Resea ch A icle
Robus Scheduling o Be h Alloca ion and Quay C ane
Assignmen P oblem
M. Rod iguez-Molins, M. A. Salido, and F. Ba be
Ins i u o de Au om`
a ica e In o m`
a ica Indus ial, Uni e si a Poli `
ecnica de Val`
encia,CaminodeVe a,s/n,46022Val
`
encia, Spain
Co espondence should be add essed o M. A. Salido; [email p o ec ed].es
Recei ed 22 July 2014; Re ised 24 No embe 2014; Accep ed 1 Decembe 2014; Published 31 Decembe 2014
Academic Edi o : And zej Swie niak
Copy igh © 2014 M. Rod iguez-Molins e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion
License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly
ci ed.
Decision make s mus ace he dynamism and unce ain y o eal-wo ld en i onmen s when hey need o sol e he scheduling
p oblems. Di e en incidences o b eakdowns, o example, ini ial da a could change o some esou ces could become una ailable,
may e en ually cause he in easibili y o he ob ained schedule. To o e come his issue, a obus model and a p oac i e app oach
a e p esen ed o scheduling p oblems wi hou any p e ious knowledge abou incidences. This pape is based on p opo ionally
dis ibu ing ope a ional bu e s among he asks. In his pape , we conside he be h alloca ion p oblem and he quay c ane
assignmen p oblem as a ep esen a i e example o scheduling p oblems. The dynamism and unce ain y a e managed by assessing
he obus ness o he schedules. The obus ness is in oduced by means o ope a ional bu e imes o abso b hose unknown
incidences o b eakdowns. The e o e, his p oblem becomes a mul iobjec i e combina o ial op imiza ion p oblem ha aims o
minimize he o al se ice ime, o maximize he bu e imes, and o minimize he s anda d de ia ion o he bu e imes. To
his end, a ma hema ical model and a new hyb id mul iobjec i e me aheu is ic is p esen ed and compa ed wi h wo well-known
mul iobjec i e gene ic algo i hms: NSGAII and SPEA2+.
1. In oduc ion
Wi hin a con aine e minal, ope a ions ela ed o mo e con-
aine s can be di ided in o ou di e en subsys ems (ship-
o-sho e, ans e , s o age, and deli e y/ eceip ) [1]. In each
subsys em, e minal ope a o s mus deal wi h wi h di e en
complex op imiza ion p oblems ha can be o e come by
using a i icial in elligence echniques. Fo ins ance, be hing
alloca ion o s owage planning p oblems a e ela ed o he
ship- o-sho e a ea [2–5], ema shalling p oblem and ans-
po op imiza ion [6] o hes o ageand ans e subsys ems,
espec i ely, and planning and scheduling hin e land ope -
a ions ela ed o ains and ucks o he deli e y/ eceip
subsys em [7].
In his pape , we ocus on wo p oblems ela ed o he
ship- o-sho e a ea, he be h alloca ion p oblem (BAP) and
he quay c ane assignmen p oblem (QCAP). The o me is a
well-known combina o ial op imiza ion p oblem [8], which
consis s in assigning be hing posi ions and moo ing imes
o incoming essels. The QCAP deals wi h assigning a ce ain
numbe o quay c anes (QCs) o each moo ed essel such ha
all equi ed mo emen s o con aine s can be ul illed [9].
A comp ehensi e su ey o BAP and QCAP is gi en in
[9]. These p oblems ha e been mos ly conside ed sepa a ely,
wi h an in e es mainly ocused on BAP. An in e es ing
app oach o BAP is p esen ed by Kim and Moon [10]whe e
a simula ed annealing me aheu is ic is compa ed wi h a
ma hema ical model. Howe e , he e a e some s udies on he
combined BAP + QCAP conside ing di e en cha ac e is ics
o be hs and c anes [11–15].
Mos o he esea ch in scheduling has been ocused on
de e minis ic and comple e in o ma ion, bu hey a e usually
no sa is iedin eal-wo lden i onmen s.Due o he ac
ha he eal wo ld is unce ain, imp ecise, and nonde e -
minis ic, he e migh be unknown in o ma ion, b eakdowns,
incidences, o changes, which make he ini ial plans o he
ob ained schedules become in alid. Thus, he e a e new
ends o cope hese aspec s in he op imiza ion echniques:
p oac i e and eac i e app oaches [16]. In his pape , a p oac-
i e app oach is s udied wi hin he be h alloca ion and
Hindawi Publishing Co po a ion
Ma hema ical P oblems in Enginee ing
Volume 2014, A icle ID 834927, 17 pages
h p://dx.doi.o g/10.1155/2014/834927
2Ma hema ical P oblems in Enginee ing
he quay c ane assignmen p oblems. The unce ain y wi hin
hese p oblems is due o lowe mo emen s pe ime uni
han expec ed o engine ailu es in quay c anes, among
o he s. Due o he in oduc ion o his new objec i e in he
scheduling op imiza ion p oblem, a mul iobjec i e op imiza-
ion app oach needs o be aken in o conside a ion.
All he abo e s udies do no ake in o conside a ion he
unce ain y o he eal wo ld o ob ain a obus scheduling.
Robus ness is a measu e o he pe o mance cha ac e iza-
ion o an algo i hm in he p esence o unce ain ies [17].
Howe e , he e a e some s udies ha add ess he obus
scheduling. In [18], a obus op imiza ion model o cyclic
be hing o a con inuous and dynamic BAP is s udied by
minimizing he maximal c ane capaci y o e di e en a i al
scena ios o a bounded unce ain y gi en by hei a i al
ag eemen s. In [19], a p oac i e app oach o a disc e e and
dynamic model o he BAP is p esen ed aking in o accoun
unce ain ies in he a i al and handling imes gi en hei
p obabili y densi y unc ions. They p opose a mixed in ege
p og amming model and a gene ic algo i hm (GA) o bo h
p oblems: disc e e be h alloca ion and QC assignmen . The
objec i e is o minimize he sum o expec ed alue, he
s anda d de ia ion o he se ice ime, and he a diness o
he incoming essels.
Robus scheduling based on ope a ional bu e s has
al eady been in oduced as a p oac i e app oach in he
BAP. An app oach o obus BAP is p esen ed in [20]. They
p esen ed a eedback p ocedu e o he BAP ha i e a i ely
imp o es he obus ness o he ini ial schedule. This eedback
p ocedu e de e mines he ime bu e s o each essel by
means o adjus men ules.
In [21], ano he app oach o he obus BAP is sol ed by
a scheduling algo i hm ha in eg a es simula ed annealing
and b anch-and-bound algo i hms. This s udy in oduces he
obus ness as an objec i e o be maximized and an e alua ion
is ca ied ou by a ying he weigh s o hese unc ions. The
obus ness is achie ed by a cons an bu e ime assigned o
all essels.
In [22], he obus BAP p oblem is s udied as a p oac i e
s a egy as a mul iobjec i e op imiza ion p oblem. They
sol ed his p oblem wi h he squeaky wheel op imiza ion
(SWO) me aheu is ic. The i s objec i e is o minimize he
la e depa u es and he de ia ion om he desi ed posi ion;
he second objec i e is o maximize he obus ness o he
schedule. They ackle he obus ness measu e as a diminish-
ing e u n, speci ically he exponen ial unc ion, o cap u e
he dec easing ma ginal p oduc i i y o slacks in a be hing
schedule.
Howe e , mos o he abo e app oaches conside disc e e
be hs o p e ious knowledge abou he unce ain y in
a i al o handling imes o p oduce obus schedules, bu
usually his knowledge is no a ailable. Fu he mo e, o he
app oaches p opose how o ob ain obus schedules by
means o ope a ional bu e imes, bu hese bu e s a e se
independen ly o he handling (o p ocessing) ime o he
essels.
O e coming he abo e app oaches, hyb id me aheu is ics
o bo h single and mul iobjec i e combina o ial op imiza-
ion p oblems ha e ecei ed a signi ican in e es om he
esea ch communi y [23,24],andalso heyha ebeenused
in a wide ange o eal-wo ld applica ions [25].
In his pape , we in oduce a obus model o deal wi h
limi ed incidences wi h no p e ious knowledge abou hem
(Sec ion 3)aswellasamul iobjec i eapp oach o ace his
p oblem (Sec ion 4). A o mal mixed in ege p og amming
(MIP)isp esen ed o hedynamicandcon inuous obus
BAP + QCAP ha ex ends he model p esen ed in [10]
(Sec ion 5). Sec ion 6p esen s ou p oposed hyb id mul iob-
jec i e gene ic algo i hm based on he scheme NSGAII [26]
in o de o ob ain nea -op imal solu ions in an e icien way.
This hyb id algo i hm is used o sol e he BAP + QCAP wi h
a con inuous quay and dynamic a i als as well as o p o ide
obus solu ions by using ope a ional bu e s. As he e is no
p e ious knowledge abou he incidences, hese ope a ional
bu e s a e p opo ionally dis ibu ed among he asks o
be able o abso b as many incidences as possible. The eby,
a new objec i e unc ion (s anda d de ia ion o obus ness
measu es) was in oduced o pu sue his goal. This algo i hm
is compa ed wi h he ma hema ical model p esen ed and wo
well-known mul iobjec i e gene ic algo i hms: NSGAII and
SPEA2+ [27](Sec ion7). The de elopmen o he echnique
p esen ed in his pape will p o ide he e minal ope a o s
wi h di e en obus be hing plans which a e able o abso b
limi ed incidences.
The o e all collabo a ion goal o ou g oup a he Uni-
e si a Poli `
ecnica de Val`
encia (UPV) wi h he Valencia
Po Founda ion and he ma i ime con aine e minal MSC
(Medi e anean Shipping Company S.A.) is o o e assis ance
and help in he planning and scheduling o asks such as
he alloca ion o spaces o ou bound con aine s, o iden i y
bo lenecks, o de e mine he consequences o changes, o
p o ide suppo in he esolu ion o inciden s, and o p o ide
al e na i e be hing plans. Thus, he de elopmen o he
echnique p esen ed in his pape will p o ide he e minal
ope a o s wi h di e en obus be hing plans which a e able
o abso b limi ed incidences.
2. Be hing Alloca ion and Quay C ane
Assignmen : BAP + QCAP
Le 𝑉be a se o incoming essels; BAP + QCAP consis s
in ob aining an op imal (o nea -op imal) schedule o he
essels 𝑉by assigning moo ing imes, be hing posi ions, and
QCs o each essel. Ou BAP + QCAP model is classi ied,
acco ding o he classi ica ion gi en by Bie wi h and Meisel
[9] as ollows.
(i) Spa ial A ibu e: Con inuous Layou . We assume ha
he quay is a con inuous line, so he e is no pa i ion-
ing o he quay and he essel can be h a a bi a y
posi ions wi hin he bounda ies o he quay. I mus
be aken in o accoun ha , o a con inuous layou ,
be h planning is mo e complica ed han o a disc e e
layou , bu i be e u ilizes he quay space [9].
(ii) Tempo al A ibu e: Dynamic A i al. Fixed a i al
imes a e gi en o he essels, so ha essels canno
be h be o e hei expec ed a i al imes.
Ma hema ical P oblems in Enginee ing 3
Quay (m)
L
600
500
400
300
200
100
2468101214
aimi
𝓁ili
wi
𝜂i
𝜂i
pi
di
hi
i
Time (h )
Figu e 1: Da a ela ed o one essel.
(iii) Handling Time A ibu e: Unknown in Ad ance.The
handling ime o a essel depends on he numbe o
assigned QCs (QCAP)and hemo es equi ed.
(i ) Pe o mance Measu e: Wai and Handling Times.The
objec i e is o minimize he sum o he wai ing and
handling imes o all essels 𝑉.
Following, we in oduce he no a ion used o each essel
𝑖∈𝑉(Figu e 1). The in ege da a a iables a e
(i) 𝐾: o al numbe o QCs in he con aine e minal.
We assume all QCs ca y ou he same numbe o
mo emen s pe ime uni (mo sQC),gi enby he
con aine e minal,
(ii) 𝐿: o al leng h o he be h in he con aine e minal,
(iii) 𝑎𝑖: a i al ime o he essel 𝑖a po ,
(i ) 𝑐𝑖:numbe o equi edmo emen s oloadandunload
con aine s o essel 𝑖,
( ) ℓ𝑖: essel leng h.
The decision a iables a e
(i) 𝑚𝑖:moo ing imeo 𝑖.Thus,wai ing ime(𝑤𝑖)o essel
𝑖is calcula ed as (𝑤𝑖=𝑚𝑖−𝑎𝑖),
(ii) 𝑝𝑖: be hing posi ion whe e essel 𝑖moo s,
(iii) 𝑞𝑖: numbe o assigned QCs o essel 𝑖,
(i ) 𝑢𝑖𝑘: indica es whe he he QC 𝑘(1≤𝑘≤𝐾)wo ks
(1)o no (0) on he essel 𝑖,
( ) 𝑛𝑖𝑘: deno es ha he numbe o QCs assigned o essel
𝑖is 𝑘QCs (𝑛𝑖𝑘 =1), Fo ins ance, i essel 3 has been
assigned 4 QCs, hen 𝑛34 =1and he o he s QCs
𝑛3𝑘 =0,∀𝑘=1,...,𝐾,𝑘 =4.
The a iables de i ed om he p e ious ones a e
(i) 𝐻𝑖𝑘: loading and unloading ime a quay (handling
ime) o essel 𝑖using 𝑘QCs (1≤𝑘≤𝐾). This
handling ime depends on 𝑐𝑖and i is de ined by
𝐻𝑖𝑘 =⌈ 𝑐𝑖
𝑘∗mo sQC ⌉ ∀𝑖∈𝑉,∀𝑘=1,...,𝐾, (1)
(ii) ℎ𝑖: equi ed handling ime o essel 𝑖when 𝑞𝑖QCs a e
assigned oi .This alueisse bymeanso 𝐻𝑖𝑞𝑖,
(iii) 𝑡𝑖𝑘: wo king ime o he 𝑘 h QC (1≤𝑘≤𝐾) ha is
assigned o essel 𝑖,
(i ) 𝑑𝑖: depa u e ime o essel 𝑖(𝑑𝑖=𝑚𝑖+ℎ𝑖),
( ) 𝑠𝑖and 𝑒𝑖: indexes o he i s and las QC assigned o
essel 𝑖, espec i ely.
In his s udy, he ollowing assump ions a e conside ed.
(i) All he in o ma ion ela ed o he wai ing essels
is known in ad ance (a i al, p io i y, mo es, and
leng h).
(ii) E e y essel has a d a ha is lowe han o equal o
he d a o he quay.
(iii) Mo emen s o QCs along he quay as well as be hing
and depa u e imes o essels a e no conside ed
since i supposes a cons an penal y ime o all
essels.
(i ) Simul aneous be hing is allowed, subjec o he
leng h o he be h.
Usually in con aine e minals, he numbe o QCs could
a y du ing execu ion a he quay. This issue has been s udied
in Rod iguez-Molins e al. [5]. Howe e , in his pape and
wi hou loss o gene ali y, we s udy he obus ness o he
schedules assuming ha he numbe o QCs assigned o one
essel does no a y along he moo ed ime. Once a QC s a s
a askina essel,i mus comple ei wi hou anypauseo
shi (nonp eemp i e asks). Thus, all QCs assigned o he
same essel 𝑖ha e hesamewo king imeon he essel(𝑡𝑖𝑘 =
ℎ𝑖,∀𝑘=1,...,𝐾,𝑢𝑖𝑘 =1).
The ollowing cons ain s mus be accomplished.
(i) Moo ed ime o essel 𝑖mus be a leas he same ha
i s a i al ime (𝑚𝑖≥𝑎𝑖).
(ii) The e is a sa e dis ance be ween wo moo ed ships.
We assume ha each essel 𝑖has a 2.5% o his leng h
a each side as a sa e dis ance (𝜂𝑖)(Figu e 1). This sa e
dis ance is added o he leng h o each essel 𝑖:𝑙𝑖:=
ℓ𝑖+2𝜂𝑖.
(iii) The e mus be enough con iguous space a be h o
moo a essel 𝑖o leng h (𝑙𝑖).
(i ) The e mus be a leas one QC assigned o each essel.
Fu he mo e, he e is a maximum numbe o QCs
ha can be assigned o essel 𝑖(QC+
𝑖). This alue,
(QC+
𝑖), depends on he leng h o each essel (ℓ𝑖), since
a sa e dis ance is equi ed be ween wo con iguous
4Ma hema ical P oblems in Enginee ing
QCs (sa eQC)and he maximum numbe o QCs
ha he con aine e minal allows pe essel (maxQC)
(equ aion (2)). Bo h sa eQC and maxQC pa ame e s
a egi enby hecon aine e minal:
QC+
𝑖=min (maxQC,max (1,⌊ ℓ𝑖
sa eQC ⌋)) ∀𝑖∈𝑉. (2)
Ou objec i e is o alloca e all essels acco ding o se e al
cons ain s minimizing he o al wai ing (𝑇𝑤) and handling
o p ocessing ime (𝑇ℎ), known as he se ice ime (𝑇𝑠), o
all essels: 𝑇𝑤=∑
𝑖∈𝑉𝑤𝑖,(3)
𝑇ℎ=∑
𝑖∈𝑉ℎ𝑖,(4)
𝑇𝑠=𝑇𝑤+𝑇𝑠.(5)
3. Robus BAP + QCAP Model
Unce ain y and nonde e minism o eal-wo ld en i on-
men s may cause di icul ies in he ini ial plans made by he
decision make s. In con aine e minals, he ini ial ob ained
schedules o he BAP + QCAP p oblem migh become
in alid due o di e en easons: b eakdowns in QCs, la e
a i als o he essels, ex eme wea he e en s, a lowe a io
o mo emen s pe QC han expec ed, and so o h.
The obus ness concep means ha , gi en a schedule,
his ini ial schedule emains easible when mino incidences
occu in i s ac ual scena io.
The usual dis up ions o be conside ed in BAP + QCAP
a e he ollowing:
(i) ea ly o la e a i al o a essel 𝑖 om i s expec ed
a i al ime (𝑎𝑖);
(ii) he handling ime o a essel 𝑖is la ge han i s
expec ed handling ime (ℎ𝑖).
In his pape , we ocus jus on he dis up ions a ec ing
he handling ime which e en ually delay he depa u e ime.
In case o incidences ela ed o la e a i als, hey could also be
modeled as delays in he handling ime o he essels which
e en ually also delay hei depa u e ime.
De ini ion 1. Gi en he possible dis up ions, we conside ha
ascheduleis obus i adis up ioninone esseldoesno a ec
o al e he moo ing imes o he o he essels.
The obus ness o a schedule o BAP + QCAP migh
be gua an eed h ough wo pe iods o ime ela ed o each
essel: wai ing ime o a essel (𝑤𝑖)andbu e imea e he
depa u e o each essel (𝑏𝑖)[28]. Wi hou loss o gene ali y,
ea ly a i als a e no aken in o accoun since hey only
inc ease wai ing imes bu hey do no al e moo ing imes.
Theschedulecouldabso bdelayso b eakdowns ha do
no exceed he sum o hose wo pe iods (𝑤𝑖+𝑏𝑖). The e o e,
bo h imes should be maximized in o de o achie e he
Quay (m)
L
2
4
3
1
h1
h2b2
b1
h3
h4
b3
b4
Time (h )
024681012141618202224
Figu e 2: Bu e imes 𝑏𝑖gi enanexampleschedule.
maximum obus ness and ensu e ha he e is no need o
eschedule hein ol ed essels.Howe e ,i shouldbekep
in mind ha he i s objec i e o he BAP + QCAP is o
minimize he o al se ice ime o he incoming essels (𝑤𝑖+
ℎ𝑖). The e o e, ollowing he p oposal gi en by Da enpo e
al. [29], we ocus on maximizing only he second pe iod o
ime, bu e imes (∑𝑏𝑖), oob ain obus schedules.
Le 𝜑𝑖be he essels ha succeed essel 𝑖and occupy some
be h space o essel 𝑖(𝜑𝑝
𝑖)o useanyo QCsassigned o
essel 𝑖(𝜑𝑞
𝑖). The bu e ime o essel 𝑖(𝑏𝑖)is heminimum
di e ence (𝜏𝑖𝑗) be ween he depa u e ime o essel 𝑖(𝑑𝑖)and
he moo ing ime o essel 𝑗(𝑚𝑗,𝑗∈𝜑𝑖). In case he e is no
essel in 𝜑𝑖, he maximum bu e ime is assigned o 𝑏𝑖(an
in ini e alue). Figu e 2shows an example o he bu e imes
(𝑏𝑖)assigned o each scheduled essel as an emp y ec angle:
𝜑𝑝
𝑖={∀𝑗∈𝑉,𝑚𝑗≥𝑑𝑖
∧[𝑝𝑖,𝑝𝑖+𝑙𝑖)∩[𝑝𝑗,𝑝𝑗+𝑙𝑗) =⌀} ∀𝑖∈𝑉
𝜑𝑞
𝑖={∀𝑗∈𝑉,𝑚𝑗≥𝑑𝑖∧∃𝑘,1≤𝑘≤𝐾
∧𝑢𝑖𝑘 =1∧𝑢𝑗𝑘 =1} ∀𝑖∈𝑉
𝜑𝑖=𝜑𝑝
𝑖∪𝜑𝑞
𝑖∀𝑖∈𝑉
𝜏𝑖𝑗 =𝑚𝑗−𝑑𝑖∀𝑖∈𝑉,∀𝑗∈𝜑𝑖
𝑏𝑖={
{
{+∞, 󵄨󵄨󵄨󵄨𝜑𝑖󵄨󵄨󵄨󵄨=0
min
𝑗∈𝜑𝑖(𝜏𝑖𝑗), o he wise ∀𝑖∈𝑉.
(6)
In his pape , we assume ha he mo e handling ime is,
he mo e likely i is o su e incidences. The e o e, in gene al,
he la ge he bu e s a e, he mo e obus he schedules a e.
Ne e heless, ega ding he concep o dec easing p oduc-
i i y (o diminishing e u ns) p esen ed in [22], he e is a
ce ain bu e size beyond which no mo e obus ness is added
o he schedule. The eby, he e is no need o assign la ge
bu e imes o each essel. Fo ins ance, in Figu e 2, essel
1 would no need 8 ime uni s o bu e ime (𝑏1)since i s
handling ime is only 3 ime uni s. I is no likely ha his
essel would su e a delay o ha magni ude. Howe e , essel
2, wi h a handling ime o 8 ime uni s, has only 2 ime uni s o
bu e ime (𝑏2). In his case, i is highly likely ha his essel
Ma hema ical P oblems in Enginee ing 5
3
(4)
1
(4)
7
(5) 9
(5)
5
(5)
4
(2)
25
25
24
37 39
21
52
30
6
(5)
8
(2)
2
(3)
700
600
500
400
300
200
100
50 100 150 200 250 300
(a) Robus schedule
3
1
(5)
5
5
6
(5)
9
(5) 5
(5)
8
(5)
17
6
(5)
4
(4)
7
(2)
2
(4)
700
600
500
400
300
200
100
50 100 150 200 250 300
(b) Op imal schedule acco ding o 𝑇𝑠
Figu e 3: Two possible schedules gi en he same incoming essels.
su e s some b eakdown o delay and so i becomes in alid
his schedule.
Fu he mo e, we conside ha he magni ude o he
incidence is ela ed o he handling ime o he essel. Thus,
he obus ness measu e o each essel 𝑖(𝑟𝑖∈[0,1])is ela ed
o he bu e ime 𝑏𝑖and he a e age handling ime ℎ∗
𝑖(equa-
ion (7)). I should be men ioned ha o he unc ions, o
example, exponen ial unc ion, could be adop ed o de ine he
obus ness o each essel:
ℎ∗
𝑖=𝑐𝑖
((1+QC+
𝑖)/2)mo sQC .(7)
Gi en he obus ness o each essel, he obus ness o a
schedule 𝑅∈[0,|𝑉|]is de ined by (9),whe e𝜔𝑖is a weigh ing
ac o (𝜔𝑖≥1) which depends on his o ical da a, i a ailable.
A𝜔𝑖=1 alue ep esen s ha essel 𝑖used o inish i s asks
as expec ed, and 𝜔𝑖>1 alue deno es ha essel 𝑖used o be
delayed:
𝑟𝑖=min (1, 𝑏𝑖
𝜔𝑖ℎ∗
𝑖), ∀𝑖∈𝑉, (8)
𝑅=∑
𝑖∈𝑉𝑟𝑖.(9)
In his pape , we add ess he BAP + QCAP p oblem
wi hou knowledge o he incidences; hus, he weigh ing
ac o is he same o all he essels (𝜔𝑖=1,∀𝑖∈𝑉).
Example 2. Figu e 3shows wo di e en schedules gi en he
same se o 9 incoming essels. Each essel is labeled wi h i s
essel’s ID and he assigned QC numbe in b acke s. Fu -
he mo e, he bu e ime be ween essels is also showed.
On he one hand, Figu e 3(a) ep esen s a obus schedule
since limi ed incidences o e any essel could be abso bed.
On he o he hand, Figu e 3(b) shows a schedule wi h he
op imal solu ion acco ding o he objec i e unc ion 𝑇𝑠.The
la e schedule will be highly likely un easible i any incidence
occu s.
Figu es 3(a) and 3(b) a e an example o he well-known
ade-o be ween op imali y and obus ness. Howe e , a
obus schedule is no only achie ed by ex ending an op imal
schedule o e he ime. A obus schedule mus also conside
an op imized alloca ion o essels o achie e he maximum
sum o bu e sizes wi h a p ope dis ibu ion among all
essels. No e ha he op imali y is no di ec ly he makespan
o heschedulebu he o alse ice ime(wai ingand
handling imes).
An impo an issue in his pape is ha he e is no a ail-
able in o ma ion abou how likely he incidences o b eak-
downs occu . The e o e, i is in e es ing ha hese bu e s a e
p opo ionally dis ibu ed among all he essels. The eby, a
hi d objec i e is in oduced in o he model in o de o
imp o e he obus ness o a schedule: minimizing he s an-
da d de ia ion (𝜎) o he obus ness measu es o all essels
(𝑟𝑖∀𝑖∈𝑉):
𝜎=√1
|𝑉|∑
𝑖∈𝑉(𝑟𝑖−𝑟), (10)
whe e 𝑟is he a e age o he bu e s o he schedule and |𝑉|is
he numbe o incoming essels.
Bo h measu es p esen ed abo e, obus ness o a schedule
(𝑅) and s anda d de ia ion o hese alues (𝜎), ep esen

6Ma hema ical P oblems in Enginee ing
5
1
(3)
36
328
(5)
9
(4) 8
(4)
10
(5)
4
(5)
93
20
50
6
(5)
3
(4) 7
(3)
2
(2)
700
600
500
400
300
200
100
50 100 150 200 250 300 350
4
00
(a) High s anda d de ia ion
5
1
(3)
25
28 19
(5)
9
(4) 8
(4)
6
(4)
4
(5)
70
13
42
30
30
3
(4) 7
(3)
2
(2)
10
(3)
700
600
500
400
300
200
100
50 100 150 200 250 300 350
4
00
(b) Low s anda d de ia ion
Figu e 4: Two di e en schedules wi h simila obus ness and di e en s anda d de ia ion.
he ac ual obus ness o a schedule R o be maximized (see
(11)). This measu e gua an ees he abso p ion o incidences
ha imply a mos a delay o a R%o heweigh eda e age
handling ime (𝜔𝑖ℎ∗
𝑖):
R=𝑅−𝜎. (11)
Example 3. Figu e 4shows wo di e en schedules wi h a
simila obus ness alue (𝑅 = 0.7) bu di e en s anda d
de ia ions, 𝜎1=0.17and 𝜎2=0.45. Wi h hese alues, he
i s schedulehasanac ual obus ness alueo R1=0.7−
0.17=0.53.Thus,ina e age, hisschedulegua an ees ha
i could abso b incidences ha imply a mos a delay o he
53% o he a e age handling ime o he essels. In con as ,
hesecondschedulehasanac ual obus ness alueo R2=
0.7−0.45=0.25; hus, i is able o abso b only incidences
ha imply a mos a 25% o he a e age handling ime o he
essels.
Su ico e al. [30] p esen ed a close unc ion o measu e he
obus ness o a schedule (a g(𝑤𝑖)−𝛼𝜎(𝑤𝑖),∀𝑖∈𝑉). a g(𝑤𝑖)
and 𝜎(𝑤𝑖)deno e he a e age and he s anda d de ia ion
o he wai ing imes, espec i ely; 𝛼is a cons an weigh ing
ac o ha mus be se . Howe e , his measu e does no
e lec he ela ionship be ween he handling o p ocessing
ime o he ask and he bu e imes. Thus, o ou bes
knowledge, he e is no o he s udy which, conside ing he
BAP+QCAPwi hacon inuousquayanddynamica i als,
ackles he obus ness wi hou any p e ious knowledge abou
he incidences.
Figu e 4shows wodi e en scheduleso 10 esselswi h
he same high alue o he obus ness measu e. Howe e ,
schedule o Figu e 4(a) has a g ea e alue o he s anda d
de ia ion (𝜎) hanscheduleo Figu e4(b).The eby,i is
impo an ono e ha bu e imes omscheduleinFig-
u e 4(a) a e no equally dis ibu ed and his schedule will ail
i an incidence which delays he depa u e ime jus 1 ime
uni o e essels 4 o 6 occu s o mo e han 3 ime uni s o e
essel 1. Howe e , in he schedule in Figu e 4(b),i ishighly
unlikely o be in alid since all essels ha e enough bu e ime
a e i s schedule depa u e ime.
4. Mul iobjec i e App oach o
he Robus BAP + QCAP
Th ee di e en objec i es mus be op imized o sol e he
obus BAP + QCAP: he se ice ime (𝑇𝑠)(equa ion (5)), he
obus ness (𝑅)(equa ion (9)), and he s anda d de ia ion o
he obus ness measu es 𝜎(𝑅)(equa ion (10)). These objec i e
unc ions mus be no malized in o de o apply he sea ch
p ocess co ec ly.
Equa ion (14) shows how o no malize he se ice ime
objec i e in o he in e al [0,1](
𝑇𝑠) and i implies o no mal-
ize bo h he wai ing ime 
𝑇𝑤(equa ion (12))and hehandling
ime 
𝑇𝑠(equa ion (13)). On he one hand, he handling ime
is jus a linea no maliza ion since he maximum (ℎ+
𝑖)and
minimum (ℎ−
𝑖) imes a e known by assigning he minimum
(1)and hemaximumnumbe o QCs o essel𝑖(QC+
𝑖).
On he o he hand, no malizing he wai ing ime equi es o
de e mine a maximum o al wai ing ime (𝑊𝐹). In his case,
𝑊𝐹 alue is he o al wai ing ime o he incoming essels
when a i s -come, i s -se ed (FCFS) policy is applied,
assigning 2 QCs o each essel, and jus one essel is allowed
in he be h a he same ime (see, o example, Figu e 5). The
maximum o al wai ing ime (𝑊𝐹)couldalsobeob ainedby
assigning jus one QC o each incoming essel, bu in ha
case, 𝑊𝐹 alue would be oo la ge and all he no malized
wai ing imes would be close o ze o:

𝑇𝑤=1
𝑊𝐹∑
𝑖∈𝑉(𝑚𝑖−𝑎𝑖)
𝑇𝑤∈[0,1],(12)
Ma hema ical P oblems in Enginee ing 7
1
(2)
(2)(2)
(2)(2)
23
45
50 100 150 200 250 300 350 400 450 500 550
700
600
500
400
300
200
100
Figu e 5: Schedule gene a ed o ob ain he maximum alue o wai -
ing ime 𝑊𝐹.

𝑇ℎ=1
|𝑉|∑
𝑖∈𝑉(ℎ𝑖−ℎ−
𝑖
ℎ+
𝑖−ℎ−
𝑖)
𝑇ℎ∈[0,1],(13)

𝑇𝑠=
𝑇𝑤+
𝑇ℎ
2
𝑇𝑠∈[0,1].(14)
Robus ness objec i e unc ion mus also be no malized
in o he in e al [0,1](
𝑅) as de ined by (15).The hi dobjec-
i e, s anda d de ia ion o obus ness measu es, is al eady
no malized due o he ac ha 𝑟𝑖 alues a e al eady in he
in e al [0,1](equa ion (10)):

𝑅=𝑅
|𝑉|
𝑅∈[0,1].(15)
The eby, he objec i e unc ion o he obus BAP +
QCAP is o minimize he unc ion 𝐹(equa ion (16)). Each
coe icien 𝜆𝑖(0 ≤ 𝜆𝑖≤1)assigns di e en weigh s o
each componen o objec i e unc ion in o de o es ablish
an agg ega e unc ion:
𝐹=𝜆1
𝑇𝑠−𝜆2
𝑅+𝜆3𝜎. (16)
These coe icien s 𝜆𝑖a e subjec o ∑𝑖𝜆𝑖=1.
In a mul iobjec i e op imiza ion p oblem, usually he e
is no single solu ion whe ein all i s objec i es a e simul a-
neously op imized. Howe e , he e may exis a se o Pa e o
op imal solu ions wi h di e en ade-o s be ween hei
objec i e unc ions. Pa e o e iciency, o Pa e o op imali y, is
a solu ion in which i is impossible o make any one c i e ion
be e o wi hou making a leas one c i e ion wo se o
[31]. Pa e o op imal solu ions a e de ined by means o he
dominance concep . Conside ing he obus BAP + QCAP, le
𝑥and 𝑦be wo di e en solu ions; 𝑥domina es 𝑦i a leas
one o he ollowing condi ions is sa is ied:

𝑇𝑠(𝑥)<
𝑇𝑠(𝑦) ∧
𝑅(𝑥)≥
𝑅(𝑦) ∧𝜎(𝑥)≤𝜎(𝑦),

𝑇𝑠(𝑥)≤
𝑇𝑠(𝑦) ∧
𝑅(𝑥)>
𝑅(𝑦) ∧𝜎(𝑥)≤𝜎(𝑦),

𝑇𝑠(𝑥)≤
𝑇𝑠(𝑦) ∧
𝑅(𝑥)≥
𝑅(𝑦) ∧𝜎(𝑥)<𝜎(𝑦). (17)
Gi en a se o easible solu ions 𝐷,asolu ion𝑥∈𝐷is
Pa e oop imalsolu ioni i isnondomina edbyanyo he
solu ion 𝑥󸀠∈𝐷.ThePa e oop imalse is hese o all he
solu ions ha a e Pa e o op imal solu ions [31].
In gene al, gene a ing he Pa e o op imal se is expensi e
compu a ionally and i is o en imp ac icable. The e o e,
algo i hms y o ind a good app oxima ion o he Pa e o
op imalse .In hiswo k,we e e ha eachapp oxima ion
as Pa e o on , which con ains solu ions ha , al hough a e
nondomina ed among hem, could be domina ed by o he
solu ions no ound by ou algo i hms.
5. Ma hema ical Fo mula ion
A mixed in ege p og amming (MIP) model is p esen ed o
sol e he obus BAP+QCAP.Theobjec i e unc iono his
model is o minimize (16). This ma hema ical model is based
on hemodelp esen edin[10,28].
In he p oposed model, 𝑀deno es a su icien ly la ge
numbe (as i is used in MIP). Fu he mo e, he e a e ou
auxilia y bina y a iables. 𝑧𝑥
𝑖𝑗 is a decision a iable ha
indica es i essel 𝑖is loca ed o he le o essel 𝑗on he be h
(𝑧𝑥
𝑖𝑗 =1); 𝑧𝑦
𝑖𝑗 =1indica es ha essel 𝑖is moo ed be o e essel
𝑗in ime. The auxilia y a iable 𝑢𝑖𝑘 indica es whe he he QC
𝑘wo ks (1)o no (0) on essel 𝑖;𝑛𝑖𝑘 =1deno es ha he
numbe o QCs assigned o essel 𝑖is 𝑘:
∀𝑖,𝑗∈𝑉
𝑖 =𝑗 ∀𝑘=1,...,𝐾 𝑧𝑥
𝑖𝑗,𝑧𝑦
𝑖𝑗,𝑢𝑖𝑘,𝑛𝑖𝑘0/1in ege .(18)
In he p oposed model, he e a e ou auxilia y bina y
a iables. 𝑧𝑥
𝑖𝑗 is a decision a iable ha indica es i essel 𝑖is
loca ed o he le o essel 𝑗on he be h (𝑧𝑥
𝑖𝑗 =1); 𝑧𝑦
𝑖𝑗 =1
indica es ha essel 𝑖is moo ed be o e essel 𝑗in ime. The
auxilia y a iable 𝑢𝑖𝑘 indica es whe he he QC 𝑘wo ks (1)o
no (0) on essel 𝑖;𝑛𝑖𝑘 =1deno es ha he numbe o QCs
assigned o essel 𝑖is 𝑘.
The cons ain s o his ma hema ical model a e de ailed
below. Cons ain (19) ensu es ha essels mus moo a e
hey a i e a he e minal:
∀𝑖∈𝑉 𝑚𝑖≥𝑎𝑖.(19)
Cons ain s (20) and (21) es ablish he wai ing and depa u e
imes acco ding o 𝑚𝑖and ℎ𝑖:
∀𝑖∈𝑉 𝑤𝑖=𝑚𝑖−𝑎𝑖,(20)
∀𝑖∈𝑉 𝑑𝑖=𝑚𝑖+ℎ𝑖.(21)
Cons ain (22) gua an ees ha a moo ed essel does no
exceed he leng h quay:
∀𝑖∈𝑉 𝑝𝑖+𝑙𝑖≤𝐿. (22)
The numbe o QCs o he essel 𝑖a e assigned by means o
cons ain s (23)–(28) as ollows:
∀𝑖∈𝑉 𝑞𝑖=𝐾
∑
𝑘=1𝑢𝑖𝑘,(23)
∀𝑖∈𝑉
𝐾
∑
𝑘=1𝑛𝑖𝑘 =1, (24)
8Ma hema ical P oblems in Enginee ing
∀𝑖∈𝑉
𝐾
∑
𝑘=1𝑛𝑖𝑘𝑘=𝑞𝑖,(25)
∀𝑖∈𝑉 1≤𝑞𝑖≤QC+
𝑖,(26)
∀𝑖∈𝑉 1≤𝑠𝑖≤𝑒𝑖≤𝐾, (27)
∀𝑖∈𝑉 𝑞𝑖=𝑒𝑖−𝑠𝑖+1. (28)
Cons ain s (29)–(31) es ablish he minimum handling ime
needed o load and unload hei con aine s acco ding o he
numbe o assigned QCs:
∀𝑖∈𝑉
𝐾
∑
𝑘=1𝑡𝑖𝑘 mo sQC ≥𝑐𝑖(29)
∀𝑖∈𝑉
𝐾
∑
𝑘=1𝑛𝑖𝑘𝐻𝑖𝑘 =ℎ𝑖(30)
∀𝑖∈𝑉 ℎ𝑖=max
∀𝑘=1,...,𝐾𝑡𝑖𝑘.(31)
Cons ain (32) ensu es ha QCs ha a e no assigned o
essel 𝑖ha e 𝑡𝑖𝑘 =0:
∀𝑖∈𝑉∀𝑘=1,...,𝐾 𝑡𝑖𝑘 −𝑀𝑢𝑖𝑘 ≤0. (32)
Cons ain (33) o ces all assigned QCs o essel 𝑖wo king he
same numbe o hou s:
∀𝑖∈𝑉∀𝑘=1,...,𝐾 ℎ𝑖−𝑀(1−𝑢𝑖𝑘)−𝑡𝑖𝑘 ≤0. (33)
Cons ain (34) a oids ha one QC is assigned o wo
di e en essels a he same ime:
∀𝑖,𝑗∈𝑉∀𝑘=1,...,𝐾 𝑢𝑖𝑘 +𝑢𝑗𝑘 +𝑧𝑥
𝑖𝑗 ≤2. (34)
Cons ain s (35) and (36) o ce heQCs obecon iguously
assigned ( om 𝑠𝑖up o 𝑒𝑖):
∀𝑖∈𝑉∀𝑘=1,...,𝐾 𝑀(1−𝑢𝑖𝑘)+(𝑒𝑖−𝑘)≥0, (35)
∀𝑖∈𝑉∀𝑘=1,...,𝐾 𝑀(1−𝑢𝑖𝑘)+(𝑘−𝑠𝑖)≥0. (36)
The sa e y dis ance be ween essels is aken in o accoun by
cons ain (37) as ollows:
∀𝑖,𝑗∈𝑉
𝑖 =𝑗 𝑝𝑖+𝑙𝑖≤𝑝𝑗+𝑀(1−𝑧𝑥
𝑖𝑗). (37)
Cons ain (38) a oids ha one essel uses a QC which should
c oss h ough he o he s QCs:
∀𝑖,𝑗∈𝑉
𝑖 =𝑗 𝑒𝑖+1≤𝑠𝑗+𝑀(1−𝑧𝑥
𝑖𝑗). (38)
Cons ain (39) a oids ha essel 𝑗moo s while he p e ious
essel 𝑖is s ill a he quay:
∀𝑖,𝑗∈𝑉
𝑖 =𝑗 𝑑𝑖≤𝑚𝑗+𝑀(1−𝑧𝑦
𝑖𝑗). (39)
Cons ain (40) es ablishes he ela ionship be ween each pai
o essels a oiding o e laps:
∀𝑖,𝑗∈𝑉
𝑖 =𝑗 𝑧𝑥
𝑖𝑗 +𝑧𝑥
𝑗𝑖 +𝑧𝑦
𝑖𝑗 +𝑧𝑦
𝑗𝑖 ≥1. (40)
Cons ain (41) ensu es ha he o al wai ing ime o he
schedule does no exceed he maximum o al wai ing ime
(𝑊𝐹):∑
𝑖∈𝑉𝑤𝑖≤𝑊𝐹.(41)
Cons ain s (42)–(44) assign he ime be ween he depa u e
ime o essel 𝑖and he moo ing ime o essel 𝑗.Fo hose
essels 𝑗so ha 𝑧𝑡
𝑖𝑗 =1, heya eassigned𝑀as a alue
ep esen ing an unbounded ime o he obus ness:
∀𝑖,𝑗∈𝑉
𝑖 =𝑗 𝑧𝑡
𝑖𝑗 =𝑧𝑥
𝑖𝑗 +𝑧𝑥
𝑗𝑖 +𝑧𝑦
𝑖𝑗,(42)
∀𝑖,𝑗∈𝑉
𝑖 =𝑗∧(𝑧𝑡
𝑖𝑗=0∨𝑧𝑡
𝑖𝑗=2) 𝜏𝑖𝑗 =𝑀, (43)
∀𝑖,𝑗∈𝑉
𝑖 =𝑗∧𝑧𝑡
𝑖𝑗=1 𝑑𝑖+𝜏𝑖𝑗 =𝑚𝑗+𝑀(1−𝑧𝑦
𝑖𝑗). (44)
Cons ain s (45) and (46) se he alue o he a ailable bu e
ime a e essel 𝑖and i s obus ness alue, espec i ely:
∀𝑖∈𝑉 𝑏𝑖=min (min
𝑗∈𝑉
𝑖 =𝑗 (𝜏𝑖𝑗),ℎ∗
𝑖),(45)
∀𝑖∈𝑉 𝑟𝑖ℎ∗
𝑖=𝑏𝑖.(46)
The decision a iable 𝑧𝑡
𝑖𝑗 (see cons ain (47)) indica es i a
essel 𝑗moo s la e han 𝑖and, a he same ime, he essel 𝑗
in e sec s wi h he be h leng h occupied by essel 𝑖(𝑧𝑡
𝑖𝑗):
∀𝑖,𝑗∈𝑉
𝑖 =𝑗 0≤𝑧𝑡
𝑖𝑗 ≤2 (47)
6. Mul iobjec i e Gene ic
Algo i hms: MOGA + SA
Commonly app oxima ions o he Pa e o op imal se s o a
mul iobjec i e op imiza ion p oblem a e ob ained by means
o mul iobjec i e e olu iona y algo i hms [31]. Fu he mo e,
nowadays, me aheu is ics a e usually hyb idized wi h o he
echniques o algo i hms in o de o enhance hei e ec i e-
ness and pe o mance [23,24]. One o he mos common
o ms o hyb id gene ic algo i hm in ol es inco po a ing
local sea ch o a canonical gene ic algo i hm. Gene ic algo-
i hm is used o pe o m global explo a ion among he pop-
ula ion, and local sea ch is used o pe o m local exploi a ion
a ound he ch omosomes. Because o he complemen a y
p ope ies o gene ic algo i hms and local sea ch me hods,
he hyb id app oach o en ou pe o ms ei he me hods ope -
a ing alone [32].
The eby, a hyb id mul iobjec i e gene ic algo i hm
(MOGA) has been implemen ed in his pape . The NSGAII
Ma hema ical P oblems in Enginee ing 9
Vessel iden i ie
(i)
Numbe o c anes
(qi)
Bu e size
(bi)
Figu e 6: S uc u e o one gene o a ch omosome.
schema has been ex ended wi h a mul iobjec i e local sea ch
based on he mul iobjec i e simula ed annealing p oposed
by Bandyopadhyay e al. [33] (AMOSA), he eina e named
as MOGA + SA (see Algo i hm 1). Mo eo e , wo di e en
schemes om he li e a u e ha e been assessed NSGAII [26]
and SPEA2+ [27].
The same ch omosome s uc u e is used in hese h ee
MOGAs. This ch omosome has as many genes as incoming
essels (|𝑉|). Each gene consis s o h ee alues (see Figu e 6):
(1) he ID o he nex essel o dispa ch (𝑖); (2) he numbe o
QCs assigned (𝑞𝑖); (3) he bu e size a e his essel (𝑏𝑖).
I shouldbeno ed ha eachgenemus becomposedo
easible alueswi h espec o essel𝑖. Tha is, acco ding o
he p oblem cons ain s, each essel 𝑖canbeassigneda mos
QC+
𝑖c anes. The e o e, 1≤𝑞𝑖≤QC+
𝑖. Likewise, i he be h
leng h is 𝐿, hen𝜂𝑖≤𝑝𝑖≤𝐿−𝑙𝑖−𝜂𝑖.
In he ollowing subsec ions, gene ic ope a o s ha a e
used by he implemen a ions o NSGAII and SPEA2+ a e
desc ibed.
6.1. Decoding and E alua ion o One Ch omosome/Solu ion.
The s uc u e o he ch omosome, speci ically he o de o
he essels, is used as a dispa ching ule. Hence, we use he
ollowing decoding algo i hm: he genes a e isi ed om le
o igh in he ch omosome sequence. Fo each gene (𝑖,𝑞𝑖,and
𝑏𝑖), he essel 𝑖is scheduled a he ea lies moo ing ime wi h
𝑞𝑖consecu i e QCs a ailable, so ha none o he cons ain s
a e iola ed. In case he e a e se e al posi ions a ailable a he
ea lies moo ing ime, he one closes o he be h ex emes is
selec ed. A e he depa u e o he essel 𝑖(𝑑𝑖), i is ensu ed
ha he e a e 𝑏𝑖 ime uni s whe e no o he essel 𝑗(∀𝑗∈𝑉,𝑗 =
𝑖) uses he QCs assigned o essel 𝑖no moo s whe e essel 𝑖
does [𝑝𝑖,𝑝𝑖+𝑙𝑖).
Once a alid moo ing ime (𝑚𝑖) and ini ial posi ion
(𝑝𝑖)ha ebeenassigned oeach essel𝑖, he i nesso
he ch omosome (equa ion (16))isob ainedbycompu ing
each one o he objec i e unc ions: o al se ice ime (
𝑇𝑠),
obus ness (
𝑅),ands anda dde ia iono he obus ness(𝜎).
6.2. Gene a ion o Ini ial Popula ion. Cons uc ion o ini ial
popula ion (𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛p ocedu e) is pe -
o med so ha he se ice ime o a pe cen age o he ini ial
popula ion (GA pa ame e ) is a leas as good as he solu ion
p o ided by he FCFS policy. The o he ch omosomes (o
solu ions) a e cons uc ed by ins an ia ing each gene in he
ollowing way.
(i) Vessel iden i ie (𝑖): an in ege , be ween 1 and 𝑁,is
andomly chosen. Two genes o he same ch omo-
some canno ha e he same essel iden i ie .
(ii) Numbe o QCs (𝑞𝑖): an in ege , be ween 1 and QC+
𝑖,
is andomly chosen.
(iii) Bu e size (𝑏𝑖): heini ialbu e sizeis0 o allgenes
o he ini ial popula ion.
Once all ch omosomes in he ini ial popula ion ha e been
ins an ia ed, hei i ness alues a e ob ained as desc ibed
in Sec ion 6.1. Fu he mo e, he Pa e o on Xis upda ed
conside ing all hese ch omosomes. Le 𝑥be a ch omosome
(o solu ion); 𝑥is added o he Pa e o on Xi he e is no
o he solu ion 𝑦∈Xsuch ha 𝑦domina es 𝑥.I 𝑥is added
o X, hen all solu ions domina ed by 𝑥a e emo ed om X.
6.3. E olu ion o One Popula ion. In each i e a ion o he
MOGA, a new popula ion is buil om he p e ious one
(o he ini ial) by applying he gene ic ope a o s o selec ion,
ep oduc ion, and eplacemen . The p oposed app oach ol-
lows he scheme:
(i) selec ion: all ch omosomes in he ac ual popula ion
a e andomly g ouped in o pai s;
(ii) ep oduc ion: (1)each one o hese pai s is ma ed
o no acco ding o he c osso e p obabili y 𝑃𝑐
gene a ing wo o sp ing; (2)each o sp ing, o pa en
i he pa en s we e no ma ed, unde goes mu a ion in
acco dance wi h he mu a ion p obabili y 𝑃𝑚;
(iii) eplacemen : a e e alua ing he ch omosomes p e-
iously gene a ed, a ou namen selec ion (4 : 2) is
ca ied ou among each pai o pa en s and hei o -
sp ing as a eplacemen .
6.4. C osso e . Thec osso e ope a o ecei esonepai o
ch omosomes (𝑃1and 𝑃2), which a e in he cu en popu-
la ion 𝑝𝑜𝑝andha ebeen andomlyselec ed.Theobjec i e
o his ope a o is o cons uc wo o sp ing ch omosomes
(𝑂1and 𝑂2). Fo ha , each ime he c osso e ope a ion is
pe o med, he ollowing s eps a e made.
(1) Two c oss poin s a e andomly chosen, 𝑘1and 𝑘2(1≤
𝑘1<𝑘2≤𝑁).
(2) Each gene in ch omosomes 𝑃1and 𝑃2which is in
posi ion 𝑝,𝑘1≤𝑝<𝑘2,iscopied o hesameposi ion
in ch omosomes 𝑂1and 𝑂2, espec i ely.
(3) Each gene in ch omosomes 𝑃1and 𝑃2which is in
posi ion 𝑝,1≤𝑝<𝑘1,iscopied o hesameposi ion
in ch omosomes 𝑂1and 𝑂2, espec i ely.
(4) Each gene in ch omosomes 𝑃1and 𝑃2which is in
posi ion 𝑝,𝑘2≤𝑝≤𝑁,iscopied o hesameposi ion
in ch omosomes 𝑂1and 𝑂2, espec i ely.
Figu e 7is a g aphical ep esen a ion o he p ocedu e
ha is used o pe o m he c osso e ope a ion, which is
basedon he echniquegene alizedposi ionc osso e [34]
ha is commonly used in pe mu a ion based encodings.
In one ch omosome he e canno be wo genes wi h he
same essel iden i ie . The e o e, i he essel iden i ie in he
gene ha will be copied al eady exis s in he o sp ing (𝑂1/𝑂2)
16 Ma hema ical P oblems in Enginee ing
Table 4: Pe cen ages o incidences abso bed in schedules o 100
essels.
(a) Delays applied o schedules wi h di e en le els o 
𝑅
Range 𝑅min 𝑅𝑖𝑅Max
𝑑∈[1,0.2ℎ𝑖]21.78 95.51 99.95
𝑑∈[1,0.5ℎ𝑖]18.73 93.58 99.85
𝑑∈[1,0.8ℎ𝑖]16.64 90.49 98.96
𝑑∈[1,1.0ℎ𝑖]13.92 87.10 98.01
𝑑∈[1,1.2ℎ𝑖]12.93 85.31 97.15
(b) Delays applied o schedules wi h di e en le els o 
𝑇𝑠
Range 𝑇𝑠min 𝑇𝑠𝑖𝑇𝑠Max
𝑑∈[1,0.2ℎ𝑖]20.17 79.91 99.94
𝑑∈[1,0.5ℎ𝑖]16.16 73.88 99.75
𝑑∈[1,0.8ℎ𝑖]13.89 65.17 98.94
𝑑∈[1,1.0ℎ𝑖]11.85 60.25 98.08
𝑑∈[1,1.2ℎ𝑖]10.32 56.76 96.74
(c) Delays applied o schedules wi h di e en le els o 𝜎(
𝑅)
Range 𝜎(𝑅)min 𝜎(𝑅)𝑖𝜎(𝑅)Max
𝑑∈[1,0.2ℎ𝑖]99.96 75.48 61.85
𝑑∈[1,0.5ℎ𝑖]99.95 72.23 54.42
𝑑∈[1,0.8ℎ𝑖]99.34 67.88 48.72
𝑑∈[1,1.0ℎ𝑖]98.87 65.34 47.77
𝑑∈[1,1.2ℎ𝑖]97.39 62.38 45.75
Table 5: Pe cen ages o incidences abso bed in schedules o 100
essels ob ained using o no LS ( imeou 30 secs).
Range 𝑅Max no LS 𝑅Max wi h LS
𝑑∈[1,0.2ℎ𝑖]99.53 99.88
𝑑∈[1,0.5ℎ𝑖]99.40 99.70
𝑑∈[1,0.8ℎ𝑖]97.62 98.51
𝑑∈[1,1.0ℎ𝑖]95.33 97.34
𝑑∈[1,1.2ℎ𝑖]94.04 96.00
and hus he hi d objec i e managed in his way is o mini-
mize he s anda d de ia ion o he obus ness measu emen s.
In his pape , a mixed in ege p og amming (MIP) model
and a new hyb id mul iobjec i e gene ic algo i hm (MOGA +
SA) we e de eloped o he dynamic and con inuous obus
BAP + QCAP. They we e compa ed wi h wo well-known
mul iobjec i e gene ic algo i hms (MOGAs): NSGAII and
SPEA2+. In mul iobjec i e op imiza ion p oblems he e is
no a unique op imal solu ion, and i is necessa y o assess
he ade-o be ween all he objec i es by using he Pa e o
on . Visualizing Pa e o on s p o ides con aine e minals
ope a o s wi h a help ul sys em o decide which schedule is
be e depending on he ac ual s a e o he con aine e minal.
The esul s showed ha he MIP model was able o
ob ain obus and e icien schedules up o 10 incoming
essels. Howe e , MOGA + SA achie ed be e Pa e o on s
han he MIP model o queues o incoming essels g ea e
han o equal o 20 essels. The eby, he schedules ob ained
by MOGA + SA we e mo e e icien and obus han he
schedules ob ained by he MIP model. Fu he mo e, he
MIP model was unable o ind any solu ion wi h a gi en
imeou o a queue o 50 incoming essels. Addi ionally,
di e ences be ween he MOGAs ha e been assessed by
means o nonpa ame ic s a is ical es s. I u ned ou o be
ha MOGA + SA ob ained be e Pa e o on s acco ding
o he hype olume measu es. Fu he mo e, di e en se s o
incidences we e simula ed in o he schedules ob ained by he
NSGAII and he MOGA + SA. The esul s e u ned ha he
schedules ob ained by MOGA+SA we e mo e obus due o
he ac ha hey could abso b mo e incidences.
Con lic o In e es s
The au ho s decla e ha he e is no con lic o in e es s
ega ding he publica ion o his pape .
Acknowledgmen s
This wo k has been pa ially suppo ed by by he Spanish
Go e nmen unde esea ch p ojec MINECO TIN2013-
46511-C2-1-P, he p ojec PIRSES-GA-2011-294931 (FP7-
PEOPLE-2011-IRSES), and he p edoc o al FPU ellowship
(AP2010-4405).
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