load om he black ehicle a 𝑡𝑝2,
deli e s he load o g een cus ome 𝑔𝑐2and e u ns o he g een depo .
I should be no ed ha he black ehicle ans e s wo loads, one o
each g een ehicle, a ans e poin 𝑡𝑝2.
4. SISR o he CVRP
The me hodology o sol ing ou p oblem is he Slack Induc ion by
S ing Remo als (SISR) me aheu is ic (Ch is iaens and Vanden Be ghe,
2020), which has p o en o be s a e-o - he-a o he CVRP. I is
based on he ALNS amewo k, ini ially p oposed in Ropke and Pisinge
(2006). Acco dingly, SISR also exploi s he uin-and- ec ea e p inciple
whe e, a each i e a ion, a numbe o nodes a e i s emo ed om
he ou es o he cu en solu ion ( uin) and einse ed ( ec ea e) o
p oduce a new solu ion. A simula ed annealing-based c i e ion is hen
applied o decide i he new solu ion should be accep ed o no as
he cu en solu ion. This amewo k has been used wi h success o
add ess many di e en ehicle ou ing p oblems, see Pisinge and
Ropke (2019). In he ollowing, we i s desc ibe he o iginal SISR
me aheu is ic o he CVRP. This desc ip ion is qui e de ailed o allow
he eade o ully unde s and la e he modi ica ions ha we ha e
pe o med o his algo i hm o add ess ou p oblem.
The basic idea o SISR is o emo e s ings o consecu i e cus ome s
om a solu ion, wi h a mos one s ing emo ed om any gi en ou e.
Algo i hm 1shows he pseudo-code o SISR o he CVRP. Fi s , wo pa-
ame e alues a e se : 𝐿𝑚𝑎𝑥, which is used o de e mine he maximum
leng h o a s ing, and 𝑐, which co esponds o he a e age numbe
o cus ome s o be emo ed om a solu ion. By app op ia ely se ing
hese alues, many s ings o small leng h o only a ew s ings o
la ge leng h can be emo ed. Gi en ha simula ed annealing p inciples
guide he sea ch h ough an exponen ial cooling schedule, he s a ing
empe a u e 𝜏0, inal empe a u e 𝜏𝑓,𝜏0> 𝜏𝑓>0, and numbe o
i e a ions 𝑓a e de ined, wi h he cu en empe a u e 𝜏ini ially se
o 𝜏0, see s a emen s 2 and 3. Then, he cooling ac o 𝜌is de ined
in s a emen 4 in such a way ha 𝑓 uin-and- ec ea e i e a ions a e
pe o med.
An adjacency lis 𝑎𝑑𝑗(𝑖)is hen c ea ed o each cus ome 𝑖in s a e-
men 5. This lis con ains all cus ome s o de ed om closes o a hes
in dis ance om 𝑖, wi h 𝑖as i s i s elemen . This adjacency lis is used
o a o he emo al o s ings ha a e ela i ely close o each o he ,
e en i hey come om di e en ou es. Be o e p oceeding wi h he
main loop, an ini ial solu ion is c ea ed in s ep 6 in a s aigh o wa d
way, by c ea ing an indi idual ou e o each cus ome . This ini ial
solu ion becomes he cu en solu ion 𝑠as well as he bes solu ion
known o da e 𝑠𝑏𝑒𝑠𝑡.
The main loop co esponds o s a emen s 8 o 18. A each i e a ion,
a uin ope a o and a ec ea e ope a o a e applied o a copy 𝑠o
cu en solu ion 𝑠, see s a emen s 9 and 10. No e ha he se 𝐴−is
used o s o e he emo ed cus ome s. The esul ing solu ion o he
uin-and- ec ea e p ocess 𝑠is accep ed as he new cu en solu ion 𝑠
i i sa is ies he simula ed annealing-based c i e ion in s a emen 11
(see Ch is iaens and Vanden Be ghe (2020)). I also eplaces 𝑠𝑏𝑒𝑠𝑡 i i
is he bes solu ion ound hus a . The cu en empe a u e 𝜏is hen
upda ed be o e he nex i e a ion s a s. A e 𝑓i e a ions o he main
loop, he whole p ocedu e s ops and e u ns he bes solu ion ound.
Algo i hm 1 SISR o CVRP
1: Se 𝐿𝑚𝑎𝑥 and 𝑐
2: Se 𝜏0,𝜏𝑓and 𝑓
3: 𝜏←𝜏0
4: 𝜌←(𝜏𝑓
𝜏0)1∕𝑓
5: Gene a e adjacency lis 𝑎𝑑𝑗(𝑖) o each cus ome 𝑖
6: Gene a e ini ial solu ion 𝑠(wi h se o ou es 𝑅𝑠)
7: 𝑠𝑏𝑒𝑠𝑡 ←𝑠
8: o 𝑓i e a ions do
9: 𝑠, 𝐴−←𝑅𝑢𝑖𝑛(𝑠)
10: 𝑠←𝑅𝑒𝑐𝑟𝑒𝑎𝑡𝑒(𝑠, 𝐴−)
11: i 𝐶𝑜𝑠𝑡(𝑠)<(𝐶𝑜𝑠𝑡(𝑠) − 𝜏ln(𝑈(0,1))) hen
12: 𝑠←𝑠;
13: end i
14: i 𝐶𝑜𝑠𝑡(𝑠)< 𝐶𝑜𝑠𝑡(𝑠𝑏𝑒𝑠𝑡) hen
15: 𝑠𝑏𝑒𝑠𝑡 ←𝑠
16: end i
17: 𝜏←𝜌𝜏
18: end o
19: Re u n 𝑠𝑏𝑒𝑠𝑡
4.1. Ruin
In he uin p ocedu e desc ibed in Algo i hm 2, he maximum leng h
o a s ing o be emo ed 𝑙𝑚𝑎𝑥
𝑠is i s se o he minimum o 𝐿𝑚𝑎𝑥
and he a e age numbe o nodes in a ou e o he cu en solu ion
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
4
F.O.G. Reyes e al.
Fig. 1. An example o a solu ion o he TDVRPTWTPRN.
Algo i hm 2 Ruin(𝑠)
1: 𝑙𝑚𝑎𝑥
𝑠←min{𝐿𝑚𝑎𝑥, 𝐴𝑣𝑔𝑁𝑜𝑑𝑒𝑠𝐼𝑛𝑅𝑜𝑢𝑡𝑒𝑠(𝑠)}
2: Calcula e 𝑛𝑚𝑎𝑥
𝑠wi h 𝑙𝑚𝑎𝑥
𝑠and 𝑐
3: 𝑛𝑠←⌊𝑈(1, 𝑛max
𝑠+ 1)⌋
4: 𝑅−←∅
5: 𝐴−←∅
6: 𝑠←𝑠 𝑅𝑠←𝑅𝑠
7: Selec andomly a seed cus ome 𝑖𝑠𝑒𝑒𝑑 in 𝑠
8: o 𝑖∈𝑎𝑑𝑗(𝑖𝑠𝑒𝑒𝑑 )and |𝑅−|< 𝑛𝑠do
9: 𝑟← ou e o cus ome 𝑖
10: i 𝑖∉𝐴−and 𝑟∉𝑅− hen
11: 𝑙max
𝑟←min{𝑙max
𝑠,|𝑟|}
12: 𝑙𝑟←⌊𝑈(1, 𝑙max
𝑟+ 1)⌋
13: 𝑅𝑢𝑖𝑛𝑂𝑝 ←𝑅𝑎𝑛𝑑𝑜𝑚(S ing,Spli -S ing)
14: 𝐴−←𝐴−∪𝑅𝑢𝑖𝑛𝑂𝑝(𝑠, 𝑟, 𝑙𝑟, 𝑖)
15: 𝑅−←𝑅−∪ {𝑟}
16: i 𝑟is emp y hen
17: 𝑅𝑠←𝑅𝑠∖{𝑟}
18: end i
19: end i
20: end o
21: Re u n 𝑠,𝐴−
Algo i hm 3 Rec ea e(𝑠,𝐴−)
1: So (𝐴−)⊳Rec ea e
2: o 𝑖∈𝐴−do
3: 𝑝𝑏𝑒𝑠𝑡 ←𝑁𝑈𝐿𝐿;𝐶𝑜𝑠𝑡𝐼𝑛𝑠𝑒𝑟𝑡𝑏𝑒𝑠𝑡 ←∞
4: o 𝑟∈𝑅𝑠and 𝑟 easible wi h inse ion o 𝑖do
5: o 𝑝𝑟in 𝑟do
6: i 𝑈(0,1) <1 − 𝛾 hen
7: i 𝑝𝑏𝑒𝑠𝑡 =𝑁𝑈𝐿𝐿 o 𝐶𝑜𝑠𝑡𝐼𝑛𝑠𝑒𝑟𝑡(𝑖, 𝑝𝑟)< 𝐶𝑜𝑠𝑡𝐼𝑛𝑠𝑒𝑟𝑡𝑏𝑒𝑠𝑡 hen
8: 𝑝𝑏𝑒𝑠𝑡 ←𝑝𝑟
9: 𝐶𝑜𝑠𝑡𝐼𝑛𝑠𝑒𝑟𝑡𝑏𝑒𝑠𝑡 ←𝐶𝑜𝑠𝑡𝐼𝑛𝑠𝑒𝑟𝑡(𝑖, 𝑝𝑟)
10: end i
11: end i
12: end o
13: end o
14: i 𝑝𝑏𝑒𝑠𝑡 =𝑁𝑈𝐿𝐿 hen
15: 𝑅𝑠←𝑅𝑠∪ {new emp y ou e 𝑟}
16: 𝑝𝑏𝑒𝑠𝑡 ← i s posi ion in 𝑟
17: end i
18: Inse 𝑖in posi ion 𝑝𝑏𝑒𝑠𝑡
19: end o
20: Re u n 𝑠
𝐴𝑣𝑔𝑅𝑜𝑢𝑡𝑒𝑁𝑜𝑑𝑒𝑠(𝑠), see s a emen 1. Then, in s a emen 2, he maximum
numbe o emo ed s ings 𝑛𝑚𝑎𝑥
𝑠is calcula ed using 𝑙𝑚𝑎𝑥
𝑠and 𝑐, see he
exac o mula in Ch is iaens and Vanden Be ghe (2020). The ac ual
numbe o emo ed s ings 𝑛𝑠is chosen om a con inuous uni o m
dis ibu ion de ined be ween 1 and 𝑛𝑚𝑎𝑥
𝑠+ 1, as indica ed in s a emen
3. The se o uined ou es 𝑅−and he se o emo ed cus ome s 𝐴−a e
hen ini ialized wi h he emp y se . A e c ea ing a copy 𝑠o he cu en
solu ion 𝑠, a andom seed cus ome 𝑖𝑠𝑒𝑒𝑑 is chosen and i s adjacency
lis is p ocessed ( om closes o a hes cus ome s) un il all cus ome s
ha e been conside ed o he numbe o uined ou es is eached, see
he main loop in s a emen s 8–20. No e ha he numbe o uined
ou es is he same as he numbe o emo ed s ings 𝑛𝑠, since each
s ing is emo ed om a di e en ou e. I he cu en cus ome 𝑖in
he adjacency lis o 𝑖𝑠𝑒𝑒𝑑 has no been p e iously emo ed and i he
ou e 𝑟 ha se es 𝑖has no been p e iously uined (s a emen 10)
hen he uin ope a o is applied o ou e 𝑟. In s a emen s 11 and 12,
he ac ual leng h o he emo ed s ing 𝑙𝑟is chosen om a con inuous
uni o m dis ibu ion de ined be ween 1 and 𝑙max
𝑟+ 1, whe e 𝑙max
𝑟is he
minimum o 𝑙max
𝑠and ca dinali y o 𝑟(since he leng h o he emo ed
s ing canno exceed he numbe o nodes in 𝑟).
Then, a andom choice be ween wo uin ope a o s akes place in
s a emen 13. These ope a o s a e:
•S ing: A andom s ing o leng h 𝑙𝑟 ha con ains he cu en
cus ome 𝑖in he adjacency lis o 𝑖𝑠𝑒𝑒𝑑 is emo ed om ou e
𝑟. This is illus a ed in Fig. 2(a) o a s ing o leng h ou wi h
he g ay node 𝑖3as cu en cus ome 𝑖;
•Spli -S ing: A andom s ing o leng h 𝑙𝑟+𝑚 ha con ains he
cu en cus ome 𝑖in he adjacency lis o 𝑖𝑠𝑒𝑒𝑑 is chosen in
ou e 𝑟(whe e he p ocedu e o selec a alue o 𝑚is p ecisely
desc ibed in Ch is iaens and Vanden Be ghe (2020)). Then, a
andom subs ing o 𝑚consecu i e cus ome s wi hin he chosen
s ing is kep in he ou e, so ha only 𝑙𝑟cus ome s a e emo ed.
The subs ing o leng h 𝑚cu s he s ing o leng h 𝑙𝑟+𝑚in wo
pa s, unless he subs ing is a he e y beginning o e y end o
he s ing o leng h 𝑙𝑟+𝑚. An example is p o ided in Fig. 2(b) o
a s ing o leng h i e wi h he g ay node 𝑖3as cu en cus ome 𝑖.
In his example, 𝑚= 2, so ha only h ee cus ome s a e emo ed
om he ou e.
The emo ed cus ome s a e hen added o 𝐴−and he uined ou e
o 𝑅−in s a emen s 14 and 15. I ou e 𝑟becomes emp y, hen i is
dele ed om he se o ou es in he solu ion, as indica ed in s a emen s
16 and 17. A he end, he uined solu ion 𝑠and he se o emo ed
cus ome s a e e u ned.
4.2. Rec ea e
A new comple e solu ion is hen p oduced wi h he ec ea e ope -
a o by einse ing he emo ed cus ome s. This ope a o is desc ibed
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
5
F.O.G. Reyes e al.
Fig. 2. Examples o he wo SISR uin ope a o s : (a) S ing (b) Spli -S ing.
in Algo i hm 3. In s a emen 1, he emo ed cus ome s in 𝐴−a e i s
so ed using a so ing c i e ion chosen wi h a pa icula dis ibu ion
p obabili y among: andom, dec easing demand, inc easing dis ance
om he depo , dec easing dis ance om he depo . Based on he
chosen o de , he cus ome s in 𝐴−a e conside ed one by one and
einse ed in he se o ou es 𝑅𝑠o solu ion 𝑠, see he main loop in
s a emen s 2–19. Each inse ion place in each ou e ha can accommo-
da e he demand o cus ome 𝑖is conside ed and he bes encoun e ed
inse ion place 𝑝𝑏𝑒𝑠𝑡 is iden i ied. I should be no ed, howe e , ha he
chosen inse ion place is no necessa ily he bes one among all easible
inse ion places, due o blinks ha co espond o a small p obabili y 𝛾
o skipping a posi ion, see s a emen 6. In pa icula , i he bes posi ion
is skipped hen only he second bes posi ion can be chosen (as long as
his posi ion is no skipped oo). S a emen s 14–17 co e he si ua ion
when no easible inse ion place is ound o cus ome 𝑖. In his case,
a new ou e is c ea ed o ha cus ome . A he end, he ec ea ed
solu ion 𝑠is e u ned.
5. Time dependency
The SISR me aheu is ic o he CVRP, as desc ibed in he p e ious
sec ion, needs o be conside ably modi ied o add ess he
TDVRPTWTPRN. In pa icula , he ime dimension mus now be aken
in o accoun ; u he mo e, ou es a e no independen anymo e since
hey in e ac h ough ans e poin s. In his sec ion, we in oduce he
basics o ou ime-dependen a el ime model and explain how ime
bounds can be de i ed a each node along he ou es o black and g een
ehicles.
5.1. Time-dependen a el imes
The IGP model p oposed in Ichoua e al. (2003) is used o model
ime dependency. In his model, he ime ho izon [0, 𝑇 ]is pa i ioned
in o a numbe 𝑙o ime pe iods [0, 𝑡1),[𝑡1, 𝑡2), ...., [𝑡𝑙−2, 𝑡𝑙−1),[𝑡𝑙−1, 𝑇 ],
whe e 𝑡1,𝑡2, ..., 𝑡𝑙−1 a e ime bounda ies be ween wo pe iods. Fo
any gi en a c, a a el speed is associa ed wi h each pe iod and a speed
change occu s when a ehicle c osses a ime bounda y. The algo i hmic
p ocedu e o compu e he a el ime along an a c o a gi en depa u e
ime based on his model is p o ided in Ichoua e al. (2003). Al hough
speed is modeled as a s ep unc ion o ime, he co esponding a el
ime unc ion is a piecewise linea unc ion. Fig. 3 shows an example
o a a el speed unc ion on a gi en a c (𝑖, 𝑗)and he co esponding
a el ime unc ion, assuming ha he a c is o leng h 4.
5.2. Dominan sho es -pa h s uc u e
The dominan sho es -pa h s uc u e (DSPS), as desc ibed in Gmi a
e al. (2021), is use ul o quickly iden i y he as es pa h be ween any
gi en pai o nodes (ei he cus ome s, depo s o ans e poin s) in he
oad ne wo k o any gi en depa u e ime. Fi s , a numbe o good
pa hs be ween wo gi en nodes 𝑖and 𝑗a e iden i ied by applying a
ime-dependen Dijks a’s algo i hm (Gmi a e al.,2021) using di e en
depa u e imes om 𝑖, like ime bounda ies be ween wo pe iods.
The a el ime unc ion o each one o hose pa hs is ob ained by
combining he a el ime unc ions o all a cs along ha pa h (which
also p oduces a piecewise linea unc ion). Fig. 4 shows an example o a
DSPS based on h ee di e en as es pa hs be ween wo nodes. In his
igu e, he a i al ime is ep esen ed as a unc ion o he depa u e
ime, so ha he co esponding a el ime is simply he di e ence
be ween a i al and depa u e imes. Since he IGP model sa is ies
he FIFO p ope y, his piecewise linea unc ion is non dec easing. By
o e lapping he h ee pa hs, i is possible o iden i y he as es among
he h ee pa hs o any gi en depa u e ime. I is wo h no ing ha he
DSPS is exac only i he ime-dependen Dijks a’s algo i hm is applied
wi h a su icien ly la ge numbe o depa u e imes o co e all as es
pa hs be ween wo nodes, which is a ely he case in p ac ice. Bu be e
accu acy is ob ained wi h mo e depa u e imes.
Two di e en a el ime unc ions a e associa ed wi h each a c,
depending i a black o a g een ehicle ollows ha a c, because hey
do no ha e he same speed. This leads o wo di e en DSPSs be ween
each pai o nodes made o ei he cus ome s, depo s o ans e poin s.
Acco dingly, he ollowing no a ion will be used:
•𝐴𝑇 𝑏(𝑖, 𝑗, 𝑑𝑡)is he a i al ime a 𝑗when a black ehicle depa s
om 𝑖a ime d and ollows he as es pa h o each 𝑗, as
de e mined by he DSPS o nodes 𝑖and 𝑗 o a black ehicle;
•𝐷𝑇 𝑏(𝑖, 𝑗, 𝑎𝑡)is he in e se o 𝐴𝑇𝑏(𝑖, 𝑗, 𝑑𝑡)and is he depa u e ime
a 𝑖 ha allows a black ehicle o a i e a 𝑗a ime a ;
•𝐴𝑇 𝑔(𝑖, 𝑗, 𝑑𝑡)is he a i al ime a 𝑗when a g een ehicle depa s
om 𝑖a ime d and ollows he as es pa h o each 𝑗, as
de e mined by he DSPS o he pai o nodes 𝑖and 𝑗 o a g een
ehicle;
•𝐷𝑇 𝑔(𝑖, 𝑗, 𝑎𝑡)is he in e se o 𝐴𝑇 𝑔(𝑖, 𝑗, 𝑑𝑡)and is he depa u e ime
a 𝑖 ha allows a g een ehicle o a i e a 𝑗a ime a .
5.3. Synch oniza ion a a ans e poin
Le us conside 𝑡𝑝𝑘𝑏
⋅a copy o ans e poin p ∈TP in he ou e o
black ehicle 𝑘𝑏∈𝐾𝐵, whe e he subsc ip ⋅co esponds o a pa icula
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
6
F.O.G. Reyes e al.
Fig. 3. (a) T a el speed unc ion o a c (𝑖,𝑗) (b) Co esponding a el ime unc ion assuming ha a c (𝑖,𝑗) is o leng h 4.
Fig. 4. (a) Th ee di e en as es pa hs be ween wo nodes ob ained a di e en ime poin s using a ime-dependen Dijks a’s algo i hm (b) Co esponding dominan sho es pa h
s uc u e.
copy ( isi ) index o ans e poin p in he ou e o ehicle 𝑘𝑏. Le us
also conside 𝑡𝑝𝑘𝑔
1
⋅,𝑡𝑝𝑘𝑔
2
⋅, ..., 𝑡𝑝𝑘𝑔
ℎ
⋅,ℎcopies o ans e poin p in he
ou es o g een ehicles 𝑘𝑔
1,𝑘𝑔
2, ..., 𝑘𝑔
ℎ∈𝐾𝐺. We assume ha black
ehicle 𝑘𝑏needs o be synch onized wi h g een ehicles 𝑘𝑔
1,𝑘𝑔
2, ..., 𝑘𝑔
ℎ
a hese copies o ans e poin p.
We i s accoun o he a i al ime o he las g een ehicle:
𝑎𝑡𝑚𝑎𝑥 = max
𝑙=1,…,ℎ{𝑎𝑡
𝑡𝑝𝑘𝑔
𝑙
⋅
}(1)
Then, he depa u e ime o black ehicle 𝑘𝑏 om 𝑡𝑝𝑘𝑏
⋅is :
𝑑𝑡𝑡𝑝𝑘𝑏
⋅
= max{𝑎𝑡𝑡𝑝𝑘𝑏
⋅
, 𝑎𝑡𝑚𝑎𝑥}(2)
Tha is, i all g een ehicles a i e a he ans e poin be o e black
ehicle 𝑘𝑏, hen he la e can depa immedia ely (gi en ha he ime
o ans e loads om he black ehicle o g een ehicles is null) wi h
𝑑𝑡𝑡𝑝𝑘𝑏
⋅
=𝑎𝑡𝑡𝑝𝑘𝑏
⋅
. O he wise, ehicle 𝑘𝑏will depa a he a i al ime 𝑎𝑡𝑚𝑎𝑥
o he las g een ehicle.
The depa u e o each g een ehicle 𝑘𝑔
𝑙 om 𝑡𝑝𝑘𝑔
𝑙
⋅,𝑙=1, ..., ℎ, is:
𝑑𝑡
𝑡𝑝𝑘𝑔
𝑙
⋅
= max{𝑎𝑡𝑡𝑝𝑘𝑏
⋅
, 𝑎𝑡
𝑡𝑝𝑘𝑔
𝑙
⋅
}, 𝑙 = 1,…, ℎ (3)
Tha is, i g een ehicle 𝑘𝑔
𝑙a i es a he ans e poin be o e black
ehicle 𝑘𝑏, i mus wai o he a i al o ehicle 𝑘𝑏be o e i can depa
om 𝑡𝑝𝑘𝑔
𝑙
⋅. O he wise, i can depa immedia ely wi h 𝑑𝑡
𝑡𝑝𝑘𝑔
𝑙
⋅
=𝑎𝑡
𝑡𝑝𝑘𝑔
𝑙
⋅
.
5.4. Time bounds
In he ollowing, we de ine ea lies and la es ime bounds o he
a i al a and depa u e om each node in he ou e o a black o
g een ehicle, whe e a node can be a cus ome , a ans e poin o a
depo . Tha is, a ehicle mus a i e a (depa om) a node be o e i s
la es a i al (depa u e) ime o gua an ee ha he es o he ou e
sa is ies he ime cons ain s. Fo simpli ica ions pu poses, he o wa d
and backwa d p opaga ion p ocedu es desc ibed below ocus on a
single black o g een ou e and do no accoun o possible complex
in e ac ions among ou es (see Sec ion 5.5) .
5.4.1. G een ou e
He e, we explain how o p opaga e he ea lies and la es a i al
and depa u e imes in a g een ou e. Fo his pu pose, le us conside
he ou e o g een ehicle 𝑘𝑔∈𝐾𝐺which is made o (1) a copy 𝑑𝑔
0o
he g een depo o s a he ou e, (2) a sequence o copies o one o
mo e ans e poin s 𝑡𝑝𝑘𝑔
𝑙⋅,𝑙=1, ..., 𝑝, each ollowed by a g een (o
neu al) cus ome 𝑖𝑙∈𝐶𝐺∪𝐶𝐸,𝑙=1, ..., 𝑝and (3) a copy 𝑑𝑔
𝑝+1 o he
g een depo o end he ou e. Tha is, he ou e o g een ehicle 𝑘𝑔is
𝑑𝑔
0,𝑡𝑝𝑘𝑔
1⋅,𝑖1,𝑡𝑝𝑘𝑔
2⋅,𝑖2, ..., 𝑡𝑝𝑘𝑔
𝑝⋅,𝑖𝑝,𝑑𝑔
𝑝+1.
Ea lies a i al and depa u e imes
Fi s , he ea lies depa u e ime om 𝑑𝑔
0is se equal o 0. Then, we
go o wa d by i s compu ing he ea lies a i al ime ea a ans e
poin 𝑡𝑝𝑘𝑔
1⋅, using unc ion 𝐴𝑇 𝑔:
𝑒𝑎𝑡𝑡𝑝𝑘𝑔
1⋅
=𝐴𝑇 𝑔(𝑑𝑔
0, 𝑡𝑝𝑘𝑔
1⋅,0) (4)
Now, o de e mine he ea lies depa u e ime ed , we need o
accoun o he co esponding black ehicle 𝑘𝑏 om which g een
ehicle 𝑘𝑔should ecei e a load. Acco dingly, i 𝑒𝑎𝑡𝑡𝑝𝑘𝑔
1⋅
< 𝑒𝑎𝑡𝑡𝑝𝑘𝑏
1⋅
, hen
𝑒𝑑𝑡𝑡𝑝𝑘𝑔
1⋅
=𝑒𝑎𝑡𝑡𝑝𝑘𝑏
1⋅
, since g een ehicle 𝑘𝑔canno depa om he ans e
poin be o e he ea lies a i al ime o black ehicle 𝑘𝑏. O he wise,
𝑒𝑑𝑡𝑡𝑝𝑘𝑔
1⋅
=𝑒𝑎𝑡𝑡𝑝𝑘𝑔
1⋅
, gi en ha he ime o ans e a load is null.
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
7
F.O.G. Reyes e al.
S ill going o wa d, we now conside cus ome 𝑖𝑖and compu e i s
ea lies depa u e ime as :
𝑒𝑎𝑡𝑖1=𝐴𝑇 𝑔(𝑡𝑝𝑘𝑔
1⋅, 𝑖1, 𝑒𝑑𝑡𝑡𝑝𝑘𝑔
1⋅
)(5)
Now, i 𝑒𝑎𝑡𝑖1< 𝛼𝑖1, hen he ea lies depa u e ime 𝑒𝑑𝑡𝑖1=𝛼𝑖1+𝑠𝑡𝑖1,
o he wise 𝑒𝑑𝑡𝑖1=𝑒𝑎𝑡𝑖1+𝑠𝑡𝑖1.
This o wa d p ocedu e is epea ed un il he end depo 𝑑𝑔
𝑝+1 is
eached and i s ea lies a i al ime is de e mined.
La es a i al and depa u e imes
We s a by se ing he la es a i al ime la a he end depo 𝑑𝑔
𝑝+1 o
be he end o ime ho izon 𝑇, ha is 𝑙𝑎𝑡𝑑𝑔
𝑝+1 =𝑇. Then, we go backwa d
by i s compu ing he la es depa u e ime ld a cus ome 𝑖𝑝 ha
allows ehicle 𝑘𝑔 o a i e a 𝑑𝑔
𝑝+1 a ime 𝑙𝑎𝑡𝑑𝑔
𝑝+1 , using unc ion 𝐷𝑇 𝑔:
𝑙𝑑𝑡𝑖𝑝=𝐷𝑇 𝑔(𝑖𝑝, 𝑑𝑔
𝑝+1, 𝑙𝑎𝑡𝑑𝑔
𝑝+1 )(6)
Now, i 𝑙𝑑𝑡𝑖𝑝> 𝛽𝑖𝑝+𝑠𝑡𝑖𝑝, hen 𝑙𝑑𝑡𝑖𝑝is ese o 𝛽𝑖𝑝+𝑠𝑡𝑖𝑝, because
ehicle 𝑘𝑔canno depa om 𝑖𝑝la e han 𝛽𝑖𝑝+𝑠𝑡𝑖𝑝wi hou iola ing
he ime window cons ain (i.e., he a i al ime canno exceed 𝛽𝑖𝑝).
Then, he la es a i al ime a cus ome 𝑖𝑝is simply compu ed as
𝑙𝑎𝑡𝑖𝑝=𝑙𝑑𝑡𝑖𝑝−𝑠𝑡𝑖𝑝.
S ill going backwa d, we now conside he ans e poin 𝑡𝑝𝑘𝑔
𝑝⋅and
compu e i s la es depa u e ime as :
𝑙𝑑𝑡𝑡𝑝𝑘𝑔
𝑝⋅
=𝐷𝑇 𝑔(𝑡𝑝𝑘𝑔
𝑝⋅, 𝑖𝑝, 𝑙𝑎𝑡𝑖𝑝)(7)
To de e mine he la es a i al ime o he g een ehicle 𝑘𝑔, we mus
accoun o he co esponding black ehicle 𝑘𝑏 ha ans e s a load
o ehicle 𝑘𝑔. Tha is, he g een ehicle canno a i e a e he la es
depa u e ime o black ehicle 𝑘𝑏 h ough he ollowing o mula :
𝑙𝑎𝑡𝑡𝑝𝑘𝑔
𝑝⋅
= min{𝑙𝑑𝑡𝑡𝑝𝑘𝑔
𝑝⋅
, 𝑙𝑑𝑡𝑡𝑝𝑘𝑏
𝑝⋅
}(8)
This backwa d p ocedu e is applied un il he s a ing depo 𝑑𝑔
0is
eached and i s la es depa u e ime is de e mined. I should be no ed
ha a o wa d p opaga ion s a ing om he la es depa u e ime a
𝑑𝑔
0, un il 𝑑𝑔
𝑝+1 is eached, would p oduce he la es easible schedule
(i.e., la es possible a i al and depa u e imes a each node along he
g een ou e).
5.4.2. Black ou e
He e, we explain how o p opaga e he ea lies and la es a i al
and depa u e imes in a black ou e. Fo his pu pose, le us conside
he ou e o black ehicle 𝑘𝑏∈𝐾𝐵which is made o (1) a copy 𝑑𝑏
0o
he black depo o s a he ou e, (2) an a bi a y sequence o leng h
𝑝o black (o neu al) cus ome s and copies o one o mo e ans e
poin s and (3) a copy 𝑑𝑏
𝑝+1 o he black depo o end he ou e.
Ea lies a i al and depa u e imes
The p ocedu e o compu e he ea lies a i al and depa u e imes
in a black ou e is simila o he one desc ibed o he g een ou e,
bu wo di e ences a e no ewo hy: (1) he unc ion 𝐴𝑇 𝑏is used o
compu e he a i al ime a a gi en node om he ea lies depa u e
ime o he p e ious node and (2) he ea lies depa u e ime a a copy
o a ans e poin is compu ed di e en ly, because g een ehicles ha
isi he same ans e poin o ge a load om he black ehicle mus
be accoun ed o .
Conside ing case (2), le us suppose ha black ehicle 𝑘𝑏 isi s copy
𝑡𝑝𝑘𝑏
⋅o ans e poin p ∈TP and ha ℎg een ehicles 𝑘𝑔
1, 𝑘𝑔
2,…, 𝑘𝑔
ℎ
isi copies 𝑡𝑝𝑘𝑔
1
⋅, 𝑡𝑝𝑘𝑔
2
⋅,…, 𝑡𝑝𝑘𝑔
ℎ
⋅o he same ans e poin and ha syn-
ch oniza ion is equi ed (i.e., black ehicle 𝑘𝑏mus ans e a load o
each g een ehicle). To compu e he ea lies depa u e ime o ehicle
𝑘𝑏a he ans e poin , we i s conside he maximum ea lies a i al
ime o e all g een ehicles, ha is:
𝑒𝑎𝑡𝑚𝑎𝑥 = max
𝑙=1,…,ℎ{𝑒𝑎𝑡
𝑡𝑝𝑘𝑔
𝑙
⋅
}(9)
Then, he ea lies depa u e ime o ehicle 𝑘𝑏a 𝑡𝑝𝑘𝑏
⋅can be
compu ed om i s ea lies a i al ime as ollow:
𝑒𝑑𝑡𝑡𝑝𝑘𝑏
⋅
= max{𝑒𝑎𝑡𝑚𝑎𝑥, 𝑒𝑎𝑡𝑡𝑝𝑘𝑏
⋅
}(10)
Tha is, black ehicle 𝑘𝑏canno depa ea lie han he ea lies
a i al ime o he las g een ehicle, o he wise one o mo e g een
ehicles will no ge hei load.
La es a i al and depa u e imes
The p ocedu e o compu e he la es a i al and depa u e imes
o a black ehicle is simila o he one desc ibed o a g een ehicle,
al hough wo di e ences a e no ewo hy: (1) he unc ion 𝐷𝑇 𝑏is used
o compu e he depa u e ime om a gi en node o each he nex
node a i s la es a i al ime and (2) he la es a i al ime a a copy
o a ans e poin is compu ed di e en ly, because g een ehicles ha
isi he same ans e poin o ge a load om he black ehicle mus
be accoun ed o .
Conside ing case (2), le us suppose ha black ehicle 𝑘𝑏 isi s copy
𝑡𝑝𝑘𝑏
⋅o ans e poin p ∈TP and ha ℎg een ehicles 𝑘𝑔
1, 𝑘𝑔
2,…, 𝑘𝑔
ℎ
isi copies 𝑡𝑝𝑘𝑔
1
⋅, 𝑡𝑝𝑘𝑔
2
⋅,…, 𝑡𝑝𝑘𝑔
ℎ
⋅o he same ans e poin and ha syn-
ch oniza ion is equi ed (i.e., black ehicle 𝑘𝑏mus ans e a load o
each g een ehicle). To compu e he la es a i al ime o ehicle 𝑘𝑏a
he ans e poin , we i s conside he minimum la es depa u e ime
o e all g een ehicles, ha is:
𝑙𝑑𝑡𝑚𝑖𝑛 = min
𝑙=1,…,ℎ{𝑙𝑑𝑡
𝑡𝑝𝑘𝑔
𝑙
⋅
}(11)
Then, he la es a i al ime o ehicle 𝑘𝑏a 𝑡𝑝𝑘𝑏
⋅can be compu ed
om i s la es depa u e ime as ollow:
𝑙𝑎𝑡𝑡𝑝𝑘𝑏
⋅
= min{𝑙𝑑𝑡𝑚𝑖𝑛, 𝑙𝑑𝑡𝑡𝑝𝑘𝑏
⋅
}(12)
Tha is, black ehicle 𝑘𝑏canno a i e a he ans e poin la e
han he minimum la es depa u e ime o e all g een ehicles ha
equi e synch oniza ion, o he wise one o mo e g een ehicles will no
ge hei load.
5.5. In e ac ion among ou es
In he p e ious sec ion, ou desc ip ion o o wa d and backwa d
p opaga ion p ocedu es o de i e ime bounds has ocused on a single
black o g een ou e. Howe e , complex in e ac ions may occu when
mul iple black and g een ou es a e in ol ed.
Fig. 5 shows an example whe e cus ome 𝑖is inse ed be ween
nodes p e and nex in black ou e 𝑘𝑏
2(as i occu s du ing he ec ea e
p ocedu e o SISR). Fo wa d p opaga ion is illus a ed in Fig. 5(a).
Fi s , he ea lies a i al and depa u e imes o he newly inse ed
cus ome 𝑖a e calcula ed om he ea lies depa u e ime a ans e
poin p e . Then, o wa d p opaga ion is igge ed along he black
ou e. Howe e , a g een ou e ha connec s he h ee black ou es is
encoun e ed a he ans e poin jus a e cus ome nex . Thus, ano he
o wa d p opaga ion is igge ed a his ans e poin along he g een
ou e which, in u n, leads o a ans e poin ha connec s he g een
ou e o black ou e 𝑘𝑏
3, hus igge ing ano he o wa d p opaga ion
along ha black ou e. I should be no ed ha his illus a ion is a wo s
case, because o wa d p opaga ion along a ou e e mina es as soon as
he ea lies depa u e ime om a node does no change.
Backwa d p opaga ion is illus a ed in Fig. 5(b). Fi s , la es a i al
and depa u e imes a he newly inse ed cus ome 𝑖a e calcula ed
om he la es a i al ime a cus ome nex . Then, backwa d p opa-
ga ion is igge ed along black ou e 𝑘𝑏
2. Since node p e is a ans e
poin , i igge s ano he backwa d p opaga ion along he co espond-
ing g een ou e. S ill going backwa d along he black ou e, ano he
ans e poin is me ha igge s backwa d p opaga ion along ano he
g een ou e. Finally, bo h g een ou es connec o black ou e 𝑘𝑏
1a he
same ans e poin , hus igge ing a backwa d p opaga ion along ha
black ou e also. Once again, his illus a ion is a wo s case, because
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
8
F.O.G. Reyes e al.
Fig. 13. Example o an ins ance gene a ed wi h oad ne wo k 𝑅𝑁3. (Fo colo s o he nodes, he eade is e e ed o he web e sion o his a icle).
only. Fo his eason, black cus ome s a e loca ed in he ou side
and bounda y egions, while g een cus ome s a e loca ed in he
down own and bounda y egions. Neu al cus ome s and ans e
poin s a e only ound in he bounda y egion, since hey can be
isi ed by bo h black and g een ehicles.
•The black depo is andomly loca ed in he ou side egion, while
he g een depo is andomly loca ed in he down own egion.
The e a e also 10 ans e poin s ha a e andomly loca ed in he
bounda y egion.
•A cs a e cha ac e ized by he egions whe e hey a e ound. Le
(𝑖, 𝑗)be an a c in he ne wo k. I bo h 𝑖and 𝑗a e in he bounda y
(o on ie ) egion, hen he a c is o ype F (and is accessible
o bo h black and g een ehicles). I bo h 𝑖and 𝑗a e in he
down own egion, o one is in he down own egion and he
o he in he bounda y egion, hen he a c is o ype D (and is
accessible only o g een ehicles). I bo h 𝑖and 𝑗a e in he ou side
egion, o i one is in he ou side egion and he o he is in he
bounda y egion, hen he a c is o ype O (and is accessible only
o black ehicles). I should be no ed ha no a cs connec he
down own and ou side egions. Gi en ha he egions a e he
same o a gi en ne wo k, hen he a c ype also s ays he same
o all ins ances gene a ed om a gi en oad ne wo k.
•The es ins ances a e duplica ed by conside ing wo di e en se s
o a el speed mul iplie s (scena ios), whe e a speed mul iplie
depends on he ime pe iod, ehicle (black o g een) and a c ype
(D, F, O), see Tables 1 and 2. In he wo scena ios 𝑆𝐼and 𝑆𝐼𝐼 , he
second and ou h pe iods co espond o ush hou s. The speed
mul iplie s o g een ehicles a e ixed a 1 e e ywhe e, which
means ha hey a e no a ec ed by conges ion since hey a e
small (e.g., bicycles). Black ehicles a e as e han g een ehicles
when he e is no conges ion. Howe e , hey a e slowe han g een
ehicles in he bounda y egion du ing ush hou s in scena io 𝑆𝐼
while hey ha e he same speed han g een ehicles in scena io
𝑆𝐼𝐼 . The second scena io is aimed a e alua ing he impac o
inc easing he speed o black ehicles when compa ed o g een
ehicles.
•The capaci y o black ehicles is se o 40.
•The demand o each cus ome is andomly selec ed om {1,…,
5}.
O e all, he e a e 4 oad ne wo ks ×3 numbe s o cus ome s (𝑛𝑐)×
4 cus ome dis ibu ions ×2 ypes o ime windows ×2 scena ios o a
o al o 192 ins ances, ha is, 96 ins ances o each scena io.
Table 1
Speed mul iplie s o (a) black ehicles and (b) g een ehicles unde scena io 𝑆𝐼.
(a) Black ehicles
A c ype Time pe iod
𝜏1= [0,20) 𝜏2= [20,30) 𝜏3= [30,70) 𝜏4= [70,80) 𝜏5= [80,100)
𝑇 𝑦𝑝𝑒 F 1.2 0.8 1.2 0.8 1.2
𝑇 𝑦𝑝𝑒 O 1.5 1.0 1.5 1.0 1.5
(b) G een ehicles
A c ype Time pe iod
𝜏1= [0,20) 𝜏2= [20,30) 𝜏3= [30,70) 𝜏4= [70,80) 𝜏5= [80,100)
𝑇 𝑦𝑝𝑒 D 1.0 1.0 1.0 1.0 1.0
𝑇 𝑦𝑝𝑒 F 1.0 1.0 1.0 1.0 1.0
Table 2
Speed mul iplie s o (a) black ehicles and (b) g een ehicles unde scena io 𝑆𝐼𝐼 .
(a) Black ehicles
A c ype Time pe iod
𝜏1= [0,20) 𝜏2= [20,30) 𝜏3= [30,70) 𝜏4= [70,80) 𝜏5= [80,100)
𝑇 𝑦𝑝𝑒 F 1.5 1.0 1.5 1.0 1.5
𝑇 𝑦𝑝𝑒 O 2.0 1.5 2.0 1.5 2.0
(b) G een ehicles
A c ype Time pe iod
𝜏1= [0,20) 𝜏2= [20,30) 𝜏3= [30,70) 𝜏4= [70,80) 𝜏5= [80,100)
𝑇 𝑦𝑝𝑒 D 1.0 1.0 1.0 1.0 1.0
𝑇 𝑦𝑝𝑒 F 1.0 1.0 1.0 1.0 1.0
7.2. Pa ame e uning
Fou pa ame e s ha e a signi ican impac on he pe o mance o
ou SISR, namely, 𝑐(a e age numbe o emo ed cus ome s), 𝑛𝑝𝑜𝑠
(numbe o bes inse ion posi ions, based on he app oxima ion), 𝑛𝑓𝑡𝑝
(numbe o nea es easible ans e poin s) and 𝐿𝑚𝑎𝑥 (maximum leng h
o emo ed s ings). To adjus hei alues, we selec ed a subse o
16 uning ins ances wi h 100 cus ome s by andomly selec ing only
one o he ou ne wo ks, o each possible con igu a ion o cus ome
dis ibu ion (𝐷1,𝐷2,𝐷3,𝐷4), ime window (NTW, WTW) and scena io
(𝑆1,𝑆2). Gi en ha solu ion quali y ends o imp o e wi h inc easing
alues o 𝑛𝑝𝑜𝑠 and 𝑛𝑓𝑡𝑝, a he expense o compu a ion ime, hese wo
pa ame e s we e i s se o high alues, ha is, 𝑛𝑝𝑜𝑠 =7 and 𝑛𝑓𝑡𝑝 =
10 ( he la e alue canno be la ge , since he e a e only 10 ans e
poin s in each ins ance). In o he wo ds, we did no ca e a his poin
abou compu a ion ime. Then, we ocused on pa ame e s 𝑐and 𝐿𝑚𝑎𝑥
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
15
F.O.G. Reyes e al.
Fig. 14. Con e gence cu es obse ed o ins ances wi h 200 cus ome s gene a ed wi h oad ne wo k 𝑅𝑁4unde scena io 𝑆𝐼. (Fo colo s o he cu es, he eade is e e ed o
he web e sion o his a icle).
Table 3
Impac o pa ame e alues on solu ion quali y and compu a ion imes.
Pa ame e alues
𝑐=9 11 13 15 17 19
A g. cos 2552.1 2532.7 2523.3 2520.3 2520.1 2520.0
A g. ime 1.29 1.54 1.81 2.10 2.36 2.56
𝑛𝑝𝑜𝑠 =12345678910
A g. cos 2537.9 2524.6 2521.4 2521.9 2520.3 2521.0 2520.3 2520.2 2519.6 2520.6
A g. ime 1.76 1.92 1.93 1.99 2.00 2.05 2.1 2.16 2.16 2.21
𝑛𝑓𝑡𝑝 =12345678910
A g. cos 2531.2 2525.5 2522.0 2520.2 2520.6 2518.9 2520.1 2520.3 2520.4 2520.3
A g. ime 0.83 1.15 1.41 1.61 1.75 1.88 1.97 2.02 2.06 2.10
𝐿𝑚𝑎𝑥 =12345678910
A g. cos 2542.9 2522.1 2520.2 2519.7 2520.4 2520.0 2520.1 2520.3 2519.8 2520.5
A g. ime 2.02 2.14 2.09 2.09 2.08 2.10 2.11 2.10 2.08 2.08
and uned hem wi h he IRACE so wa e (López-Ibánez e al.,2016),
using 𝑐= {5,…,17} and 𝐿𝑚𝑎𝑥 = {3,…,13}. The bes alues e u ned
by IRACE we e 𝑐= 15 and 𝐿𝑚𝑎𝑥 = 8. Based on he de aul con igu a ion
𝑐= 15,𝑛𝑝𝑜𝑠 =7, 𝑛𝑓𝑡𝑝 =10 and 𝐿𝑚𝑎𝑥 =8, we hen modi ied he
alue o one pa ame e a a ime, keeping he o he pa ame e s a
hei de aul alue. The alues conside ed o each pa ame e we e:
𝑐= {9,11,13,15,17,19},𝑛𝑝𝑜𝑠 = {1,…,10},𝑛𝑓𝑡𝑝 = {1,…,10} and 𝐿𝑚𝑎𝑥 =
{1,…,10}. Since ou algo i hm is non de e minis ic, we show he
a e age esul s (solu ion quali y, compu a ion ime in hou s) ob ained
o e 10 uns on each uning ins ance in Table 3.
As expec ed, inc easing he alues o pa ame e s 𝑛𝑝𝑜𝑠 and 𝑛𝑓𝑡𝑝
leads o an inc ease in compu a ion ime, al hough he impac is
mo e signi ican in he case o 𝑛𝑓𝑡𝑝, since i inc eases he numbe o
possible inse ions o g een and neu al cus ome s in Phase III and Phase
IV o he Rec ea e me hod ( hese inse ions a e qui e complica ed).
The compu a ion imes inc ease e en mo e wi h inc easing alues
o pa ame e 𝑐because mo e emo ed cus ome s simply mean mo e
cus ome s o be einse ed. On he o he hand, pa ame e 𝐿𝑚𝑎𝑥 has
no impac on compu a ion ime. Wi h ega d o solu ion quali y, we
obse e an imp o emen in solu ion quali y o he i s alues o each
pa ame e , bu hen some kind o s agna ion is obse ed. Acco dingly,
he pa ame e se ing 𝑐=15, 𝑛𝑝𝑜𝑠 = 3,𝑛𝑓𝑡𝑝 = 4 and 𝐿𝑚𝑎𝑥 = 4 was
chosen o he expe imen s epo ed in he ollowing sec ions. We also
checked ha his pa icula combina ion o pa ame e alues led o
good solu ions on he uning ins ances, which u ned o be ue wi h
an a e age solu ion cos o 2522.2 and a e age compu a ion ime o
1.47 h.
Some expe imen s we e also pe o med wi h ega d o he numbe
o i e a ions. We obse ed ha con e gence is ob ained, e en on he
la ges ins ances wi h 200 cus ome s, a e a maximum o 300,000
i e a ions. Tha is, a pla eau is eached and no u he signi ican im-
p o emen in solu ion quali y is obse ed. Fig. 14 shows an example o
con e gence cu es o he bes solu ions ound on ins ances wi h 200
cus ome s gene a ed wi h oad ne wo k 𝑅𝑁4, using he ou cus ome
dis ibu ions, and bo h na ow and wide ime windows, unde scena io
𝑆𝐼. In his igu e, black, g een, g ay and blue cu es a e associa ed wi h
cus ome dis ibu ions 𝐷1,𝐷2,𝐷3and 𝐷4, espec i ely, while ull lines
and b oken lines a e associa ed wi h ins ances wi h na ow and wide
ime windows, espec i ely. Based on he esul s ob ained, he numbe
o i e a ions was se o 300,000 o all ins ances (which is admi edly
oo much o ins ances wi h 50 and 100 cus ome s).
7.3. Resul s on es ins ances
We epo in his sec ion he esul s p oduced by ou algo i hm on
he whole se o es ins ances, based on 10 di e en uns on each
ins ance. Table 8 in he Appendix epo s he bes and a e age cos s,
as well as he a e age compu a ion imes in hou s o each ins ance.
Each line o his able co esponds o a pa icula ype o ins ance using
he no a ion 𝑅𝑁𝑥_𝑛𝑦_𝐷𝑧, whe e 𝑥is he oad ne wo k index, 𝑦is he
numbe o cus ome s and 𝑧is he cus ome dis ibu ion index. Fo each
ype, we show he esul s ob ained on ins ances associa ed wi h na ow
ime windows (NTW) and wide ime windows (WTW) unde scena ios
𝑆𝐼and 𝑆𝐼𝐼 .Table 4 in his sec ion is a educed e sion, whe e a e ages
a e aken o e he ou oad ne wo ks. Tha is, 𝑛𝑦_𝐷𝑧 in Table 4
encompasses 𝑅𝑁1_𝑛𝑦_𝐷𝑧,𝑅𝑁2_𝑛𝑦_𝐷𝑧,𝑅𝑁3_𝑛𝑦_𝐷𝑧 and 𝑅𝑁4_𝑛𝑦_𝐷𝑧, so
ha he numbe s in Table 4 co espond o he bold A g. lines in he
ull able in he Appendix.
Since he e a e no simila ins ances in he li e a u e ha we could
e e o o compa ison pu poses, Table 5 epo s bo h he a e age bes
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
16
F.O.G. Reyes e al.
Table 4
Solu ion cos and compu a ion ime in hou s o each subse o ins ances.
Ins ances NTW WTW
A g. bes cos A g. cos A g. ime A g. bes cos A g. cos A g. ime
𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼
n50_D1 1397.8 1125.3 1398.9 1126.6 0.27 0.28 1219.0 929.5 1220.5 930.0 0.26 0.26
n50_D2 2002.5 1841.0 2004.5 1844.5 0.77 0.87 1801.4 1608.4 1805.2 1615.5 0.76 0.81
n50_D3 1368.4 1112.9 1374.8 1118.0 0.66 0.69 1117.0 899.2 1117.0 901.8 0.59 0.54
n50_D4 1748.9 1517.1 1752.0 1520.4 0.53 0.56 1542.4 1279.1 1543.8 1280.3 0.51 0.51
n100_D1 2498.1 2030.9 2505.7 2034.1 0.64 0.70 2128.9 1702.6 2140.4 1710.0 0.64 0.65
n100_D2 3684.7 3298.5 3696.0 3308.6 2.18 2.37 3314.9 2909.0 3326.5 2922.0 2.06 2.13
n100_D3 2368.0 1976.9 2372.9 1979.9 1.86 1.92 1916.3 1564.2 1921.6 1572.9 1.42 1.39
n100_D4 3138.3 2734.4 3148.2 2756.3 1.34 1.55 2743.1 2303.3 2754.8 2316.5 1.35 1.34
n200_D1 4186.9 3461.1 4221.5 3489.7 1.71 1.88 3582.6 2822.7 3610.5 2855.8 1.42 1.57
n200_D2 6980.9 6356.7 7028.4 6408.2 5.76 6.47 6321.5 5691.5 6380.1 5758.7 5.47 5.69
n200_D3 4399.9 3638.8 4439.4 3672.7 4.54 5.03 3636.4 2918.6 3663.4 2941.5 3.44 3.52
n200_D4 5613.9 4894.5 5642.2 4930.7 3.70 4.21 4927.8 4255.0 4967.9 4294.4 3.41 3.57
O e all A g. 3282.4 2832.3 3298.7 2849.1 2.00 2.21 2854.3 2407.0 2871.0 2425.0 1.78 1.83
Table 5
A e age bes imp o emen s and a e age imp o emen s o each subse o ins ances.
Ins ances NTW WTW
A g. bes Imp . A g. Imp . A g. bes Imp . A g. Imp .
𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼
n50_D1 13.02 14.14 10.48 11.19 16.03 18.38 13.48 15.03
n50_D2 13.28 15.33 11.21 13.15 16.75 19.36 14.28 16.51
n50_D3 33.53 41.27 30.95 38.51 35.42 43.14 32.02 39.05
n50_D4 11.20 14.87 9.49 11.64 15.06 18.21 12.62 15.52
n100_D1 17.59 20.28 15.96 18.46 22.32 24.56 20.33 22.70
n100_D2 17.55 19.68 15.34 17.67 19.48 22.70 17.63 20.68
n100_D3 41.76 46.49 39.68 44.93 44.31 48.55 42.14 45.87
n100_D4 14.67 17.62 13.27 15.28 20.16 22.18 17.98 20.56
n200_D1 25.57 27.73 24.00 25.46 29.57 31.24 27.73 29.54
n200_D2 18.76 22.39 17.32 21.05 21.01 23.97 19.50 22.20
n200_D3 46.53 52.75 45.27 51.05 48.17 54.24 46.51 52.07
n200_D4 19.75 22.61 18.46 20.89 23.95 25.71 21.95 23.81
O e all A g. 22.77 26.26 20.95 24.11 26.02 29.35 23.85 26.96
imp o emen s and a e age imp o emen s p o ided by ou algo i hm
o e he ini ial solu ions, o measu e i s op imiza ion powe . Deno ing
𝑠𝑖and 𝑠𝑓 he ini ial and inal solu ions p oduced by ou algo i hm on
a gi en ins ance, wi h 𝑐𝑜𝑠𝑡(𝑠𝑖)and 𝑐𝑜𝑠𝑡(𝑠𝑓) hei espec i e cos , he
pe cen age o imp o emen o he inal solu ion o e he ini ial one
is calcula ed as ollow:
𝐼𝑚𝑝𝑟 = 100 (𝑐𝑜𝑠𝑡(𝑠𝑖) − 𝑐𝑜𝑠𝑡(𝑠𝑓)
𝑐𝑜𝑠𝑡(𝑠𝑖))(14)
Tables 4 and 5will be e e ed o in he ollowing sec ions when
we analyze in mo e de ail he beha io o ou algo i hm o di e en
cha ac e is ics o he es ins ances. Fo now, we can obse e he
ob ious inc ease in solu ion cos and compu a ion ime wi h ins ance
size in Table 4. We also no e he o e all a e age imp o emen s o e he
ini ial solu ions in Table 5 ha s and be ween 20% and 30%, which
is subs an ial. I is clea ha he g eedy inse ion heu is ic is limi ed
wi h ega d o solu ion quali y, bu s ill, hese pe cen ages o imp o e-
men show ha ou algo i hm can ake ad an age o op imiza ion
oppo uni ies.
7.4. Synch oniza ion
In his sec ion, we examine i synch oniza ion be ween black and
g een ehicles a ans e poin s is e icien . Fo his pu pose, we de ine
he pe cen age o solu ion cos (du a ion) ha co esponds o he o al
ime ha black and g een ehicles spend a ans e poin s. Fo a gi en
solu ion 𝑠, his pe cen age is deno ed as 𝜌𝑠. In Eq. (15), his pe cen age
Table 6
Minimum, maximum and a e age alues o 𝜌𝑠 o each subse o ins ances.
Ins ances Min Max A g.
n50_D1 0.12 2.18 0.77
n50_D2 1.05 3.71 2.08
n50_D3 0.03 2.55 1.02
n50_D4 0.18 3.23 1.60
A g. 0.34 2.92 1.37
n100_D1 0.01 3.03 0.90
n100_D2 0.82 3.11 1.91
n100_D3 0.19 1.97 0.95
n100_D4 0.45 3.13 1.48
A g. 0.37 2.81 1.31
n200_D1 0.45 2.45 1.25
n200_D2 0.84 3.37 1.91
n200_D3 0.25 2.20 0.86
n200_D4 0.53 2.85 1.63
A g. 0.52 2.71 1.41
O e all A g. 0.01 3.71 1.36
is de ined using 𝛥TP𝑏
𝑠and 𝛥TP𝑔
𝑠, which a e he o al ime spen a
ans e poin s by black and g een ehicles, espec i ely.
𝜌𝑠= 100 (𝛥TP𝑏
𝑠+𝛥TP𝑔
𝑠
𝑐𝑜𝑠𝑡(𝑠))(15)
Table 6 epo s he minimum, maximum and a e age alues o 𝜌𝑠
on di e en subse s o ins ances. The o ma o his able is educed
when compa ed o Tables 4 and 5by also a e aging o e he wo
ypes o ime windows and he wo scena ios. We obse e ha he 𝜌𝑠
alues a e smalle o cus ome dis ibu ions 𝐷1and 𝐷3. In ac , i we
compu e he a e age 𝜌𝑠 alues o 𝐷1,𝐷2,𝐷3and 𝐷4, we ob ain 0.97%,
1.97%, 0.94% and 1.57%, espec i ely. The ac ha 𝐷1and 𝐷3lead o
be e synch oniza ion han 𝐷2and 𝐷4can be explained by hei small
pe cen age o g een cus ome s (20%), since hey a e he only ones o
which synch oniza ion a a ans e poin is manda o y. In any case,
he di e ences obse ed a e small in absolu e e ms. The ac is ha
only 1.36% (o e all a e age) o solu ion cos is due o synch oniza ion,
which indica es ha synch oniza ion is well achie ed.
7.5. Cus ome dis ibu ions
Table 4 shows ha cus ome dis ibu ions 𝐷1and 𝐷3a e he bes
o solu ion cos , while 𝐷1ou pe o ms he h ee o he dis ibu ions
o compu a ion ime. Bo h 𝐷1and 𝐷3ha e only 20% o g een cus-
ome s, which is bene icial o solu ion cos because hese cus ome s
equi e a de ou a a ans e poin o bo h a black and a g een
ehicle. Fu he mo e, 𝐷1has a la ge pe cen age o 60% o black
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
17
F.O.G. Reyes e al.
cus ome s ( e sus only 20% o neu al cus ome s), which is help ul
o compu a ion ime because only simple inse ions in black ou es
need o be conside ed o black cus ome s. Al hough 𝐷3is compe i i e
wi h 𝐷1 o solu ion cos , his is no he case o compu a ion ime.
Dis ibu ion 𝐷3has 60% o neu al cus ome s ( e sus only 20% o black
cus ome s) and hei po en ial inse ion in bo h black and g een ou es
need o be conside ed. As opposed o black ou es, inse ion in g een
ou es is mo e complica ed and mo e compu a ionally expensi e. Wi h
he la ges pe cen age o 60% o g een cus ome s, dis ibu ion 𝐷2is
consequen ly he wo s o solu ion cos and compu a ion ime.
When conside ing imp o emen s o e ini ial solu ions in Table 5,
he la ges imp o emen s a e associa ed wi h dis ibu ion 𝐷3. This
dis ibu ion has he la ges pe cen age o neu al cus ome s (60%)
and hese cus ome s o e mo e lexibili y o op imiza ion because
hey can be inse ed ei he in black o g een ou es. In his case,
he a e age pe cen age o neu al cus ome s belonging o black ou es
in he ini ial solu ions anges be ween 51.1% and 54.4%. Howe e ,
in he inal solu ions, hese pe cen ages d as ically inc ease be ween
94.6% and 98.6%. Tha is, he op imiza ion algo i hm inds ways o
mo e a la ge p opo ion o neu al cus ome s in black ou es, which
dec eases solu ion cos . Acco dingly, mo e neu al cus ome s means
mo e oppo uni ies o imp o emen .
7.6. Time windows
Tables 4 and 5show ha be e solu ion cos s and la ge im-
p o emen s o e ini ial solu ions a e associa ed wi h ins ances wi h
wide ime windows. Clea ly, hese ime windows o e mo e easible
inse ion places o cus ome s and, consequen ly, g ea e lexibili y o
he op imiza ion p ocedu e o mo e hem a ound. Fo example, he
pe cen age o neu al cus ome s in black ou es o he ins ances wi h
wide ime windows is 98.3%, as compa ed wi h 92.8% o ins ances
wi h na ow ime windows. We also obse ed a educed numbe o
ou es in he solu ions ob ained on ins ances wi h wide ime windows,
when compa ed wi h na ow ime windows, namely 25.2% less black
ou es and 15.1% less g een ou es. Finally, black ehicles isi 2%
mo e ans e poin s on a e age in he p esence o wide ime windows.
7.7. Scena ios
Be e solu ion cos s a e obse ed in Table 4 unde scena io 𝑆𝐼𝐼 ,
when compa ed wi h scena io 𝑆𝐼. Since black ehicles now a el
as e , he ime windows a black and neu al cus ome s become easie
o sa is y. We also obse ed ha , on a e age, a la ge pe cen age o
neu al cus ome s is se ed by black ehicles in he inal solu ions
unde scena io 𝑆𝐼𝐼 (97.4%) when compa ed o scena io 𝑆𝐼(93.7%).
I is gene ally less cos ly o se e neu al cus ome s wi h black ehicles
and he la e can ge a la ge sha e o neu al cus ome s when hei
speed inc eases. In o he wo ds, hey win mo e o en he ‘‘ba le’’ o
neu al cus ome s agains g een ehicles in he bounda y egion. We
also obse ed a educ ion o 14.7% and 6.2% in he numbe o black
and g een ou es, espec i ely, unde scena io 𝑆𝐼𝐼 , wi h co esponding
inc eases o 20.5% and 3.5% in he a e age numbe o cus ome s in
black and g een ou es, espec i ely. Finally, black ehicles isi 4.4%
mo e ans e poin s on a e age, when compa ed o scena io 𝑆𝐼.
7.8. Numbe o ans e poin s
We p opose he e an expe imen whe e we g adually educe he
numbe o ans e poin s o see he impac on he solu ions ob ained.
To his end, some ans e poin s a e andomly selec ed and ans-
o med in o simple nodes ( oad junc ions). In some cases, in easible
ins ances may be c ea ed (e.g., i may no be possible o any g een
ehicle o isi a ans e poin and se e a g een cus ome be o e he
uppe bound o i s ime window).
Table 7
A e age gaps be ween a e age solu ion cos s wi h 10 ans e poin s and 𝑘= 8, 6, 4
ans e poin s.
Ins ances A g. Gap
𝑘= 8 𝑘= 6 𝑘= 4
RN1_n100_D1 −0.04 6.02 –
RN2_n100_D1 −0.10 −0.11 0.47
RN3_n100_D1 −0.02 −0.04 0.51
RN4_n100_D1 0.51 1.74 1.76
RN1_n100_D2 2.06 10.95 27.45
RN2_n100_D2 1.56 1.58 5.86
RN3_n100_D2 0.62 1.56 9.08
RN4_n100_D2 1.81 7.09 7.30
RN1_n100_D3 0.48 7.32 16.08
RN2_n100_D3 −0.05 −0.09 0.48
RN3_n100_D3 1.12 2.34 6.55
RN4_n100_D3 1.24 1.79 1.82
RN1_n100_D4 1.00 9.72 –
RN2_n100_D4 −0.86 −0.75 −0.02
RN3_n100_D4 0.19 1.21 4.45
RN4_n100_D4 0.86 1.47 1.68
He e, we ocus on he es ins ances wi h 100 cus ome s. Fi s , wo
ans e poin s a e andomly chosen and emo ed om he 10 o iginal
ones o ob ain ins ances wi h 8 ans e poin s. F om hese 8 ans e
poin s, he p ocedu e is epea ed o ge ins ances wi h 6 ans e poin s.
Finally, wo addi ional ans e poin s a e emo ed o ge ins ances wi h
4 ans e poin s. Ten uns we e pe o med on each easible ins ance
wi h a educed numbe o ans e poin s. Table 7 shows he a e age
gaps be ween he a e age cos o solu ions ob ained wi h 10 ans e
poin s and wi h 𝑘= 8,6,4 ans e poin s, based on Eq. (16). Each line
in his able is he a e age o e ou ins ances, conside ing ha he e
a e wo ypes o ime windows and wo scena ios.
𝐺𝑎𝑝 = 100 (𝑎𝑣𝑒|𝑇 𝑃 |=𝑘
𝑐𝑜𝑠𝑡 −𝑎𝑣𝑒|𝑇 𝑃 |=10
𝑐𝑜𝑠𝑡
𝑎𝑣𝑒|𝑇 𝑃 |=10
𝑐𝑜𝑠𝑡 )(16)
We obse ed no in easible ins ance wi h 𝑘=8 ans e poin s, one
in easible ins ance wi h 𝑘=6 ans e poin s and 19 in easible ins ances
(ou o 64 ins ances) wi h 𝑘=4. Thus, some a e ages a e compu ed
wi h less han ou gaps. When he e is no alue, he ou ins ances
a e in easible. A ew small nega i e alues appea in he able, which
means ha sligh ly be e solu ions a e ob ained wi h ewe ans e
poin s. This si ua ion can occu , due o he andomized na u e o ou
algo i hm, when he emo ed ans e poin s a e no o seldom used
in he o iginal ins ances, hus p oducing no o li le impac on solu ion
quali y.
Ob iously, emo ing ans e poin s gene ally lead o wo se solu-
ions and his end is mo e p onounced when mo e ans e poin s
a e emo ed. Cus ome dis ibu ion 𝐷2is mo e a ec ed han he o he
dis ibu ions due o i s la ge pe cen age o g een cus ome s (which
mus use ans e poin s). We also see ha ins ances associa ed wi h
oad ne wo k 𝑅𝑁1a e g ea ly a ec ed when 4 o 6 ans e poin s a e
emo ed. In he solu ions ob ained wi h 10 ans e poin s, we obse ed
ha some ans e poin s a e much mo e exploi ed han o he s. Clea ly,
such c i ical ans e poin s a e mo e likely o disappea when mo e
ans e poin s a e emo ed, which in u n g ea ly impac solu ion
quali y.
8. Conclusion
In his wo k, we ha e p oposed a new SISR me aheu is ic o he
TDVRPTWTPRN. To he bes o ou knowledge, his challenging p ob-
lem whe e cus ome s can be se ed ei he di ec ly by black ehicles
o indi ec ly by g een ehicles h ough ans e poin s has no been
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
18
F.O.G. Reyes e al.
Table 8
Solu ion cos and compu a ion ime in hou s o each ins ance.
Ins ances NTW WTW
Bes cos A g. cos A g. ime Bes cos A g. cos A g. ime
(10 uns) (10 uns) (h) (10 uns) (10 uns) (h)
𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼 𝑆𝐼𝑆𝐼𝐼
RN1_n50_D1 1563.47 1217.90 1563.91 1217.90 0.30 0.29 1331.90 1030.23 1332.03 1030.23 0.26 0.29
RN2_n50_D1 1365.48 1096.91 1365.48 1096.91 0.25 0.23 1159.71 898.09 1165.04 898.09 0.26 0.24
RN3_n50_D1 1430.09 1159.05 1432.17 1159.65 0.25 0.27 1292.61 950.84 1292.81 950.85 0.25 0.24
RN4_n50_D1 1232.26 1027.51 1233.85 1031.95 0.28 0.34 1091.71 839.01 1092.13 840.98 0.27 0.28
A g. 1397.83 1125.34 1398.85 1126.60 0.27 0.28 1218.98 929.54 1220.50 930.04 0.26 0.26
RN1_n50_D2 2295.75 2067.62 2295.83 2069.44 0.81 0.90 2018.20 1737.63 2018.72 1742.08 0.82 0.89
RN2_n50_D2 1929.28 1752.89 1932.68 1753.13 0.74 0.81 1664.44 1476.99 1664.84 1478.37 0.71 0.77
RN3_n50_D2 1996.08 1864.24 1996.14 1875.20 0.69 0.71 1843.62 1696.06 1855.69 1713.47 0.63 0.71
RN4_n50_D2 1788.88 1679.38 1793.19 1680.13 0.83 1.06 1679.18 1523.08 1681.66 1527.97 0.88 0.88
A g. 2002.50 1841.03 2004.46 1844.48 0.77 0.87 1801.36 1608.44 1805.23 1615.47 0.76 0.81
RN1_n50_D3 1440.17 1183.37 1449.07 1184.69 0.65 0.68 1234.84 983.12 1234.86 985.14 0.61 0.53
RN2_n50_D3 1365.98 1087.79 1378.58 1105.24 0.72 0.66 1059.19 845.14 1059.19 853.41 0.57 0.53
RN3_n50_D3 1353.17 1120.44 1356.42 1121.83 0.57 0.61 1094.61 931.62 1094.61 931.62 0.59 0.53
RN4_n50_D3 1314.31 1059.94 1314.97 1060.22 0.70 0.78 1079.52 836.87 1079.52 837.14 0.60 0.58
A g. 1368.41 1112.88 1374.76 1118.00 0.66 0.69 1117.04 899.19 1117.05 901.83 0.59 0.54
RN1_n50_D4 1786.66 1587.00 1786.66 1587.00 0.49 0.53 1679.56 1365.57 1679.56 1365.71 0.56 0.47
RN2_n50_D4 1944.47 1629.15 1947.70 1630.98 0.54 0.51 1622.76 1328.29 1622.76 1330.34 0.57 0.61
RN3_n50_D4 1535.58 1376.85 1537.07 1386.52 0.48 0.47 1355.06 1195.54 1355.06 1197.90 0.41 0.45
RN4_n50_D4 1728.69 1475.37 1736.48 1477.26 0.61 0.72 1512.20 1227.06 1517.89 1227.06 0.49 0.51
A g. 1748.85 1517.09 1751.98 1520.44 0.53 0.56 1542.39 1279.12 1543.82 1280.25 0.51 0.51
RN1_n100_D1 2526.91 2085.89 2528.87 2088.53 0.64 0.70 2189.86 1662.22 2198.75 1675.03 0.65 0.56
RN2_n100_D1 2559.89 2067.83 2570.32 2071.26 0.53 0.65 2161.96 1726.10 2177.80 1727.87 0.56 0.59
RN3_n100_D1 2540.37 2113.93 2540.48 2115.17 0.64 0.65 2200.53 1889.07 2202.66 1891.56 0.70 0.78
RN4_n100_D1 2365.21 1856.08 2383.25 1861.46 0.73 0.81 1963.41 1533.17 1982.42 1545.58 0.63 0.69
A g. 2498.09 2030.93 2505.73 2034.11 0.64 0.70 2128.94 1702.64 2140.41 1710.01 0.64 0.65
RN1_n100_D2 3804.11 3452.62 3812.33 3461.66 2.07 2.22 3426.28 2988.77 3434.23 3006.73 2.03 1.99
RN2_n100_D2 3537.95 3237.98 3543.50 3246.02 1.84 2.37 3256.69 2866.68 3265.31 2873.14 2.05 2.03
RN3_n100_D2 3919.34 3520.71 3936.62 3534.99 2.12 2.18 3538.40 3158.21 3551.07 3168.52 1.79 1.98
RN4_n100_D2 3477.32 2982.62 3491.33 2991.65 2.69 2.72 3038.18 2622.38 3055.51 2639.41 2.36 2.53
A g. 3684.68 3298.48 3695.95 3308.58 2.18 2.37 3314.89 2909.01 3326.53 2921.95 2.06 2.13
RN1_n100_D3 2480.34 1996.41 2488.51 1997.98 1.83 1.7 2124.82 1635.38 2127.03 1645.98 1.58 1.29
RN2_n100_D3 2315.24 1990.81 2319.78 1996.14 1.70 1.71 1923.52 1612.65 1930.14 1618.16 1.37 1.44
RN3_n100_D3 2435.62 2051.60 2440.28 2054.78 2.09 2.03 1851.77 1569.09 1856.39 1573.41 1.32 1.36
RN4_n100_D3 2240.82 1868.86 2243.17 1870.65 1.82 2.23 1764.91 1439.53 1772.65 1454.09 1.40 1.47
A g. 2368.00 1976.92 2372.94 1979.89 1.86 1.92 1916.26 1564.16 1921.55 1572.91 1.42 1.39
RN1_n100_D4 3534.04 2929.31 3538.32 2940.50 1.56 1.65 3044.64 2398.28 3048.74 2415.50 1.43 1.31
RN2_n100_D4 2807.13 2505.56 2827.05 2573.06 1.09 1.27 2444.41 2141.93 2456.91 2151.48 1.07 1.16
RN3_n100_D4 3161.54 2876.90 3168.41 2882.75 1.28 1.64 2742.29 2428.80 2765.12 2448.75 1.48 1.50
RN4_n100_D4 3050.61 2625.61 3058.81 2628.78 1.43 1.65 2741.18 2244.28 2748.61 2250.31 1.41 1.38
A g. 3138.33 2734.35 3148.15 2756.27 1.34 1.55 2743.13 2303.32 2754.84 2316.51 1.35 1.34
RN1_n200_D1 4586.99 3673.59 4607.01 3698.37 1.97 1.84 3890.26 2964.06 3911.65 3002.76 1.59 1.63
RN2_n200_D1 4006.30 3338.27 4037.61 3380.21 1.41 1.76 3420.59 2774.14 3444.88 2805.14 1.32 1.62
RN3_n200_D1 4235.06 3605.87 4287.61 3632.18 1.83 1.97 3632.22 3011.05 3649.54 3032.49 1.27 1.43
RN4_n200_D1 3919.20 3226.55 3953.88 3247.98 1.62 1.95 3387.47 2541.72 3436.05 2582.96 1.52 1.59
A g. 4186.89 3461.07 4221.53 3489.68 1.71 1.88 3582.63 2822.74 3610.53 2855.84 1.42 1.57
RN1_n200_D2 7428.48 6656.70 7468.67 6667.52 5.64 6.09 6650.45 5893.97 6692.08 5936.45 5.23 5.60
RN2_n200_D2 6883.99 6320.66 6938.59 6428.07 5.29 6.62 6220.31 5650.36 6267.01 5712.54 5.40 5.85
RN3_n200_D2 7143.03 6595.11 7201.80 6636.35 6.35 6.69 6615.83 6020.32 6681.27 6079.96 5.38 5.24
RN4_n200_D2 6468.03 5854.25 6504.63 5900.71 5.76 6.48 5799.34 5201.53 5879.89 5306.00 5.86 6.07
A g. 6980.88 6356.68 7028.42 6408.16 5.76 6.47 6321.48 5691.54 6380.06 5758.74 5.47 5.69
RN1_n200_D3 4333.66 3631.80 4385.00 3671.93 4.12 4.57 3647.47 2941.04 3684.66 2960.16 2.97 2.99
RN2_n200_D3 4336.39 3638.83 4385.59 3681.60 4.46 5.27 3622.43 2916.56 3644.45 2943.40 3.56 4.12
RN3_n200_D3 4600.69 3735.66 4628.11 3767.51 4.49 4.84 3818.68 3096.20 3835.78 3112.43 3.53 3.35
RN4_n200_D3 4328.94 3548.88 4358.82 3569.77 5.06 5.45 3457.12 2720.75 3488.66 2749.87 3.69 3.63
A g. 4399.92 3638.79 4439.38 3672.70 4.54 5.03 3636.43 2918.64 3663.39 2941.46 3.44 3.52
RN1_n200_D4 5863.31 5023.25 5872.71 5048.40 4.19 4.35 5056.61 4284.61 5081.22 4323.29 3.37 3.26
RN2_n200_D4 5302.13 4717.90 5314.75 4770.30 2.82 3.42 4661.26 4074.33 4685.65 4125.05 2.93 3.15
RN3_n200_D4 5842.90 5220.16 5902.48 5258.46 4.38 4.93 5141.24 4588.17 5213.82 4629.61 3.86 3.95
RN4_n200_D4 5447.38 4616.55 5478.77 4645.50 3.42 4.17 4851.89 4073.05 4890.97 4099.56 3.50 3.93
A g. 5613.93 4894.47 5642.18 4930.67 3.70 4.21 4927.75 4255.04 4967.92 4294.38 3.41 3.57
O e all a g. 3282.36 2832.34 3298.69 2849.13 2.00 2.21 2854.27 2406.95 2870.99 2424.95 1.78 1.83
p e iously add essed in he li e a u e. Compu a ional esul s on es in-
s ances wi h di e en cha ac e is ics show ha ou algo i hm pe o ms
well, in pa icula by inding ways o ans e mo e neu al cus ome s
in o black ou es, which lead o solu ions o be e quali y. Fu he -
mo e, we obse ed ha he ime spen by ehicles a ans e poin s is
e y low, hus indica ing ha good synch oniza ion is achie ed.
Fo he u u e, new uin ope a o s could be de eloped o enhance
he pe o mance o ou algo i hm and sol e o he di icul a ian s o
ehicle ou ing p oblems. I would also be in e es ing o conside he
in eg a ion o lea ning in o ou algo i hm, o example o iden i y he
mos p omising ans e poin s, based on he opology o he ne wo k
and dis ibu ion o cus ome s.
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
19
F.O.G. Reyes e al.
CRediT au ho ship con ibu ion s a emen
Fe nando O. Guillen Reyes: Da a cu a ion, In es iga ion, Me hod-
ology, So wa e, Valida ion, W i ing – o iginal d a . Michel Gen-
d eau: Concep ualiza ion, Funding acquisi ion, Me hodology, P ojec
adminis a ion, Resou ces, Supe ision, Valida ion, W i ing – e iew
& edi ing. Jean-Y es Po in: Concep ualiza ion, Funding acquisi ion,
Me hodology, Supe ision, W i ing – e iew & edi ing, P ojec admin-
is a ion, Resou ces.
Decla a ion o compe ing in e es
The au ho s decla e he ollowing inancial in e es s/pe sonal ela-
ionships which may be conside ed as po en ial compe ing in e es s:
Jean-Y es Po in epo s inancial suppo was p o ided by Na u al
Sciences and Enginee ing Resea ch Council o Canada. I he e a e o he
au ho s, hey decla e ha hey ha e no known compe ing inancial in-
e es s o pe sonal ela ionships ha could ha e appea ed o in luence
he wo k epo ed in his pape .
Acknowledgmen s
The wo k epo ed in his pape was inancially suppo ed by he
Na u al Sciences and Enginee ing Resea ch Council (NSERC) o Canada.
This suppo is g a e ully acknowledged.
Appendix
See Table 8.
Re e ences
Ande luh, A., Hemmelmay , V., Nolz, P., 2017. Synch onizing ans and ca go bikes in
a ci y dis ibu ion ne wo k. CEJOR Cen . Eu . J. Ope . Res. 25, 345–376.
Balsei o, S., Loiseau, I., Ramone , J., 2011. An an colony algo i hm hyb idized wi h
inse ion heu is ics o he ime dependen ehicle ou ing p oblem wi h ime
windows. Compu . Ope . Res. 38, 954–966.
Beasley, J., 1981. Adap ing he sa ings algo i hm o a ying in e -cus ome a el
imes. Omega 9, 658–659.
Ben Ticha, H., 2017. Vehicle Rou ing P oblems wi h Road-Ne wo k In o ma ion (Ph.D.
hesis). Uni e si é Cle mon Au e gne.
Ben Ticha, H., Absi, N., Feille , D., Quillio , A., 2017. Empi ical analysis o he VRPTW
wi h a mul ig aph ep esen a ion o he oad ne wo k. Compu . Ope . Res. 88,
103–116.
Ben Ticha, H., Absi, N., Feille , D., Quillio , A., 2018. Vehicle ou ing p oblems wi h
oad-ne wo k in o ma ion: S a e o he a . Ne wo ks 72, 393–406.
Ben Ticha, H., Absi, N., Feille , D., Quillio , A., 2019. Mul ig aph modeling and adap i e
la ge neighbo hood sea ch o he ehicle ou ing p oblem wi h ime windows.
Compu . Ope . Res. 104, 113–126.
Ben Ticha, H., Absi, N., Feille , D., Quillio , A., Van Woensel, T., 2019. A b anch-
and-p ice algo i hm o he ehicle ou ing p oblem wi h ime windows on a oad
ne wo k. Ne wo ks 73, 401–417.
Ben Ticha, H., Absi, N., Feille , D., Quillio , A., Van Woensel, T., 2021. The
ime-dependen ehicle ou ing p oblem wi h ime windows and oad-ne wo k
in o ma ion. SN Ope . Res. Fo um 2, 4.
Bou os, P., Sacha idis, D., Dalamagas, T., Sellis, T., 2011. Dynamic pickup and deli e y
wi h ans e s. In: Ge z, M., Renz, M., Zhou, X., Hoel, E., Ku, W.-S., Voisa d, A.,
Zhang, C., Chen, H., Tang, L., Huang, Y., Lu, C.-T., Ra ada, S. (Eds.), Ad ances
in Spa ial and Tempo al Da abases. In: Lec u e No es in Compu e Science 10411,
pp. 112–129.
Ch is iaens, J., Vanden Be ghe, G., 2020. Slack induc ion by s ing emo als o ehicle
ou ing p oblems. T ansp. Sci. 54, 417–433.
C ainic, T.G., Mancini, S., Pe boli, G., Tadei, R., 2011. Mul i-s a heu is ics o he
wo-echelon ehicle ou ing p oblem. In: Me z, P., Hao, J.-K. (Eds.), E olu iona y
Compu a ion in Combina o ial Op imiza ion. In: Lec u e No es in Compu e Science
6622, Sp inge , pp. 179–190.
C ainic, T.G., Riccia di, N., S o chi, G., 2009. Models o e alua ing and planning ci y
logis ics sys ems. T ansp. Sci. 43, 432–454.
Dabia, S., Röpke, S., Van Woensel, T., de Kok, T., 2013. B anch and p ice o he
ime-dependen ehicle ou ing p oblem wi h ime windows. T ansp. Sci. 47,
380–391.
Dona i, A.V., Mon emanni, R., Casag ande, N., Rizzoli, A.E., Gamba della, L.M., 2008.
Time dependen ehicle ou ing p oblem wi h a mul i an colony sys em. Eu opean
J. Ope . Res. 185, 1174–1191.
Ga aix, T., A igues, C., Feille , D., Josselin, D., 2010. Vehicle ou ing p oblems wi h
al e na i e pa hs: An applica ion o on-demand anspo a ion. Eu opean J. Ope .
Res. 204, 62–75.
Gend eau, M., Ghiani, G., Gue ie o, E., 2015. Time-dependen ou ing p oblems: A
e iew. Compu . Ope . Res. 64, 189–197.
Gmi a, M., Gend eau, M., Lodi, A., Po in, J.-Y., 2021. Tabu sea ch o he ime-
dependen ehicle ou ing p oblem wi h ime windows on a oad ne wo k.
Eu opean J. Ope . Res. 288, 129–140.
G angie , P., Gend eau, M., Lehuédé, F., Rousseau, L.-M., 2016. An adap i e la ge
neighbo hood sea ch o he wo-echelon mul iple- ip ehicle ou ing p oblem wi h
sa elli e synch oniza ion. Eu opean J. Ope . Res. 254, 80–91.
Haghani, A., Jung, S., 2005. A dynamic ehicle ou ing p oblem wi h ime-dependen
a el imes. Compu . Ope . Res. 32, 2959–2986.
Hill, A.V., Ben on, W.C., 1992. Modelling in a-ci y ime-dependen a el speeds o
ehicle scheduling p oblems. J. Ope . Res. Soc. 43, 343–351.
Ichoua, S., Gend eau, M., Po in, J.-Y., 2003. Vehicle dispa ching wi h ime-dependen
a el imes. Eu opean J. Ope . Res. 144, 379–396.
Jia, S., Deng, L., Zhao, Q., Chen, Y., 2022. An adap i e la ge neighbo hood sea ch
heu is ic o mul i-commodi y wo-echelon ehicle ou ing p oblem wi h sa elli e
synch oniza ion. J. Ind. Manage. Op im. 19, 1187–1210.
Ka le, N., Zou, B., Lin, J., 2017. Design and modeling o a c owdsou ce-enabled sys em
o u ban pa cel elay and deli e y. T ansp. Res. B 99, 62–82.
Le ch o d, A.N., Nasi i, S.D., Oukil, A., 2014. P icing ou ines o ehicle ou ing wi h
ime windows on oad ne wo ks. Compu . Ope . Res. 51, 331–337.
López-Ibánez, M., Dubois-Lacos e, J., Pé ez Cáce es, L., S ü zle, T., Bi a a i, M., 2016.
The IRACE package: I e a ed acing o au oma ic algo i hm con igu a ion. Ope .
Res. Pe spec . 3, 43–58.
Maland aki, C., Daskin, M., 1992. Time dependen ehicle ou ing p oblems:
Fo mula ions, p ope ies and heu is ic algo i hms. T ansp. Sci. 26, 185–200.
Minic, S., Lapo e, G., 2006. The pickup and deli e y p oblem wi h ime windows and
ansshipmen . INFOR 44, 217–227.
Pan, B., Zhang, Z., Lim, A., 2021a. A hyb id algo i hm o ime-dependen ehicle
ou ing p oblem wi h ime windows. Compu . Ope . Res. 128, 105193.
Pan, B., Zhang, Z., Lim, A., 2021b. Mul i- ip ime-dependen ehicle ou ing p oblem
wi h ime windows. Eu opean J. Ope . Res. 291, 218–231.
Pe boli, G., Tadei, R., Tadei, R., 2010. New amilies o alid inequali ies o he
wo-echelon ehicle ou ing p oblem. Elec on. No es Disc e e Ma h. 36, 639–646.
Pe boli, G., Tadei, R., Vigo, D., 2009. The wo-echelon capaci a ed ehicle ou ing
p oblem: Models and ma h-based heu is ics. T ansp. Sci. 45, 364–380.
Pisinge , D., Ropke, S., 2019. La ge neighbo hood sea ch. In: Gend eau, M., Po in, J.-
Y. (Eds.), Handbook o Me aheu is ics, Thi d Edi ion. In: In e na ional Se ies in
Ope a ions Resea ch & Managemen Science 272, Sp inge , pp. 99–127.
Ropke, S., Pisinge , D., 2006. An adap i e la ge neighbo hood sea ch heu is ic o he
pickup and deli e y p oblem wi h ime windows. T ansp. Sci. 40, 455–472.
Sampaio, A., Sa elsbe gh, M., Veelen u , L., Van Woensel, T., 2020. Deli e y sys-
ems wi h c owd-sou ced d i e s: A pickup and deli e y p oblem wi h ans e s.
Ne wo ks 76, 232–255.
Spe anza, M., Guas a oba, G., Vigo, D., 2016. In e media e acili ies in eigh
anspo a ion planning: A su ey. T ansp. Sci. 50, 763–789.
Van Belle, J., Valckenae s, P., Ca ysse, D., 2012. C oss-docking: S a e o he a .
Omega 40, 827–846.
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100143
20
