Stability metrics for a maritime inventory routing problem under sailing time uncertainty

Shaabani, Homayoun; H a um, La s Magnus; Lapo e, Gilbe ; Ho , A ild
A icle
S abili y me ics o a ma i ime in en o y ou ing p oblem
unde sailing ime unce ain y
EURO Jou nal on T anspo a ion and Logis ics (EJTL)
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Sugges ed Ci a ion: Shaabani, Homayoun; H a um, La s Magnus; Lapo e, Gilbe ; Ho , A ild (2024) :
S abili y me ics o a ma i ime in en o y ou ing p oblem unde sailing ime unce ain y, EURO
Jou nal on T anspo a ion and Logis ics (EJTL), ISSN 2192-4384, Else ie , Ams e dam, Vol. 13, Iss. 1,
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h ps://doi.o g/10.1016/j.ej l.2024.100146
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S abili y me ics o a ma i ime in en o y ou ing p oblem unde sailing
ime unce ain y
Homayoun Shaabani
a,*
, La s Magnus H a um
a
, Gilbe Lapo e
a,b
, A ild Ho
a
a
Facul y o Logis ics, Molde Uni e si y College, PO Box 2110, NO 6402, Molde, No way
b
HEC Mon ´
eal, Mon ´
eal, H3T 2A7, Canada
ARTICLE INFO
Keywo ds:
Reop imiza ion
Unce ain y
S abili y me ics
Ma i ime in en o y ou ing
ABSTRACT
We s udy a mul i-p oduc ma i ime in en o y ou ing p oblem (MIRP) wi h sailing ime unce ain y. We
explici ly conside he eplanning ha happens a e unce ain y is e ealed. The objec i e is o de e mine he
s abili y o he adjus ed plans a e he occu ence o an unce ain e en and o e alua e he e ec o inco po-
a ing di e en s abili y me ics in he escheduling p ocess. Fi e s abili y me ics a e in oduced, and ma he-
ma ical o mula ions o he MIRP inco po a ing each me ic a e p esen ed. A eop imiza ion amewo k is hen
used o analyze he impac o each s abili y me ic. Calcula ions a e pe o med using 360 ins ances. The main
esul is ha adjus men s o he o iginal plan occu a no addi ional cos almos 50% o he ime. I decision
make s wan a mo e s able plan, hey should accep a 5% cos de e io a ion, esul ing in 20% mo e s able
solu ions.
1. In oduc ion
In 2022, mo e han 80% o he olume o goods in in e na ional ade
was ca ied by ma i ime anspo , co esponding o 12.03 billion ons.
I is expec ed ha he olume o ma i ime ade will g ow by mo e han
2% annually be ween 2024 and 2028 (UNCTAD, 2023), and he e o e
op imized ma i ime anspo a ion is o g ea impo ance. We s udy he
ma i ime in en o y ou ing p oblem (MIRP) which is a pa icula
ma i ime anspo a ion planning p oblem (Papageo giou e al., 2014).
The MIRP is a a ian o he in en o y ou ing p oblem (IRP) in a
ma i ime con ex . The IRP in eg a es in en o y managemen decisions
wi h ou ing decisions unde a endo -managed in en o y (VMI) sys-
em, whe e he supplie is esponsible o de e mining he deli e y
schedule o a gi en cus ome , he deli e y quan i y o ha cus ome ,
and he assignmen o cus ome s o ehicle ou es.
In he MIRP he e a e i e ime elemen s, shown in Fig. 1 along wi h
six e en poin s labelled om “a" o “ ". The ou ing is be ween poin s “a"
and “b" and a e “ ". Al hough he essel is s a ioned a a po be ween
poin s “b" and “ ", he empo al s a us o his in e al a ec s he ou ing
a e “ " o each “b" a he nex po .
The e a e always se e al unce ain pa ame e s in ma i ime ade.
Acco ding o UNCTAD (2021), supply chain dis up ions, changes in
globaliza ion pa e ns, anspo a ion cos s, po conges ion, and pan-
demics a e he main unce ain elemen s. In he MIRP, he e a e se e al
p oblem ea u es ha could be in luenced by unce ain y, some o which
a e lis ed below.
•The sailing ime can be unce ain due o easons such as bad wea he
condi ions (Rod igues and Ag a, 2022), mechanical ailu e o essels
(Rod igues and Ag a, 2022), o he ice condi ions in he A c ic egion
(Choi e al., 2015).
•The wai ing ime can be unce ain due o po conges ion (Ag a e al.,
2015).
•The po delay ime can be unce ain o easons such as s ikes and
equipmen ailu e a po s (Ch is iansen and Nyg een, 2005).
•Demand, which is he main ea u e wi h unce ain y in inland IRPs
(Touzou e al., 2021), can also be unce ain in a ma i ime se ing
(Cheng and Du an, 2004;So oush and Al-Yakoob, 2018).
P e ious esea ch on he MIRP unde unce ain y has mos ly ocused
on sailing ime as an unce ain pa ame e . Papageo giou e al. (2014)
s a ed ha he sailing ime is one o he p ima y ea u es in luenced by
unce ain y in ma i ime applica ions. Acco dingly, sailing ime is
conside ed as he only sou ce o unce ain y in he cu en s udy.
* Co esponding au ho .
E-mail add esses: [email p o ec ed] (H. Shaabani), [email p o ec ed] (L.M. H a um), [email p o ec ed] (G. Lapo e), A ild.Ho @himolde.
no (A. Ho ).
Con en s lis s a ailable a ScienceDi ec
EURO Jou nal on T anspo a ion and Logis ics
jou nal homepage: www.sciencedi ec .com/jou nal/eu o-jou nal-on- anspo a ion-and-logis ics
h ps://doi.o g/10.1016/j.ej l.2024.100146
Recei ed 2 Janua y 2023; Recei ed in e ised o m 14 June 2024; Accep ed 21 Sep embe 2024
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100146
A ailable online 25 Sep embe 2024
2192-4376/© 2024 The Au ho s. Published by Else ie B.V. on behal o Associa ion o Eu opean Ope a ional Resea ch Socie ies (EURO). This is an open access
a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
Th ee di e en app oaches o dealing wi h unce ain y in an op i-
miza ion p oblem ha e been p oposed in he li e a u e (Rod igues and
Ag a (2022);De Maio e al. (2021);Ay ug e al. (2005)), de ined in
Table 1.
In a MIRP wi h eac i e app oaches, equen adjus men s o he
o iginal plan can lead o ine iciencies om he pe spec i e o he po
planne (Liu e al., 2017), since hese adjus men s can igge a se ies o
changes in subsequen decisions such as s a scheduling and con aine
s o age (Xu e al., 2012). I is he e o e impo an o know how s able he
adjus ed plans a e, i.e., how la ge he de ia ion om he o iginal plan is
when eac i e ac ions a e applied. To inco po a e he equency in o -
ma ion, Cui e al. (2022) in oduced a dis ibu ionally obus op imi-
za ion aimed a ensu ing he obus ness o in en o y eplenishmen and
ou ing decisions agains he impac o dis ibu ional ambigui y. Hence,
he esea ch ques ion o he cu en s udy is: how o measu e he s a-
bili y o solu ions o a MIRP unde he unce ain y o sailing ime? We
in oduce s abili y me ics ha examine he sequence o ou es, which
po isi s a e made, and he quan i ies loaded and unloaded in each
isi .
The only pape ha ing used a eac i e app oach o he MIRP is by
Dong e al. (2018) who sol ed an unce ain MIRP using a mixed in ege
linea p og amming model and hen e iewed he in o ma ion e ealed
in each pe iod. Whene e he solu ion ob ained a e conside ing his
in o ma ion is in easible, a eop imiza ion is pe o med o he en i e
planning ho izon, igno ing he amoun o de ia ion om he o iginal
plan. A e he eop imiza ion, he ho izon is olled o wa d and he
p ocedu e is i e a ed un il he end o he ho izon.
Touzou e al. (2021) a emp ed o measu e he s abili y o solu ions
o he IRP unde unce ain demand using eop imiza ion models. They
s a ed ha hei me hod could be ex ended o o he applica ions and
p oposed o conside o he sou ces o unce ain y. In his ega d, he
cu en s udy aims o in oduce a eop imiza ion amewo k o he
MIRP unde sailing ime unce ain y in which s abili y me ics a e
in oduced. The main con ibu ion o his pape is h ee old.
1. Unlike Dong e al. (2018) and Touzou e al. (2021), who pe o med
eop imiza ion a speci ic ime in e als, in he cu en s udy eop-
imiza ion can occu a any poin in ime. This allows us o espond o
unce ain ies whene e hey a e e ealed. This is explained in Sec-
ion 5.
2. Rele an s abili y me ics o he MIRP a e iden i ied, and ma he-
ma ical o mula ions o each o hese me ics a e p oposed. This is
explained in Sec ion 6.
3. Each o he o mula ions is es ed by pe o ming compu a ional ex-
pe imen s o de e mine he impac o each s abili y me ic. This is
explained in Sec ion 7.
The emainde o he pape is o ganized as ollows. Sec ion 2 e iews
he li e a u e on MIRPs wi h unce ain ies and classi ies he pape s ac-
co ding o unce ain pa ame e s, app oaches, and models. Sec ion 3
p o ides a desc ip ion o he p oblem. Ma hema ical no a ions a e
explained in Sec ion 4. Sec ion 5is de o ed o he eop imiza ion
amewo k. S abili y me ics a e in oduced in Sec ion 6, ollowed by
hei analysis in Sec ion 7, whe e nume ical esul s and indings a e
p esen ed. Finally, Sec ion 8p o ides concluding ema ks.
2. Li e a u e e iew
The mos ecen e iew o he MIRP was p esen ed by Papageo giou
e al. (2014), who s udied a de e minis ic single-p oduc MIRP. The
au ho s s a ed ha obus ness is a challenge o MIRP and he e o e
ecommended he de elopmen o app oaches ha can deal wi h un-
ce ain y. Ksciuk e al. (2022) p o ided a e iew o unce ain y in
ma i ime ship ou ing and scheduling, examining unce ain y in eigh
di e en p oblems, including he MIRP. The au ho s men ioned ha in
he MIRP, he e a e no ixed pickup and deli e y po pai s, and no
p ede e mined numbe o po calls. The e o e, hey concluded ha his
makes he MIRP a challenging p oblem e en wi hou unce ain y.
The cu en sec ion ocuses on MIRPs wi h unce ain y. The e iewed
pape s a e summa ized in Table 2, which indica es he unce ain pa-
ame e s, he app oaches used o deal wi h unce ain y, and he
employed model. The emainde o his sec ion i s in oduces some o
he commonly used modelling echniques in an unce ain en i onmen
and hen discusses each o he app oaches used o deal wi h unce ain y.
Some o he commonly used echniques o modelling o op imiza-
ion p oblems unde unce ain y a e he ollowing.
•S ochas ic p og amming is a modeling amewo k o op imiza ion
p oblems unde unce ain y (Klein Hane eld e al., 2020), in which
he unce ain pa ame e s a e assumed o ollow known (o pa ially
known) p obabili y dis ibu ions (Rod igues and Ag a, 2022).
•Recou se models a e a class o models in s ochas ic p og amming,
including wo-s age and mul is age models. When he ue alue o
an unce ain pa ame e is obse ed, co ec i e ac ions can be aken
in ecou se models (Klein Hane eld e al., 2020).
•Chance-cons ained p og amming, in oduced by Cha nes and
Coope (1959), p o ides a ool o sol ing op imiza ion p oblems
Fig. 1. Fi e ime elemen s.
Table 1
Th ee app oaches o dealing wi h unce ain y.
App oaches Which decisions a e made be o e unce ain y is
e ealed
Which decisions a e made a e
unce ain y is e ealed
No es
Conside ing unce ain
in o ma ion explici ly
Conside ing
de e minis ic
pa ame e s
P oac i e All decisions –No adjus men These app oaches a e be e sui ed o p oblems wi h low unce ain y
and whe e he o iginal plan can be main ained wi hou any adjus men .
Reac i e –An ini ial plan All he decisions a e ecou se
ac ions
These app oaches a e be e sui ed o p oblems wi h high unce ain y.
Mixed An ini ial plan –Some o he decisions a e
ecou se ac ions
This is known as a p io i op imiza ion, a concep in oduced by
Be simas e al. (1990)
H. Shaabani e al. EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100146
2
unde unce ain y. This me hod op imizes he p oblem in such a way
ha he cons ain s a e sa is ied wi h a gi en p obabili y. The min-
imum equi ed eliabili y should be se by he decision make o a
alue be ween ze o and one. I his alue is ze o, he decision make
is ex emely isk seeking, and i i is one, i indica es an ex emely
conse a i e a i ude ( isk a e se).
•Robus op imiza ion accoun s o unce ain y se s whe e he p oba-
bili y dis ibu ion is unknown o does no exis (Rod igues and Ag a,
2022). The decision make cons uc s a solu ion ha is easible o
each ealiza ion o unce ain y in he gi en se (Be simas e al.,
2011). In o he wo ds, i op imizes he p oblem based on he wo s
possible ou come wi hin he unce ain y se . Unlike s ochas ic p o-
g amming, whe e he unce ain pa ame e is assumed o be a
andom a iable ha ollows a known (o pa ially known) p oba-
bili y dis ibu ion, he unce ain y model in obus op imiza ion is
usually de e minis ic and se -based (Be simas e al., 2011). The e-
o e, he unce ain pa ame e s can ake any alue wi hin he un-
ce ain y se .
•Reop imiza ion can be used o deal wi h unce ain y o in si ua ions
whe e he planning ho izon is sho e han he ho izon o he ac ual
p oblem (Dong e al., 2018).
•The in eg a ion o simula ion and op imiza ion can be used o deal
wi h unce ain y. Zhou e al. (2021) s udied di e en ypes o in e-
g a ion app oaches in ma i ime logis ics.
Touzou e al. (2021) exp essed ha a p io i app oaches p oac i ely
add ess unce ain ies by o mula ing obus eplenishmen plans. In
e ms o p oac i e app oaches, Cheng and Du an (2004) conside ed a
decision suppo sys em ha uses a simula ion model and an op imiza-
ion model. The simula ion model ep esen s he in en o y and ans-
po a ion sys em, and he op imiza ion model is o mula ed as a
disc e e- ime Ma ko decision p ocess ha deals wi h he unce ain y
o sailing ime and o demand. A de e minis ic model wi h penal y cos s
is used as a p oac i e app oach in wo s udies. Fi s , Ch is iansen and
Nyg een (2005) conside ed he unce ain y o sailing ime and wai ing
ime o a single-p oduc MIRP. They applied so in en o y cons ain s,
whe e le els should lie wi hin a ce ain in e al, and in oduced lowe
and uppe ala m in e als wi h a i icial penal y cos s o inc ease he
obus ness o hei model. Second, Rakke e al. (2011) in oduced a
de e minis ic model wi h penal y cos s o de ia ing om long- e m
cus ome con ac s, maximizing e enue based on he spo ma ke
p ice and he quan i y o sales in ha ma ke .
In ano he p oac i e app oach by So oush and Al-Yakoob (2018) o
a single-p oduc MIRP, demand was assumed o be a no mally dis ib-
u ed andom a iable, and penal ies o unde s ocking o o e s ocking
we e conside ed. The au ho s p oposed a s ochas ic op imiza ion model
wi h linea cons ain s and a con ex objec i e unc ion. They used
DICOPT as a comme cial sol e o sol e he p oblem.
Zhang e al. (2018) employed ime windows o model sailing ime
unce ain y o a single-p oduc MIRP. They de ined lexible solu ions as
hose ha can accommoda e unplanned dis up ions by adjus ing ou ing
solu ions whe e deli e y da es and o al deli e y quan i ies canno be
changed. Fu he mo e, a Lag angian heu is ic was implemen ed o ind
lexible solu ions using so cons ain s, and a simula o was in oduced
ha gene a es a dis up ion in each simula ion un o e alua e he
obus ness o he solu ions. Diz e al. (2019) conside ed he unce ain y
o he o al ime essels spend in po s due o delays in essel ope a ions
o a single-p oduc MIRP. They de eloped a obus op imiza ion
scheme using mo e essels o p o ec he solu ion agains delays. The
isk o in easibili y was quan i ied o di e en le els o obus ness and
Gu obi used o sol e he p oblem.
Rega ding he mixed app oaches, h ee s udies ha e conside ed
ecou se models ha ake in o accoun ou ing, he quan i ies o be
loaded and unloaded, he o de o po isi s in he i s s age, as well as
Table 2
Summa y o MIRP pape s wi h unce ain y.
Yea Au ho s Unce ain pa ame e s App oach Models
P
a
R
b
M
c
2004 Cheng &Du an •Sailing imes
•Demand
✓  Simula ion and op imiza ion
2005 Ch is iansen &
Nyg een
•Sailing imes
•Po delay imes
✓  De e minis ic model wi h penal y cos o in en o y iola ion
2011 Rakke e al. •Spo ma ke p ice ✓  De e minis ic model wi h penal y cos o de ia ion om he cus ome long- e m
con ac s
2015 Ag a e al. •Sailing imes
•Wai ing imes
  ✓S ochas ic p og amming
2016 Ag a e al. •Sailing imes   ✓S ochas ic p og amming
2018 So oush &Al-Yakoob •Demand ✓  Chance-cons ained p og amming
2018 Ag a e al. •Sailing imes
•Po delay imes
  ✓Robus op imiza ion
2018 Cho e al. •Sailing imes   ✓S ochas ic p og amming
2018 Zhang e al. •Sailing imes ✓  S ochas ic p og amming
2018 Dong e al. •Vessel a ailabili y
•T ip delays
•Pick-up window in o ma ion
•Consump ion and p oduc ion
a es
✓Reop imiza ion
2019 Diz e al. •Wai ing imes
•Po delay imes
✓  Robus op imiza ion
2019 Rod igues e al. •Sailing imes ✓✓•De e minis ic models wi h in en o y bu e s
•Robus op imiza ion
•S ochas ic p og amming
•Condi ional alue-a - isk
2021 Liu e al. •Sailing imes
•Wai ing imes
  ✓Two-s age dis ibu ionally obus op imiza ion
2023 Nikolaisen e al. •Depa u e imes
•Sailing imes
  ✓Op imiza ion and simula ion
2024 Cu en s udy •Sailing imes ✓Reop imiza ion including s abili y me ics
a
P: P oac i e app oaches.
b
R: Reac i e app oaches.
c
M: Mixed app oaches.
H. Shaabani e al. EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100146
3
isi imes o po s and in en o y decisions in he second s age, which
can be adjus ed o he scena io. The i s s udy, by Ag a e al. (2015),
in oduced a wo-s age s ochas ic p og amming model wi h ecou se
and sol ed his model using a decomposi ion algo i hm in which op i-
mali y cu s a e added dynamically. The second s udy, by Ag a e al.
(2016), used a model simila o ha o he p e ious s udy, bu sol ed i
wi h a combina ion o a comme cial sol e and local sea ch heu is ics. In
he hi d s udy, by Ag a e al. (2018), obus op imiza ion was used and
a decomposi ion algo i hm was sugges ed. Also, an i e a ed local sea ch
heu is ic was in oduced o imp o e he decomposi ion algo i hm.
Se e al echniques o handle unce ain y in MIRPs we e compa ed
by Rod igues e al. (2019), who conside ed unce ain sailing imes o a
single-p oduc MIRP and employed di e en models and algo i hms o
handle unce ain y. They disco e ed ha h ee me hods p o ide a good
ade-o be ween he amoun and p obabili y o in en o y limi iola-
ions and ou ing cos s. These me hods a e 1) de e minis ic modeling
wi h in en o y bu e s, 2) s ochas ic p og amming wi h high penal ies
o in en o y bounds iola ions, and 3) a hyb id algo i hm ha sol es a
de e minis ic app oach wi h in en o y bu e s de i ed om a condi-
ional alue-a - isk app oach.
Ano he mixed me hodology applied o MIRP unde unce ain y
comes om Cho e al. (2018), who p oposed a wo-s age s ochas ic
p og amming model in which p oduc ion in en o y schedule decisions
a e made in he i s s age and he p oduc ion a e is adjus ed o each
scena io in he second s age. Liu e al. (2021) applied a wo-s age dis-
ibu ionally obus op imiza ion algo i hm in which he ou ing de-
cisions a e made in he i s s age, while decisions ega ding quan i ies
o be loaded and unloaded, isi ime o po s, and in en o y le els a e
made in he second s age a e obse ing unce ain ies.
A eac i e me hodology was applied once in he con ex o MIRP by
Dong e al. (2018). The au ho s de eloped s ochas ic simula ions o
accoun o se e al sou ces o unce ain y, p esen ed in Table 2, and an
algo i hm ha in eg a es eop imiza ion and s ochas ic simula ion e-
sul s. They eop imized he model a a speci ied equency, ypically
once pe day. A each s age, he pa ame e s a e upda ed as unce ain ies
a e obse ed, and he op imiza ion p oblem is sol ed. This p ocedu e is
epea ed o each day o he ime ho izon o he cu en p oblem.
3. P oblem desc ip ion
The MIRP conside s he anspo a ion o p oduc s be ween mul iple
po s while mee ing in en o y equi emen s. Di e en po s p oduce
and consume mul iple p oduc s a a gi en p oduc ion and consump ion
a e. Ini ial in en o ies, minimum in en o y le els, and maximum in-
en o y le els a e speci ied o each po .
A he e ogeneous lee o essels wi h a gi en capaci y, a ixed speed,
and a daily ope a ing cos is gi en. The posi ion o a essel a he
beginning o he planning ho izon is e e ed o as i s o igin, which can
be a po o any loca ion a sea. Sailing imes om he o igin o each po
and be ween each pai o po s a e de e mined based on he gi en dis-
ance and speed o he essel. The sailing cos s a e also de i ed om he
sailing ime mul iplied by he daily cos o a essel. The maximum
unloading quan i ies a e de e mined by he consump ion po s based on
he essel capaci y and he maximum in en o y o he po . The
maximum numbe o isi s o each po is p ede e mined. The holding
cos and penal y cos o each p oduc in each po a e known. The
objec i e o he p oblem is o minimize he sum o h ee componen s:
sailing cos s, in en o y holding cos s, penal y cos s o backlogs and
o e s ocks.
The sailing imes a e assumed o be subjec o unce ain y due o
wea he condi ions. Al hough a planning ho izon is speci ied, he un-
ce ain y in sailing imes may cause he planning ho izon o be excee-
ded. The p oblem is sol ed unde de e minis ic condi ions and
whene e he unce ain y is e ealed, eop imiza ion is pe o med.
4. Ma hema ical no a ions
This sec ion explains some o he mos equen ly used no a ions
h oughou his pape , whe eas he comple e lis o no a ions can be
ound in Appendix A. The p oblem consis s o some po s ep esen ed by
i,jand h, and each po can be isi ed a mos m imes. The e is a se o
p oduc s deno ed by Kand a se o essels deno ed by V. We de ine a
ne wo k in which he nodes a e ep esen ed by (i,m), deno ing he isi
m o po i. The essels mo emen om node (i,m) o node (j,n)a e
ep esen ed by (i,m,j,n). The se o possible po a i als (i,m)is de ined
as SAand he se o po a i als ha may be made by essel is de ined
as SA
. The se o all possible essel mo emen s (i,m,j,n)is de ined as SX
and he se o all possible mo es o essel is de ined as SX
.
The bina y a iable oim k is one i and only i p oduc kis loaded on o
o unloaded om essel a he po isi (i,m). The amoun o p oduc k
loaded on o o unloaded om essel a po isi (i,m)is deno ed by
qim k. The amoun o p oduc k ha essel anspo s om po isi
(i,m) o po isi (j,n)is deno ed by imjn k. Le simk ep esen he in-
en o y le el o p oduc ka he s a o po isi (i,m)and sE
imk ep esen
he in en o y le el o p oduc ka he end o po isi (i,m).
The sailing o essel om po a i al (i,m)di ec ly o po a i al
(j,n)is deno ed by ximjn , sailing o essel om i s ini ial posi ion o
po a i al (i,m)is deno ed by xO
im , he po isi (i,m)is deno ed by yim,
he isi o po iby essel a po a i al (i,m)is deno ed by wim . Le
im be he s a ime o po a i al (i,m)and E
im be he end ime o po
a i al (i,m).
Fig. 2 depic s he ou e o one essel as an example o his ne wo k,
whe e O is he o igin o .
5. Reop imiza ion amewo k
The ma hema ical o mula ion o he MIRP gi en in Appendix A is
he same as ha o Shaabani e al. (2023). I is used as he basic model o
he eop imiza ion amewo k. The solu ion o his de e minis ic model
is conside ed as he ini ial plan o he eop imiza ion.
Pe iodic eop imiza ion conside s he p oblem pe iodically a ixed
ime in e als. Con inuous eop imiza ion, on he o he hand, sol es he
p oblem h oughou he day and whene e da a change; a p ocedu e
collec s he in o ma ion up o ha poin and hen s a s he eop im-
iza ion (Pillac e al., 2013). In he cu en s udy, unlike Dong e al.
(2018) and Touzou e al. (2021), a con inuous- ime model is used ha
eac s o unce ain ies as soon as hey appea , hence con inuous eop-
imiza ion is pe o med. In his con ex , TUis de ined as he ime a
which an unce ain e en occu s. The e o e, he nominal sailing imes
a e used un il TUand hen he sailing imes a e changed due o he
unce ain e en . Since TUis an unce ain e en , i can occu a any ime
Fig. 2. Example o he ne wo k.
H. Shaabani e al. EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100146
4

in [0,T]. We assume ha he e is only one TU alue in each planning
ho izon. Whene e he TU alue is e ealed, he ollowing changes a e
made o he basic model o p epa e he model o eop imiza ion.
•The se SAin he basic model, is eplaced wi h SB, excluding po
a i als isi ed be o e TU. In he same manne , he se SA
is eplaced
wi h SB
, he se SXis eplaced wi h SY, he se SX
is eplaced wi h SY
.
•The solu ion o he de e minis ic p oblem is ex ac ed and de ined as
he da a se o he eop imiza ion model. The sailing o essel om
po a i al (i,m)di ec ly o po a i al (j,n)is deno ed by Ximjn , he
sailing o essel om i s ini ial posi ion o po a i al (i,m)is
deno ed by XO
im , he isi o po a i al (i,m)is deno ed by Yim, he
isi o po iby essel a po a i al (i,m)is deno ed by Wim , and
he amoun o p oduc kloaded on o o unloaded om essel a
po isi (i,m)is deno ed by Qim k.
•Due o he occu ence o he unce ain e en , wo new ime con-
s ain s a e de ined:
im ≥TU(i,m)∈SB(1)
E
im ≥TU(i,m)∈SB.(2)
I a essel was on ou e om (i,m) o (j,n)when ime hi TU, hen he
essel is o ced o isi (j,n), bu he planned a i al ime may be a ec ed
by he upda ed sailing imes.
•The ini ial in en o y le els a e upda ed when he new p oblem s a s
a e TU. I Jik = − 1, he amoun o in en o y consumed up o TUis
sub ac ed om he ini ial in en o y, and i Jik =1, he amoun o
in en o y p oduced up o TUis added o he ini ial in en o y.
•The alues o he decision a iables isi ed be o e TUa e ixed. These
decision a iables a e as ollows: ximjn ,xO
im ,oim k,qim k, imjn k,simk,
sE
imk, E
im.
•Cons ain s (A28) and (A29) a e dele ed because he unce ain y o
he sailing ime is conside ed, which may lead o exceeding he
planning ho izon.
Now he modi ied model is eady, and we call i “Model 0”, which
ep esen s he me ic “cos ”. The eop imized solu ion ep esen s he
sailing cos s and he po ope a ion cos s, plus penal y cos s o backlogs
and o e s ocks wi hou s abili y me ics. The sailing cos s and po
ope a ion cos s a e called C*. The e o e, he eop imized solu ion may
di e om he ini ial solu ion. In his con ex , s abili y me ics a e
in oduced in he nex sec ion o educe his disc epancy.
6. S abili y me ics
In his sec ion we p esen i e s abili y me ics o he MIRP and new
cons ain s added o he model a e hen gi en o each me ic. The
objec i e unc ion o he ma hema ical model o each o he s abili y
me ics consis s o wo pa s: i s i minimizes he iola ion o he
me ics, second i minimizes he penal y cos o backlogs and o e -
s ocks. Based on he alues o consump ion and p oduc ion a es, essel
capaci y, and minimum and maximum in en o ies, he sizes o he
di e en elemen s in he objec i e unc ion a e such ha we implici ly
p io i ize he i s pa o he objec i e unc ion be o e he second pa .
6.1. Sequence p ese a ion
The sequence p ese a ion me ic, called SP, means ha he
sequence o he eop imized solu ion should no di e signi ican ly om
ha o he o iginal solu ion (De enbach and Ubbe , 2015). Applica ions
o he sequence p ese a ion me ic mos ly belong o ou ing and
scheduling p oblems (Touzou e al., 2021).
In he MIRP, a eling imes a e ypically much longe han in an
inland IRP, and because he unce ain e en can occu a any ime,
changing he sequence and e ou ing may be cos ly. I he sequence o
shipmen s has changed, mo e li ing ope a ions a e equi ed a he new
po in o de o each he unscheduled p oduc unloads, esul ing in
highe cos s. Ano he case whe e sequence p ese a ion is c i ical in
ma i ime anspo occu s on ansshipmen ou es whe e ano he essel
is wai ing a a po o ansshipmen .
The ma hema ical o mula ion o he SP me ic con ains wo new
bina y a iables. The bina y a iable zSP
imjn is de ined o indica e whe he
o no he e is a sequence change, and zSPO
im is a bina y a iable equal o
one i and only i he e is a change in he i s isi made by he essel.
The e o e, wo new cons ain s a e de ined as ollows:
zSP
imjn =Ximjn −ximjn  ∈V,(i,m,j,n) ∈ SY
(3)
zSPO
im =XO
im −xO
im  ∈V,(i,m)∈SB
.(4)
Cons ain s (3) and (4) a e nonlinea bu can be linea ized in o
cons ain s (6) o (9) as shown by Touzou e al. (2021). Cons ain s (3)
coun a sequence change when an a c om (i,m) o (j,n)is isi ed by
essel in he o iginal solu ion bu no in he eop imized solu ion, and
ice e sa. Cons ain s Eq. (4) coun a sequence change i essel sails
di ec ly om i s ini ial posi ion o po a i al (i,m)in he o iginal so-
lu ion bu does no in he eop imized solu ion, and ice e sa. The
ma hema ical o mula ion o he SP me ic is as ollows:
Model 1: Reop imiza ion based on sequence p ese a ion (SP) me ic
Minimize∑ ∈V∑(i,m,j,n)∈SX
zSP
imjn +∑ ∈V∑(i,m)∈SA
zSPO
im +
∑(i,m)∈SA∑k∈K |Jik=− 1CP
ik( imk + E
imk)+∑i∈N∑k∈K |Jik =− 1CP
ik T
ik +
∑i∈N∑k∈K |Jik=1CPP
ik PT
ik
(5)
subjec o  
(1) and (2)  
(A2) o (A27) and (A30) o (A45)  
zSP
imjn ≥Ximjn −ximjn ∈V,(i,m,j,n) ∈ SY
(6)
zSP
imjn ≥ximjn −Ximjn ∈V,(i,m,j,n) ∈ SY
(7)
zSPO
im ≥XO
im −xO
im ∈V,(i,m) ∈ SB
(8)
zSPO
im ≥xO
im −XO
im ∈V,(i,m) ∈ SB
(9)
zSP
imjn ∈ {0,1} ∈V,(i,m,j,n) ∈ SY
(10)
zSPO
im ∈ {0,1} ∈V,(i,m) ∈ SB
.(11)
6.2. Sequence p ese a ion wi h essel eplacemen
The sequence p ese a ion wi h essel eplacemen me ic, called
SPV, is simila o he SP me ic wi h one di e ence. Touzou e al. (2021)
p oposed he SP me ic whe e he sequence o an o iginal solu ion mus
be p ese ed in he eop imized solu ion; in compa ison, he SPV me ic
measu es he p ese a ion o he sequence o he o iginal solu ion o he
eop imized solu ion, e en i he essels a e eplaced wi h o he s. Fo
example, an a c om (i,m) o (j,n) ha was a e sed by essel 1 in he
o iginal solu ion can be a e sed by essel 3 in he eop imized solu-
ion. A sequence he e is mean o be he sequence o deli e ies o po s
ega dless o essel numbe .
Wi h espec o he SP me ic, he SPV me ic places g ea e emphasis
on main aining he deli e y sequence, ensu ing he o de o deli e ies o
po s is p ese ed e en wi h a new essel, which is mo e s ingen
compa ed o he SP me ic. This me ic aims o enhance ope a ional
e iciency and logis ical coo dina ion a po s. Po s ypically alloca e
schedules and esou ces based on he expec ed sequence o a i als.
Dis up ing his sequence can lead o delays and inc eased wai ing imes
o essels. Addi ionally, ca go handling and dis ibu ion a e o en
planned acco ding o a speci ic sequence. By main aining his sequence,
he igh esou ces and pe sonnel can be ensu ed o be a ailable a he
igh ime and place. Besides, he SPV me ic may make a co ec
measu emen when he lee is homogeneous, whe eas SP me ic may
make a co ec measu emen when he lee is he e ogeneous.
The SPV implici ly conside s he a i al ime because p ese ing he
H. Shaabani e al. EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100146
5
sequence can also p ese e he isi ing imes o he po s. This is shown
in Fig. 3 which also shows he di e ence o SP and SPV. Fig. 3 shows an
example o a pa o a ep esen a ion o a solu ion o he o iginal plan
(3a) and wo examples o he eop imized plans (3b,3c). The alues o SP
and SPV o each example a e calcula ed and p esen ed in Figu e (3). In
Figu e (3b), he eop imized solu ion main ains he same essel as he
o iginal plan bu changes he sequence o isi s. In con as , Figu e (3c)
shows a eop imized solu ion ha employs di e en essels compa ed o
he o iginal plan while p ese ing he o iginal sequence. When
compa ing he solu ions in Figu e (3b) and Figu e (3c), he al e ed
sequence in Figu e (3b) can esul in signi ican changes o he a i al
imes o mos po isi s, whe eas he solu ion in Figu e (3c) p ese es
he o iginal sequence and, consequen ly, he same a i al imes as he
o iginal plan.
The ma hema ical o mula ion o he SPV me ic uses wo new bi-
na y a iables. The bina y a iable zSPV
imjn is de ined o indica e whe he o
no he e is sequence change wi h essel eplacemen , and zSPVO
im is a
bina y a iable equal o one i and only i he e is sequence change wi h
essel eplacemen o he ini ial posi ion o he essels. The e o e, wo
new cons ain s a e de ined as ollows:
zSPV
imjn =∑
∈V
Ximjn −∑
∈V
ximjn 
(i,m,j,n) ∈ SY(12)
zSPVO
im =∑
∈V
XO
im −∑
∈V
xO
im 
(i,m) ∈ SB.(13)
Like cons ain s (3) and (4), cons ain s (12) and (13) can be line-
a ized, as is he case in cons ain s (15) o (18). Cons ain s (12) coun a
change o sequence wi h essel eplacemen i an a c om (i,m) o (j,n)
was isi ed in he o iginal solu ion bu no in he eop imized solu ion
and ice e sa. Cons ain s (13) coun a change o sequence wi h essel
eplacemen i a essel sails di ec ly om i s ini ial posi ion o po
a i al (i,m)in he o iginal solu ion bu no in he eop imized solu ion
Table 3
P obabili y dis ibu ions o sailing ime.
P obabili y dis ibu ions Sailing ime
×1×1.5×2
P=1 0.85 0.10 0.05
P=2 0.50 0.30 0.20
P=3 0.15 0.45 0.40
Fig. 3. Example o he calcula ion o he alues o he SP and SPV.
H. Shaabani e al. EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100146
6
and ice e sa. The ma hema ical o mula ion o he SPV me ic is as
ollows:
Model 2: Reop imiza ion based on sequence p ese a ion wi h essel eplacemen
(SPV) me ic
Minimize∑(i,m,j,n)∈SXzSPV
imjn +∑(i,m)∈SAzSPVO
im +∑(i,m)∈SA∑k∈K |Jik=− 1CP
ik( imk +
E
imk)+∑i∈N∑k∈K |Jik =− 1CP
ik T
ik +∑i∈N∑k∈K |Jik =1CPP
ik PT
ik
(14)
subjec o  
(1) and (2)  
(A2) o (A27) and (A30) o (A45)  
zSPV
imjn ≥∑ ∈VXimjn −∑ ∈Vximjn (i,m,j,n) ∈ SY(15)
zSPV
imjn ≥∑ ∈Vximjn −∑ ∈VXimjn (i,m,j,n) ∈ SY(16)
zSPVO
im ≥∑ ∈VXO
im −∑ ∈VxO
im (i,m) ∈ SB(17)
zSPVO
im ≥∑ ∈VxO
im −∑ ∈VXO
im (i,m) ∈ SB(18)
zSPV
imjn ∈ {0,1} (i,m,j,n) ∈ SY(19)
zSPVO
im ∈ {0,1} (i,m) ∈ SB.(20)
6.3. Visi de ia ion
The isi de ia ion me ic, called VD, compa es po isi s in he
eop imized solu ion o hose in he o iginal solu ion and coun s isi
iola ions which should be minimized (Touzou e al., 2021). This
me ic does no conside he essel numbe . As an example, po ia isi
m ha is isi ed by essel 1 in he o iginal solu ion, may be isi ed by
essel 3 in he eop imized solu ion. This me ic is help ul in si ua ions
whe e i is e y cos ly o miss a scheduled isi . I a planned po isi is
omi ed in he eop imized solu ion, his may esul in was ed ime and
esou ces, and i a new po isi occu s in he eop imized solu ion ha
was no planned in he o iginal solu ion, his may esul in highe
ope a ing cos s due o he una ailabili y o esou ces a a po .
The ma hema ical o mula ion o he VD me ic has a new bina y
a iable, zVD
im which is de ined o deno e whe he o no he e is a isi
de ia ion o po a i al (i,m). Thus, he new cons ain is de ined as
ollows:
zVD
im = |Yim −yim| (i,m) ∈ SB.(21)
Like cons ain s (3) and (4), cons ain (21) can be linea ized,
esul ing in cons ain s (23) and (24). Cons ain s (21) coun a isi
iola ion i po a i al (i,m)was isi ed in he o iginal solu ion and no
in he eop imized solu ion, and ice e sa. The ma hema ical o mu-
la ion o he VD me ic is as ollows:
Model 3: Reop imiza ion based on isi de ia ion (VD) me ic
Minimize∑(i,m)∈SAzVD
im +∑(i,m)∈SA∑k∈K |Jik =− 1CP
ik( imk + E
imk)+
∑i∈N∑k∈K |Jik =− 1CP
ik T
ik +∑i∈N∑k∈K |Jik=1CPP
ik PT
ik
(22)
subjec o  
(1) and (2)  
(A2) o (A27) and (A30) o (A45)  
zVD
im ≥Yim −yim (i,m) ∈ SB(23)
zVD
im ≥yim −Yim (i,m) ∈ SB(24)
zVD
im ∈ {0,1} (i,m) ∈ SB.(25)
6.4. Visi de ia ion wi hou essel eplacemen
The isi de ia ion wi hou essel eplacemen me ic, called VDV,
compu es he numbe o po s ha a e no isi ed in he eop imized
solu ion bu a e isi ed in he o iginal solu ion wi h a ce ain essel
numbe . This me ic di e s wi h p e ious me ic in e ms o essel
numbe . Fo example, i W221 =1 and w223 =1 hen VDV me ic is equal
o 2 bu VD me ic is equal o 0 because Y22 =1 and y22 =1.
We ha e in oduced he new VDV me ic, which di e s om he VD
me ic desc ibed by Touzou e al. (2021). Since hese au ho s conside a
single-p oduc p oblem, all he ehicles anspo he same p oduc .
The e o e, i he same cus ome is isi ed in he eop imized solu ion
compa ed o he o iginal solu ion, hen i is less impo an by which
ehicle i has been isi ed. Howe e , since he cu en s udy conside s a
mul i-p oduc p oblem, i could be impo an o use he same essel in
he eop imized solu ion as in he o iginal solu ion because each essel
can ca y a di e en p oduc mix.
When de e mining which essel will be assigned o a pa icula ca go,
i is impo an o plan o any addi ional equipmen o se ices needed o
po ope a ions, such as pilo , ugboa s, and po se ices. These a -
angemen s can be made in ad ance. Howe e , he lee o essels is
he e ogeneous, and i a ca go is escheduled and assigned o a di e en
essel, i may necessi a e he p ocu emen o addi ional equipmen o
se ices. In such cases, he exis ing a angemen s mus be cancelled, and
new ones mus be es ablished. Making hese changes, e en i possible, can
be a labou -in ensi e p ocess (Fage hol e al., 2009).
The ma hema ical o mula ion o he VDV me ic con ains a new
bina y a iable. Le zVDV
ideno e whe he he e is a isi de ia ion
wi hou essel eplacemen o po io no . Hence, he new cons ain is
de ined as ollows:
zVDV
im = |Wim −wim | ∈V,(i,m) ∈ SB
.(26)
Cons ain s (26) coun a iola ion o isi wi hou essel eplacemen
i he numbe o isi s o po iin he o iginal solu ion is no he same as
in he eop imized solu ion. Like cons ain s (3) and (4), cons ain s (26)
can be linea ized as shown in cons ain s (28) and (29). The ma he-
ma ical o mula ion o he VDV me ic is as ollows:
Model 4: Reop imiza ion based on isi de ia ion wi hou essel eplacemen (VDV)
me ic
Minimize∑(i,m)∈SA
∑ ∈VzVDV
im +∑(i,m)∈SA∑k∈K |Jik=− 1CP
ik( imk +
E
imk)+∑i∈N∑k∈K |Jik =− 1CP
ik T
ik +∑i∈N∑k∈K |Jik =1CPP
ik PT
ik
(27)
subjec o  
(1) and (2)  
(A2) o (A27) and (A30) o (A45)  
zVDV
im ≥Wim −wim ∈V,(i,m) ∈ SB
(28)
zVDV
im ≥wim −Wim ∈V,(i,m) ∈ SB
(29)
zVDV
im ∈ {0,1} ∈V,(i,m) ∈ SB
.(30)
6.5. Quan i y de ia ion
The quan i y de ia ion me ic, called QD, calcula es he di e ence
be ween he quan i y o p oduc loaded on o o unloaded om a essel
a a po in he o iginal solu ion and he loaded o unloaded quan i y in
he eop imized solu ion. Minimizing his me ic leads o ewe planning
issues (Touzou e al., 2021) and i is signi ican because i is he only
me ic ha add esses he in en o y componen o he MIRP.
The ma hema ical o mula ion o he QD me ic con ains a new
a iable. Le zQD
im k be he di e ence in he quan i y o p oduc kloaded
on o o unloaded om essel upon a i al a po (i,m)in he o iginal
solu ion and he eop imized solu ion. Thus, he new cons ain is
de ined as ollows:
zQD
im k = |Qim k −qim k| ∈V,(i,m) ∈ SB
,k∈K :Jik ∕= 0.(31)
Cons ain s (31) calcula e he di e ence be ween he loaded o
unloaded quan i y in he o iginal solu ion and he eop imized solu ion.
Like cons ain s (3) and (4), cons ain s (31) can be linea ized as in
cons ain s (33) and (34). The ma hema ical o mula ion o he QD
me ic is as ollows:
Model 5: Reop imiza ion based on quan i y de ia ion (QD) me ic
Minimize∑(i,m)∈SA
∑ ∈V∑k∈K
Jik∕=0zQD
im k +∑(i,m)∈SA∑k∈K |Jik=− 1CP
ik( imk +
E
imk)+∑i∈N∑k∈K |Jik =− 1CP
ik T
ik +∑i∈N∑k∈K |Jik =1CPP
ik PT
ik
(32)
subjec o  
(1) and (2)  
(A2) o (A27) and (A30) o (A45)  
zQD
im k ≥Qim k −qim k ∈V,(i,m) ∈ SB
,k∈K :Jik ∕= 0(33)
zQD
im k ≥qim k −Qim k ∈V,(i,m) ∈ SB
,k∈K :Jik ∕= 0(34)
zQD
im k ≥0 ∈V,(i,m) ∈ SB
,k∈K :Jik ∕= 0.(35)
H. Shaabani e al. EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100146
7
7. Analysis o he s abili y me ics
In his sec ion, each s abili y me ic is analyzed. Fi s , he p oblem
ins ances a e p esen ed in Sec ion 7.1. The e alua ion p ocedu e is
explained in Sec ion 7.2 and nume ical esul s and indings a e p e-
sen ed in Sec ion 7.3.
7.1. P oblem ins ances
The e a e 360 ins ances in his pape . Ins ances a e di ided in o h ee
g oups called A, B and C. G oup A consis s o ins ances wi h one p oduc ,
h ee essels and eigh po s; G oup B consis s o ins ances wi h wo
p oduc s, ou essels and 16 po s; and g oup C is simila o g oup A bu
wi h ou p oduc s. In g oup A and B, each po is limi ed o a mos one
p oduc , bu in g oup C, he e a e some imes mo e han one p oduc in a
gi en po . The inpu pa ame e s o ins ances a e de i ed om base
cases I1 o I10 o Shaabani e al. (2023) whe e each case di e s by he
ini ial in en o y o p oduc ka po i, he maximum in en o y o
p oduc ka po i, and he demand a e o po i o p oduc k. Two
di e en ime ho izons (T=30,60 days)a e conside ed o he p oblem.
The ins ances a e a ailable upon eques .
The sailing imes a e assumed o be subjec o unce ain y due o
wea he condi ions. As was done by Ag a e al. (2015), in he cu en
s udy we in oduce wo possible changes in sailing imes, conside ing
ha sailing imes can also emain unchanged. In he i s change, sailing
imes a e inc eased o 1.5 imes he o iginal alue, and in he second
change, hey a e inc eased o wice he o iginal alue. Based on any o
hese changes, he sailing imes o each po may change. Since
unce ain y may a ec an a ea a sea, sailing imes a e selec ed based on
he combina ion o a i al and depa u e po s. The e o e, o example,
i an e en occu s in he a ea o po 1, all sailing imes om po 1 o
o he po s and om o he po s o po 1 a e a ec ed. Table 3 shows he
h ee p obabili y dis ibu ions conside ed, whe e each column ep e-
sen s one o he possible changes ha can occu o sailing imes and
each ow ep esen s a p obabili y dis ibu ion and gi es he p obabili y
o each o he h ee possible changes. I all sailing imes a e assigned o
he i s column, hen no unce ain y has occu ed. Howe e , since he
cu en p oblem examines unce ain y in sailing imes, his is no
conside ed, and ano he change is c ea ed based on he same p obabili y
dis ibu ion un il a leas one sailing ime is assigned o he second o
hi d column. Besides, ou scena ios a e gene a ed o each p obabili y
dis ibu ion.
Table 4 shows he cha ac e is ics o he p oblem ins ances whe e he
o al numbe o ins ances is 360 (3 g oups ×5 base cases ×
2 ime ho izons ×3 p obabili y dis ibu ions ×4 scena ios). Le g be a
g oup o ins ances and G= {A,B,C}be he se o g oups. Also, le ube a
base case and Ua se o base cases, hen
U={I1,…,I5,g∈ {A,B}
I6,…,I10,g∈ {C}.
Acco ding o Shaabani e al. (2023), he s uc u e o ins ances can
make he MIRP di icul o sol e; he e o e, hese ins ances a e selec ed
such ha he op imal solu ions can be ound in less han 21,600 s,
he eby enabling a ai analysis o he s abili y me ics.
7.2. E alua ion p ocedu e
A se o me ics which ep esen s he ʹʹcos ʹʹ me ic and i e in o-
duced s abili y me ics in Sec ion 6is de ined as Θ= {θ0,…,θ5}whe e
θ0=cos ,θ1=SP,θ2=SPV,θ3=VD,θ4=VDV,θ5=QD.
All ins ances a e sol ed di ec ly by CPLEX 20.1. Fi s he de e min-
is ic model gi en in Sec ion 5is sol ed acco ding o Algo i hm 1. Then,
he e alua ion p ocedu e o an ins ance is gi en in Algo i hm 2 and
ma ix R ep esen s he s uc u e o he ou comes o all me ics o an
ins ance and is shown in Table 5.
The de ails o he e alua ion p ocedu e a e as ollows. A e sol ing
he de e minis ic model he modi ica ions o he basic model explained
in Sec ion 5a e applied. Now he modi ied model can be sol ed, mini-
mizing he ʹʹcos ʹʹ me ic, and he eop imized solu ion is ob ained,
which includes he sailing cos s and po ope a ion cos s, plus penal y
cos s o backlogs and o e s ocks, bu only he sailing cos s and po
ope a ion cos s, which is called C*, is epo ed. Since all models include
penal y cos e ms and he alues ob ained o hese e ms a e mos ly
iden ical, he solu ions o all models a e epo ed wi hou he alue o
Table 4
Cha ac e is ics o he p oblem ins ances.
G oups gNumbe o p oduc s
|K|
Numbe o essels |V|Numbe o po s
|N|
Base cases
U
Time ho izon
T
P obabili y dis ibu ions PScena ios
S
A1 3 8 {I1,…,I5} {30,60} {1,2,3} {1,2,3,4}
B2 4 16 {I1,…,I5}
C4 3 8 {I6,…,I10}
Table 5
Ma ix Rshowing he s uc u e o ou comes o an ins ance.
Cos SP SPV VD VDV QD
Cos C*SP*(C*)SPV*(C*)VD*(C*)VDV*(C*)QD*(C*)
SP C*(SP*)SP*SPV*(SP*)VD*(SP*)VDV*(SP*)QD*(SP*)
SPV C*(SPV*)SP*(SPV*)SPV*VD*(SPV*)VDV*(SPV*)QD*(SPV*)
VD C*(VD*)SP*(VD*)SPV*(VD*)VD*VDV*(VD*)QD*(VD*)
VDV C*(VDV*)SP*(VDV*)SPV*(VDV*)VD*(VDV*)VDV*QD*(VDV*)
QD C*(QD*)SP*(QD*)SPV*(QD*)VD*(QD*)VDV*(QD*)QD*
Table 6
Addi ional cons ain s.
Addi ional
cons ain s
indica o
Added cons ain s
α
0∑ ∈V∑(i,m,j,n)∈SX
CT
ij ximjn +∑ ∈V∑(i,m)∈SA
CTO
oi xO
im +
∑ ∈V∑(i,m)∈SA
∑k∈K |Jik∕=0CO
ikoim k =C*
α
1Cons ain s (6) o (11),
and ∑ ∈V∑(i,m,j,n)∈SX
zSP
imjn +
∑ ∈V∑(i,m)∈SA
zSPO
im =SP*
α
2Cons ain s (15) o (20),
and ∑(i,m,j,n)∈SXzSPV
imjn +∑(i,m)∈SAzSPVO
im =
SPV*
α
3Cons ain s (23) o (25),
and ∑(i,m)∈SAzVD
im =VD*
α
4Cons ain s (28) o (30),
and ∑(i,m)∈SA
∑ ∈VzVDV
im =VDV*
α
5Cons ain s (33) o (35),
and ∑(i,m)∈SA
∑ ∈V∑k∈K
Jik∕=0zQD
im k =QD*
H. Shaabani e al.
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100146
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Diz, G.S., dos, S., Hamache , S., Oli ei a, F., 2019. A obus op imiza ion model o he
ma i ime in en o y ou ing p oblem. Flex. Se . Manu . J. 31, 675–701. h ps://doi.
o g/10.1007/s10696-018-9327-9.
Dong, Y., Ma a elias, C.T., Je ome, N.F., 2018. Reop imiza ion amewo k and policy
analysis o ma i ime in en o y ou ing unde unce ain y. Op im. Eng. 19 (4),
937–976. h ps://doi.o g/10.1007/s11081-018-9383-8.
Fage hol , K., Ko s ik, J.E., Løkke angen, A., 2009. Ship ou ing scheduling wi h
pe sis ence and dis ance objeci es. Lec . No es Econ. Ma h. Sys . 89–107. h ps://
doi.o g/10.1007/978-3-540-92944-4.
Klein Hane eld, W.K., an de Vle k, M.H., Romeijnde s, W., 2020. S ochas ic
P og amming. Sp inge In e na ional Publishing, Cham. h ps://doi.o g/10.1007/
978-3-030-29219-5.
Ksciuk, J., Kuhlemann, S., Tie ney, K., Kobe s ein, A., 2022. Unce ain y in ma i ime ship
ou ing and scheduling: a Li e a u e e iew. Eu . J. Ope . Res. h ps://doi.o g/
10.1016/j.ejo .2022.08.006.
Liu, B., Zhang, Q., Yuan, Z., 2021. Two-s age dis ibu ionally obus op imiza ion o
ma i ime in en o y ou ing. Compu . Chem. Eng. 149. h ps://doi.o g/10.1016/j.
compchemeng.2021.107307. A icle 107307.
Liu, C., Xiang, X., Zheng, L., 2017. Two decision models o be h alloca ion p oblem
unde unce ain y conside ing se ice le el. Flex. Se . Manu . J. 29 (3–4), 312–344.
h ps://doi.o g/10.1007/s10696-017-9295-5.
Nikolaisen, J.B., Vågen, S.S., Schü z, P., 2023. Sol ing a ma i ime in en o y ou ing
p oblem unde unce ain y using op imiza ion and simula ion. Compu . Manag. Sci.
20 (1). h ps://doi.o g/10.1007/s10287-023-00459-x.
Papageo giou, D.J., Nemhause , G.L., Sokol, J., Cheon, M.S., Keha, A.B., 2014. MIRPLib -
a lib a y o ma i ime in en o y ou ing p oblem ins ances: su ey, co e model, and
benchma k esul s. Eu . J. Ope . Res. 235 (2), 350–366. h ps://doi.o g/10.1016/j.
ejo .2013.12.013.
Pillac, V., Gend eau, M., Gu´
e e , C., Medaglia, A.L., 2013. A e iew o dynamic ehicle
ou ing p oblems. Eu . J. Ope . Res. 225 (1), 1–11. h ps://doi.o g/10.1016/j.
ejo .2012.08.015.
Rakke, J.G., S ålhane, M., Moe, C.R., Ch is iansen, M., Ande sson, H., Fage hol , K.,
No s ad, I., 2011. A olling ho izon heu is ic o c ea ing a lique ied na u al gas
annual deli e y p og am. T anspo . Res. C Eme g. Technol. 19 (5), 896–911.
h ps://doi.o g/10.1016/j. c.2010.09.006.
Rod igues, F., Ag a, A., 2022. Be h alloca ion and quay c ane assignmen /scheduling
p oblem unde unce ain y: a su ey. Eu . J. Ope . Res. 303 (2), 501–524. h ps://
doi.o g/10.1016/j.ejo .2021.12.040.
Rod igues, F., Ag a, A., Ch is iansen, M., H a um, L.M., Requejo, C., 2019. Compa ing
echniques o modelling unce ain y in a ma i ime in en o y ou ing p oblem. Eu .
J. Ope . Res. 277 (3), 831–845. h ps://doi.o g/10.1016/j.ejo .2019.03.015.
Shaabani, H., Ho , A., H a um, L.M., Lapo e, G., 2023. A ma heu is ic o he mul i-
p oduc ma i ime in en o y ou ing p oblem. Compu . Ope . Res. 154. h ps://doi.
o g/10.1016/j.co .2023.106214. A icle 106214.
So oush, H.M., Al-Yakoob, S.M., 2018. A ma i ime scheduling anspo a ion-in en o y
p oblem wi h no mally dis ibu ed demands and ully loaded/unloaded essels.
Appl. Ma h. Model. 53, 540–566. h ps://doi.o g/10.1016/j.apm.2017.08.015.
Touzou , F.A., Ladie , A.-L., Hadj-Hamou, K., 2021. Modelling and compa ison o
s abili y me ics o a e-op imisa ion app oach o he in en o y ou ing p oblem
unde demand unce ain y. EURO J. T anspo . Logis . 10. h ps://doi.o g/10.1016/
j.ej l.2021.100050.
UNCTAD, 2021. Re iew o Ma i ime T anspo . Uni ed Na ions, Gene a. Re ie ed om.
h ps://unc ad.o g/web lye / e iew-ma i ime- anspo -2021.
UNCTAD, 2023. Re iew o Ma i ime T anspo . Uni ed Na ions, Gene a. Re ie ed om.
h ps://unc ad.o g/mee ing/launch- e iew-ma i ime- anspo -2023.
Xu, Y., Chen, Q., Quan, X., 2012. Robus be h scheduling wi h unce ain essel delay
and handling ime. Ann. Ope . Res. 192 (1), 123–140. h ps://doi.o g/10.1007/
s10479-010-0820-0.
Zhang, C., Nemhause , G.L., Sokol, J., Cheon, M.S., Keha, A., 2018. Flexible solu ions o
ma i ime in en o y ou ing p oblems wi h deli e y ime windows. Compu . Ope .
Res. 89, 153–162. h ps://doi.o g/10.1016/j.co .2017.08.011.
Zhou, C., Ma, N., Cao, X., Lee, L.H., Chew, E.P., 2021. Classi ica ion and li e a u e e iew
on he in eg a ion o simula ion and op imiza ion in ma i ime logis ics s udies. IISE
T ansac ions 53 (10), 1157–1176. h ps://doi.o g/10.1080/
24725854.2020.1856981.
H. Shaabani e al. EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100146
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