On the optimization of green multimodal transportation: a case study of the West German canal system

Bins eld, Tom; Hamdan, Sadeque; Jouini, Oualid; Gas , Johannes
A icle — Published Ve sion
On he op imiza ion o g een mul imodal anspo a ion: a
case s udy o he Wes Ge man canal sys em
Annals o Ope a ions Resea ch
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Sugges ed Ci a ion: Bins eld, Tom; Hamdan, Sadeque; Jouini, Oualid; Gas , Johannes (2024) : On he
op imiza ion o g een mul imodal anspo a ion: a case s udy o he Wes Ge man canal sys em,
Annals o Ope a ions Resea ch, ISSN 1572-9338, Sp inge US, New Yo k, NY, Vol. 351, Iss. 1, pp.
667-726,
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Annals o Ope a ions Resea ch (2025) 351:667–726
h ps://doi.o g/10.1007/s10479-024-06075-5
ORIGINAL RESEARCH
On he op imiza ion o g een mul imodal anspo a ion: a
case s udy o he Wes Ge man canal sys em
Tom Bins eld1,3 ·Sadeque Hamdan2·Oualid Jouini1·Johannes Gas 3
Recei ed: 30 May 2023 / Accep ed: 22 May 2024 / Published online: 4 June 2024
© The Au ho (s) 2024
Abs ac
In his s udy, we add ess a biobjec i e mul imodal ou ing p oblem ha consis s o selec -
ing anspo a ion modes and hei espec i e quan i ies, op imizing ansshipmen loca ions,
and alloca ing po o de s. In he objec i e unc ions, we minimize o al anspo a ion cos s
and use he EcoT ansi me hodology o minimize o al g eenhouse gas emissions. The op i-
miza ion model selec s he anspo a ion mode and ansshipmen po whe e quan i ies a e
ansshipped om one mode o ano he . We compa e inland wa e way anspo a ion and
ucks encoun e ing in as uc u e ailu es ha equi e e ou ing o modal shi ing in a eal-
li e case s udy on he supply o goods o he chemical indus y in he Wes Ge man canal
sys em. We p opose a popula ion-based heu is ic o sol e la ge ins ances in a easonable com-
pu a ion ime. A sensi i i y analysis o demand, o a ying lock imes, and o in as uc u e
ailu e scena ios was conduc ed. We show ha compa ed wi h inland wa e way anspo a-
ion, mul imodal anspo a ion educes cos s by 23% because o longe lock imes. Ou
analysis shows ha he use o inland wa e way anspo a ion only du ing in as uc u e ail-
u es imposes nea ly 28% highe cos s pe day depending on he ailu e loca ion compa ed
o ha o he case o no ailu es. We also show ha he use o a mul imodal anspo a ion
sys em helps o educe his cos inc ease in lock ailu e scena ios.
Keywo ds Mul imodal anspo a ion ·Inland wa e way anspo ·G eenhouse gas
emissions ·Sus ainabili y ·Vehicle ou ing ·Modal shi ·Op imiza ion
BSadeque Hamdan
s.hamdan@bango .ac.uk
Tom Bins eld
.bins eld@4 low.com
Oualid Jouini
[email p o ec ed]
Johannes Gas
[email p o ec ed]
1Labo a oi e Genie Indus iel, Uni e si é Pa is-Saclay, Cen aleSupélec, 3 Rue Jolio -Cu ie,
Gi -su -Y e e 91190, Île-de-F ance, F ance
2Bango Business School, Bango Uni e si y, College Rd, Bango , Gwynedd LL57 2DG, UK
34 low Resea ch, 4 low, Halle s aße 1, 10587 Be lin, Ge many
123
668 Annals o Ope a ions Resea ch (2025) 351:667–726
1 In oduc ion
While he need o ene gy secu i y has induced p essu e on economies and socie ies in 2024,
inland wa e ways ensu e a scalable supply o ene gy eeds ock. Unde ypical ope a ing
condi ions, wa e ways a e eliable and lexible anspo sys ems based on an e icien in as-
uc u e (Fede al Minis y o T anspo and Digi al In as uc u e, 2019). Howe e , ex eme
wea he and dilapida ed in as uc u e h ea en he a ailabili y o wa e ways o eigh ans-
po in Ge many. App oxima ely 18 million ons o goods a e anspo ed mon hly on Ge man
inland wa e ways depending on hei a ailabili y. This olume equa es o mo e han wo mil-
lion long-haul uckloads (Fede al O ice o S a is ics, 2019). In gene al, public au ho i ies
aspi e o u he u ilize exis ing capaci y ese es o his en i onmen ally iendly mode o
anspo ; he plan is o shi a ic om oads o inland wa e ways: The Eu opean Union
is pu suing he a ge o doubling hei modal shi sha e up o 9% in 2030, as aligned wi h
he Ge man "Mas e plan Binnenschi " om he Ge man Fede al Minis y o T anspo and
Digi al In as uc u e (Sims e al., 2014).
O e all, inland wa e way anspo ep esen s an elemen a y componen o he Ge man
andEu opeanlogis icalsupplychains.None heless,inlandba gescanno se eandsa is y he
logis ical equi emen s o e e y indus y. O he anspo a ion modes, such as ucks, p omise
g ea e lexibili y and a ailabili y while no equi ing dedica ed in as uc u e (i.e., po s and
canals). E ol ing isks, such as in as uc u e ailu es o clima e change, among o he s,
also impac anspo a ion mode choice. Hence, mul imodal anspo is o en es ablished o
exploi he ad an ages o each mode.
In as uc u e ailu es, such as lock ailu es o b idge damage, a e he main easons o
he nona ailabili y o whole canal sec ions, esul ing in speci ic po s becoming empo a ily
una ailable (Gas e al., 2020). This una ailabili y has caused signi ican (economic) damage
o companies. Fo example, he low amoun o wa e on he Rhine Ri e in 2018 induced a
cos o e245 million o a chemical company in Ge many (Reu e s, 2019). Mo eo e , hese
isks lead o highe cos s, highe emissions, and missing he ime schedule; such issues can
incu addi ional cos s in he downs eam supply chain. These obse a ions highligh he need
o a no el mul imodal concep o ensu e good low along he supply chain, ega dless o he
a ailabili y o p ima y anspo a ion in as uc u e a he ime.
In his s udy, we o mula e an op imiza ion p oblem ha add esses hese issues. In his
o mula ion, di e en anspo a ion modes can be used ei he di ec ly om he depo o a
any po ha ac s as a ansshipmen po , whe e quan i ies unloaded by one mode a e shi ed
o ano he mode o anspo a ion. The p oblem is o mula ed as a biobjec i e ma hema ical
model in which we minimize he o al anspo a ion cos and he g eenhouse gas emissions
o he ne wo k sys em. The p oposed mixed-in ege linea model helps decision-make s o
iden i y he op imal ou e o each anspo a ion mode and each ehicle, he quan i y ans-
po ed by each ehicle and each anspo a ion mode, he loca ion o he ansshipmen po ,
and he quan i y shi ed om one mode o ano he . We demons a e how his mul imodal con-
side a ion can be educed o a single-mode op imiza ion model. We combine he ad an ages
o di e en anspo a ion modes and de elop an op imiza ion ool o de e mine he op imal
anspo a ion mode. The p oblem is NP-ha d, as we show in Sec . Ao he Appendix. The e-
o e, we p opose a popula ion-based heu is ic o sol e his p oblem. The heu is ics gene a e
a se o easible indi iduals, and he model a emp s o imp o e hese indi iduals in each
i e a ion by using 18 ope a o s. We also compa e di e en scena ios by using single and
mul iple modes. Fu he mo e, we analyze he impac o di e en scena ios on he anspo
o chemical goods in he Wes Ge man canal sys em in a case s udy.
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Annals o Ope a ions Resea ch (2025) 351:667–726 669
This s udy p o ides he ollowing con ibu ions. We op imized he selec ion o ansship-
men po s ha can be used o bo h loading and unloading in his se ing, which is inspi ed by
a eal-wo ld p oblem, gi en he mul imodal choice o ei he uck o inland wa e way ans-
po a ion. This ype o p oblem has ecei ed limi ed a en ion in he li e a u e. Fu he mo e,
we allow o di e en ypes o ehicles o be chosen; in he li e a u e, he ocus has been on
a single- ehicle ype in mul imodal p oblems. By adding a i ual po o ac as a depo , ou
app oach main ains model linea i y; his app oach a oids u he complexi y in his NP-ha d
p oblem. We use a biobjec i e o mula ion o model he po en ially con lic ing objec i es o
cos educ ion and educ ion o g eenhouse gas emissions, wi h emissions calcula ed by using
he EcoT ansi me hodology. Al hough his p oblem can s ill be sol ed in an accep able un-
ime o small o medium ins ances, sol ing la ge ins ances op imally becomes in easible. To
add ess his issue, we de eloped a pa icula popula ion-based heu is ic ha pe o ms well,
wi h less han 5% e o om he bes exac solu ion ound and mo e han 83% ime sa ings
in mos cases. We de i e se e al manage ial insigh s by using a p ac ical case s udy om
he esea ch p ojec P e iew. In his p ojec , we assess he ulne abili y and he esilience o
supply chains ha depend on he in as uc u e o he Wes Ge man canal sys ems by mainly
equi ing inland wa e way anspo . We highligh he bene i s o mul imodal anspo a ion
o e single-mode anspo a ion in e ms o cos and emissions unde a ious scena ios. The
scena ios a y in e ms o lock ime, demand, and whe he in as uc u e ailu e occu s. In e -
es ingly, i appea s ha he use o mul imodal anspo a ion educes he impac o inc easing
lock imes; mo eo e , mul imodal anspo a ion allows o sa ings in he case o in as uc-
u e ailu e. This demons a es he economic e ec i eness o e ou ing and modal shi s as
isk-mi iga ion s a egies o supply chains. We calcula ed he cos o educing emissions by
using each anspo a ion model, and we showed ha he mul imodal anspo a ion mode
p o ides a as e educ ion a e o 1.12% o emissions o each 1% inc ease in he cos s.
The emainde o his s udy is o ganizedas ollows. In Sec .2, we e iew he ele an wo k
ela ed o his s udy. In Sec .3, we desc ibe he me hodology used in his s udy. Sec ion4
con ains he solu ion app oach. In Sec .5, we p esen a case s udy, and in Sec .6, we desc ibe
he nume ical expe imen s ela ed o he p oposed heu is ic. Sec ion7p o ides manage ial
insigh s. Finally, Sec .8concludes he pape , and we highligh u u e esea ch a enues.
2 Li e a u e e iew
The mul imodal anspo a ion model is a ou ing p oblem a ian ha uses mul iple ehicles
and di e en anspo a ion modes. The use o mul iple ehicles is simila o he mul iple
a eling salesman p oblem (m-TSP), which is a gene aliza ion o he TSP p oblem, which
o iginally includes only one ehicle (salespe son). One impo an ea u e o he mul imodal
p oblem is he use o a ansshipmen poin , whe e quan i ies a e ans e ed om one mode o
ano he . The e o e, in his sec ion we e iew he ela ed li e a u e. Fi s , we discuss di e en
ou ing model a ian s. Then, we e iew wo ks ha ocus on mul imodal anspo a ion and
p o ide con ex o ou wo k wi hin he li e a u e.
2.1 Rou ing model a ian s and solu ion algo i hms
The TSP is a undamen al ou ing challenge o ind he sho es op imal ou es o minimize
a el cos s. Fi s in oduced in he 1930s and ex ensi ely analyzed since hen, he TSP
equi es inding a sequence o a salespe son o isi a se o nodes exac ly once, s a ing
123
670 Annals o Ope a ions Resea ch (2025) 351:667–726
and ending a a depo (Mille e al., 1960). Gu in and Punnen (2006) o e ed an o e iew
up o 2006, highligh ing he b oad applicabili y o he TSP and he eme gence o many
a ian s. The m-TSP, in ol ing mul iple salespeople, is de e mine he op imal sequence o
mul iple ehicles, wi h each salesman isi ing a subse o nodes exac ly once (Mille e
al., 1960). Rao (1980) explo ed bo h symme ic and asymme ic m-TSPs and compa ed hei
ans o ma ions.TheVRP,whichisclosely ela ed o heTSP,di e sin ha i seekssequences
o ehicles while conside ing hei capaci y. Va ian s include simul aneous pick-up and
deli e y (Min, 1989), spli pick-up m-VRP (Lee e al., 2006), and mul idepo m-VRP wi h
uel cons ain s (Sunda e al., 2016). A wo-echelon mul i ehicle loca ion- ou ing p oblem
wi h ime windows was in oduced by Go indan e al. (2014), and a hyb id mul iobjec i e
mul idepo VRP was de eloped by Londoño e al. (2024) o minimize he dis ance and he
con ol ou e leng h s anda d de ia ion. O he in e es ing a ian s include he ailway TSP,
whe e salespeople u ilize ailways o minimize a el ime, conside ing ains’ schedules and
nons op ou es (Hadjicha alambous e al., 2007), and he colo ed TSP and colo ed bo leneck
TSP,which add ess mul imachine enginee ingsys em planning p oblems (Dong& Cai, 2019;
Dong e al., 2023).
The a eling pu chase p oblem (TPP), a ou ing and pu chasing challenge, in ol es
isi ing supplie s o buy p oduc s a a ying p ices o sa is y demand a he lowes cos . The
TPP is dis inguished by he need o minimize bo h a eling and pu chasing cos s, making he
p oblem mo e complex (Cheai ou e al., 2021b). Va ian s o he TPP include de e minis ic,
biobjec i e, and budge cons ain s o o al quan i y discoun s (Mane ba e al., 2017;Ra i
&Salman,1999; Rie a-Ledesma & Salaza -González, 2005; Mane ba & Mansini, 2012).
Finally, he Family TSP is ocused on minimizing cos s o isi a p ede e mined numbe o
ci ies, equi ing decisions on which ci ies o isi wi hin each g oup (Be na dino & Paias,
2018). Fo u he explo a ion o a ian s such as quo a, p o i -based, and ime window TSPs,
eade s can e e o Ila a asi and Joseph (2014), Pop e al. (2024).
The TSP and i s a ian s, which a e s ongly NP-ha d, ha e p omp ed esea che s o
de elop a ious heu is ics o e icien solu ion inding. Xing and Tu (2020)employeda
Mon e Ca lo ee sea ch o o e an al e na i e o adi ional exac me hods o he TSP. Fo
clus e ed ci ies, Ja a zadeh e al. (2017) in oduced an enhanced gene ic algo i hm ha was
based on nea es neighbo sea ch, while Smi h and Imeson (2017) de eloped a compe i i e
la ge neighbo hoodsea ch heu is ic o ha d-ins ance andnonclus e ed p oblems. Mahmoud-
inazlou and Kwon (2024) p oposed a hyb id gene ic algo i hm o he m-TSP o sho en he
longes ou leng h. In add essing he TPP, s a egies such as abu sea ch (El-Dean, 2008;
Mansini e al., 2005), simula ed annealing (Voß, 1996), and gene ic algo i hms (Almeida e
al., 2012; Goldba g e al., 2009) ha e been u ilized. Roy e al. (2023) s udied a mul i ehicle
clus e ed TPP wi h a a iable-leng h gene ic algo i hm o minimize sys em cos s by op i-
mizing clus e selec ion, ma ke isi s, p ocu emen quan i ies, and ou ing. Fo he amily
TSP, Be na dino and Paias (2021) applied popula ion-based heu is ics combined wi h local
sea ch me hods. An o e iew o exac and heu is ic algo i hms o TPP a ian s is gi en by
Mane ba e al. (2017); hey highligh he di e se solu ion app oaches in he ield.
2.2 Mul imodal anspo a ion
The use o di e en modes o anspo a ion, such as ships, ains o ucks, can lead o a ade-
o be ween ansi ime and cos . E e y combina ion is possible in heo y, bu only some o
he possible combina ions a e common in logis ics sys ems o mul imodal anspo , such as
uck/ essel- ain- uck and uck/ essel-ship- uck, depending on he leg conside ed in he
123

Annals o Ope a ions Resea ch (2025) 351:667–726 671
supply chain. ViaDonau (2012) in oduced loading uni s o accele a e unloading p ocesses
ha occu in mul imodal anspo a ion ne wo ks. Cu en ly, he e a e s anda dized load uni s
suchas con aine s, swap bodies, and semi aile s.When ansshipping om inlandwa e ways
o ucks, o example, p oduc s a e ans e ed o con aine s (Ghiani e al., 2004).
In ecen s udies, esea che s ha e combined ucks and d ones in mul imodal models
(Jeong e al., 2019). An example o a uck-ai model is a mul imodal hub loca ion and hub
ne wo k design p oblem, such as ha de eloped by Alumu e al. (2012). S eadieSei i e al.
(2014) men ioned he di e en de ini ions o e minologies ha ci cula e in he li e a u e,
such as mul imodal, in e modal, comodal, and, mo e ecen ly, synch omodal anspo a ion.
A e e ising hese de ini ions, we e e o his model as a mul imodal anspo a ion model.
In an e e al. (2009) de eloped a ship- uck TPP whe e he aim is o de ine he sequence wi h a
ocus on whe e ucks should dis ibu e goods o inal wa ehouses. Sun e al. (2018) p oposed
a biobjec i e nonlinea uck- ail ou ing model ha minimizes he o al cos and o al CO2
emi ed. In con as o hei wo k, ou linea model does no equi e speci ic se ice se s o
each mode o anspo a ion because mode exchange can occu a any node. Fazayeli e al.
(2018) de eloped a mul imodal ou ing-loca ion model in which mode changes a e allowed
a p ede ined nodes. Again, he model p esen ed in his manusc ip op imizes ansshipmen
loca ions and does no equi e speci ying ansshipmen loca ions, unlike hei wo k.
HaoandYue(2016) used dynamic p og amming o sol e a mul imodal anspo a ion
model. They did no conside he quan i ies anspo ed in hei model. Ins ead, hey assumed
ha he same model deli e s all he quan i ies. Compa ed o hei model, ou model conside s
henumbe o ehicles,deli e yquan i y,andpossibili yo deli e y ia womodes o hesame
node. Zhang e al. (2011) sol ed a mul imodal uncapaci a ed ou ing p oblem. The model
selec ed one mode o each ci y pai . Howe e , i did no conside he anspo quan i ies
o he possibili y o ha ing mul iple ehicles wi h di e en capaci ies. Xiong and Wang
(2014) s udied a biobjec i e mul imodal ou ing p oblem wi h a ime window. The model
minimizes he o al anspo a ion cos and o al a eling ime. Howe e , hey do no conside
he capaci y o each anspo a ion mode. They in eg a ed he k-sho es pa h and a gene ic
algo i hm o add ess his p oblem. Moccia e al. (2011) s udied he mul imodal ou ing
p oblem wi h shipmen consolida ion op ions and ime windows. Zameni and Razmi (2015)
p oposed an uncapaci a ed mixed-in ege mul imodal hub loca ion- ou ing p oblem wi h
simul aneous pickups and deli e ies. The p oposed model alloca es a anspo a ion mode
o each ou e be ween he hubs and minimizes he o al ne wo k cos . Riessen e al. (2015)
conside ed ba ge and ail anspo a ion modes in he con aine anspo a ion p oblem in
no hwes Eu ope. In hei model, hey did conside he penal y o o e due deli e ies. Demi
e al. (2016) de eloped an in e modal se ice ne wo k design p oblem ha educes cos s and
emissions while using inland wa e way anspo a ion, ail, and ucks. Assadipou e al.
(2016) p oposed a bile el biobjec i e ma hema ical model o egula e hazma shipmen s by
using oad and ail anspo a ion modes. The model was sol ed by using a pa icle swa m
algo i hm. Qu e al. (2016) conside ed emission and ans e cos s in se ice ne wo k design
in a mul imodal anspo a ion p oblem. Taw ik and Limbou g (2019) p oposed a bile el
pa h-based in e modal ma hema ical model ha maximizes p o i and minimizes disu ili y.
Wang and Qi (2019) s udied he design o a ime-dependen se ice ne wo k wi h mul iple
se ice ypes.
Mo e ecen ly, Ni senko e al. (2020) s udied he isks o mul imodal anspo a ion by
using uzzy logic. Kaew ak e al. (2021) de eloped a decision suppo model o de e mine
op imal mul imodal ou es. The p oposed mul iobjec i e model conside ed anspo a ion
cos s, anspo a ion ime, and se en anspo a ion isks o imp o e logis ics and anspo a-
ion pe o mance. In addi ion, he analy ic hie a chy p ocess and ze o–one goal p og amming
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672 Annals o Ope a ions Resea ch (2025) 351:667–726
me hods we e used o de e mine mul imodal ou e selec ion. He e al. (2021) p oposed an
app oach o a mul imodal ne wo k model and obus ness assessmen o eigh anspo
ne wo ks. They analyzed he in e dependencies be ween anspo modes and conside ed he
dis up ions o single nodes. The obus ness o he ne wo k was conside ed based on he a el
ime esul ing om pe u ba ions in he ne wo k. Thei model helps o schedule main enance
ope a ions by p io i izing he c i ical elemen s in he ne wo k. The app oach di e s based on
he p oposed algo i hm and he mul iobjec i e unc ions. P zys upa e al. (2021) de eloped
a mul iobjec i e op imiza ion model o sol e a mul ic i e ia anspo p oblem. Thei algo-
i hm is o la ge-da a case s udies wi h any numbe o ypes o anspo and op imiza ion
c i e ia. The case s udy conside ed minimizes he anspo a ion cos s and anspo a ion isk
le els. Real e al. (2021) p oposed a mixed in ege p og amming model o sol e mul imodal
hub design p oblems wi h lexible ou es, and hey de eloped me aheu is ics based on an
adap i e la ge neighbo hood sea ch. Ye e al. (2021) p oposed a bile el ma hema ical model
o de e mine he ans e loca ion and he in as uc u e capaci y o a mul imodal ans-
po a ion ne wo k design wi h elas ic demand. Reade s may e e o Elbe e al. (2020) o
a sys ema ic e iew o he opic. To he bes o ou knowledge, none o he models in he
e iewed li e a u e op imize he selec ion o he ansshipmen po in a capaci y-cons ained
linea p oblem o mula ion. In addi ion, we o mula ed a biobjec i e model by using he Eco-
T ansi emission calcula o , applied he model o a eal case s udy, and analyzed he e ec
o in as uc u e ailu e on he mul imodal o mula ion. Real-li e analysis allows o a be e
unde s anding o he op imiza ion p oblem, as well as mo e insigh ul esul s.
3 Me hodology
Ou main a ge is o op imally de e mine he anspo a ion mode and he ou e by consid-
e ing bo h he o al cos and he emissions. Addi ionally, we s udy he impac o conside ing
di e en modes on he o al cos and emissions. To achie e hese goals, we compa ed h ee
anspo a ion modes, namely, inland wa e way anspo a ion, ucks, and mul imodal ans-
po a ion (Fig.1). The mul imodal mode allows swi ching om one mode o ano he , whe e
quan i ies a e unloaded a a ansshipmen po and loaded in ano he mode. We conside
he scena io o deli e ing chemical p oduc s om a depo o di e en po s by using h ee
anspo a ion modes. We lis all he se s, pa ame e s, and decision a iables in Sec .3.1.
Then, we desc ibe he p oblem, and we p esen he modeling concep in Sec .3.2.Thecos
and emission calcula ions equi ed o he objec i e unc ion o mula ion a e p esen ed in
Sec s.3.3 and 3.4. Finally, we p esen he biobjec i e model and i s solu ion app oach in
Sec s.3.5 and 3.6, espec i ely.
3.1 No a ions
The se s used a e as ollows:
•L: Se o po s included in he ne wo k indexed by iand j. The indices i0,i1and i1
ep esen he i ual, ac ual and duplica ed ac ual depo s, espec i ely.
•M: Se o anspo a ion modes indexed by m.
•Km: Se o ehicles belonging o anspo a ion mode m, indexed by k.
Table 1lis s he ehicle pa ame e s used; hey we e classi ied as inland wa e way anspo a-
ion and ucks. In addi ion, we ha e:
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Fig. 1 The conside ed anspo a ion modes and objec i es
•Di: Demand a po i[ /day].
•Sis a small numbe .
The decision a iables o he op imiza ion model de ined in Sec .3.5 a e:
•yk,m
i: Decision a iable ha equals 1 i po iis isi ed by ehicle ko mode m.
•Xk,m
i,j: Decision a iable ha equals 1 i ehicle ko mode m a els om po i o jby
using ehicle ko mode m.
•Qk,m
i: Quan i y o po iloaded a ehicle ko mode m.
•uk,m
i: In ege a iable ep esen ing he sequencing o isi s o po iand ehicle ko
mode m.
• k,m
i,j: Con inuous (nonnega i e) a iable ha gi es he o al quan i y loaded in ehicle k
o mode m om i o j.
•UQk.m
i: T ansshipmen quan i ies o ans e ca go unloaded a po iby using ehicle k
in mode m. These quan i ies a e hen anspo ed by ano he ehicle(s) and mode m o
o he po s.
•Tk,m
i: Decision a iable (bina y) equals 1 i ehicle ko mode mis used o anspo
quan i ies om ansshipmen po i.
3.2 P oblem desc ip ion and modeling idea
Goods a e o be anspo ed om a depo po (i1=Po 1, as shown in Fig.2) oo he
po s by using inland wa e way anspo a ion, ucks, o bo h modes o ul ill he demand a
each po i. T ucks a e a ailable a all po s. Thus, hey can be used o ei he he main se ice
o anspo ing goods om he depo o o he po s o he seconda y se ice o anspo ing
goods om a ansshipmen po o o he po s. Figu e2shows one main se ice s a ing
om he depo (Po 1), ep esen ed by solid a ows, and isi ing Po s 2, 3 and 4, a e which
i e u ns o he depo . Figu e2also shows one seconda y se ice ( ansshipmen ) deno ed
by dashed a ows s a ing om Po 3 and isi ing Po s 5 and 6. Because inland wa e way
anspo a ion is only a ailable a he depo po , i can be used only o he main se ice.
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674 Annals o Ope a ions Resea ch (2025) 351:667–726
Table 1 Desc ip ion o he pa ame e s o he a ious anspo a ion modes
No a ion Desc ip ion Uni Inland Wa e way anspo a ion T uck
αk,mVehicle uni cos o co e
main enance and
dep ecia ion
[e/h]†,[e/km]†† xx
βk,mHou ly pe sonnel cos pe
wo ke
[e/h] x x
μk,mUni uel cos [e/km] x x
pPo cha ges paid by inland
wa e ways upon unloading
[e/ ] x
γk,mCos o handling
ansshipmen (cos o
loading ca go a ano he
ehicle when changing he
anspo a ion mode)
[e/ ] x x
k,mCos o handling deli e ies
(unloading cos s a
des ina ion po )
[e/ ] x x
k,mRen cos o con aine [e/h] x
k,mhou ly uck ixed cos o
co e insu ance, pa king
and capi al e u n
[e/h] x
Tk,mToll a e on highways [e/km] x
nk,mNumbe o wo ke s needed o
ope a e he ehicle
−x∗
dk,m
i,jDis ance be ween po s iand
jusing anspo a ion mode
mo ehiclek
[km] x x
qi,jNumbe o locks be ween
po s iandj
−x
hDocking ime a he po s [h] x
τLock ime a he locks [h] x
Ck,mCapaci y o anspo a ion
mode mo ehiclek
[ ] x x
ηHandling pe o mance o po
du ing loading and
unloading ac i i ies
[ /h] x
sk,mSpeed o anspo a ion mode [km/h] x x
κk,m
i,jEmp y pe cen age ac o o
anspo a ion mode
−xx
zcon e sion ac o om MJ o
kWh
x
k,mLoad ac o (u iliza ion o
anspo mode)
−xx
ρEne gy ac o ( ank- o-wheel) [g/MJ] x x
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s a s om hispo o deli e o o he po s doesno includeadeli e y o hispo .Cons ain s
(25)and(26) a e Mille –T ucke –Zemlin (MTZ) sub ou elimina ion cons ain s ha ensu e
ha a ehicle does no sub ou on a ou e. Cons ain (27) de ines he low om i o jas he
low ha en e s i om all possible j alues minus he quan i ies. Cons ain (28) de ines he
i s low as he o al loaded quan i y. Cons ain (29) links k,m
i,jwi h Xk,m
i,jso ha k,m
i,jis ze o
i he co esponding Xk,m
i,jis ze o. Cons ain (30) ensu es ha he o al quan i y unloaded
by mul iple ehicles a one po is equal o he demand. Cons ain (31) ensu es ha he o al
deli e ed and unloaded quan i ies o ehicle kin mode mdo no exceed he ehicle capaci y
i he ehicle is used. Cons ain (32) links Tk,m
jwi h Xk,m
i,jsuch ha i Tk,m
j=1, he ehicle
mo es di ec ly om he i ual depo o he ansshipmen po j. Cons ain (33) s a es ha
i ehicle ko anspo a ion mode mis used om he i ual depo , he ehicle mus ei he
use he a c om he i ual depo o he ac ual depo o be used a a ansshipmen po .
Cons ain (34) ensu es ha he quan i y deli e ed o he po s om ansshipmen po
jdoes no exceed he o al quan i y unloaded by o he ehicles om o he modes a po
j. Cons ain (35) ensu es ha unloaded quan i ies a e allowed i and only i he ehicle
mo es om he i ual depo o he ac ual depo (i.e., he ehicle s a s om he ac ual
depo ). Cons ain (36) ensu es ha he numbe o ehicles and hei capaci ies a e su icien
o anspo he unloaded quan i y a po i. Cons ain (37) ensu es ha all he unloaded
quan i ies a e deli e ed. Cons ain (38) ensu es ha i a ehicle is used a he exchange po ,
hen i mus deli e some quan i ies; ha is, he ini ial low should be posi i e. Cons ain (39)
s a es ha he ehicle mus ca y some nonze o quan i ies i i a els om he i ual po o
he ac ual po . Cons ain (40) ensu es ha i a ehicle mo es om he i ual depo o he
o iginal depo , i mo es om he o iginal depo o he i ual depo . Consequen ly, he ou
is comple e. Cons ain (41) p e en s unloading a he ac ual and i ual depo s. Sec . Bo
he Appendix shows he s eps equi ed o con e he mul imodal model in o a single-mode
model.
3.6 Biobjec i e op imiza ion app oach
Dealing simul aneously wi h cos minimiza ion and emission educ ion leads o a biobjec i e
op imiza ion wi h wo con lic ing goals. Imp o ing cos e iciency may wo sen emissions and
ice e sa. Fo ins ance, in inland wa e way anspo a ion, p io i izing cos minimiza ion
migh emphasize o al docking and eigh - ela ed cos s o e o al dis ance and uel cos s.
This app oach can esul in be e cos pe o mance bu can p oduce highe emissions due o a
sligh inc ease in uel consump ion and dis ance a eled. Con e sely, ocusing on emission
educ ion ends o educe a el dis ance and, consequen ly, uel consump ion, leading o
lowe emissions while inducing highe cos s associa ed wi h docking and eigh handling.
Se e al s udies in he li e a u e ha e explo ed hese con lic ing cos s and emission objec i es
ac oss di e en modes o anspo a ion, such as g ound anspo a ion (Cheai ou e al.,
2021b; Demi e al., 2016; Molina e al., 2014) and ma i ime anspo a ion (Dulebene s,
2018;Zhaoe al.,2019). As a esul , he e is a se o nondomina ed, e icien , op imal
solu ions ha a e Pa e o-op imal (Fonseca & Fleming, 1998). Pa e o poin s can be ob ained
by using se e al me hods, such as he u ili y unc ion me hod, lexicog aphic me hod, goal
p og amming, no mal bounda y in e sec ion me hod and e olu iona y algo i hms (Ghane-
Kana i & Kho am, 2015; Singh Yada , 2023). The me hod choice depends on he p e e ence
a ailabili y om decision-make s (no p e e ence me hod, a p io i, a pos e io i and in e ac i e
me hods). In his wo k, an a p io i scala iza ion me hod is used because he decision-make
p e e ence is known in ad ance. The weigh ed comp ehensi e c i e ion me hod is used o
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682 Annals o Ope a ions Resea ch (2025) 351:667–726
sol e he biobjec i e p oblem because o i s simplici y and sui abili y o heu is ics because
i does no in oduce addi ional cons ain s.
In his me hod, he mul iobjec i e unc ion mus be di ided in o wo single unc ions ha
dispose o di e en uni s and o de s o magni ude. On he one hand, he cos unc ion is in
e, and on he o he hand, he emission unc ion is in g. Single objec i e unc ions a e de ined
as objec i e unc ions and a e subjec o he same cons ain s in he espec i e sec ions.
Bo h single subp oblems a e sol ed o ob ain he op imal solu ions, which we call cmin and
emin. A e ecei ing bo h single op imal solu ions, we me ge hem in o a single no malized
objec i e unc ion (Dehghani e al., 2013). The equa ions used o no malize he objec i e
unc ions a e as ollows:
c=c−cmin
cmin
,(42)
e=e−emin
emin
.(43)
Equa ions (42)and(43) a e he ela i e a ia ions be ween he single objec i e unc ions
o he anspo a ion cos s and emission cos s and hei espec i e op imal alues. We hen
mul iply each no malized single-objec i e unc ion by a ela i e weigh , depending on he
decision-make . This mul iplica ion leads o he ollowing objec i e unc ion.
min δ=a1×c+a2×e,(44)
whe e a1and a2a e he weigh s ha depend on he decision-make , and a1+a2=1. This
objec i e unc ion is minimized and is subjec o he same cons ain s as he ini ial p oblem.
Fo a gi en se o a1and a2, he e is only one Pa e o-op imal solu ion. Howe e , i hese
weigh s a e changed, di e en Pa e o-op imal solu ions may esul (Ma le & A o a, 2004).
The use o he scala iza ion echnique may no esul in all Pa e o op imal poin s, as his
depends on con exi y, among o he ac o s (Ghane-Kana i & Kho am, 2015).
4 Solu ion app oach
This p oblem is a gene aliza ion o he a eling pu chase p oblem, which is an NP-ha d
p oblem (see Sec . Ain he appendix). The e o e, he p oblem canno be sol ed by using an
exac app oach o la ge ins ances. We use a popula ion-based heu is ic o o e come his issue
by designing sea ch ope a o s o align wi h he s udied p oblem. An in e es ing ad an age
o popula ion-based heu is ics is he possibili y o adap ing hem h ough he design o
sea ch ope a o s, numbe o i e a ions and popula ion size o con ol solu ion quali y and
compu a ion ime. Popula ion-based heu is ics ha e been used success ully in he li e a u e
o sol e simila p oblems, such as TSP wi h p ocessing ime (Bo˙zejko & Wodecki, 2009),
he e ogeneous VRP (Liu, 2013), TPP wi h speed op imiza ion (Cheai ou e al., 2021b),
dynamic VRP wi h unknown cus ome s (C épu e al., 2012; Saba e al., 2021), capaci a ed
elec ic VRP (Wang e al., 2023), pe iodic VRP (Vidal e al., 2012; Bo hen e al., 2018)
and line shipping (Ca iou e al., 2018; Cheai ou e al., 2021a). In addi ion, a ian s o his
heu is ic ha e been success ully used in o he domains, such as supplie selec ion (Hamdan e
al., 2023), main enance s a egy op imiza ion (Alsha qawi e al., 2021), machine scheduling
(Nea chou, 2010) and ai c a mo ion planning (Wu, 2021).
The heu is ic used o sol e he p oblem is p esen ed in Algo i hm 1. I s a s by calling a
popula ion gene a ion sub ou ine (see Sec . Cin he appendix) o c ea e a popula ion (P)
o  easible indi iduals (po en ial solu ions) ha should be mul iples o he o al numbe
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o ope a o s (=18). The andomly gene a ed easible indi iduals sa is y all he model
cons ain s(20)–(41).Thes uc u eo eachindi idualis de ailedinSec .D.1in heappendix.
The global cos , he emission, and he a ia ion alues a e ini ialized as ollows: cglobal =∞,
eglobal =∞and δglobal =∞. The algo i hm a emp s o enhance he quali y o indi iduals
in each i e a ion unde a p ede ined o al numbe o i e a ions o gene a ions (G).
In each i e a ion, g=1, ..., G, he algo i hm i s calcula es he cos and emission objec-
i e unc ion (cψand eψ) by using Equa ions (18)and(19) o each indi idual (Algo i hm
2). Based on he op imiza ion se ing (ObjType), ei he a single objec i e o a biobjec i e
op imiza ion is execu ed. In he case o single objec i e se ing, he algo i hm compa es
he bes cos (emission) alue o all indi iduals wi h he global cos (emission) alue and
upda es cglobal (eglobal) acco dingly. The algo i hm e u ns cglobal as he bes objec i e alue
and he co esponding solu ion de ails (i.e., ou e sequence, quan i y alloca ion and ehicle
assignmen ). In addi ion, i e u ns he co esponding o al emission (cos ) alue and upda es
ele an heu is ic pa ame e s, such as . In he case o he biobjec i e se ing and be o e cal-
cula ing he a ia ions, he global a ia ion is ecalcula ed i he global cos (cglobal)o global
emissions (eglobal) a e upda ed. This p ocedu e ensu es he use o he mos ecen e e ence
poin s o he a ia ion calcula ion. The a ia ion o each indi idual (δψ) is calcula ed by
using Equa ion (44). Then, he algo i hm iden i ies he bes objec i e unc ion alue in he
cu en i e a ion (minψ∈{1,...,}δψ) and compa es i wi h he cu en global objec i e alue
δglobal. The algo i hm hen upda es he global objec i e alue i he bes - ound objec i e alue
is be e han he cu en global objec i e alue. The algo i hm e u ns he bes objec i e alue
δglobal, which co esponds o he solu ion de ails, he co esponding o al cos and emission
o he bes indi idual and he upda ed heu is ic pa ame e s, such as ,cglobal and eglobal,as
shown in Algo i hm 2.
Finally, he algo i hm andomly di ides all indi iduals () in o subse s, called ϕυ, each o
which is indi iduals. These subse s a e cons uc ed andomly. In each subse , he algo i hm
iden i ies and selec s he bes indi idual in e ms o he alue o Equa ion (18), (19), o (44)
depending on he op imiza ion se ing (ObjType) in he subse hen uses one ope a o a a
ime o p oduce a new indi idual om each ope a o , which esul s in a o al o indi iduals
s o ed in a empo a y popula ion P mp.Theope a o s used in he algo i hm a e as ollows:
•Ope a o 1: Keep he bes indi idual in he subg oup unchanged “Do no hing”.
Rou e ope a o s ( e e o Sec . D.2):
•Ope a o 2: Randomly selec one ehicle and lip i s ou e.
•Ope a o 3: Randomly selec mul iple ehicles hen lip he ou e o each ehicle.
•Ope a o 4: Randomly selec one ehicle and swap i s ou e.
•Ope a o 5: Randomly selec mul iple ehicles hen swap he ou e o each ehicle.
•Ope a o 6: Randomly selec one ehicle and slide i s ou e.
•Ope a o 7: Randomly selec mul iple ehicles hen slide ou es o each ehicle.
S a ing poin ope a o ( e e o Sec . D.3):
•Ope a o 8: Randomly selec a ehicle ha s a s om a ansshipmen po and o ce i
o s a om he depo .
Vehicle ype ope a o s ( e e o Sec . D.4):
•Ope a o 9: Change he ehicle ype o a andom ehicle o a lowe capaci y.
•Ope a o 10: Randomly change he ehicle ype o a andom ehicle.
The quan i y exchange ope a o s a e as ollows ( e e o Sec . D.5):
•Ope a o 11: Randomly exchange wo po s be ween wo andom ehicles.
•Ope a o 12: Conduc mul iple andom po exchanges be ween wo andom ehicles.
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684 Annals o Ope a ions Resea ch (2025) 351:667–726
•Ope a o 13: Randomly selec wo ehicles hen exchange unique po s wi h simila
quan i ies.
•Ope a o 14: Selec mul iple andom ehicles and exchange unique po s wi h simila
quan i ies.
The quan i y ans e ope a o s a e as ollows ( e e o Sec . D.6):
•Ope a o 15: Randomly ans e quan i ies om he ehicle wi h he lowes u iliza ion
o he ehicle wi h he highes u iliza ion.
•Ope a o 16: Randomly ans e quan i ies om mul iple ehicles wi h low u iliza ion
o he ehicle wi h high u iliza ion un il i is ully u ilized.
•Ope a o 17: Randomly ans e quan i ies om one po o ano he be ween andom
ehicles.
•Ope a o 18: Randomly ans e quan i ies om mul iple po s se ed by one andom
ehicle o ano he andom ehicle.
Algo i hm 1 Main heu is ic
1: Inpu : Con ex - ela ed: (L,M,Km,Di), ehicle- ela ed: (αk,m,Ck,m,βk,m,nk,m,dk,m
i,j,qi,jh,τ,η,
μk,m,p,k,m,γk,m,sk,m,k,mk,m,Tk,m,κk,m
i,j,z, k,m,ρ,φ,ξ,ωk,m,χk,m,Ek,m), and heu is ic-
ela ed: (G,,,, ObjType).
2: Ou pu : The bes o al cos , he bes o al emission, he bes ou e o each ehicle, quan i ies deli e ed by
each ehicle.
3: se cglobal ←∞,eglobal ←−∞,δglobal ←∞and =0
4: P←Popula ion Gene a ion(Con ex - ela ed, ehicle- ela e and heu is ic- ela ed pa ame e s, )
5: o g←1 o Gdo
6: {Bes alue and co esponding de ails, ,cglobal,eglobal,δglobal }←Ge Objec i e Value(Con ex -
ela ed, ehicle- ela e and heu is ic- ela ed pa ame e s, ,cglobal,eglobal,δglobal)
7: di ide P andomly and equally in o ϒsub g oups (ϕυ)o Indi iduals.
8: o υ←1 o ϒdo
9: selec he bes indi idual om ϕυ:
10: =⎧
⎪
⎨
⎪
⎩
a gminψ∈ϕυcψ,i ObjType =‘Cos ’
a gminψ∈ϕυeψ,i ObjType =‘Emission’
a gminψ∈ϕυδψ,i ObjType =‘Bi’
11: pe o m ope a ions on he bes indi idual (ϕ
υ)
12: s o e new indi iduals in P mp
13: end o
14: i > hen
15: disca d indi iduals in Pand P mp
16: P←Popula ion Gene a ion(Con ex - ela ed, ehicle- ela e and heu is ic- ela ed pa ame e s,)
17: else
18: P←P mp
19: end i
20: end o
21: e u n he heu is ic bes solu ion.
The ope a o s a e designed such ha he easibili y o he indi idual ope a o is no
a ec ed. Since he ini ial indi iduals a e easible, he ope a o s main ain his easibili y,
whe e any quan i y mo emen is pe mi ed only a e ensu ing ha enough capaci y is a ail-
able (capaci y is ne e iola ed in his case). In addi ion, since he ini ial ch omosomes sa is y
he demand equi emen s and none o he ope a o s c ea e o gene a e addi ional quan i ies,
demand cons ain s a e no impac ed. Rou e and o he cons ain s a e espec ed h ough ch o-
mosome design. The algo i hm hen eplaces all old indi iduals () wi h newly p oduced
indi iduals in he empo a y popula ion P mp c ea ed by he ope a o s. No ably, one o he
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ope a o s pe o ms no changes on he solu ion. Thus, he bes indi idual in each subse is
kep unchanged and mo ed o he nex i e a ion. Mo eo e , he heu is ic disca ds all indi id-
uals and p oduces a new gene a ion i he numbe o i e a ions wi hou imp o emen in he
solu ion coun e () eaches a p ede ined limi . The p ocess con inues un il he maximum
numbe o i e a ions (G) is eached.
Algo i hm 2 Ge objec i e alue
1: Inpu : Con ex - ela ed, ehicle- ela ed and heu is ic- ela ed pa ame e s, cglobal,eglobal,δglobal,and
2: Ou pu : The bes objec i e alue and he co esponding solu ion de ails.
3: o ψ←1 o do
4: e alua e Equa ions (18)and(19)ands o e he alues in cψand eψ, espec i ely
5: end o
6: i TypeOPT = ’Cos ’ hen
7: i minψ∈{1,...,}cψ<cglobal hen
8: s o e he solu ion o (a gminψ∈{1,...,}cψ) indi idual
9: ←0
10: else
11: ←+1
12: end i
13: e u n he upda ed cglobal and i s co esponding solu ion de ails and o al emission alue, and he
upda ed .
14: end i
15: i TypeOPT = ’Emission’ hen
16: i minψ∈{1,...,}eψ<eglobal hen
17: s o e he solu ion o (a gminψ∈{1,...,}eψ) indi idual
18: ←0
19: else
20: ←+1
21: end i
22: e u n he upda ed eglobal and i s co esponding solu ion de ails and o al cos alue, and he upda ed
.
23: end i
24: i TypeOPT = ’Bi’ hen
25: i minψ∈{1,...,}cψ<cglobal and g= 1 hen
26: ecalcula e δglobal using minψ∈{1,...,}cψ
27: end i
28: i minψ∈{1,...,}eψ<eglobal and g= 1 hen
29: ecalcula e δglobal using minψ∈{1,...,}eψ
30: end i
31: o ψ←1 o do
32: e alua e Equa ion (44)ands o e he alue in δψ
33: end o
34: i minψ∈{1,...,}δψ<δ
global hen
35: s o e he solu ion o (a gminψ∈{1,...,}δψ) indi idual
36: ←0
37: else
38: ←+1
39: end i
40: e u n he upda ed δglobal and i s co esponding solu ion de ails, o al cos and emission alues, he
and he upda ed cglobal,eglobal and .
41: end i
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686 Annals o Ope a ions Resea ch (2025) 351:667–726
Fig. 3 Ne wo k o he Wes Ge man canal sys em. Da a: Fachse ie Binnenschi ah . (S a is ischeBundesam ,
2019) Layou : OpenS ee Map
5 Real-li e case s udy
In his sec ion, we conside he eal case s udy o he Wes Ge man canal sys em. The case
s udy is used o e alua e heu is ic pe o mance in Sec .6.3 and o de i e he manage ial
insigh s p esen ed in Sec .7.
We examine his o ical goods and inland wa e way anspo a ion lows ac oss he canal
sys em o he esea ch p ojec P e iew. We also e alua e he s ake-holding indus ies and
hei expec ed cos impac on anspo due o dis up ions, such as queues a locks o un o e-
seen in as uc u e ailu es. The expec ed cos impac on he en i e supply chain is mo e
se e e because, among o he s, hese cos s include he managemen e o o modal shi
o ucks, ime-sensi i e ma ke p emiums, and subsequen cos s i supply chain ope a ions
a e delayed. Since he in as uc u e o he Wes Ge man canal sys em is old, c i ical poin s
om he logis ics pe spec i e mus be iden i ied, and he need o a mul imodal spli will
be analyzed. Weh le e al. (2020) p o ided mo e insigh s in o he esiliency issues o he
Wes Ge man canal sys em and no ed he impo ance o inland wa e way anspo a ion in
Ge many. Thei model helps o de ec c i ical in as uc u e o schedule he main enance o
he in as uc u e o he sys em.
Figu e3p o ides a de ailed iew o he di e en canals. The e a e ou canals ha belong
o heWes Ge man canalsys em.Duisbu g was he s a ing poin o he base scena io. Inland
wa e way anspo a ion mus include canals. Consequen ly, i one po is no a ailable, he
dis ance om one po o he o he is o en g ea e han is he ela i e ailu e o oads because
he e a e mo e al e na i es o ucks in he e en o any ailu e.
We used 16 po s and 14 locks. The wa e le el is 1.9ms in he baseline scena io. All
locks and po s we e a ailable, and he lock ime was 40min. Because o he high demand
o indus ies in ha egion, we compa ed inland wa e way anspo a ion and a package
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Annals o Ope a ions Resea ch (2025) 351:667–726 687
Table 2 Po s and locks in he
case s udy Index Po Lock
0 Emsland Meide ich
1 Müns e Obe hausen
2 Do mund Gelsenki chen
3 Rhein-Lippe Wanne-Eickel
4 Ma l He ne-Os
5 Lünen Hen ichenbu g
6 Be gkamen Da eln
7 Hamm Ahsen
8 Schmehausen Flaesheim
9 Bo op Do s en
10 Essen Hünxe
11 Coelln-Neuessen F ied ichs eld
12 Ruh Öl Hamm
13 Gelsenki chen Müns e
14 Wanne-Eickel –
15 Duisbu g (depo ) –
o en ucks in ou case s udy as a easonable elaxa ion. E ec s such as pla ooning ha e
no ye been conside ed, e en hough hey ha e been e alua ed in p ac ice o educe uel
cos s and emissions. The e o e, ou ships wi h di e en capaci ies and inpu pa ame e s
we e included. T ucks and inland wa e way anspo a ion use diesel engines. We ocused on
he chemis y indus y. The in e na ional s anda d o measu ing g eenhouse gas emissions,
ISO14083, is he new equi alen o he Ge man DIN EN 16,258 s anda d o epo ing
and o quan i ying g eenhouse gas emissions om anspo a ion ope a ions (ISO, 2019).
The demand is based on his o ical da a o he las en yea s o eigh spending in canal
sys ems and po s published in he “Fachse ie Binnenschi ah ” (S a is ischeBundesam ,
2019). The dis ances a e calcula ed wi h he He e-API and he websi e o Ins i u ü Ene gie-
und Umwel o schung Heidelbe g gGmbH (2023) o inland wa e ways because he He e-
API does no include canal na iga ion. In Manage ial Insigh s 1–3, we conside si ua ions
whe e he decision-make decides ha he wo objec i e unc ions a e equally impo an o
he p ojec . Consequen ly, we se he impo ance weigh s (a1and a2) o 0.5. Le us no e
ha a decision-make may u ilize a mul ic i e ia decision-making ool, such as he analy ic
hie a chy p ocess, o assess he impo ance o each objec i e. Table 2lis s he po s and locks
conside ed in his case s udy, whe e Duisbu g is he ac ual depo o he model. Tables 9,10,
11,12,13 and 14 in Appendix Ep o ide he da a used in his s udy. The eal-li e case s udy
p o ides answe s o he ollowing ques ions:
•Wha si ua ions make he use o mul imodal anspo a ion mo e a ac i e han unimodal
anspo a ion?
•Wha e ec s do a iable lock imes ha e on he o al cos ?
•Wha a e he ad an ages o mul imodal anspo a ion in he e en o in as uc u e ail-
u es?
•Wha is he ade-o be ween cos sand emissions when usingmul imodal anspo a ion?
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6 Nume ical expe imen s
In his sec ion, we s udy he p oblem complexi y o unde s and he impac o he numbe o
po s, he demand and he numbe o ehicles on he complexi y o he exac solu ion. We
also analyze he impac o heu is ic pa ame e s (maximum numbe o i e a ions G,maximum
numbe o indi iduals and he numbe o i e a ions wi hou imp o emen in he solu ion
) on he solu ion quali y and ime. Finally, we p esen he heu is ic pe o mance agains
he exac app oach. All expe imen s we e conduc ed by using a compu e equipped wi h an
In el(R) Co e(TM) i7-9750H CPU @ 2.6 GHz and 16 GB o RAM unning he Windows
10 Home 64-bi ope a ing sys em. Le us no e ha since he heu is ic uses a andomized
p ocess o gene a e i s ini ial popula ion and in applying ope a o s, we sol ed each ins ance
en imes and compu ed he a e age o de e mine he pe o mance and he obus ness. We
used CPLEX 20.1.0 o ob ain he exac solu ion; we se a ime limi o h ee hou s as he
s opping c i e ion o he sol e .
6.1 Impac o sys em pa ame e s on he exac solu ion complexi y
This biobjec i e mul imodal anspo a ion p oblem is NP-ha d since i can be educed o
a TSP p oblem, which is also p o en o be NP-ha d. We conduc a complexi y s udy o
unde s and he con ibu ions o pa ame e s o he p oblem complexi y. We a y he numbe
o po s |L|={8,13,18,23}. Le us no e ha o each p oblem size, he i s h ee po s a e
he i ual depo , he depo and i s duplica e. We also inc ease he demand by a ac o FD,
FD={1,1.3,1.5,1.7}. In addi ion o unde s and he impac o ehicles, we inc ease he
numbe o ehicles FKm,FKm={1,1.5,2,2.5}. Le us no e ha we conside he ins ance
wi h |L|=5, FD=1, and FKm=1 as a baseline case. In each expe imen , we eco d he
exac sol ing ime and he sol e ’s gap, and hen, we combine he wo o o m a complexi y
sco e (complexi y sco e = 0.5×Compu a ion ime
Time limi +0.5×MIP gap). Le us no e ha in hese
expe imen s, we a e age he complexi y sco e o he h ee objec i e unc ions (cos only,
emission only and he biobjec i e).
A sco e be ween 0 and 0.5 means ha an exac op imal solu ion (MIP gap = 0) is ound
wi hin he ime limi . A complexi y sco e highe han 0.5 means ha he ime limi (compu-
a ion ime = ime limi ) is eached and ha he inc eased pa ep esen s he MIP gap, and
a sco e g ea e han 1 means ha he MIP gap when he ime limi is eached is g ea e han
100%. Figu e4shows he complexi y sco e unde he expe imen al se up om he baseline
case. Figu e4a shows he complexi y sco e as a unc ion o he demand ac o and he num-
be o po s. I is clea ha inc easing bo h he demand and he numbe o po s inc ease he
complexi y. Howe e , i is di icul o say ha inc easing demand esul s in only g ea e com-
plexi y o a gi en numbe o po s (e.g., a |L|=20, see FD=1.3andFD=1.5). This is
due o he high a iabili y in he MIP Gap o he di e en ins ances. Howe e , inc easing he
numbe o po sonly inc eases hecomplexi y(see, o ins ance, he di e en p oblem sizes a
FD=1). Figu e4b illus a es he e ec o he numbe o ehicles ( ep esen ed by he ehicle
ac o , FKm) and he numbe o po s on he p oblem complexi y om he baseline case. A
g ea e numbe o ehicles inc eases he complexi y sco e; al hough in some ins ances, he
inc eased numbe o ehicles sligh ly educes he complexi y (see |L|= 18, om FKm=2 o
FKm=2.5). Ins ances wi h mo e han 13 po s and high demand (FD≥1.5) and mo e han
23 po s and a high numbe o ehicles (FKm≥2) expe ience high complexi y and canno be
sol ed by using he exac app oach (Fig.4, o which he sco e is g ea e han 0.5). Th ee-way
analysis o a iance (ANOVA) was used o unde s and he impac o he h ee pa ame e s.
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Annals o Ope a ions Resea ch (2025) 351:667–726 689
Fig. 4 The a e age complexi y sco e using he mul imodal model
The es esul s indica e ha he numbe o po s only and he numbe o ehicles only ha e
signi ican impac s on he complexi y sco e (p alues = 0 and 0.002, espec i ely), while
he demand ac o does no ha e a s a is ically signi ican impac on he complexi y sco e (p
alue = 0.1771). In addi ion, he combina ion o bo h he numbe o po s and he numbe o
ehicles and he combina ion o demand and numbe o ehicles and he combina ion o all
h ee ac o s we e ound o ha e a s a is ically signi ican impac on he complexi y sco e (p
alues = 0.007, 0 and 0, espec i ely).
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690 Annals o Ope a ions Resea ch (2025) 351:667–726
Table 3 Cha ac e is ics o he 21 ins ances
Ins ance Size |L|FD|Km|Ins ance Size |L|FD|Km|
1 Small 8 1 11 12 Medium 23 1.546
281.313 13 281 40
381.514 14 281.350
4 13 1 17 15 28 1.560
5131.3 21 16 La ge 33 1 46
6131.523 17 331.361
7 Medium 18 1 37 18 33 1.569
8181.345 19 381 55
9181.555 20 381.367
10 23 1 32 21 38 1.580
11 23 1.341
6.2 Impac o heu is ic pa ame e s
We gene a ed 21 ins ances wi h di e en cha ac e is ics o analyze he impac o heu is ic
pa ame e s on he solu ion quali y and ime o he mul imodal model. Table 3shows he
cha ac e is ics o hese ins ances. In he ollowing expe imen s, we a ied he numbe o
i e a ions om 1000 o 10,000 wi h a s ep o 1000.
Fi s , we ixed he numbe o indi iduals  o 54 (i.e., 3 mul iplies o ), and we a ied
he maximum numbe o i e a ions wi h imp o emen as ollows: =100, 500, 1000, 1500,
and 2000.
Figu e5shows he a e age impac o all ins ances ( en uns o each ins ance and he h ee
objec i e unc ions; cos , emission and he biobjec i e o mula ion). Figu e5a shows he
a e age ela i e di e ence in he objec i e alue wi h espec o he bes heu is ic objec i e
alue ound (i.e., he smalles among all i e a ions and  alues) unde di e en  alues as
a unc ion o he numbe o i e a ions. Figu e5b shows he a e age compu a ion ime sa ings
(wi h espec o he la ges compu a ion ime) as a unc ion o he numbe o i e a ions. In
e ms o quali y, se ing  o 500 p o ides, on a e age, he bes solu ion quali y a (and
beyond) 5000 i e a ions. Howe e , in e ms o ime sa ings, choosing =2000 o 1500
esul s in he as es algo i hm se ing, wi h 83% a 5000 i e a ions compa ed o 96% a 1000
i e a ions. I is wo h no ing ha se ing =500 a 5000 i e a ions is 3.5% slowe han ha
o =2000. Gi en ha he di e ence is small and ha he solu ion quali y di e ence is
mo e a o able, we chose o pe o m ou expe imen s by using =500. Le us no e ha
he highe he alue o is, he less he e is a need o he algo i hm o gene a e a new
popula ion, hus making i as e . Howe e , his may lead o ewe oppo uni ies o explo e
new po en ial indi iduals ( h ough gene a ing comple ely new popula ions) bu a g ea e
possibili y o explo ing modi ica ions o he cu en indi iduals.
Nex , we explo e he impac o he numbe o indi iduals in he popula ion as ollows: 
= 54, 108 and 162 (3, 6 and 9 mul iplies o he numbe o ope a o s ). In hese expe imen s,
we se  o 500, as i was ound o pe o m well in e ms o he solu ion quali y. Figu e6
shows he a e age ela i e di e ence o he objec i e alue wi h espec o he bes solu ion
ound and he a e age ime sa ings wi h espec o he slowes case. Figu e6a shows ha he
di e ence be ween he bes solu ion ound when using = 54 and when using = 108 is
0.02% a 5000 i e a ions, while he algo i hm becomes slowe as he ime sa ings dec ease
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Annals o Ope a ions Resea ch (2025) 351:667–726 697
Fig. 7 Objec i e unc ion alues o inland wa e ways, ucks and mul imodal anspo a ion unde a ious
demand le els
Obse a ion 2: Al hough he inc ease in he lock ime inc eases he o al cos , he
mul imodal model educes his impac by 23% compa ed o he inland wa e way
anspo a ion model.
We analyzed he e ec o longe lock imes due o a ic jams on he canal sys em. We a ied
he lock imes om 40 o 120min in 20-minu e s eps. An inc ease in he numbe o lock
imes equi es mo e inland wa e way anspo a ion when demand is oo high; in p ac ice, his
app oach inc eases he likelihood o long queues a he lock, u he inc easing he numbe
o lockage imes. We also illus a e how he mul imodal model minimizes cos s. The esul s
a e shown in Fig.8. Di e en scena ios o longe lock imes do no in luence he single-
uck model because hey in luence only he op imal solu ions o models in ol ing inland
wa e way anspo a ion. Emissions a e also no a ec ed. Howe e , cos s a e a ec ed. By
compa ing he esul s while minimizing cos s, we can see ha he cos s o inland wa e way
anspo a ion inc ease as e han hose o mul imodal anspo a ion sys ems (Fig.8).
The o al cos s o inland wa e way anspo a ion inc ease by e165.7 pe minu e o
inc ease in he lock ime (linea i , wi h a coe icien o de e mina ion o 0.999) compa ed
wi h e126.2 pe minu e when using mul imodal anspo a ion (linea , wi h a coe icien o
de e mina ion o 0.992). This is an app oxima ely 23.8% educ ion in he cos pe minu e
o lock ime. The mul imodal model abso bs he impac o he inc ease in lock ime due o
he g ea e lexibili y in ou ing decisions p o ided by he a ailabili y o ucks as an op ion.
The indings demons a e he cos -sa ing po en ial o he di e en modes. Figu e8bshows
he ela i e di e ence in he cos wi h espec o he baseline case o 40min o lock ime.
The igu e shows ha he cos inc eases, on a e age, by 5.05% o inland wa e ways and by
3.71% o mul imodal anspo a ion o each 20-minu e inc ease in lock ime. We conclude
ha mul imodal anspo a ion p o ides a s a is ically signi ican dec ease in cos compa ed
o inland wa e way anspo a ion (no mal wi h p alue = 0.67, one- ailed pai ed es wi h
p alue = 0.03).
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698 Annals o Ope a ions Resea ch (2025) 351:667–726
Fig. 8 E ec o a ying lock imes on o al cos s
Fig. 9 E ec s o lock ailu e scena ios on o al cos s while minimizing only cos s
Obse a ion 3: In as uc u e ailu es impose highe cos s o nea ly 28% pe day; in his
case, mul imodal anspo a ion is mo e e icien han a single anspo a ion mode.
We in es iga ed he e ec o he ailu e o each lock in he sys em on pe o mance. This anal-
ysis allows decision-make s o know he e ec s o hei espec i e po loca ions i one lock
ails. Only some scena ios allow inland wa e way anspo a ion o ul ill he o al demand
(see Table 11 in he Appendix). O he s mus be se ed by inland wa e way anspo a ion
and ucks.
Figu e9shows he o al cos o inland wa e way anspo a ion and mul imodal ans-
po a ion unde he a ious lock ailu e scena ios. Le us no e ha scena io 0 ep esen s
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Annals o Ope a ions Resea ch (2025) 351:667–726 699
Fig. 10 E ec s o lock ailu es o he biobjec i e model
he baseline case wi h no lock ailu es. Lock ailu e unde scena ios 4 and 7–12 esul s in
in easible se ice due o accessibili y issues (see Table 11). The o al cos unde mul imodal
anspo a ion is signi ican ly lowe han ha unde inland wa e way anspo a ion (no mal
wi h p alue = 0.2, one- ailed pai ed es wi h p alue = 0.045). As Fig.9shows, minimiz-
ing he o al cos s leads o a maximum inc ease o 27.59% o he cos s when using inland
wa e way anspo a ion (see he inc ease in he o al cos o scena io 1). This inc ease is
wi h espec o he baseline scena io (scena io 0 - no lock ailu e). The locks in scena ios 1,
2, and 3 a e he mos c i ical due o he high inc ease in he cos and equi e a highe p io i y
o mode nize he in as uc u e o main enance ope a ions.
Scena io 5 was o in e es . We see ha aking he al e na i e no he n ou e ia Rhein-
Lippe is mo e bene icial e en i he locks ail. The eason o his is he high demand o
Ma l and he numbe o locks ha ha e o be passed o each Ma l. To achie e cos e iciency,
decision-make s should selec a ou e ia Rhein-Lippe o accessing he po , conside ing
he high demand o chemical p oduc s in his egion. The mul imodal model allows sa ings
o up o 78% compa ed wi h a single model. In as uc u e dependency demons a es he
bene i s o ha ing an addi ional mode. Figu e9shows ha inland wa e way anspo a ion
esul s in a 10.72% inc ease in he o al cos on a e age in Scena ios 1, 2, 3, 5 and 6 compa ed
o an a e age inc ease o 5.72% when using mul imodal anspo a ion.
Figu e10 shows he e ec o minimizing bo h cos s and emissions ( he biobjec i e o -
mula ion). We can again obse e he g ea es di e ence o scena ios 1 o 3. The o al cos
and emissions when using he mul imodal model a e lowe han hose o inland wa e way
anspo a ion. The mean o al cos di e ence is no signi ican ly di e en be ween he wo
models (no mal wi h p alue = 0.54, wo- ailed pai ed es wi h p alue = 0.87). The mean
o al emission was ound o be signi ican ly lowe in he mul imodal model (no no mal wi h
p alue = 0.006, one- ailed sign es wi h p alue = 0.03). The ade-o be ween emissions
and cos s is clea . Fo some scena ios, highe cos s a e compensa ed o by he use o ewe
emissions. These esul s show he a ia ions in bo h objec i es, which a e u he discussed
in he nex sec ion. The inc ease in emissions unde he mul imodal model in Fig.10 is
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700 Annals o Ope a ions Resea ch (2025) 351:667–726
wi h espec o he baseline scena io (no lock ailu e). Howe e , he inc ease in emissions
unde hese scena ios is s ill lowe han ha unde inland wa e way anspo a ion. Decision-
make s should assign di e en weigh s o bo h objec i es depending on he impo ance and
p io i iza ion o he decision.
Obse a ion 4: Mul imodal anspo a ion p o ides a as e emission educ ion a e
when he o al cos inc eases by 1%.
He e, we calcula ed he cos o educing emissions using inland wa e way ships, ucks, and
mul imodal anspo a ion. The analysis is pe o med by a ying a1and a2in Equa ion (44),
which esul sindi e en Pa e oop imalsolu ions.Weemphasize ha hechosenscala iza ion
me hod was selec ed due o he a ailabili y o decision-make p e e ences, i s simplici y, and
i s sui abili y o he heu is ic app oach. Al hough his me hod migh no yield e e y Pa e o
op imal poin , i aligns well wi h ou analysis, gi en ha he decision-make ’s p e e ences
a e p ede ined. Mo eo e , since he decision-make p ede e mines he impo ance o each
objec i e unc ion, his app oach adequa ely p o ides he necessa y insigh s o in o med
decision-making. The ela i e di e ence be ween any poin on he Pa e o op imal se and
i s subsequen poin (nex poin ound on he on ) is hen calcula ed as (ci−ci+1
ci,ei−ei+1
ei),
whe e iis he cu en poin , i+1 is he nex (subsequen ) poin on he on , and ciand eia e
he cos and emission alues o poin i, espec i ely. All he esul ing ela i e di e ences a e
a e aged. Figu e11 shows he o al cos and he o al emission when a ying he impo ance
weigh s om 0 o 1 wi h an inc emen o 0.1 when using he h ee models (inland wa e ways,
ucks and mul imodal). All solu ions we e checked, and only nondomina ed unique solu ions
we e e ained. Figu e11 illus a es he ade-o be ween he o al cos and o al emissions
o each model. A 1% educ ion in emissions inc eased he o al cos by 1.32%, 2.93%,
and 0.90%, espec i ely, when using inland wa e way anspo a ion, ucks, and mul imodal
anspo a ion. This highligh s ha using mul imodal anspo a ion helps educe emissions
while incu ing lowe cos s when compa ed o o he op ions. The indings also highligh ha
i decision-make s gi e less p e e ence o he en i onmen , a educ ion in cos is associa ed
wi h a g ea e a e o inc ease in emissions (compa ed wi h he ideal case o minimizing
emissions) when using mul imodal anspo a ion. Tha is, educing cos s by 1% esul s in
inc eases in emissions o 0.76%, 0.34%, and 1.12% o inland wa e way ships, ucks, and
mul iple modes, espec i ely. I is also ue ha inc easing cos s by 1% educes emissions by
1.12% when using mul imodal anspo a ion. These pe cen ages e lec he a e o change
o how quickly cos s o emissions inc ease and dec ease.
We u ilized he Wilcoxon ank sum es ( he da a we e no no mally dis ibu ed, p alue
<0.05) o compa e he Pa e o poin s o each anspo a ion mode and o assess whe he hese
di e ences we e s a is ically signi ican . We ound ha he di e ences in cos and emissions
be ween inland wa e ways and mul imodal anspo a ion a e no s a is ically signi ican (p
alue>0.08). This means ha al hough mul imodal wa e ways p o ide sligh ly be e ade-
o s han do inland wa e ways, hese di e ences a e no s a is ically signi ican and ha bo h
modes ha e compa able e iciencies in e ms o balancing cos and emissions. In con as ,
he di e ences be ween mul imodal anspo a ion and ucks a e s a is ically signi ican ,
indica ing ha mul imodal anspo a ion o inland wa e way anspo a ion o e s a mo e
cos -e ec i e solu ion o educing emissions han ucks.
To assess he heu is ic pe o mance in gene a ing he biobjec i e solu ions, we used he
hype olume indica o o quan i y he olume o he objec i e unc ion space co e ed by
he Pa e o poin s wi h espec o a e e ence poin . This measu e indica es how close he
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Annals o Ope a ions Resea ch (2025) 351:667–726 701
Fig. 11 To al cos e sus o al emissions o each anspo a ion mode. The ci cles ’o’ and ’x’ ep esen he
solu ions ob ained by using he exac and heu is ic app oaches, espec i ely
Table 8 Hype olume esul s o he exac and heu is ic Pa e o op imal se s
Hype olume RD (%)
T anspo a ion mode Exac Heu is ic
Inland wa e ways 767,895,800 763,730,000 0.54
T uck 25,730,000 25,730,000 0.00
Mul imodal 10,322,413,400 10,256,591,500 0.64
heu is ic Pa e o se is o he exac app oach Pa e o se . We used he “ec ” package in R
S udio o calcula e he nondomina ed hype olume o he Pa e o op imal se by using he
exac and heu is ic app oaches (Table 8). The ela i e di e ence in he hype olume be ween
he wo se s was e y small (less han 1%), meaning ha he wo se s had e y simila esul s.
8 Conclusion
In his s udy, we gained insigh s in o how mul imodal anspo a ion can be used in he
Wes Ge man egion, mo e p ecisely, wha impac he choice o anspo a ion mode has
on o al cos s and emissions. Fu he mo e, he iabili y o his choice was e alua ed unde
di e en wa e way in as uc u e ailu e scena ios, which a ec ed he mul imodal balance.
While using a cos unc ion p oposed by he Ge man Fede al Ins i u ion and an emission
unc ion based on ISO14083, he esul s a e eliable ega ding his o ic ou ing p oblems in
he company and cos es ima ions o anspo a ion by he Ge man Fede al Ins i u ion. To
c ea e di e en scena ios, we u ilized ac ual da a om he Wes Ge man canal sys em and
i s 16 po s and 14 locks. Manage ial insigh s, such as he e ec o in as uc u e ailu es on
he logis ics low in he egion, allow decision-make s o know he mone a y and emission-
ela ed cos s o hei companies. Mo eo e , he e ec o a ying demand on he ne wo k
123
702 Annals o Ope a ions Resea ch (2025) 351:667–726
was analyzed, and decision-make s we e able o unde s and how inc eased demand olumes
impac he cos and emissions o di e en anspo a ion modes. We also iden i ied addi ional
cos s incu ed when he numbe o lock imes inc eased and demons a ed he ad an ages o
using he wo modes. We also desc ibe he e ec s o using only one objec i e op imiza ion
compa ed o ha o he biobjec i e unc ion and i s e ec on he objec i e alues. The
p oposed heu is ic app oach o ou model leads o an a e age ime sa ings o 82% and
accep able accu acy wi h a ela i e di e ence o less han 5% when compa ed o ha o he
bes exac solu ion ound. This o mula ion allowed us o gene a e good esul s wi hin an
accep able ime.
This s udy was based on he li e a u e on anspo a ion and sus ainable ou ing p ob-
lems. To he bes o ou knowledge, he e is no algo i hm ha selec s ansshipmen nodes
in he models, as we do he e. The biobjec i e app oach allows decision-make s o conside
he u u e p icing o emissions. Single biobjec i e and mul imodal biobjec i e anspo a ion
modelscanbeapplied oanywa e waysys emwi hmul imodalhubssubjec ocompu a ional
bounda ies,suchas heinlandwa e waysys emo heNe he lands.Businessdecision-make s
bene i om ob aining anspa ency abou he expec ed cos inc ease in hei supply ela ions
ega ding he cu en s a e o in as uc u e a ailabili y. Public decision-make s bene i om
he possibili y o p io i izing he ope a ion o in as uc u e ega ding he mone a y and he
emission cos s o in as uc u e ailu es in he sys em. The model de e mines he quan i a-
i e e alua ion o modal shi and o e ou ing in esponse o in as uc u e ailu e as a isk
mi iga ion s a egy ha can be in eg a ed in o a b oade isk managemen o supply chain
esilience pe spec i e by decision-make s. The analysis in his s udy allows a be e unde -
s anding o he impac o a ying demands on anspo a ion mode choice, and he e ec s
o a ying lock imes and in as uc u e ailu e can be in e p e ed by decision-make s wi h
his model. The p oposed o mula ion allows he decision-make o op imize he ansship-
men po , deli e y ou es, and quan i ies, as well as he numbe o ehicles used in each
mode, based on a easonable se ing o pa ame e s. Mo eo e , o he da a on s ake-holding
indus ies in he Wes Ge man egion, such as he coal, a c, and s one indus ies, can be
used o ob ain insigh s in o he op imal anspo a ion mode choice. This analysis allows
decision-make s o gauge he locks and b idges ha can ha e he mos subs an ial impac
on cos s and emissions in he e en o ailu e. Sys em cos le els a e minimized unde he
assump ion ha shippe s coope a e and consolida e hei shipmen s o op imal cos -e icien
u iliza ion. This beha io , namely, amp shipping, is e iden in sho -sea shipping and inland
wa e way anspo . Howe e , he eal cos s o he anspo sys em and he ad e se e ec s
o in as uc u e ailu e a e g ea e in p ac ice, as he sys em con ains mo e unde u ilized
poin - o-poin anspo . The algo i hm is a single-p oduc , mul i ehicle model. In he u u e,
conside ing mul iple p oduc s o simila indus ies can be ad an ageous o decision-make s.
Howe e , conside ing a bundle o 10 ucks can be c i icized because i leads o highe cos s
i he capaci y is no ully used. The o mula ion complexi y is signi ican ly a ec ed by he
p oblem size because he un ime exponen ially inc eases wi h he addi ion o new po s and
ehicles. As ansshipmen emissions (e.g., hose s emming om p olonged s o age imes)
a e di icul o accoun o , his aspec was no conside ed in he case s udy. This aspec can
be explo ed in u u e s udies. An ad an age o he model is ha i allows he addi ion o
new anspo a ion modes wi hou changing cons ain s; he e o e, adding ail can bene i
decision-make s because i is a sus ainable subs i u ion, especially o ucks. Fu he mo e,
speed op imiza ion can be added o educe emissions and cos s. I is o in e es o implemen
he algo i hm in o he case s udies, such as he coal, o e, and s one indus ies. Mo eo e , i
is o in e es o e alua e he algo i hm wi h ano he en ance po , ha is, he depo , in he
model o analyze i s in luence on he ne wo k. We also pe o med a sensi i i y analysis o
123

Annals o Ope a ions Resea ch (2025) 351:667–726 703
he ehicles and hei e ec on he op imal solu ion and un ime. Ano he scena io may be
he de elopmen o new echnologies in he uck ma ke wi h g eene p opulsion echnolo-
gies. Le us no e ha decision-make s can use he esul s o his s udy o he alloca ion o
p oduc ion si es a a s a egic le el because o he ulne abili y o he exis ing in as uc u e.
Appendix A P oblem complexi y
The p oblem complexi y is cha ac e ized in Lemma 1:
Lemma 1 The mul imodal anspo a ion p oblem wi h ansshipmen po alloca ion de ined
in Equa ions (18)–(41)is an NP-ha d p oblem.
P oo We p o e ha his p oblem is an NP-ha d by educing i o a well-known NP-ha d
p oblem (A o a & Ba ak, 2009). Le m=1andm∈MKm=1, ha is, one ehicle
wi h unlimi ed capaci y is a ailable in he se ice. Thus, Cons ain s (26)and(31) can be
elimina ed. Ha ing one ehicle equi es ha one se ice be possible and ha a seconda y
se ice ( ansshipmen ) is impossible, esul ing in he emo al o decision a iables Tk,m
iand
UQk,m
iand hei associa ed cons ain s. Since all demand mus be sa is ied (Cons ain (30)),
Cons ain (21) becomes edundan as all po s a e o ced o be isi ed by he same ehicle.
Thus, decision a iables yk,m
i,Qk,m
iand k,m
i,jbecome unnecessa y, and hei associa ed
cons ain s can be emo ed. This educes his p oblem o he well-known a eling salesman
p oblem, which is NP-ha d; see (Ga ey & Johnson, 1979; Jungnickel, 1999), among o he s.
Since he simpli ied case is NP-ha d, he gene al case p esen ed in his wo k is also NP-ha d,
which comple es he p oo . 
Appendix B Reducing mul imodal anspo a ion o a single-mode
model
The mul imodal model can be educed o a single model o inland wa e way anspo a ion
o ucks. In he single model, we no longe ha e a i ual depo bu a he an ac ual o dina y
depo o he ne wo k. Inpu da a, such as he dis ances o inland wa e way anspo a ion,
ucks, and demand, mus be adap ed. The decision a iables UQk,m
iand Tk,m
iwe e no
needed. Mo eo e , some cons ain s emain unchanged, o he s need o be adap ed, and o he s
a e no needed. The ollowing equa ions a e subjec o he same objec i e unc ion and a e
unchanged: (20), (21), (22), (25), (26), (29)and(30). The cons ain s (23), (27), (28), and
(31) a e changed o (B1), (B2), (B3), and (B4), espec i ely.
S×yk,m
i≤Qk,m
i≤Di×yk,m
i∀i∈L {i0},∀k∈Km∀m∈M,(B1)

j∈L
k
i,j=
j∈L
k,m
j,i−Qk,m
i,∀i∈L,∀k∈Km,∀m∈M,(B2)

j∈L
k
i0,j=
j∈L
Qk,m
j,∀k∈Km,∀m∈M,(B3)

i∈L
Qk,m
i≤Ck,m∀k∈Km∀m∈M.(B4)
Cons ain (B1) ensu es ha i we do no selec he po , hen he quan i y equals ze o. I
we selec he po o be isi ed, hen he quan i y is nonze o, less han o equal o he demand.
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704 Annals o Ope a ions Resea ch (2025) 351:667–726
Fig. 12 Main algo i hm and sub ou ines connec ions
Cons ain (B2) de ines he low om i o jas he low ha en e s i om all possible j alues
minus he quan i ies. Cons ain (B3) de ines he i s low, which is he o al loaded quan i y,
and Cons ain (B4) ensu es ha he o al quan i ies loaded in each ehicle do no exceed i s
capaci y.
Appendix C Heu is ic sub ou ines
In his sec ion, we de ail he a ious sub ou ines used in he p oposed heu is ic. The connec-
ions be ween di e en sub ou ines a e illus a ed in Fig.12.
C.1 Popula ion gene a ion Sub ou ine
The Popula ion Gene a ion sub ou ine (shown in Algo i hm 3) gene a es andomly easible
ini ial indi iduals. The popula ion gene a ion sub ou ine c ea es wo ch omosomes o each
indi idual (ψ), called he “ ou e ch omosome” and he “quan i y ch omosome.” The ou e
ch omosome ca ies in o ma ion abou he ehicle ype and he sequence o he po isi
ou e, and he quan i y ch omosome ca ies in o ma ion abou he quan i y deli e ed o each
po and he ansshipmen quan i ies o be anspo ed by using ano he anspo a ion model.
Fi s , ehicle in o ma ion is gene a ed by andomly deciding he numbe o ehicles (NV
ψ)
and he ype o each ehicle (m∈M). The ou e is hen gene a ed by andomly choosing he
numbe and sequence o po s isi ed (NP eh
ψandRou e eh
ψ)by each ehicle.The isi edpo s
a e c ea ed andomly such ha a ehicle canno isi he same po mo e han once, sa is ying
Cons ain (20). Likewise, he sequence in he ou e ch omosome implies he sa is ac ion o
Cons ain (20). Finally, each ehicle is loaded wi h a andom quan i y o each isi ed po ,
and hein o ma ioniss o edon hequan i ych omosome.Once he ehiclein o ma ion, ou e
sequence, and quan i y deli e ed a e gene a ed o each indi idual, he algo i hm compa es
he o al loaded quan i y o each ehicle wi h i s capaci y. I a iola ion is de ec ed, i is called
he Fix Vehicle Capaci y sub ou ine, which educes he o al loaded quan i y and ensu es
he ul illmen o he model cons ain (31). Subsequen ly, he o al quan i y ca ied by all
ehicles is compa ed wi h he o al demand (Cons ain (30)). I a iola ion is de ec ed, hen
he Fix To al Quan i y sub ou ine is called. Finally, i ini ializes he ansshipmen quan i y
(s o ed in he quan i y ch omosome) and checks whe he he o al quan i y deli e ed o each
po ul ills he demand o he po . The Fix Indi idual Quan i y sub ou ine is used o ix
any iola ions in Cons ain (30).
Algo i hm 3 Popula ion gene a ion sub ou ine
1: Inpu : Con ex - ela ed, ehicle- ela ed, heu is ic- ela ed pa ame e s, 
2: Ou pu : P: an ini ial easible popula ion o indi iduals Gene a e ini ial indi iduals
3: o ψ←1 o do
4: decide andomly he numbe o ehicles o be used (NV
ψ); NV
ψ≤m∈M|Km|
5: o eh ←1 o NV
ψdo
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Annals o Ope a ions Resea ch (2025) 351:667–726 705
6: decide andomly he ehicle ype (m∈M)
7: decide andomly he numbe o po s (NP eh
ψ) o be included in he ehicle ou e (Rou e eh
ψ)
8: selec andomly NP eh
ψpo s and add hem o Rou e eh
ψ
9: o j←1 o NP eh
ψdo
10: load ehicle eh wi h a andom quani i y (Q eh
ψ,Rou e eh
ψ,j
) o po Rou e eh
ψ,j, calcula ed as
11: Q eh
ψ,Rou e eh
ψ,j
= and() ×DRou e eh
ψ,j
whe e and() ep esen s a andomly gene a ed numbe
be ween 0 and 1.
12: end o
13: calcula e he o al quna i y (NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,j
) ca ied by ehicle eh and compa e i wi h
he ehicle capaci y (C eh)
14: i NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,j
>C eh hen
15: Fix Vehicle Capaci y sub ou ine
16: end i
17: end o
18: calcula e he o al quan i y NV
ψ
eh=1NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,jand compa e i wi h he o al demand
(|L|
j=1Dj)
19: i NV
ψ
eh=1NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,j
= |L|
j=1Dj hen
20: Fix To al Quan i y sub ou ine
21: end i
22: ini ialize unloaded quan i ies UQ eh
ψ
23: o j←1 o NP eh
ψdo
24: i NV
ψ
eh=1Q eh
ψ,Rou e eh
ψ,j
−NV
ψ
eh=1UQ eh
ψ,Rou e eh
ψ,j
−DRou e eh
ψ,j
= 0 hen
25: Fix Indi idual Quan i y sub ou ine
26: end i
27: end o
28: end o
C.2 Fix ehicle capaci y sub ou ine
The Fix Vehicle Capaci y sub ou ine (Algo i hm 4) is called du ing easible popula ion
ini ializa ion when a ehicle wi hin an indi idual (po en ial solu ion) has a quan i y g ea e
han i s capaci y. The sub ou ine calcula es he excess quan i y ca ied (Q+) as he di e ence
be ween he o al quan i y loaded and he ehicle capaci y. The algo i hm hen andomly
selec s a po om he isi ed po lis o he ehicle (Rou e eh
ψ, nd) such ha he quan i y
anspo ed o ha po is nonze o. The selec ed po quan i y (Q eh
ψ,Rou e eh
ψ, nd
) is educed by
Q+.I Q+is la ge han Q eh
ψ,Rou e eh
ψ, nd
,Q eh
ψ,Rou e eh
ψ, nd
is se equal o ze o o a oid a nega i e
quan i y. The algo i hm e mina es when he ehicle capaci y cons ain is no iola ed.
Algo i hm 4 Fix ehicle capaci y sub ou ine
1: Inpu : NP eh
ψ,Q eh
ψ,Rou e eh
ψ,j
,C eh
2: Ou pu : Q eh
ψ,Rou e eh
ψ,j
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706 Annals o Ope a ions Resea ch (2025) 351:667–726
3: while NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,j
>C eh do calcula e excess Q
4: Q+←NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,j
−C eh
5: selec andomly ( nd = andi([1NP eh
ψ])) a po (Rou e eh
ψ, nd) such ha Q eh
ψ,Rou e eh
ψ, nd
>0
whe e andi([1NP eh
ψ]) e u ns a andom in ege be ween 1 and NP eh
ψ ha ep esen s a po wi hin he
ou e
6: Q eh
ψ,Rou e eh
ψ, nd
←max{0,Q eh
ψ,Rou e eh
ψ, nd
−Q+}
7: end while
8: e u n Q eh
ψ,Rou e eh
ψ,j
.
C.3 Fix o al quan i y sub ou ine
A e he Popula ion Gene a ion sub ou ine ixes he ehicle capaci y issues by calling he
Fix Vehicle Capaci y sub ou ine, he o al demand is compa ed wi h he o al loaded quan i y
o all ehicles. Oncea iola ion isde ec ed, he Fix To al Quan i y sub ou ine(Algo i hm 5)
is called. This algo i hm is based on wo cases. The i s case is he case o an excess quan i y
(Q+) ha exceeds he o al demand. The algo i hm selec s he ehicle wi h he lowes load
and hen selec s a isi ed po on he ou e o he selec ed ehicle. Subsequen ly, i educes
he loaded quan i y o ha po o ei he ze o o by Q+. The selec ion o he ehicle wi h he
lowes load helps educe he numbe o ehicles used. Le us no e ha i a ehicle ca ies no
quan i y, i is excluded om he sea ch. The second case ep esen s he si ua ion in which he
o al quan i y loaded is less han he o al demand (called he sho age case), in which case
he missing quan i y (Q−) is calcula ed as he di e ence be ween he o al demand and he
o al quan i y loaded on he ehicle. The emaining capaci y o each ehicle was calcula ed,
and ehicles wi h no emaining capaci y we e excluded. This decision led o wo possible
scena ios. In he i s scena io, whe e all used ehicles ha e no emaining capaci y, a new
ehicle is added o a andomly a ailable ype, and andom po s a e assigned. The o al load
o his new ehicle can be calcula ed as he maximum be ween he ehicle’s capaci y and
Q−, which is hen dis ibu ed andomly among he assigned po s. In he second scena io,
whe e some ehicles a e no ully u ilized, selec he ehicle wi h he lowes u iliza ion and
ans e o a andom po a minimum quan i y be ween he ehicle’s emaining capaci y and
Q−. The algo i hm e mina es when he o al demand is sa is ied and e u ns he upda ed
indi idual ch omosomes o he Popula ion Gene a ion sub ou ine.
Algo i hm 5 Fix o al quan i y sub ou ine
1: Inpu : Rou eψ,Qψ,NP
ψ,NV
ψ,C,D,L
2: Ou pu : Rou eψ,Qψ,NP
ψ,NV
ψ
3: while NV
ψ
eh=1NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,j
= |L|
j=1Djdo
4: while NV
ψ
eh=1NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,j
>|L|
j=1Djdo
5: Q+←NV
ψ
eh=1NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,j
−|L|
j=1Dj
6: Exclude a ehicle eh i NP eh
ψ
j=1Q eh
ψ,Rou e eh
ψ,j
=0
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Annals o Ope a ions Resea ch (2025) 351:667–726 713
Fig. 17 Illus a ion o Ope a o 11
1 and 3 a e chosen andomly, and hen Po 3 on Vehicle 1 and Po 4 on Vehicle 3 a e
chosen andomly. Exchanging he quan i ies does no iola e he capaci y cons ain ; hus,
i is pe mi ed. Consequen ly, Vehicle 1 deli e s 150 o Po 4 (p e iously, i was assigned
o deli e 50), and Vehicle 3 deli e s 100 o Po 3. Le us no e ha Po 4 is emo ed om
Vehicle 3 and ha Po 3 is emo ed om Vehicle 1 hen added o Vehicle 3 since i was no
se ed be o e. Ope a o 12 is simila o Ope a o 11; howe e , ins ead o doing one change in
each gene a ion (i e a ion), i ies o exchange mo e han one po be ween he wo ehicles,
whe e he numbe o exchanges is decided andomly based on he smalles numbe o po s
isi ed by he ehicles.
Ope a o 13 conside s exchanging unique po s be ween wo ehicles (Fig.18). Ope a o
13 andomly iden i ies wo ehicles (Vehicles 1 and 3 in Fig.18). Then, i checks he ou e
o each ehicle and iden i ies unique po s on each ehicle (i.e., po s no isi ed by he
o he ehicle). In Fig.18, he unique po s a e 2 and 3 o Vehicle 1 and 8 o Vehicle 3. I
hen selec s one po se ed by he i s ehicle and ies o exchange i wi h one o mo e
unique po s se ed by he second ehicle based on he emaining capaci y. In Fig.18,Po
3 se ed by he i s ehicle (Vehicle 1) is chosen as a unique po , and since he second
ehicle (Vehicle 3) has only one unique po (po 8), he ope a o ies o exchange 3 and 8.
Fi s , he emaining capaci y o Vehicle 1 and he quan i y deli e ed o po a e calcula ed as
(C−Q−Q3), which ep esen s he allowable exchange limi o Vehicle 1 a e emo ing
Po 3. Then, o he second ehicle (Vehicle 3), he algo i hm sea ches o an equi alen
exchange (i.e., a unique po wi h a quan i y close o he allowable exchange limi o Vehicle
1). I no po exis s wi h a ma ching quan i y, he algo i hm sea ches o mul iple unique
po s no exceeding he allowable exchange limi o Vehicle 1. Then, i swaps he quan i ies
be ween he wo ehicles and adds he new po s o he ehicles’ ou es i capaci y limi
allows his exchange. In Fig.18, he second ehicle (Vehicle 3) has only one unique po
(Po 8). Thus, i checks he emaining capaci y as C−Q−Q8and compa es i wi h
he quan i y o Po 3. The exchange o Po s 3 and 8 is pe mi ed since he capaci y is no
iola ed. Ope a o 14 has a simila unc ionali y o Ope a o 13, excep ha i ies o swap
123

714 Annals o Ope a ions Resea ch (2025) 351:667–726
Fig. 18 Illus a ion o Ope a o 13
po s be ween Vehicle 1 and mul iple o he ehicles, whe e he numbe o ehicles in ol ed
is chosen andomly based on he numbe o ehicles in ol ed in he se ice.
D.6 Quan i y ans e ope a o s
Ope a o s 15–18 a e unidi ec ional quan i y mo emen ope a o s. Ope a o 15, as shown
in Fig.19, iden i ies ehicles ha s a om he depo and a e no ully u ilized (i.e., ha e
emaining capaci ies) and selec s wo andom ehicles. I iden i ies he ehicle wi h he
highes u iliza ion (smalles emaining capaci y) o ecei e quan i ies, called he ecei ing
ehicle, and he ehicle wi h he lowes u iliza ion, named he sending ehicle. In Fig.19, he
ecei ing and sending ehicles a e assumed o be Vehicles 1 and 3, espec i ely. I calcula es
he emaining capaci y o he ecei ing ehicle and hen andomly selec s a subse o he
po s se ed by he sending ehicle. Figu e19 shows an example o a subse o one po (Po
7). I hen ans e s quan i ies om each chosen po on he sending ehicle (Po 7 in his
example) as long as he ans e able quan i ies a e wi hin he ehicle’s emaining capaci y.
The ans e p ocess s ops when ei he he emaining capaci y eaches ze o o a andom
123
Annals o Ope a ions Resea ch (2025) 351:667–726 715
Fig. 19 Illus a ion o Ope a o 15
numbe o po s is eached. Le us no e ha i he ecei ing ehicle does no o iginally isi
a ce ain po , i is hen added o he ou e andomly. We no e also ha his ope a o does no
necessa ily esul in a ully u ilized ecei ing ehicle (e.g., i he o al quan i y ans e ed
due o he numbe o selec ed po s is less han he emaining capaci y). The ou e sequence
ch omosome is upda ed acco dingly. Ope a o 16 ollows he same concep as Ope a o 15
bu ensu es ha he ecei ing ehicle is ully u ilized by allowing mo e han one ehicle o be
a sending ehicle. In he case whe e he ecei ing ehicle ecei es all quan i ies anspo ed
by a sending ehicle, he sending ehicle is emo ed om he se ice.
Ope a o 17, shown in Fig.20, ep esen s a simpli ied e sion o Ope a o 11, whe e he
quan i y mo emen occu s in one di ec ion. Tha is, one po se ed by he sending ehicle
is chosen andomly (Po 3 o Vehicle 1), and he possibili y o mo ing i s quan i y o he
ecei ing ehicle (Vehicle 3 in Fig.20) is checked. I he emaining capaci y is nonnega i e,
hen he po is mo ed o he ecei ing ehicle. Ope a o 18 is he mul iple po ans e
e sion o Ope a o 17.
123
716 Annals o Ope a ions Resea ch (2025) 351:667–726
Fig. 20 Illus a ion o Ope a o 17
123
Annals o Ope a ions Resea ch (2025) 351:667–726 717
Appendix E Case s udy da a
Table 9 Inpu da a o he di e en anspo a ion modes
No a ion Inland wa e way anspo a ion T uck Uni
αk,m58.88 −103.45 0.1506 [e/h], [e/km]
Ck,m900 −2000 26 [ ]
βk,m37.52 −44.95 19.33 [e/h]
nk,m2−2.43 −−
dk,m
i,jsee Table 13 see Table 14 [km]
qi,jsee Table 10 −−
h1 −[h]
τ0.67 −[h]
η230 −[ /h]
μk,m20.36 −24.25 0.3208 [e/km]
p0.44 −[e/ ]
k,m3,25 2 [e/ ]
γk,m2,30 3,25 [e/ ]
sk,m10 65 [km/h]
k,m−0.08 [e/h]
k,m−13.54 [e/h]
Tk,m−0.155 [e/km]
κk,m
i,j0,15 see Table 12 −
z3.6
k,m0,45 k,m
i,j/Ck,m−
ρ23,2 23,2 [g/MJ]
φ0,03−1,4 0,03 −1,4 [g/km]
ξ0,0957 0,277 [g/MJ]
ωk,m152 −181 −[kW]
χk,m−−[g]
Ek,m−10,9 [MJ/km]
123
718 Annals o Ope a ions Resea ch (2025) 351:667–726
Table 10 Ma ix o he numbe o locks be ween po iand j o he single inland wa e way anspo a ion model
Po s0123456789101112131415
0 0127411224443336
1 1016300113332225
2 2107411224443336
3 7670366772223330
4 43430334466 6 5 5 5 8
5 1016300113332225
6 1016300113332225
7 2127411004443336
8 2127411004443336
9 4342633440001112
10 43426334400 0 1 1 1 2
11 43426334400 0 1 1 1 2
12 32335223311 1 0 0 0 3
13 32335223311 1 0 0 0 3
14 32335223311 1 0 0 0 3
15 65608556622 2 3 3 3 0
123

Annals o Ope a ions Resea ch (2025) 351:667–726 719
Table 11 In as uc u e ailu e scena ios
Scena io In as uc u e ailu e Accessibili y Scena io In as uc u e ailu e Accessibili y
1 Meide ich Yes 7 Hamm No
Obe hausen Yes 8 Müns e No
2 Gelsenki chen Yes 9 Do mund-Ems-Kanal No
3 Wanne Eickel Yes
He ne Os
4 Hen ichenbu g No 10 Da eln-Hamm-Kanal No
5 Da eln Yes 11 Rhein-He ne-Kanal No
Ahsen
Flaesheim
6 Do s en Yes 12 Wesel-Da eln Kanal No
Hünxe
F ied ichs eld
123
720 Annals o Ope a ions Resea ch (2025) 351:667–726
Table 12 Emp y pe cen age o
ucks om po i o jPo s01234567
0 – 0.38 0.30 0.30 0.38 0.30 0.38 0.30
1 0.38 – 0.38 0.30 0.38 0.38 0.44 0.38
2 0.30 0.38 – 0.38 0.44 0.44 0.44 0.38
3 0.30 0.30 0.38 – 0.38 0.38 0.38 0.30
4 0.38 0.38 0.44 0.38 – 0.44 0.38 0.38
5 0.30 0.38 0.44 0.38 0.44 – 0.44 0.44
6 0.38 0.44 0.44 0.38 0.38 0.44 – 0.44
7 0.30 0.38 0.38 0.30 0.38 0.44 0.44 –
8 0.30 0.38 0.38 0.30 0.38 0.44 0.44 0.44
9 0.30 0.38 0.44 0.44 0.44 0.44 0.38 0.38
10 0.30 0.38 0.44 0.44 0.44 0.44 0.38 0.38
11 0.30 0.38 0.44 0.44 0.44 0.44 0.38 0.38
12 0.30 0.38 0.44 0.38 0.44 0.44 0.38 0.38
13 0.30 0.38 0.44 0.44 0.44 0.44 0.38 0.38
14 0.30 0.38 0.44 0.38 0.44 0.44 0.38 0.38
15 0.30 0.38 0.38 0.44 0.44 0.38 0.38 0.30
Po s8 9 101112131415
0 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
1 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.30
2 0.38 0.44 0.44 0.44 0.44 0.44 0.44 0.38
3 0.30 0.44 0.44 0.44 0.38 0.44 0.38 0.44
4 0.38 0.44 0.44 0.44 0.44 0.44 0.44 0.44
5 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.38
6 0.44 0.38 0.38 0.38 0.44 0.38 0.44 0.38
7 0.44 0.38 0.38 0.38 0.38 0.38 0.38 0.30
8 – 0.38 0.38 0.38 0.38 0.38 0.38 0.30
9 0.38 – 0.44 0.44 0.44 0.44 0.44 0.44
10 0.38 0.44 – 0.44 0.44 0.44 0.44 0.44
11 0.38 0.44 0.44 – 0.44 0.44 0.44 0.44
12 0.38 0.44 0.44 0.44 – 0.44 0.44 0.44
13 0.38 0.44 0.44 0.44 0.44 – 0.44 0.44
14 0.38 0.44 0.44 0.44 0.44 0.44 – 0.44
15 0.30 0.44 0.44 0.44 0.44 0.44 0.44 –
123
Annals o Ope a ions Resea ch (2025) 351:667–726 721
Table 13 Dis ance ma ix o inland wa e way anspo a ion om po i o j
Po s01234567
0040.15 104.74 144.39 107.29 98.33 110.35 131.92
140.15 0 64.58 104.24 67.14 58.18 70.20 91.77
2 104.74 64.58 0 78.54 41.43 28.87 40.89 62.46
3 144.39 104.24 78.54 0 37.10 72.13 84.15 105.72
4 107.29 67.14 41.43 37.10 0 35.03 47.05 68.62
598.33 58.18 28.87 72.13 35.03 0 12.02 33.59
6 110.35 70.20 40.89 84.15 47.05 12.02 0 21.57
7 131.92 91.77 62.46 105.72 68.62 33.59 21.57 0
8 134.41 94.25 64.94 108.21 71.10 36.07 24.06 11.46
9 120.70 80.55 44.47 54.17 57.40 44.84 56.85 78.43
10 120.70 80.55 44.47 54.17 57.40 44.84 56.85 78.43
11 120.70 80.55 44.47 54.17 57.40 44.84 56.85 78.43
12 108.80 68.65 32.57 66.39 45.50 32.94 44.95 66.53
13 112.50 72.35 36.27 62.37 49.20 36.64 48.66 70.23
14 105.86 65.70 29.62 69.02 42.55 29.99 42.01 63.58
15 135.70 95.55 59.47 40.17 72.40 59.83 71.85 93.42
Po s8 9 101112131415
0 134.41 120.70 120.70 120.70 108.80 112.50 105.86 135.70
194.25 80.55 80.55 80.55 68.65 72.35 65.70 95.55
264.94 44.47 44.47 44.47 32.57 36.27 29.62 59.47
3 108.21 54.17 54.17 54.17 66.39 62.37 69.02 40.17
471.10 57.40 57.40 57.40 45.50 49.20 42.55 72.40
536.07 44.84 44.84 44.84 32.94 36.64 29.99 59.83
624.06 56.85 56.85 56.85 44.95 48.66 42.01 71.85
711.46 78.43 78.43 78.43 66.53 70.23 63.58 93.42
8080.91 80.91 80.91 69.01 72.71 66.06 95.91
980.91 0 0 0 12.22 8.20 14.85 15.00
10 80.91 0 0 0 12.22 8.20 14.85 15.00
11 80.91 0 0 0 12.22 8.20 14.85 15.00
12 69.01 12.22 12.22 12.22 0 4.02 2.94 27.21
13 72.71 8.20 8.20 8.20 4.02 0 6.65 23.19
14 66.06 14.85 14.85 14.85 2.94 6.65 0 29.84
15 95.91 15.00 15.00 15.00 27.21 23.19 29.84 0
123
722 Annals o Ope a ions Resea ch (2025) 351:667–726
Table 14 Dis ance ma ix o ucks om po i o j
Po s01234567
00.00 57.28 115.24 136.53 96.09 102.96 83.63 113.78
157.23 0.00 75.09 108.84 55.75 62.80 43.48 73.62
2 121.79 80.36 0.00 76.92 46.99 10.62 45.78 62.97
3 135.66 110.02 77.41 0.00 55.82 77.96 97.69 114.87
497.42 56.31 46.76 54.69 0.00 47.30 67.04 84.22
5 104.85 63.42 10.52 76.78 46.85 0.00 13.44 46.03
687.29 45.86 23.46 91.43 61.50 13.44 0.00 36.83
7 113.34 71.90 63.90 114.22 84.29 45.21 37.33 0.00
8 114.34 72.90 64.90 115.22 85.29 46.21 38.33 3.05
9 134.29 80.17 40.91 44.16 25.97 48.66 68.40 85.58
10 135.93 81.82 40.79 44.04 27.62 47.12 66.86 84.04
11 134.28 80.17 40.13 43.37 25.97 48.66 68.39 85.57
12 115.08 73.98 28.62 51.46 24.38 36.26 55.99 73.18
13 138.86 81.00 33.66 46.18 23.24 39.99 59.72 76.91
14 114.99 73.89 27.04 51.04 30.51 36.17 55.90 73.08
15 145.67 99.65 59.20 29.93 45.45 67.58 87.32 104.50
Po s8 9 101112131415
0 114.77 135.14 136.78 135.14 115.63 121.06 113.89 149.81
174.62 79.56 81.20 79.56 75.29 80.72 73.54 108.45
263.96 41.33 41.21 40.54 28.57 34.00 26.83 58.38
3 115.86 42.55 43.71 43.04 50.92 48.23 51.65 43.06
485.21 25.40 27.05 25.40 30.94 23.06 29.19 48.49
547.02 48.35 46.80 46.13 34.17 39.60 32.42 70.58
637.83 63.00 61.45 60.79 48.82 54.25 47.08 85.23
73.05 85.79 84.24 85.79 71.61 77.04 69.87 108.03
80.00 86.79 85.24 86.79 72.61 78.04 70.87 109.02
986.57 0.00 2.43 0.79 14.37 11.68 17.08 30.29
10 85.03 2.43 0.00 1.65 14.25 11.56 16.96 30.17
11 86.57 0.79 1.65 0.00 13.58 10.89 16.29 29.50
12 74.17 15.81 15.69 15.02 0.00 5.47 5.90 37.59
13 77.90 10.53 10.41 9.75 5.47 0.00 9.83 32.31
14 74.08 17.46 17.34 16.68 4.71 10.14 0.00 39.24
15 105.49 25.57 25.45 24.78 32.65 29.96 35.36 0.00
Funding The p ojec (FKZ: 13N14700) is pa ially unded by he Ge man Fede al Minis y o Educa ion and
Resea ch (BMBF).
A ailabili y o da a and ma e ials A ailable upon eques .
Code A ailabili y A ailable upon eques .
123