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In i ed e iew
T anspo a ion and deli e y in low-shop scheduling p oblems: A sys ema ic
e iew
Vic o Fe nandez-Viagas
Indus ial Managemen , School o Enginee ing, Uni e si y o Se ille, Camino de los Descub imien os s/n, 41092 Se ille, Spain
ARTICLE INFO
Keywo ds:
Scheduling
Flow shop
T anspo
Vehicle
Exac delay
Time lag
Rou ing
Se e
Dis ibu ion
Hois scheduling
Coupled ope a ions
ABSTRACT
This pape p esen s a li e a u e e iew o low-shop scheduling p oblems wi h anspo a ion o deli e y
o jobs. Flow-shop scheduling p oblems a e one o he mos widely s udied op imisa ion p oblems in he
li e a u e on Ope a ions Resea ch. Al hough hese ha e adi ionally been s udied assuming negligible o
cons an anspo imes, his does no co espond o eal manu ac u ing scena ios in he indus y. In ac ,
he ex ensi e au oma ion and synch onisa ion demanded by Indus y 4.0 may well be a d i ing ac o in
he g owing in e es in he li e a u e on low-shop scheduling p oblems wi h anspo cons ain s. Despi e
his in e es , he li e a u e is disjoin ed, and many e ms ha e been used in e changeably. This e iew aims
o o ganise he li e a u e on he opic and p opose a new no a ion o hese p oblems. This con ibu ion
is expec ed o help s uc u e ad ancemen s in he ield, classi ying hem by p oblem ype. Fu he mo e, a
de ailed s udy is ca ied ou on he complexi y and ela ionship be ween di e en a ian s. This p o ides a
ep esen a ion o he ad ances disco e ed in he li e a u e while also demons a ing new heo e ical esul s,
be o e inally iden i ying he mos p omising esea ch di ec ions.
1. In oduc ion
In ecen yea s, ac o ies ha e unde gone a signi ican shi owa ds
inc eased au oma ion, a mo emen p opelled by he ad ancemen s
o Indus y 4.0 (Waschneck e al.,2017), including lexible manu-
ac u ing cells, con eyo sys ems, au oma ed ehicles, obo s, CNC
machines, and mo e. Despi e he g ea po en ial shown by Indus y 4.0
(see Echchakoui & Ba ka,2020 o bene i s), especially in scheduling
(Rossi e al.,2019), a e y high in es men is equi ed o i s comple e
deploymen (De e al.,2020). In his si ua ion, p ecision in anspo
and scheduling becomes impe a i e and eme ges as a c ucial elemen
o achie e lawless synch onisa ion wi hin ac o ies. All his is com-
pounded by he ac ha he globalisa ion o ecen decades has led
o manu ac u ing companies in eg a ing p oduc ion and dis ibu ion
a eas mo e closely (Ramesh Kuma & Tiwa i,2020). This in eg a ion in
u n o ces companies o p oduce he igh amoun o p oduc s o he
indi idual cus ome s, sending hem o a speci ic place a he igh ime
(Sa aei e al.,2010). The impo ance o anspo and i s in eg a ion
a all decision-making le els o companies ha e spa ked a g owing
in e es in he scheduling li e a u e on anspo cons ain s. Among
he wide ange o manu ac u ing layou s (see Hu ink & Knus ,2005;
Lacomme e al.,2013;Li e al.,2023;Su e al.,2023), his su ey
ocuses on he low-shop scheduling p oblem (FSP), adi ionally one
o he mos ex ensi ely s udied p oblems in ope a ional esea ch. Since
he publica ion o he seminal pape by Johnson (1954), con ibu ions
E-mail add ess: [email p o ec ed].
o his p oblem ha e pionee ed esea ch on scheduling wi h a ious
cons ain s and objec i es, and many op imisa ion app oaches ha e
hei o igins in his layou (Fe nandez-Viagas e al.,2020).
When he e is a gi en se o jobs o be manu ac u ed, low shop
scheduling unde akes he p ocessing o hese jobs on se e al machines,
wi h each job ollowing he same ou e h ough all he machines. This
e iew ocuses on he p oblem including he anspo o semi- inished
o inished jobs, – commonly known as FSP – wi h anspo a ion
o deli e y cons ain s. He e, he challenge is o ind sequences o
jobs on he indi idual machines and anspo e s in o de o minimise
a gi en objec i e unc ion. This p oblem has many applica ions in
he eal wo ld. No able examples include: he s eel ingo eeming,
hea ing and olling p ocess (Tang e al.,2010;Wang e al.,2022;
Yuan e al.,2021); he pipe-making p ocess (Yuan e al.,2020); ca -
assembly business (Fab i e al.,2019); sani a y-wa e p oduc ion and
dis ibu ion (Rahman e al.,2021); elec onic de ice manu ac u ing
(Tonizza Pe ei a & Seido Nagano,2022); su ace ea men in elec-
opla ing plan s and ai c a indus ies (Paul e al.,2007); ci cui
boa d manu ac u ing sys em (Yih,1994); p oduc ion o connec ing ods
o engines (Lu e al.,2017); scheduling ba ges in seapo s (Zhang &
an de Velde,2015); semiconduc o manu ac u ing (in bu n in ope a-
ion Behnamian e al.,2012a, o in wa e ab ica ion p ocess, Geige
e al.,1997); manu ac u ing o connec ing od o gings (Sekkal &
h ps://doi.o g/10.1016/j.ejo .2024.11.034
Recei ed 11 Ma ch 2024; Accep ed 19 No embe 2024
Eu opean Jou nal o Ope a ional Resea ch 325 (2025) 1–19
A ailable online 30 No embe 2024
0377-2217/© 2024 The Au ho . Published by Else ie B.V. 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/ ).
V. Fe nandez-Viagas
Belkaid,2023); and book digi isa ion se ice (Villa inho e al.,2021).
On occasions, ins ead o emaining s a ic he machine y is anspo ed
elsewhe e o use in he ask. This phenomenon, known as he ou ing
low shop scheduling p oblem, is obse ed when jobs a e oo big o
hea y. Examples o his include when wo king wi h plane o ship
componen s (A e bakh & Be man,1999), when he epai enginee s
o machines mus p o ide epai se ices (Yu e al.,2011), and jobs in
he cons uc ion sec o (Che nykh e al.,2023).
The i s pape o examine he issue o anspo a ion is Mi en
(1959). While his pape does no explici ly iden i y i as pe aining
o model anspo a ion in he shop, i add esses he wo-machine
pe mu a ion low shop wi h ime lags, which can be iewed as he
anspo a ion o jobs when conside ing unlimi ed capaci y (e.g., using
a con eyo bel ). Since hen, nume ous con ibu ions ha e a emp ed
o analyse di e en aspec s ela ing o anspo a ion cons ain s in
he low shop. Howe e , despi e he g owing in e es in his ype o
cons ain , he li e a u e emains unclea . Nume ous a ian s o he
p oblem ha e been s udied independen ly, and di e en no a ions ha e
been used o issues ha a e closely ela ed and e en iden ical. This
e iew aims o consolida e, summa ise and s uc u e all hese indings
ela ing o he anspo o i ems in he low shop. Fu he mo e, unde
speci ic condi ions, ce ain ela ed scheduling p oblems can be iewed
as pa icula ins ances o he FSP wi h anspo . The eby, scheduling
p oblems conside ing, e.g., ime lags, se up imes, se e s, and elease
imes can also be used o model anspo in FSP. Howe e , a de ailed
analysis is equi ed gi en ha he co ela ion be ween hese p oblems
emains unclea in he li e a u e. To he bes o ou knowledge, his
is he i s e iew ha comple ely add esses anspo a ion in he low
shop scheduling p oblem. Addi ionally, his s udy aims o es ablish a
o mal classi ica ion o he p oblem wi hin he scheduling li e a u e. In
o de o do so, we p opose he ollowing con ibu ions:
•A new sys ema ic e iew o he li e a u e, co e ing all known
anspo cons ain s in he low shop layou .
•A classi ica ion and summa y o pape s in ables, and a new
no a ion o di e en ypes o anspo , (which can be in eg a ed
in he 𝛽 ield o G aham e al.,1979).
•An analysis and discussion o he complexi y o hese ypes o
p oblems.
•An in-dep h analysis o he lis o p oblems unde conside a ion,
examining scheduling p oblems and summa ising he indings
documen ed in he exis ing li e a u e.
•An in-dep h discussion o open esea ch ques ions in he li e a u e
wi h a iew o inco po a ing hem in u u e esea ch.
The pape is hus o ganised as ollows: he desc ip ion o he p ob-
lem is p esen ed in Sec ion 2. Sec ion 3p esen s he me hodology
p oposed o e iew he li e a u e. The p oposed no a ion o he p ob-
lem is desc ibed in Sec ion 4. The subsequen sec ions inco po a e a
comp ehensi e e iew o he pape s ha add ess he p oblem, dis-
cussing hose selec ed ollowing he me hodology ea u ed in Sec ion 3.
Fu he mo e, we p o ide a heo e ical analysis o he p oblem and a
discussion o he li e a u e ega ding he complexi y o he p oblem
and i s equi alence wi h ela ed scheduling p oblems. These sec ions
a e di ided by p oblem ype. Fi s ly, Sec ion 5p esen s a discus-
sion o semi- inished job app oaches including anspo a ion be ween
s ages (Sec ion 5.1); se e app oaches (Sec ion 5.2); he ou ing low
shop p oblem (Sec ion 5.3); and anspo a ion be o e p ocessing (Sec-
ion 5.4). This is hen ollowed by a discussion on pape s analysing he
deli e y o inal p oduc s in Sec ion 6, examining di ec deli e y (Sec-
ion 6.1) and in eg a ed ou ing and scheduling (Sec ion 6.2). Sec ion 7
p esen s an analysis o mixed app oaches, while Sec ion 8p esen s
quan i a i e analyses and a summa y o he main indings. Finally,
conclusions a e discussed and u u e lines o esea ch a e p esen ed in
Sec ion 9.
2. P oblem desc ip ion
In he low shop scheduling p oblem, he e a e 𝑛jobs ha a e
p ocessed on 𝑚machines. Each job 𝑗(wi h 𝑗∈ {1,…, 𝑛}) ollows
he same ou e o machines, p ocessing i s ope a ion 𝑂𝑖𝑗 on machine
𝑖(wi h 𝑖∈ {1,…, 𝑚}). This pape speci ically add esses he ans-
po a ion o jobs (o po en ially machines) be o e o a e p ocessing
a gi en ope a ion. When e e ing o he anspo p ocess, se e al
e ms ha e adi ionally been used in he li e a u e, depending on he
speci ic ield add essed in indi idual pape s. A summa y is o e ed
he e o some o he mos common e ms: ehicles, agen s, se e s,
anspo agen s, obo ic a ms, c anes, obo ic ans e de ices, obo s,
anspo e s, con eyo bel s, ca ie s, ma e ial handling de ices, ca s
o (indus ial) ucks. Fo he a oidance o con usion, his pape uses
he e ms ehicles and anspo e s o deno e he objec s which mo e
he p oduc s (al hough in a eal scena io hese could ob iously be a
ca , uck, obo ...). The numbe o ehicles conside ed is deno ed by
𝜏, while hei capaci y is deno ed by 𝑐. The anspo ime equi ed o
mo e job 𝑗 om machine 𝑖− 1 o 𝑖is indica ed by 𝑡𝑖𝑗 . Fu he mo e,
he p ocessing ime o job 𝑗on machine 𝑖is deno ed by 𝑝𝑖𝑗 . The
sequence o jobs on machine 𝑖is deno ed by 𝛱𝑖= (𝜋𝑖1, 𝜋𝑖2,…, 𝜋𝑖𝑛). The
comple ion ime o ope a ion 𝑂𝑖,𝜋𝑖𝑗 is deno ed by 𝐶𝑖𝑗 . When cla i y can
be gua an eed, indices 𝑖[𝑗]a e used o deno e ope a ion 𝑂𝑖,𝜋𝑖𝑗 , i.e. 𝑝𝑖[𝑗]
ep esen s he p ocessing ime o his ope a ion.
In o de o iden i y he pape s in his e iew, he 𝛼|𝛽|𝛾no a ion
p oposed by G aham e al. (1979) has been used, wi h 𝛼=𝐹𝑚
(co esponding o a low shop layou ). A summa y o he abb e ia ions
applied o he ields 𝛽and 𝛾in he e iewed a icles, using a no a ion
based on Pinedo (2012) as well as on p e ious li e a u e e iews
(see Komaki e al.,2019;Miya a & Nagano,2019;Rolim & Nagano,
2020;Rossi e al.,2018), can be ound in Appendix A.
3. Me hodological app oach
The me hodology p oposed in his pape is in oduced in his sec-
ion. A sys ema ic e iew o he li e a u e allows us a comp ehensi e
examina ion o he issue unde conside a ion, colla ing he maximum
numbe o con ibu ions. Th ough his app oach, we aim o explo e he
ollowing esea ch ques ions:
RQ1: How a e he di e en a ian s o he p oblem named?
RQ2: Which p oblems ha e been p o ed o be NP-ha d o o
ha e exac polynomial- ime algo i hms?
RQ3: Which p ope ies ha e been es ablished? Wha a e he
ela ionships be ween he p oblems unde conside a ion
and ela ed scheduling p oblems?
RQ4: Which echniques and models ha e been de eloped?
RQ5: How can each speci ic anspo a ion cons ain be ep e-
sen ed in he 𝛼|𝛽|𝛾no a ion?
RQ6: Wha a e he u u e esea ch lines?
A necessa y ini ial s ep in he me hodology p oposed is he iden i i-
ca ion o a icles o be e iewed. The wo le els o keywo ds ha e been
es ablished in his phase:
•Keywo d le el 1: lowshop o ‘‘ low shop’’
•Keywo d le el 2: anspo a ion o a el o ou ing o deli e y
o obo ic o se e o delay o ‘‘ ime lag’’ o anspo
These combina ions o keywo ds ha e been en e ed in h ee da abases:
Scopus, Google Schola and Sp inge Link. A icles mee ing he c i e ion
o con aining a leas one keywo d om each le el we e chosen o
inclusion. Fu he mo e, a icles hey ci e, as well as hose ci ing hem,
we e also examined (snowball sea ch). Mo e han 300 a icles we e
iden i ied in his sea ch. In o de o ensu e he quali y and hema ic
ele ance o he con ibu ions e iewed, esul s we e hen e ined, il-
e ing he a icles acco ding o he exclusion c i e ia below (see Neu eld
e al.,2023;Rolim & Nagano,2020 o simila exclusion and quali y
c i e ia):
Eu opean Jou nal o Ope a ional Resea ch 325 (2025) 1–19
2
V. Fe nandez-Viagas
Fig. 1. P oposed me hodology.
•C i e ion E1: Con e ence pape s and chap e s a e no included.
•C i e ion E2: Non-English manusc ip s a e no conside ed.
•C i e ion E3: Simula ion o di e en p oblem layou s, such as
dis ibu ed (𝛼=𝐷 𝐹𝑚), hyb id (𝛼=𝐻 𝐹𝑚), o assembly low shop
( ypically deno ed as 𝛼=𝐷 𝑃 𝑚→𝐹𝑚o 𝐴𝐹𝑚) a e no included,
hus only 𝛼=𝐹𝑚, wi h 𝑚≥2is conside ed.
•C i e ion E4: Only pape s in jou nals published in he i s and
second qua ile acco ding o he Scimago jou nal anking (SJR)
a e included. Mo e speci ically, all a icles in qua ile Q1 o Q2 o
SJR, ei he in he yea o publica ion o in he las ank (i.e., SJR
2022) ha e been conside ed. A icles wi h a yea o publica ion
be o e 1999 a e also included i he jou nal has been in Q1 o Q2
in any yea .
Following C i e ia E1 and E2, 299 pape s we e selec ed. This o al
igu e was hen educed o 209, upon excluding pape s add essing
di e en layou s (E3). Finally, a e applying C i e ion E4, 119 a icles
we e selec ed.
Once all he a icles a e iden i ied and e iewed, hei main cha -
ac e is ics a e eco ded in his pape . O he aspec s e iewed in his
pape and included in ables a e he no a ion o he p oblem using 𝛼|𝛽|𝛾
acco ding G aham e al. (1979) (P oblem); ypes o exac me hods used
o sol e he p oblems unde conside a ion (Exac ); ype o app oxima e
algo i hms used o sol e he p oposed p oblem (App oxima e); and,
inally, addi ional main con ibu ions o he pape s (O he ). A comp e-
hensi e heo e ical analysis is also ca ied ou o he p oblems unde
conside a ion. An ini ial analysis is conduc ed o eco d he complexi y
o he p oblems, hei p ope ies, and i s equi alence wi h ela ed
scheduling p oblems om he li e a u e. This is hen ollowed by new
heo ems, p oposed o ensu e be e unde s anding o he bounda y
lines o he p oblems unde conside a ion. An o e iew o he p oposed
me hodology o his e iew is p esen ed in Fig. 1.
The aim o his me hodology is o p o ide he i s li e a u e e iew
o low shop scheduling p oblems wi h anspo a ion. In e ms o
simila s udies in he li e a u e, he ecen p oposals by Be ghman
e al. (2023) and Hosseini e al. (2023) a e conside ed he wo mos
closely connec ed e iews. Despi e he signi ican in e es o bo h hese
e iews, hei scope o s udy di e s g ea ly. Bo h add ess e y speci ic
cases o anspo a ion, ocusing on scheduling in gene al (i.e. including
single machine, pa allel machines, e c.). Howe e , Be ghman e al.
(2023) add ess only he speci ic p oblem o in eg a ing ou ing and
scheduling, while Hosseini e al. (2023) examine anspo a ion be-
ween s ages. In ac , o he a icles e iewed in his s udy, only six
and 29 a e ci ed in Be ghman e al. (2023) and Hosseini e al. (2023),
espec i ely.
4. P oposed no a ion
Di e en app oaches ha e been conside ed in he li e a u e o ad-
d ess anspo in he low shop scheduling p oblem. In his e iew,
o he pu poses o classi ica ion, we p opose he no a ion 𝑇 𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡𝜏
(𝑊 𝑥, 𝑌 𝑧)o 𝑇𝜏(𝑊 𝑥, 𝑌 𝑧)(when his does no lead o con usion) o
classi y hem,1and he use o wo ields: ype o p oblem (𝑊 𝑥) and ype
o anspo e (𝑌 𝑧). As de ailed in he p e ious sec ion, 𝜏is he numbe
o ehicles conside ed. Consequen ly, when se e al jobs (indi idually
o in ba ches) can be anspo ed simul aneously (ei he by conside ing
in ini e ehicles o a con eyo bel wi h in ini e capaci y), 𝜏, is omi ed.
The la e is ypically deno ed in he li e a u e as ime lag o delay.
4.1. Type o p oblem
The i s ield, deno ed by 𝑊 𝑥, ep esen s he ype o anspo
aking place. In he p ima y ield 𝑊, a se ies o p oblems can be dis in-
guished and di ided in o anspo a ion o semi- inished and inished
jobs:
T anspo a ion o semi- inished jobs: In his case, ou ypes o p ob-
lems can be iden i ied o anspo semi- inished jobs ( ypically) o
machines:
•T anspo a ion be ween s ages (𝑊=𝑀). The e a e ehicles
( ypically deno ed agen s o obo s) ha anspo jobs be ween
machines. T adi ionally, all anspo e s can be used se e all
machines. Howe e , when anspo e s a e in ended o a spe-
ci ic subse o machines, he p oblem is indica ed by 𝑊=𝑀𝑎
and he anspo can be classi ied as alloca ed (Ahmadi-Ja id
& Hooshangi-Tab izi,2015). Fu he mo e, when anspo e s a e
guided and can only mo e along a speci ic sequence o machines
(e.g., i s machine 1, hen machine 2, ...), his is deno ed by
𝑊=𝑀𝑔.
•Se e case o mul i-i em hois scheduling (𝑊=𝑆). In his
case, du ing anspo a ion, nei he he ehicle no he pos e-
io machine can pe o m any o he ask, i.e., he machine is
blocked. This ype o si ua ion ends o a ise when he anspo e
( ypically deno ed as se e in his p oblem) in oduces he job
di ec ly in o he machine eed. I should be no ed ha in he
se e scheduling li e a u e, he se up cons ain is equi alen
o he anspo cons ain , bu only when se up imes a e non-
an icipa o y.2O he wise, he p oblem is o di e en na u e, and
as such, no conside ed ele an o his esea ch.
•Rou ing low shop p oblem, (𝑊=𝑅). When anspo a ion is no
needed be ween ope a ions o he same jobs, bu a he be ween
ope a ions on he same machine, a di e en p oblem a ian
a ises. In his case, he machines a e anspo ed o he jobs ha
a e ixed. I is gene ally assumed ha he speci ic ou es and hus,
a el imes, be ween each job pai a e known in ad ance (i.e. a
comple e g aph is conside ed). Ne e heless, when a ee g aph
is conside ed, he p oblem is shown as 𝑊=𝑅𝑡.
1Al hough his no a ion could be di ided in o wo, 𝑇𝜏and 𝑇(𝑊 𝑥, 𝑌 𝑧), o
be en e ed in o he ields 𝛼(machine en i onmen ) and 𝛽(job cha ac e is ics)
o he no a ion by G aham e al. (1979), espec i ely, we ecommend using
he p oposed no a ion di ec ly in ield 𝛽 o simpli ica ion o p ac i ione s and
academics.
2A se up is classi ied as non-an icipa o y (o equi alen ly non-sepa able)
when i canno be ini ia ed p io o job a i al. Con e sely, i is classi ied
as an icipa o y (o sepa able) when i can be s a ed be o e o a e he job
a i al.
Eu opean Jou nal o Ope a ional Resea ch 325 (2025) 1–19
3
V. Fe nandez-Viagas
•T anspo a ion o aw ma e ials o anspo a ion be o e p o-
cessing and p oduc ion, deno ed 𝑊=𝑃. In his case, ehicles
anspo aw ma e ials o p e-p ocessing p oduc s om p o ide s
o o he a eas o he company.
Deli e y o inal p oduc s: In his case, a lee o anspo e s ( ypi-
cally deno ed by ehicles, ca s o ucks) is in cha ge o deli e ing he
inal p oduc s o he clien s o o he inal p oduc bu e . Depending
on he numbe o cus ome s isi ed o each ehicle and on he deli e y
da es, he ollowing wo app oaches can be conside ed:
•Scheduling wi h di ec deli e y, 𝑊=𝐷. In his case, each ehicle
se es only one cus ome in each ip.
•In eg a ed P oduc ion and Dis ibu ion P oblem (also deno ed
as scheduling wi h ehicle ou ing o in eg a ed ou ing and
scheduling p oblem), 𝑊=𝐶. In his p oblem, he e is a ou ing
decision me hodology o ag ee he ou e o each ehicle o se e
cus ome s. Typically, each node in he ou ing decision p oblem
has been assigned a cus ome and has speci ic job demands.
In addi ion, in he seconda y ield 𝑥, wo cases can be iden i ied o
each o he p e ious p oblem ypes, ha is, 𝑊= {𝑃 , 𝑀 , 𝑆 , 𝑅, 𝐷}:
•One-way ip (𝑥=𝑜). Each ehicle anspo s he job di ec ly o a
machine ( ypically o i s bu e ). Du ing his ime, he p e ious
machine may be p ocessing a di e en job. Once he job is
anspo ed, he ehicle is a ailable o anspo a di e en job.
This ypical ep esen a ion o cases inco po a es elemen s such as
con eyo bel s, whe e he ehicle akes he ollowing job di ec ly
om he same machine, o whe e he ime ames o e u ning
he ehicles ( o he s a ing posi ion o o ano he machine) a e
negligible.
•Round ip (𝑥=𝑟). This case conside s bo h he ime o mo e he
job o a speci ic machine and he ime o e u n he emp y ehicle
o i s s a ing posi ion o o ano he machine. The e o e, ehicle
canno begin i s subsequen anspo un il he a i al ime. No e
ha he machine can s a p ocessing i s subsequen job a e he
ime o he one-way ip.
Fu he mo e, in anspo a ion om p o ide s (𝑊=𝑃) o o clien s
(𝑊=𝐷o 𝑊=𝐶), i is implici ly conside ed in he no a ion ha he
Va iable Deli e y Da e (VDD) app oach is ollowed, so ha he jobs
a e deli e ed a any ime a ehicle is a ailable. Howe e , i a Fixed
Deli e y Da e (FDD) app oach is assumed (whe e jobs, indi idually o
in ba ches, mus be deli e ed only on speci ic da es o in e als), hen
𝑓should be included in ield 𝑥.
4.2. Type o anspo e
The second ield, deno ed by 𝑌 𝑧, ep esen s he ype o anspo e
conside ed. The ollowing ypes can be dis inguished in he p ima y
ield 𝑌, depending on he numbe o jobs ha a e anspo ed:
•Vehicles anspo jobs indi idually, 𝑌=𝐼. In his case, se e al
subcases a e iden i ied in he seconda y ield 𝑧, depending on how
anspo imes a e conside ed:
–T anspo ime depends on machines’ o clien s’ loca ions,
𝑧=𝑖( o example, anspo imes depend on he dis ances
be ween machines 𝑖and 𝑖+ 1o on he dis ance be ween
machine 𝑖and he loca ion o i s aw ma e ial).
–T anspo ime depends on he job anspo ed, 𝑧=𝑗.
–T anspo ime depends on he speed o he anspo e ,
𝑧=𝑠.
–T anspo ime depends on he ac ual job being anspo ed
as well as he p e ious job anspo ed, 𝑧=𝑗 𝑘. This means
ha he e is a po en ially di e en anspo ime when job
𝑗is succeeded by job 𝑘ins ead o ano he job.
–T anspo ime depends on he speci ic ehicle 𝑣o i s
selec ed speed (in cases wi h di e en speed op ions o each
ehicle), 𝑧=𝑣.
•Vehicles mo e jobs in ba ches wi h limi ed capaci y 𝑌=𝐵. In he
case o unlimi ed capaci y o ehicles, his is deno ed by 𝑌=𝐵∞.
In addi ion, when a speci ic limi ed capaci y 𝑏is add essed, i is
indica ed by 𝑌=𝐵𝑏. Rega ding he anspo ime o ba ches,
se e al speci ic app oaches can be iden i ied in his case:
–T anspo a ion imes depend on he loca ion o he ma-
chines o clien s, 𝑧=𝑖.
–T anspo a ion imes a e di e en o each ehicle 𝑣,𝑧=𝑣.
–T anspo a ion imes depend on he jobs con ained wi hin
he ba ch 𝑗,𝑧=𝑗.
–T anspo a ion imes depend on he speed o he ans-
po e , 𝑧=𝑠.
–T anspo a ion imes depend on he size o he ba ch, 𝑧=𝑐.
Ob iously, hese p e ious no a ions in he seconda y ield 𝑧can be
combined, e.g., i he a el imes depend on he ehicle and he job, i
can be deno ed by 𝑧=𝑣𝑗. Fu he mo e, i 𝑧is omi ed, anspo imes
a e cons an . Finally, i should be no ed ha when jobs a e deli e ed
di ec ly o cus ome s in ba ches, 𝑇(𝐷 𝑥, 𝐵 𝑧), only jobs belonging o he
same o de /clien can be included wi hin he same ba ch.
The es o he pape is s uc u ed acco ding o he ypes o p oblem
explained abo e.
5. T anspo a ion o semi- inished jobs
This sec ion p esen s a comp ehensi e e iew o he pape s add ess-
ing he anspo a ion o semi- inished jobs ollowing he p ocedu e
desc ibed in Sec ion 3. A se ies o app oaches – anspo a ion be ween
s ages (Sec ion 5.1); se e app oach (Sec ion 5.2); ou ing low shop
p oblem (Sec ion 5.3; and anspo a ion o aw ma e ials (Sec ion 5.4)
– a e discussed.
5.1. T anspo a ion be ween s ages (𝑊=𝑀)
This is he mos discussed FSP wi h anspo a ion cons ain s in he
li e a u e. All selec ed pape s conside ing anspo be ween s ages a e
summa ised in Table 1. Mos o he con ibu ions add ess p oblems,
which in many cases ha e been iden i ied as NP-ha d, by applying op i-
misa ion me hods. In o de o analyse he indings o he li e a u e, his
sec ion is di ided i s ly in o Sec ion 5.1.1, analysing he heo e ical
esul s ound in he li e a u e and p oposing new esul s which add ess
he equi alence o hese p oblems wi h o he ela ed scheduling p ob-
lems, and Sec ion 5.1.2, p o iding a e iew o op imisa ion algo i hms
and ela ed con ibu ions applied o sol e he p oblem.
5.1.1. Analysis o he p oblem
The anspo a ion be ween s ages has been add essed in he li e a-
u e conside ing a limi ed o unlimi ed numbe o ehicles. In he case
o he la e , he p oblem is simila o he classical low shop wi h ime
lags. I should also be no ed ha , e en in he li e a u e, he de ini ion
o ime lags emains unclea . While he oldes li e a u e on he opic
usually conside s he ime lag as he minimum ime be ween he s a
(comple ion) o wo consecu i e ope a ions o a job (Mi en,1959;
Rinnooy Kan,1976), nowadays i is e e ed o as he minimum ime
be ween he comple ion o he ope a ion on machine 𝑖and he s a o
he ollowing ope a ion on machine 𝑖+ 1(see e.g. Mkadem e al.,2021;
Sama ghandi,2019). Howe e , ega dless o hese de ini ions, he ime-
lag cons ain can be mos ly modelled as a anspo a ion cons ain , as
indica ed in he ollowing heo ems (all p oo s o he heo ems included
in his pape a e included as supplemen a y ma e ial):
Eu opean Jou nal o Ope a ional Resea ch 325 (2025) 1–19
4
V. Fe nandez-Viagas
Table 1
Summa y o con ibu ions o anspo a ion be ween s ages.
Re e ence P oblem Exac App oxima e O he
Lan e al. (2024)𝐹2|𝑇1(𝑀 𝑟, 𝐵)|𝐶𝑚𝑎𝑥 PTAA P
Bou ellouh and Belkaid
(2023)
𝐹𝑚|𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔 , 𝑆 𝑖𝑗 𝑘, 𝑚𝑎𝑖𝑛𝑡, 𝑇𝜏(𝑀𝑔𝑟, 𝐼 𝑖𝑗 𝑠)|#(𝐶𝑚𝑎𝑥, 𝑇 𝐸 𝐶)MILP ACA
Kha ami e al. (2023)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡, 𝑜𝑟𝑑 𝑒𝑟𝑒𝑑|𝐶𝑚𝑎𝑥 DPA, EPA P
Kha ami e al. (2023)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝐶𝑚𝑎𝑥 EPA P, CA
Wang e al. (2022)𝐹2|𝑇1(𝑀 𝑟, 𝐵 𝑖), 𝑝−𝑏𝑎𝑡𝑐 ℎ|𝐶𝑚𝑎𝑥 MILP CH
Gna owski e al. (2022)𝐹𝑚|𝑇1(𝑆 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 MILP TB, CH P
Mkadem e al. (2021)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗)|𝐶𝑚𝑎𝑥 B&B P, LB
Yuan e al. (2020)𝐹2|𝑇1(𝑀 𝑟, 𝐼 𝑗), 𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔 , 𝑓 𝑎𝑚|𝐶𝑚𝑎𝑥 MILP GA CA
Agee (2020)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡, 𝑡𝑗∈ {𝑡′, 𝑡′′ }|𝐶𝑚𝑎𝑥 PTAA P, LB
Sama ghandi (2019)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |𝐶𝑚𝑎𝑥 MILP, CP TS P
Hamdi and Toumi (2019)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢|∑𝑇𝑗MILP
Dhouib e al. (2018)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑤𝑡𝑚𝑎𝑥
𝑗|∑𝑈1
𝑗, 𝐶2
𝑚𝑎𝑥 MILP CH P
Wang, Huang, and Li
(2018)
𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |𝐶𝑚𝑎𝑥 MILP CH P
Zhao e al. (2017)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |𝐶𝑚𝑎𝑥 CH, IG
Zhao e al. (2017)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |𝐶𝑚𝑎𝑥 CH, IG
Ye e al. (2017)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |𝐶𝑚𝑎𝑥 IG
Msakni e al. (2016)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢|∑𝐶𝑗B&B IG LB
Dong e al. (2016)𝐹2|𝑇1(𝑀 𝑟, 𝐵)|𝐶𝑚𝑎𝑥 PTAA P
Liou and Hsieh (2015)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼𝑖𝑗 ), 𝑝𝑟𝑚𝑢, 𝑓 𝑎𝑚, 𝑠𝑖𝑗 𝑘|𝐶𝑚𝑎𝑥 PSO LB
Ahmadi-Ja id and
Hooshangi-Tab izi (2015)
𝐹𝑚|𝑇𝜏(𝑀𝑎𝑟, 𝐼𝑖𝑗 ), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 MILP ASO LB
Zhang and an de Velde
(2015)
𝐹2|𝑇(𝑀 𝑜, 𝐼), 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙|𝑤𝑁 DPA CA
Zhong and Chen (2015)𝐹2|𝑇1(𝑀 𝑟, 𝐵)|𝐶𝑚𝑎𝑥 CH P
Hamdi and Loukil (2015a)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |∑𝑇𝑗MILP DR LB
Hamdi and Loukil (2015b)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗)|∑𝑈𝑗DR LB
Khalili (2014)𝐹𝑚|𝑇 𝑚− 1(𝑀𝑎𝑟, 𝐼), 𝑠𝑘𝑖𝑝, 𝑝𝑚|∑𝐶𝑗and ∑𝑇𝑗DR, CH, EM, SA
Dhouib e al. (2013)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |∑𝑈1
𝑗, 𝐶2
𝑚𝑎𝑥 MILP SA
Gup a e al. (2013)𝐹2|𝑇1(𝑀 𝑟, 𝐼 𝑗), 𝑠𝑖𝑗 |
𝐶𝑚𝑎𝑥 and ∑
𝐹𝑗EPA
Behnamian e al. (2012b)𝐹2|𝑇1(𝑀 𝑟, 𝐵), 𝑝−𝑏𝑎𝑡𝑐 ℎ|𝐶𝑚𝑎𝑥 MILP CH LB
Behnamian e al. (2012a)𝐹3|𝑇2(𝑀𝑎𝑜, 𝐵 𝑖), 𝑝−𝑏𝑎𝑡𝑐 ℎ|𝐶𝑚𝑎𝑥 MILP CH, GA
Khalili and
Ta akkoli-Moghaddam
(2012)
𝐹𝑚|𝑇 𝑚− 1(𝑀𝑎𝑟, 𝐼), 𝑝𝑟𝑚𝑢, 𝑠𝑘𝑖𝑝|#(𝐶𝑚𝑎𝑥,∑𝑤𝑗𝑇𝑗)EM
Gong and Tang (2011)𝐹2|𝑇1(𝑀 𝑟, 𝐵)|𝐶𝑚𝑎𝑥 CH P
Tang e al. (2010)𝐹2|𝑇1(𝑀 𝑟, 𝐼)|𝐶𝑚𝑎𝑥 B&B IH LB
Nade i, Ahmadi Ja id, and
Jolai (2010)
𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 MILP DR, CH, AIS
Nade i, Ahmadi Ja id, and
Jolai (2010)
𝐹𝑚|𝑇 𝑚− 1(𝑀𝑎𝑟, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 MILP DR, CH, AIS
Zhang and Van De Velde
(2010)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑟𝑗, 𝑠𝑘𝑖𝑝|𝐶𝑚𝑎𝑥 PTAA
Zhang and Van De Velde
(2010)
𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑠𝑘𝑖𝑝|∑𝐶𝑗PTAA
Nade i,
Ta akkoli-Moghaddam,
and Khalili (2010)
𝐹𝑚|𝑇 𝑚− 1(𝑀𝑎𝑟, 𝐼), 𝑝𝑟𝑚𝑢, 𝑠𝑘𝑖𝑝|𝐶𝑚𝑎𝑥 and ∑𝑤𝑗𝑇𝑗DR, CH, EM, SA
Tang and Liu (2009b)𝐹2|𝑇1(𝑀 𝑟, 𝐵), 𝑏𝑎𝑡𝑐 ℎ|𝐶𝑚𝑎𝑥 CH CA
Tang and Liu (2009a)𝐹2|𝑇1(𝑀 𝑟, 𝐵), 𝑏𝑎𝑡𝑐 ℎ|𝐶𝑚𝑎𝑥 MILP CH CA
Huo e al. (2009)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡|∑𝐶𝑗DR, TS, SA P, CA
Fond e elle e al. (2009)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝐿𝑚𝑎𝑥 B&B CH P, LB
Munie -Ko don and
Rebaine (2008)
𝐹3|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑝𝑖𝑗 = 1|𝐶𝑚𝑎𝑥 EPA
(con inued on nex page)
Eu opean Jou nal o Ope a ional Resea ch 325 (2025) 1–19
5
V. Fe nandez-Viagas
Table 1(con inued).
Re e ence P oblem Exac App oxima e O he
Munie -Ko don and
Rebaine (2008)
𝐹4|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑝𝑖𝑗 = 1|𝐶𝑚𝑎𝑥 EPA
Raywa d-Smi h and
Rebaine (2008)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑖𝑗 = 1|𝐶𝑚𝑎𝑥 DR P
Fond e elle e al. (2008)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |∑𝑤𝑗𝐶𝑖B&B P, CA
Agee (2008)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝1𝑗=𝑝2𝑗|𝐶𝑚𝑎𝑥 PTAA LB
Leung e al. (2007)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡, 𝑠𝑘𝑖𝑝|𝐶𝑚𝑎𝑥 PTAA CA
Leung e al. (2007)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡, 𝑠𝑘𝑖𝑝|∑𝐶𝑗EPA PTAA CA
Agee and Babu in (2007)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗 , 𝑛𝑜 −𝑤𝑎𝑖𝑡, 𝑝𝑖𝑗 = 1)|𝐶𝑚𝑎𝑥 PTAA LB
Fond e elle e al. (2006)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |𝐶𝑚𝑎𝑥 B&B CH P, LB, CA
Çe inkaya (2006)𝐹2|𝑇1(𝑀 𝑟, 𝐼), 𝑠𝑖𝑗 |𝐶𝑚𝑎𝑥 EPA P
P asad e al. (2006)𝐹𝑚|𝑇(𝑀 𝑓 𝑜, 𝐵 𝑖), 𝑝𝑟𝑚𝑢, 𝑏𝑢𝑓 𝑓 𝑒𝑟, 𝑏𝑎𝑡𝑐 ℎ|#(∑𝐶𝑗,
𝐶𝑏, 𝜎𝐶𝑗)CH, GA
Rebaine (2005)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗)|𝐶𝑚𝑎𝑥 P
Rebaine (2005)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 P
Lee and S use ich (2005)𝐹2|𝑇1(𝑀 𝑟, 𝐵∞)|𝐶𝑚𝑎𝑥 CH P, CA
B ucke e al. (2004) Se e al p oblemsaEPA CA
Yu e al. (2004)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑖𝑗 = 1|𝐶𝑚𝑎𝑥 P, CA
Hu ink and Knus (2001)𝐹𝑚|𝑇1(𝑀 𝑜, 𝐼 𝑖𝑗)|𝐶𝑚𝑎𝑥 EPA CA
Lee and Chen (2001) Se e al p oblemsbEPA CH P, CA
Yang and Che n (2000)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑓 𝑎𝑚|𝐶𝑚𝑎𝑥 EPA P
Haoua i and Ladha i
(2000)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑟𝑗|𝐿𝑚𝑎𝑥 B&B CH LB
Ganesha ajah e al. (1998)𝐹2|𝑇1(𝑀 𝑟, 𝐼 𝑗), 𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔|𝐶𝑚𝑎𝑥 CA
Riezebos and Gaalman
(1998)
𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑟𝑒𝑐 𝑟|𝐶𝑚𝑎𝑥 DR LB
S e ens and Gemmill
(1997)
𝐹2|𝑇1(𝑀 𝑟, 𝐼)|𝐿𝑚𝑎𝑥 CH
Dell’Amico and Vaessens
(1996)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝1𝑗=𝑝2𝑗|𝐶𝑚𝑎𝑥 LB, CA
Dell’Amico (1996)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗)|𝐶𝑚𝑎𝑥 CH, TS P, LB, CA
Yu (1996)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝1𝑗=𝑝2𝑗, 𝑡𝑗∈ 0, 𝑙|𝐶𝑚𝑎𝑥 LB, CA
Riezebos e al. (1995)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑟𝑒𝑐 𝑟|𝐶𝑚𝑎𝑥 B&B DR LB
Panwalka (1991)𝐹2|𝑇1(𝑀 𝑟, 𝐼), 𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔|𝐶𝑚𝑎𝑥 EPA
S e n and Vi ne (1990)𝐹2|𝑇1(𝑀 𝑟, 𝐼 𝑗), 𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔|𝐶𝑚𝑎𝑥 PTAA P, LB
Szwa c (1983)𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 LB
Maggu e al. (1982)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 EPA
Maggu e al. (1981)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 DA P
Rinnooy Kan (1976)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 EPA
Mi en (1959)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 EPA
a𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑖𝑗 =𝑝, 𝑎𝑔 𝑟𝑒𝑒𝑎𝑏𝑙 𝑒|∑𝑤𝑗𝐶𝑗,𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑖𝑗 =𝑝|∑𝐶𝑗,𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑖𝑗 = 1|∑𝑤𝑗𝐶𝑗,𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑖𝑗 = 1, 𝑟𝑗|∑𝐶𝑗.
b𝐹2|𝑇𝜏(𝑀 𝑟, 𝐼)|𝛾𝑟,𝐹2|𝑇1(𝑀 𝑟, 𝐼)|𝐶𝑚𝑎𝑥,𝐹2|𝑇 𝜏(𝑀 𝑟, 𝐵)|𝐶𝑚𝑎𝑥 ,𝐹2|𝑇1(𝐷 𝑟, 𝐵)|𝐶𝑚𝑎𝑥, and 𝐹2|𝑇1(𝐷 𝑟, 𝐼)|𝐶𝑚𝑎𝑥 .
Theo em 5.1. Le 𝑎be he 𝐹𝑚|𝑙𝑚𝑖𝑛
𝑖𝑗 |𝛾p oblem, whe e 𝑙𝑚𝑖𝑛
𝑖𝑗 ep esen s
he minimum ime lag be ween he comple ion ime o a job and he s a
ime in he subsequen s age. Le 𝑏be he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗)|𝛾p oblem, whe e
𝑡𝑖𝑗 =𝑙𝑚𝑖𝑛 is he anspo ime be ween ope a ions 𝑂(𝑖, 𝜋𝑖𝑗 )and 𝑂(𝑖, 𝜋𝑖+1,𝑗 ).
Bo h p oblems a e he e o e equi alen .
Theo em 5.2. Le 𝑎be he 𝐹𝑚|𝑙𝑆
𝑖𝑗 |𝛾p oblem, whe e 𝑙𝑆
𝑖𝑗 is he minimum
s a ime lag be ween he s a imes o ope a ions 𝑂(𝑖, 𝜋𝑖𝑗 )and 𝑂(𝑖, 𝜋𝑖+1,𝑗 ).
Le 𝑏be he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗)|𝛾p oblem, wi h 𝑡𝑖𝑗 =𝑙𝑆
𝑖𝑗 −𝑝𝑖𝜋𝑗as he
anspo ime be ween ope a ions 𝑂(𝑖, 𝜋𝑖𝑗 )and 𝑂(𝑖, 𝜋𝑖+1,𝑗 ). Bo h p oblems
a e he e o e equi alen .
Theo em 5.3. Le 𝑎be he 𝐹𝑚|𝑙𝐶
𝑖𝑗 |𝛾p oblem, whe e 𝑙𝐶
𝑖𝑗 is he minimum
s op ime lag be ween he comple ion imes o ope a ions 𝑂(𝑖, 𝜋𝑖𝑗 )and
𝑂(𝑖, 𝜋𝑖+1,𝑗 ). Le 𝑏be he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗)|𝛾p oblem, wi h 𝑡𝑖𝑗 =𝑙𝐶
𝑖𝑗 −𝑝𝑖+1,𝜋𝑗
as he anspo ime be ween ope a ions 𝑂(𝑖, 𝜋𝑖𝑗 )and 𝑂(𝑖, 𝜋𝑖+1,𝑗 ). Bo h
p oblems a e he e o e equi alen .
Theo em 5.4. Le 𝑎be he 𝐹𝑚|𝑙𝑆
𝑖𝑗 , 𝑙𝐶
𝑖𝑗 |𝛾p oblem, whe e 𝑙𝑆
𝑖𝑗 and 𝑙𝐶
𝑖𝑗 a e he
minimum s a and s op ime lag be ween he comple ion imes o ope a ions
𝑂(𝑖, 𝜋𝑖𝑗 )and 𝑂(𝑖, 𝜋𝑖+1,𝑗 ), espec i ely. Le 𝑏be he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗)|𝛾p ob-
lem, wi h 𝑡𝑖𝑗 = max{𝑙𝑆
𝑖𝑗 −𝑝𝑖𝜋𝑗, 𝑙𝐶
𝑖𝑗 −𝑝𝑖+1,𝜋𝑗}as he anspo ime be ween
ope a ions 𝑂(𝑖, 𝜋𝑖𝑗 )and 𝑂(𝑖, 𝜋𝑖+1,𝑗 ). Bo h p oblems a e he e o e equi alen .
Theo em 5.5. Le 𝑎be he 𝐹𝑚|𝑙𝑖𝑗 |𝛾p oblem, whe e 𝑙𝑖𝑗 is he exac ime
lag be ween he comple ion ime o a job and he s a ime in he subsequen
s age. Le 𝑏be he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾p oblem, wi h 𝑡𝑖𝑗 =𝑙𝑖𝑗 as he
anspo ime be ween ope a ions 𝑂(𝑖, 𝜋𝑖𝑗 )and 𝑂(𝑖, 𝜋𝑖+1,𝑗 ). Bo h p oblems
a e he e o e equi alen .
Theo em 5.6. Le 𝑎be he 𝐹𝑚|𝑙𝑚𝑖𝑛
𝑖𝑗 , 𝑙𝑚𝑎𝑥
𝑖𝑗 |𝛾p oblem, whe e 𝑙𝑚𝑖𝑛
𝑖𝑗 and 𝑙𝑚𝑎𝑥
𝑖𝑗
a e he minimum and maximum ime lag be ween he comple ion ime o a
job and he s a ime in he subsequen s age, espec i ely. Le 𝑏be he
𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡, 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |𝛾p oblem, wi h 𝑡𝑖𝑗 =𝑙𝑚𝑖𝑛
𝑖𝑗 as he anspo
ime be ween ope a ions 𝑂(𝑖, 𝜋𝑖𝑗 )and 𝑂(𝑖, 𝜋𝑖+1,𝑗 ), and 𝑤𝑡𝑚𝑎𝑥 =𝑙𝑚𝑎𝑥
𝑖𝑗 −𝑙𝑚𝑖𝑛
𝑖𝑗 .
Then, bo h p oblems a e equi alen . Bo h p oblems a e he e o e equi alen .
Eu opean Jou nal o Ope a ional Resea ch 325 (2025) 1–19
6
V. Fe nandez-Viagas
Fu he mo e, he p oblem is conside ed equi alen o o he ela ed
scheduling p oblems unde di e en condi ions. Maggu e al. (1981)
ha e shown ha comple ion imes o a schedule in he 𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗)|𝛾
p oblem can be di ec ly compu ed by sol ing he 𝐹2∥𝛾p oblem, wi h
he same schedule, and adding a cons an o ob ained comple ion imes
in his las p oblem, i.e., bo h p oblems a e equi alen i he objec i e
unc ion is egula . As esul , 𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗)|𝐶𝑚𝑎𝑥 and 𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗),
𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 p esen he same op imal solu ions (see Mkadem e al.,2021
o u he e idence o his obse a ion). A simila esul is ound by Lee
and Chen (2001) be ween 𝐹2|𝑇(𝑀 𝑜, 𝐼)|𝛾𝑟and 𝐹2|𝑇(𝑀 𝑜, 𝐵), 𝑝𝑟𝑚𝑢|𝛾𝑟(𝛾𝑟
being any egula unc ion). Rega ding he p oblem wi h maximum
wai ing ime (i.e. he classical low shop wi h minimal and maximal
ime lags), Sama ghandi (2019) es ablishes an equi alence be ween
he pe mu a ion and non-pe mu a ion a ian s when he maximum
wai ing ime o each job 𝑗,𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 , sa is ies 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 < 𝑝𝑖𝑘 +𝑝𝑖+1,𝑘 o
each 𝑖,𝑗, and 𝑘≠𝑗. In ac , o he wo-machine a ian o he
p oblem, Dhouib e al. (2018) also p o e ha e e y non-pe mu a ion
solu ion is un easible i max∀𝑗{𝑤𝑡𝑚𝑎𝑥
𝑗}is below min∀𝑗{𝑝1𝑗+𝑝2𝑗+𝑡𝑗}. They
also analyse mo e speci ic cases o equi alence and he idle ime in-
cu ed when a non-pe mu a ion solu ion is applied. Recen ly, Kha ami
e al. (2023) es ablish he equi alence be ween he 𝐹2|𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾
and 𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾p oblems.
Rega ding o he equi alences, he ollowing p ope y s a es ha
anspo imes can be omi ed i hey a e cons an and a egula
objec i e unc ion is assumed. The e o e, in he case o cons an ans-
po imes, his p ope y ex ends he p e ious heo e ical esul ound
by Maggu e al. (1981) o 𝑖 >2.
Theo em 5.7. Le 𝑎be he 𝐹𝑚∥𝛾𝑟p oblem (i 𝛾𝑟is any egula objec i e
unc ion). Le 𝑏be he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼)|𝛾𝑟p oblem. 𝑎and 𝑏a e he e o e
equi alen .
In cases whe e he numbe o ehicles is a ini e numbe , Soukhal
e al. (2005) es ablish ha he wo-machine p oblem wi h no-wai con-
s ain and a single anspo e is equi alen o a no-wai h ee-machine
low shop scheduling p oblem, while S e n and Vi ne (1990) analyse
he equi alence o he wo-machine scheduling p oblem conside ing a
blocking cons ain . In iew o he abo e, we gene alise his las inding
by assuming a single anspo e be ween each ba ch o p ocessing
machine.
Theo em 5.8. Le 𝑎be he 𝐹2𝑚−1∥𝛾p oblem. Le 𝑏be he
𝐹𝑚|𝑇𝑚−1(𝑀𝑎𝑜, 𝐼 𝑖𝑗)|𝛾p oblem whe e he e is a anspo e be ween each pai
o sequen ial machines. Bo h p oblems a e he e o e equi alen .
Co olla y 5.1. Le 𝑎be he 𝐹2𝑚−1|𝑏𝑎𝑡𝑐 ℎ|𝛾p oblem whe e machines 𝑖=
1,3,…,2𝑚− 1a e single p ocessing machines and machines 𝑖= 2,4,…,2𝑚−
2a e 𝑝-ba ch p ocessing machines. Le 𝑏be he 𝐹𝑚|𝑇𝑚−1(𝑀𝑎𝑜, 𝐵 𝑖𝑗)|𝛾p ob-
lem whe e he e is a anspo e be ween each pai o sequen ial machines.
𝑎and 𝑏a e he e o e equi alen .
As can be obse ed, many o he a ian s o he p oblem a e
equi alen o he adi ional low shop p oblem. I is he e o e no
su p ising ha many o he esul s build upon he ad ances es ablished
in he adi ional low shop. This is he case, o example, o he
e e sibili y p ope y, whe e he same makespan alue can be ound
whe he he p oblem is cons uc i e o wa d o backwa d (see Ribas
e al.,2010 o mo e de ails). This p ope y can be applied, o example,
o he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |𝐶𝑚𝑎𝑥 p oblem (Wang, Huang, & Li,
2018). Rega ding o he p oblem p ope ies, Kha ami e al. (2023) ha e
es ablished ha a pe mu a ion schedule is no necessa ily an op imal
solu ion o he 𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾p oblem. An examina ion o
mo e cons ained p oblems can also be ound in Fond e elle e al.
(2009), Huo e al. (2009), who s udy speci ic p ope ies o he no-wai
wo machine scheduling p oblem wi h 𝑇(𝑀 𝑜, 𝐼 𝑗), and in Raywa d-
Smi h and Rebaine (2008), Rebaine (2005), Yu e al. (2004), who
analyse p ope ies conside ing uni p ocessing imes and wo machines.
A summa y o he equi alences iden i ied in he li e a u e and
in his pape o he FSP conside ing anspo a ion be ween s ages
is p esen ed in Table 2. Finally, in many cases, NP-ha dness o he
p oblem has been asce ained. A de ailed summa y o hese cases is
p esen ed in Table 3.
5.1.2. Op imisa ion algo i hms
5.1.2.1. T anspo a ion be ween s ages wi h an unlimi ed numbe o ans-
po e s. Rega ding he li e a u e using op imisa ion algo i hms o sol e
his ype o p oblem, Maggu e al. (1981,1982), Mi en (1959),
Rinnooy Kan (1976) a e he i s o add ess he p oblem o in e me-
dia e anspo a ion. S udying he p oblem wi h wo machines and
a pe mu a ion cons ain , hey assume job-dependen anspo a ion
imes and an unlimi ed numbe o anspo e s and p opose exac
algo i hms based on Johnson (1954). These algo i hms also ind he
op imum when he sum o weigh ed machine comple ion imes is
minimised (Fond e elle e al.,2008). Munie -Ko don and Rebaine
(2008) also p opose wo exac polynomial algo i hms o he h ee-
and ou -machine p oblems, al hough hey equi e p ocessing imes
o be uni s. In cases whe e p ocessing imes a e equal o he i s
and second machine o each job in he wo-machine case, Agee
(2008) p oposes a 3/2-app oxima ion algo i hm and se e al lowe
bounds. A 3/2 app oxima ion algo i hm and se e al lowe bounds a e
p oposed by Agee and Babu in (2007) o he no-wai a ian o
he p e ious p oblem wi h uni p ocessing imes. The no-wai a ian
wi h anspo imes aking wo alues is add essed by Agee (2020),
who p oposes a lowe bound and a 2-app oxima ion a algo i hm.
Rega ding o he cons ained p oblems, Yang and Che n (2000) p opose
a polynomial exac algo i hm o he pe mu a ion g oup scheduling
a ian . Fo he wo-machine p oblem wi h bo h an icipa o y and
non-an icipa o y sequence-independen se up imes, Çe inkaya (2006)
p oposes a Johnson-based exac algo i hm. The wo-machine p oblem
wi h no-wai (exac delays imes app oach) and missing ope a ions is
add essed by Leung e al. (2007), who p opose se e al exac and ap-
p oxima ion algo i hms o some a ian s o he p oblem. Also in ega d
o minimising o al comple ion imes, Msakni e al. (2016) add ess
he pe mu a ion wo-machine scheduling p oblem wi h minimum ime
lags, p oposing a b anch-and-bound and an i e a ed g eedy algo i hm
o sol e he p oblem. The same p oblem wi h no-wai cons ain (exac
delays) is sol ed in Huo e al. (2009) wi h he p oposal o se e al simple
heu is ics and wo me aheu is ics. Haoua i and Ladha i (2000) p opose
a b anch-and-bound algo i hm o he 𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑟𝑗|𝐿𝑚𝑎𝑥 p ob-
lem, including a simple heu is ic o ob ain uppe bounds and analysing
di e en lowe bounds. The wo-machine p oblem wi hou pe mu a ion
cons ain is add essed by Raywa d-Smi h and Rebaine (2008) o
makespan minimisa ion and conside ing uni p ocessing imes. They
p opose wo dispa ching ules analysing hei wo s cases. B ucke e al.
(2004) success ully p o e ha se e al a ian s o his p oblem a e
polynomially sol able (e en wi h 𝑚machines) o di e en objec i e
unc ions ( o al weigh ed comple ion imes, o al weigh ed numbe o
jobs, and o al weigh ed a diness). Fu he mo e, Dell’Amico (1996)
sol es he p oblem wi h non-uni p ocessing imes (𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗)|𝐶𝑚𝑎𝑥),
wi h ou app oxima e algo i hms, a abu sea ch me aheu is ic, and
se e al lowe bounds. Also in ela ion o his p oblem, Mkadem e al.
(2021) p opose a b anch-and-bound algo i hm, which is compa ed
wi h he p e ious exac algo i hm in 480 ins ances. Kha ami e al.
(2023) p o e ha he no-wai a ian o he p e ious p oblem is
polynomial and p opose se e al exac algo i hms o sol e some a ian s
o his p oblem. Dhouib e al. (2018) conside maximum wai ing imes
ins ead o he no-wai cons ain and p opose a new heu is ic by aking
in o accoun p ope ies o he p oblem. Fu he mo e, hey p opose an
MILP model (which can also be applied o he 𝑚-machine p oblem)
which ou pe o med he pe mu a ion MILP model p oposed by Dhouib
e al. (2013). The wo-machine low shop scheduling p oblem wi h
ime windows and equal a el imes is shown o be polynomial
by Zhang and an de Velde (2015) o maximisa ion o he weigh ed
Eu opean Jou nal o Ope a ional Resea ch 325 (2025) 1–19
7
V. Fe nandez-Viagas
Table 2
Equi alence be ween p oblems.
P oblem A P oblem B Re e ence
TSP(a)𝐹2|𝑇1(𝑀 𝑟, 𝐼), 𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔|𝐶𝑚𝑎𝑥 S e n and Vi ne (1990)
𝐹2||𝛾𝑟𝐹2|𝑇(𝑀 𝑜, 𝐼𝑗)|𝛾𝑟Maggu e al. (1981)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢|𝛾 𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗)|𝐶𝑚𝑎𝑥 Mkadem e al. (2021)
𝐹𝑚||𝛾𝑟𝐹𝑚|𝑇(𝑀 𝑜, 𝐼)|𝛾𝑟Theo em 5.7
𝐹𝑚|𝑙𝑚𝑖𝑛
𝑗|𝛾 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑗)|𝛾Theo em 5.1
𝐹𝑚|𝑒𝑥𝑎𝑐 𝑡−𝑙𝑗|𝛾 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾Theo em 5.5
𝐹𝑚|𝑒𝑥𝑎𝑐 𝑡−𝑙𝑗|𝛾 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾Theo em 5.5
𝐹2|𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾 𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾Kha ami e al. (2023)
𝐹2|𝑇(𝑀 𝑜, 𝐵), 𝑝𝑟𝑚𝑢|𝛾𝑟𝐹2|𝑇(𝑀 𝑜, 𝐼)|𝛾𝑟Lee and Chen (2001)
𝐹2𝑚−1||𝛾 𝐹𝑚|𝑇𝑚−1(𝑀𝑎𝑜, 𝐼 𝑖𝑗)|𝛾Theo em 5.8
𝐹(2𝑚− 1)|𝑏𝑎𝑡𝑐 ℎ|𝛾(machines 𝑖= 2,4,…a e 𝑝-ba ch) 𝐹𝑚|𝑇𝑚−1(𝑀𝑎𝑜, 𝐵 𝑖𝑗)|𝛾Co olla y 5.1
𝐹𝑚|𝑠𝑖𝑗 𝑘|𝛾(non-an icipa o y 𝑠𝑖𝑗 𝑘)𝐹𝑚|𝑇(𝑆 𝑜, 𝐼 𝑗 𝑘)|𝛾Theo em 5.9
𝐹𝑚, 𝑆 𝜏|𝑠𝑖𝑗 𝑘|𝛾(non-an icipa o y 𝑠𝑖𝑗 𝑘)𝐹𝑚|𝑇𝜏(𝑆 𝑜, 𝐼 𝑖𝑗 𝑘)|𝛾Theo em 5.10
𝐹𝑚|𝑠𝑖𝑗 𝑘|𝛾 𝐹𝑚|𝑅𝑚(𝑅𝑜, 𝐼 𝑗 𝑘)|𝛾Theo em 5.11
𝐹2|𝑅2(𝑅𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢|𝛾 𝐹2|𝑅2(𝑅𝑜, 𝐼 𝑗)|𝛾Theo em (Yu e al.,2011)
𝐹𝑚|𝑟𝑗|𝛾 𝐹𝑚|𝑇(𝑃 𝑜, 𝐼 𝑗)|𝛾Theo em 5.12
𝐹𝑚+1||𝛾 𝐹𝑚|𝑇1(𝑃 𝑟, 𝐼 𝑗)|𝛾Theo em 5.13
𝐹𝑚+1||𝛾 𝐹𝑚|𝑇1(𝑃 𝑜, 𝐼 𝑗)|𝛾Co olla y 5.2
𝐻 𝐹𝑚+1 ,(𝑃 𝜏 , 𝐹𝑚)||𝛾 𝐹𝑚|𝑇𝜏(𝑃 𝑟, 𝐼 𝑗)|𝛾Co olla y 5.3
𝐹𝑚||𝐿𝑚𝑎𝑥 𝐹𝑚|𝑇(𝐷 𝑜, 𝐼 𝑗)|𝐶𝐷
𝑚𝑎𝑥 Hall (1997)
𝐹𝑚||∑𝐶𝑗𝐹𝑚|𝑇(𝐷 𝑜, 𝐼 𝑗)|∑𝐶𝐷
𝑗Theo em 6.1
𝐹𝑚||∑𝐿𝑗𝐹𝑚|𝑇(𝐷 𝑜, 𝐼 𝑗)|∑𝐿𝐷
𝑗Co olla y 6.1
𝐹𝑚|𝑟𝑗|∑𝐹𝑗𝐹𝑚|𝑇(𝐷 𝑜, 𝐼 𝑗), 𝑟𝑗|∑𝐹𝐷
𝑗Co olla y 6.2
𝐹𝑚||𝐶𝑚𝑎𝑥 𝐹𝑚|𝑇(𝐷 𝑓 𝑜, 𝐼), 𝑡𝑗= 0|𝐶𝐷
𝑚𝑎𝑥 Hall e al. (2001)
𝐹𝑚||𝐿𝑚𝑎𝑥 𝐹𝑚|𝑇(𝐷 𝑓 𝑜, 𝐼), 𝑡𝑗= 0|𝐿𝐷
𝑚𝑎𝑥 Hall e al. (2001)
𝐹𝑚+1|𝑠𝑚+1,𝑗 |𝛾 𝐹𝑚|𝑇1(𝐷 𝑟, 𝐼 𝑗)|𝛾Theo em 6.2
𝐹𝑚+1||𝛾 𝐹𝑚|𝑇1(𝐷 𝑜, 𝐼 𝑗)|𝛾Co olla y 6.3
𝐻 𝐹𝑚+1 ,(𝐹𝑚, 𝑃 𝜏)|𝑠𝑚+1,𝑗 |𝛾 𝐹𝑚|𝑇𝜏(𝐷 𝑟, 𝐼 𝑗)|𝛾Theo em 6.3
𝐻 𝐹𝑚+1 ,(𝐹𝑚, 𝑃 𝜏)||𝛾 𝐹𝑚|𝑇𝜏(𝐷 𝑜, 𝐼 𝑗)|𝛾Co olla y 6.4
𝐹3|𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾 𝐹2|𝑇1(𝐷 𝑟, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝛾Soukhal e al. (2005)
𝐹𝑚|𝑇(𝐷 𝑟, 𝐼 𝑗)|𝛾 𝐹𝑚|𝑇(𝐶 , 𝐼 𝑗)|𝛾Theo em 6.4
𝐹𝑚|𝑇(𝐷 𝑟, 𝐵 𝑗)|𝛾 𝐹𝑚|𝑇(𝐶 , 𝐵 𝑗)|𝛾Co olla y 6.5
𝐹𝑚||𝐿𝑚𝑎𝑥 𝐹𝑚|𝑇(𝐶 , 𝐼 𝑗)|𝐶𝐷
𝑚𝑎𝑥 Co olla y 6.6
aT a elling salesman p oblem.
numbe o selec ed jobs. To sol e he p oblem, hey p opose a Dynamic
P og amming algo i hm.
Rega ding he pe mu a ion p oblem wi h 𝑚machines, Szwa c (1983)
p oposes a lowe bound o he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 p oblem. Fo
bo h his p oblem and ha wi h one speci ic anspo e be ween each
wo consecu i e machines, Nade i, Ahmadi Ja id, and Jolai (2010)
p opose 3 MILP models, an a i icial immune sys em, and adap wo dis-
pa ching ules and ou cons uc i e heu is ics om ela ed scheduling
p oblems. Fond e elle e al. (2006) also add ess his p oblem, al hough
conside ing minimal and maximal ime lags. They p opose a b anch-
and-bound algo i hm using di e en lowe bounds, which is ini ialised
wi h se e al cons uc i e heu is ics. Also o his p oblem, Wang,
Huang, and Li (2018) p opose an MILP model and a cons uc i e
heu is ic using i s e e sibili y p ope y. These p oposals a e com-
pa ed wi h he bes heu is ics o Fond e elle e al. (2006) and Hamdi
and Loukil (2011). In his p oblem, Zhao e al. (2017) p opose an
i e a ed g eedy algo i hm, which also ou pe o ms he gene ic algo-
i hm by Hamdi and Loukil (2011) and a cons uc i e heu is ic based
on Fond e elle e al. (2006). They also add essed he non-pe mu a ion
a ian o he p oblem. In con as , he pe mu a ion a ian o he
p oblem wi h o al a diness minimisa ion is add essed by Hamdi and
Loukil (2015a), Hamdi and Toumi (2019). The o me p opose and
compa e an MILP model, h ee dispa ching ules, and se e al lowe
bounds o he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢, 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |∑𝑇𝑗p oblem, while he
la e p opose di e en MILP models o he 𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢|∑𝑇𝑗
p oblem. In o de o minimise he o al numbe o jobs, Hamdi and
Loukil (2015b) p opose h ee dispa ching ules and di e en lowe
bounds, whe eas Fond e elle e al. (2008) op o he minimisa ion o
he weigh ed sum o machine comple ion imes, p oposing a b anch-
and-bound algo i hm o he 𝑚-machine p oblem. The aspec o minimi-
sa ion o maximum la eness is add essed in Fond e elle e al. (2009).
They p opose a b anch-and-bound algo i hm using di e en lowe
bounds and ini ial solu ions. The exac algo i hm is compa ed wi h
he p oposal o Fond e elle e al. (2005). In addi ion, a lexicog aphic
op imisa ion o he p oblem is add essed by Dhouib e al. (2013)
minimising he numbe o a dy jobs and he makespan, while also
de eloping an MILP model and se e al a ian s o simula ed annealing
algo i hms. Liou and Hsieh (2015) sol e he mul i-s age low shop
g oup scheduling wi h pe mu a ion and makespan minimisa ion by
p oposing a hyb id me aheu is ic ha combines bo h he Pa icle
Swa m Op imisa ion (PSO) and GA algo i hms. A a ian o he low
shop wi h limi ed in e media e bu e , pe mu a ion, ba ch, and ixed
in e media e anspo da ed is add essed by P asad e al. (2006). They
p opose a gene ic algo i hm o sol e bo h he minimisa ion o mean
comple ion imes o jobs and ba ches and ha o he s anda d de ia ion
o he comple ion imes, which is es ed agains se e al di e en
exis ing gene ic algo i hms. A compa ison be ween pe mu a ion and
Eu opean Jou nal o Ope a ional Resea ch 325 (2025) 1–19
8
V. Fe nandez-Viagas
Table 3
NP-ha d scheduling p oblems wi h anspo a ion be ween s ages.
P oblem Re e ence P oblem Re e ence
𝐹2|𝑇1(𝑀 𝑟, 𝐵𝑏)|𝐶𝑚𝑎𝑥 (𝑏≥3) (a)Lee and Chen (2001)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡|∑𝐶𝑗Leung e al. (2007)
𝐹2|𝑇1(𝑀 𝑟, 𝐼)|𝐶𝑚𝑎𝑥 (a)Lee and Chen (2001)𝐹2|𝑇(𝑀 𝑜, 𝐼), 𝑛𝑜 −𝑤𝑎𝑖𝑡|∑𝐶𝑗Leung e al. (2007)
𝐹2|𝑇1(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑝𝑖𝑗 =𝑝|𝐶𝑚𝑎𝑥 Hu ink and Knus (2001)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑡𝑗∈ {𝑡′, 𝑡′′ }|∑𝑤𝑖𝐶𝑖Fond e elle e al. (2008)
𝐹2|𝑇1(𝑀 𝑜, 𝐼), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 Hu ink and Knus (2001)𝐹2|𝑇1(𝑀 𝑟, 𝐵), 𝑏𝑎𝑡𝑐 ℎ|𝐶𝑚𝑎𝑥 Tang and Liu (2009b)
𝐹2|𝑇1(𝑀 𝑟, 𝐼 𝑗), 𝑓 𝑎𝑚, 𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔|𝐶𝑚𝑎𝑥 Yuan e al. (2020)𝐹3|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑟𝑚𝑢, 𝑛𝑜 −𝑤𝑎𝑖𝑡|𝐶𝑚𝑎𝑥 Kha ami e al. (2023)
𝐹2|𝑇1(𝑀 𝑟, 𝐼 𝑗), 𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔|𝐶𝑚𝑎𝑥 Ganesha ajah e al. (1998)𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡, 𝑜𝑟𝑑 𝑒𝑟𝑒𝑑|𝐶𝑚𝑎𝑥 Kha ami e al. (2023)
𝐹2|𝑇1(𝑀 𝑟, 𝐼), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 Kise e al. (1991b)𝐹2|𝑇2(𝑅𝑡𝑜, 𝐼 𝑗 𝑘)|𝐶𝑡𝑣
𝑚𝑎𝑥 Yu e al. (2011)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝1𝑗=𝑝2𝑗|𝐶𝑚𝑎𝑥 Dell’Amico and Vaessens (1996)𝐹2|𝑇(𝑃 𝑜, 𝐼 𝑗)|𝐶𝐷
𝑚𝑎𝑥 Lens a e al. (1977)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗)|𝐶𝑚𝑎𝑥 (una y NP-comple e) Dell’Amico (1996)𝐹2|𝑇1(𝐷 𝑟, 𝐵𝑏)|𝐶𝑚𝑎𝑥 (4≤𝑏≤𝑛∕2)Lee and Chen (2001)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑚𝑡𝑛|𝐶𝑚𝑎𝑥 (una y NP-comple e) Dell’Amico (1996)𝐹2|𝑇1(𝐷 𝑟, 𝐼)|𝐶𝑚𝑎𝑥 (a)Lee and Chen (2001)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑟𝑗, 𝑝𝑖𝑗 = 1|∑𝐶𝑗B ucke e al. (2004)𝐹2|𝑇(𝐷 𝑜, 𝐼 𝑗)|𝐶𝐷
𝑚𝑎𝑥 Lens a e al. (1977)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑖𝑗 = 1|∑𝑤𝑗𝐶𝑗B ucke e al. (2004)𝐹2|𝑇1(𝐷 𝑟, 𝐵)|𝐶𝑚𝑎𝑥 Pan e al. (2009)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝1𝑗=𝑝2𝑗, 𝑡𝑗∈ {0, 𝑙}|𝐶𝑚𝑎𝑥 Yu (1996)𝐹2|𝑇1(𝐷 𝑟, 𝐵)|∑𝐶𝑗Pan e al. (2009)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑖𝑗 = 1|∑𝐶𝑖[𝑛]Fond e elle e al. (2008)𝐹2|𝑇1(𝐷 𝑟, 𝐵2𝑗)|𝐶𝑚𝑎𝑥 Soukhal e al. (2005)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑝𝑖𝑗 = 1|𝐶𝑚𝑎𝑥 Yu e al. (2004)𝐹2|𝑇1(𝐷 𝑟, 𝐵2𝑗), 𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔|𝐶𝑚𝑎𝑥 Soukhal e al. (2005)
𝐹2|𝑇(𝑀 𝑜, 𝐼), 𝑤𝑡𝑚𝑎𝑥 |𝐶𝑚𝑎𝑥 Fond e elle e al. (2006)𝐹2|𝑇1(𝐷 𝑟, 𝐵3)|𝐶𝑚𝑎𝑥 Soukhal e al. (2005)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡, 𝑡𝑗∈ {𝑡′, 𝑡′′ }|𝐶𝑚𝑎𝑥 Leung e al. (2007)𝐹2|𝑇1(𝐷 𝑟, 𝐵3), 𝑝1𝑗=𝑝|𝐶𝑚𝑎𝑥 Yuan e al. (2007)
𝐹2|𝑇(𝑀 𝑜, 𝐼 𝑗), 𝑛𝑜 −𝑤𝑎𝑖𝑡, 𝑝𝑖𝑗 = 1|𝐶𝑚𝑎𝑥 Leung e al. (2007)𝐹2|𝑇1(𝐷 𝑟, 𝐵3), 𝑏𝑙 𝑜𝑐 𝑘𝑖𝑛𝑔|𝐶𝑚𝑎𝑥 Soukhal e al. (2005)
aI is NP-ha d e en when he a el imes in bo h di ec ions a e iden ical.
non-pe mu a ion cons ain s is pe o med by Rebaine (2005), compa -
ing he wo s -case pe o mance a io be ween he bes solu ions o
𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑝𝑟𝑚𝑢|𝐶𝑚𝑎𝑥 and 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗)|𝐶𝑚𝑎𝑥 ei he when p ocess-
ing imes a e uni o he numbe o machines is wo. In ela ion solely
wi h he non-pe mu a ion p oblem wi h 𝑚machines, Sama ghandi
(2019) p oposes an MILP model and wo cons ain p og amming algo-
i hms o exac ly sol e small ins ances o he 𝐹𝑚|𝑇(𝑀 𝑜, 𝐼 𝑖𝑗), 𝑤𝑡𝑚𝑎𝑥
𝑖𝑗 |𝐶𝑚𝑎𝑥
p oblem, and a Tabu Sea ch me aheu is ic o la ge-sized ins ances. The
la e ou pe o ms he meme ic algo i hm p oposed by Caumond e al.
(2008), which is adap ed om he jobshop layou . This algo i hm is
also ou pe o med by Ye e al. (2017), who p opose i e e sions o
he i e a ed g eedy algo i hm. Zhang and Van De Velde (2010) p opose
an app oxima e algo i hm o he p oblem wi h missing ope a ions,
minimising o al comple ion imes, and ano he ha conside s elease
imes and makespan minimisa ion. The p oblem o allowing mul i-
ple ope a ions o a job a he same s age (deno ed eci cula ion) is
add essed by Riezebos and Gaalman (1998), Riezebos e al. (1995),
who p opose se e al lowe bounds and simple cons uc i e heu is ics,
conside ing elease imes.
5.1.2.2. T anspo a ion be ween s ages wi h a ini e numbe o anspo e s.
Rega ding a ini e numbe o anspo e s, Panwalka (1991) p oposes
a Johnson-based exac algo i hm o he wo-machine p oblem wi h
cons an a el imes, blocking, and a single anspo e , while Lee
and Chen (2001) p opose an app oxima e algo i hm and di e en poly-
nomial exac algo i hms o speci ic a ian s o he p oblem. Hu ink
and Knus (2001) also p opose wo polynomial exac algo i hms o
wo a ian s o he wo-machine scheduling p oblem (wi h cons an
p ocessing imes and bina y anspo a ion imes, and uni p ocessing
imes) and o he 𝑚-machine p oblem wi h cons an p ocessing imes
wi h job-independen anspo a ion imes. Lee and S use ich (2005)
add ess he wo-machine low shop wi h in e media e anspo a ion
wi h non-negligible ime o e u n he anspo e o he i s machine.
Jobs a e mo ed using a anspo e wi h in ini e capaci y. To sol e
his p oblem, he au ho s p opose an app oxima e algo i hm ha is a
mos (3/2) imes he op imum. This p oblem wi h a ba ch is add essed
by Zhong and Chen (2015), who p opose a Johnson-based app oxima e
algo i hm whose limi s a e heo e ically analysed. Indi idual ans-
po a ion is applied by Tang e al. (2010), who p opose a cons uc i e
heu is ic. Simila ly, Gong and Tang (2011) add ess a ela ed wo-
machine low shop, al hough anspo ing jobs in ba ches (wi h a
single anspo e wi h limi ed capaci y), wi h each job occupying a
physical space wi hin he ba ch. The au ho s p opose wo app oxima e
algo i hms o he p oblem and o he speci ic case whe e all jobs
occupy he same space. This algo i hm is ou pe o med in Dong e al.
(2016) by modi ying he bin-packing algo i hm and he i s and las
ba ches o he schedule (whe e he algo i hm solu ion is a mos 11/5 o
he op imum). In u n, his algo i hm was ecen ly ou pe o med in Lan
e al. (2024) (wi h a mos 5/3 o he op imum) conside ing iden ical
sizes o all jobs. Tang and Liu (2009b) conside ha he second
machine is a ba ch machine and sol e his p oblem wi h a ma hema ical
model, and a simple heu is ic. They also analyse he wo s case o he
he simple heu is ic, compa ing i wi h wo lowe bounds. A simila
esea ch is ca ied ou by Tang and Liu (2009a) conside ing ha he
i s machine is ba ch and he second one is a single machine. The las
wo pa allel ba ch app oaches a e also add essed by Behnamian e al.
(2012b), who p opose an MILP model and a heu is ic o each o he
p e ious cases. Wang e al. (2022) ackle he wo-machine low shop
scheduling p oblem wi h a pa allel ba ch machine and a single p o-
cessing machine, conside ing anspo imes be ween bo h machines,
which can be done in ba ches. They assume a cons an ound- ip ime
as anspo ime which is he same o he ou wa d and e u n ips.
To sol e he p oblem, hey p opose an app oxima e algo i hm which
is compa ed wi h an MILP model. S e n and Vi ne (1990) add ess he
wo-machine low shop scheduling p oblem wi h non-negligible ime o
e u n he anspo be ween s ages and wi h ze o in e media e bu e .
They de elop an app oxima e polynomial algo i hm ha educes he
p oblem o an asymme ic a elling salesman p oblem, which is hen
compa ed o a p oposed lowe bound o he p oblem. Yuan e al. (2020)
conside he wo-machine g oup scheduling wi h anspo a ion imes
be ween bo h machines, whe e some jobs canno be s o ed in he bu e
(pa ial blocking o jobs), and he makespan is minimised. They p opose
bo h an MILP model and a co-e olu iona y gene ic algo i hm. The
gene ic algo i hm is compa ed wi h di e en a ian s o he algo i hm
and wi h s a e-o - he-a algo i hms o ela ed scheduling p oblems.
The p oblem wi h uzzy da a is add essed by Gup a e al. (2013) o wo
machines and conside ing se up imes, p oposing an exac algo i hm
o bo h makespan and o al low ime as single objec i es. In case o
o he objec i e unc ions, S e ens and Gemmill (1997) add ess he wo-
machine low shop scheduling p oblem wi h ound- ip and cons an
a el imes. In o de o minimise maximum la eness, hey p opose
wo cons uc i e heu is ics, which a e compa ed wi h he minimum
Eu opean Jou nal o Ope a ional Resea ch 325 (2025) 1–19
9
V. Fe nandez-Viagas
Fig. 4. P oposed con ibu ions and me hodologies.
ela ionships o he p oblems unde s udy wi h ela ed scheduling
p oblems ha e also been s udied. In ligh o his e iew and he poin s
highligh ed in he p e ious sec ion, he ollowing conclusions and
u u e esea ch lines can be iden i ied:
•F om p e ious poin s 3,4,5, and 6, i ollows ha he e is
no exhaus i e compa ison o algo i hms. In u n, his conclusion
leads o he ollowing lines o esea ch:
–An ex ensi e compu a ional e alua ion o exac and app ox-
ima e algo i hms is pe inen .
–Small, medium, and la ge-size benchma k ins ances a e
needed o compa e he p oposals.
–Fu u e p oposals o de eloping op imisa ion algo i hms
should be compa ed wi h p e ious s a e-o - he a algo-
i hms om he p oblem unde conside a ion. In case o new
p oposals o no el scheduling p oblems, hese should be
compa ed wi h algo i hms adap ed om ela ed o equi a-
len scheduling p oblems unde he same condi ions, when
possible, o wi h exac app oaches designed o he p oblem
(in case o new app oxima e app oaches).
•Despi e inc eased in e es in he p oblem in ecen yea s, he e
a e s ill se e al p oblems (see p e ious poin s 2,7,10, and 11)
which ha e nei he been analysed no sol ed.
•Almos all pape s add ess p oblems wi h anspo conside ing
de e minis ic da a. I would be o in e es o u he analyse
hese ypes o p oblems wi h s ochas ici y, especially as ega ds
anspo imes, a poin which, o he bes o ou knowledge, has
no ye been ouched upon.
•In addi ion, mos pape s a e ocused on adi ional e olu iona y
o non-popula ion-based algo i hms. Only a single e e ence used
a ein o cemen lea ning-based me hod, and none men ioned neu-
al ne wo ks o machine lea ning di ec ly. Fu he esea ch in o
he pe o mance o hese inno a i e echniques o he unde
s udy would be bene icial.
In addi ion, he ollowing signi ican challenges a e open o u u e
academics and p ac i ione s in e es ed in he p oblem.
•Al hough a heo e ical analysis o he ela ionship be ween he
p oblems unde conside a ion and ela ed ones is p esen ed in
his pape , his analysis should be empi ically ex ended. Tha is,
comple e enume a ions and op imal solu ion analyses a e needed
o es ablish o he limi s o he p oblems, such as he analysis
pinpoin ing when he in eg a ed ou ing and scheduling p oblem
is mo e closely ela ed o he scheduling associa e p oblem o o
i s equi alen ehicle ou ing p oblem.
•Despi e he impo ance o speed-up me hods in low-shop-based
scena ios (see e.g. Fe nandez-Viagas,2022;Fe nandez-Viagas
e al.,2020;Fe nandez-Viagas, Talens, & F aminan,2022;Geng &
Li,2023;Tailla d,1990;Tao e al.,2023;Wang e al.,2023), we
a e no awa e o any ela ed me hod o he low shop p oblem
wi h anspo a ion.
•The echnological ad ances b ough abou by Indus y 4.0 al-
low eal- ime in o ma ion upda es o be conside ed (p ocessing
imes, due da es, elease imes, anspo imes, ...) in he shop
(Fe nandez-Viagas & F aminan,2022). These new indus y 4.0-
based scena ios (see e.g. Fa hollahi-Fa d e al.,2024;Ghaleb &
Taghipou ,2023;Ghaleb e al.,2020;Li & Huang,2021) would
be o g ea in e es in he ield.
•Finally, al hough many applica ions in he li e a u e ha e been
sol ed compa ed o ela ed scheduling p oblem, he li e a u e
should mo e owa ds mo e ealis ic applica ions a he han he
cu en p edominan heo e ical a ian s. The e o e, heo e ical
s udies should ocus mo e on adi ional p oblems wi hou many
cons ain s, whe eas mo e cons ained p oblems should be ad-
d essed mainly as eal-wo ld applica ions.
CRediT au ho ship con ibu ion s a emen
Vic o Fe nandez-Viagas: W i ing – e iew & edi ing, W i ing
– o iginal d a , Visualiza ion, Valida ion, Supe ision, So wa e, Re-
sou ces, P ojec adminis a ion, Me hodology, In es iga ion, Funding
acquisi ion, Fo mal analysis, Da a cu a ion, Concep ualiza ion.
Acknowledgemen s
The au ho s wish o hank he e e ees o hei commen s on he
ea lie e sions o he manusc ip . This s udy has been unded by Ins i-
u o de Salud Ca los III (ISCIII) h ough he p ojec ‘‘PI22/01096’’ and
co- unded by he Eu opean Union. This esea ch was also suppo ed by
he Eu opean Commission, unde he p ojec ExPliCi ( e .101086465 -
HORIZON-MSCA-2021-SE-01-01)
Appendix A. Supplemen a y da a
Supplemen a y ma e ial ela ed o his a icle can be ound online
a h ps://doi.o g/10.1016/j.ejo .2024.11.034.
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