Speed optimization and bunkering in liner shipping in the presence of uncertain service times and time windows at ports

warwick.ac.uk/lib-publications

Aydin, Nursen, Habin, Lee and Mansouri, Afshin. (2017) Speed optimization and bunkering in

liner shipping in the presence of uncertain service times and time windows at ports.

European Journal of Operational Research, 259 (1). pp. 143-154.

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ContentslistsavailableatScienceDirect

European

Journal

of

Operational

Research

journalhomepage:www.elsevier.com/locate/ejor

Production,

Manufacturing

and

Logistics

Speed

optimization

and

bunkering

in

liner

shipping

in

the

presence

of

uncertain

service

times

and

time

windows

at

ports

N.

Aydin

a

_,

_H.

_Lee

b,∗

_,

_S.

_A.

_Mansouri

b

a_{WarwickBusinessSchool,UniversityofWarwick,Coventry,UnitedKingdom} b_{BrunelBusinessSchool,BrunelUniversityLondon,Uxbridge,UnitedKingdom}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received31August2015 Accepted2October2016 Availableonline8October2016

Keywords: Transportation ORinmaritimeindustry Fuelcost

Dynamicprograming

a

b

s

t

r

a

c

t

Recentstudiesinmaritimeshippinghaveconcentratedonenvironmentalandeconomicimpactsofships. Inthisregard,fuelisconsideredasoneoftheimportantfactorsforsuchimpacts.Inparticular,thesailing speedofthevesselsaffectsthefuelconsumptiondirectly.Inthisstudy,weconsideraspeedoptimization probleminlinershipping,whichischaracterizedbystochasticporttimesandtimewindows.The objec-tiveistominimizethetotalfuelconsumptionwhilemaintainingtheschedulereliability.Wedevelopa dynamicprogramingmodelbydiscretizingtheportarrivaltimestoprovideapproximatesolutions.A de-terministicmodelispresentedtoprovidealowerboundontheoptimalexpectedcostofthedynamic model.Wealsoworkontheeffectofbunkerpricesonthelinerserviceschedule.Weproposeadynamic programingmodel forbunkeringproblem.Ournumerical studyusingrealdatafromaEuropeanliner shippingcompanyindicatesthatthespeedpolicyobtainedbyproposeddynamicmodelperforms signif-icantlybetterthantheonesobtainedbybenchmarkmethods.Moreover,ourresultsshowthatmaking speeddecisionsconsideringtheuncertaintyofporttimeswillnoticeablydecreasefuelconsumptioncost.

© 2016TheAuthors.PublishedbyElsevierB.V. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

1. Introduction

Sailing speed is an important decision variable affecting fuel consumption and consequently, greenhouse gas emission. Many operational strategies in container shipping have focused on re-ducingfuelcosts(Christiansen,Fagerholt,Nygreen,&Ronen,2013). For instance, Maersk, one of the major liner shipping company, introduced slow steaming strategy in 2008 to cope withthe in-creasingbunker price. Althoughsailingwiththe slowestspeedis favorablewithrespect tothe fuelcost andgreenhouse gas emis-sions,itmaynotbealwaysfeasibleduetotheuncertaintiesinsea transport legsand the ports that may affect servicelevel agree-mentwithcustomers.Sailingspeeddecisionatseatransportlegs mainly depends on the port time windows and the transit time betweenports.Therefore,itisnaturaltoexpectthatthespeed de-cisionshouldbemadebyconsideringboththetimewindowsand theuncertainportservicetimes.

Stochastic port service time and travel time play important rolesonthetotalcruisetime ofvessels.Portandtravel timescan

∗ _{Correspondingauthor.}

E-mailaddresses:nursen.aydin@wbs.ac.uk(N.Aydin),habin.lee@brunel.ac.uk

(H.Lee),afshin.mansouri@brunel.ac.uk(S.A.Mansouri).

behighlyvariableduetocongestion,handlingandweather condi-tions (Notteboom, 2006). Congestion ordisruption ina port may resultindeviation fromthe plannedschedule andhence,it may causedelays at thefollowing ports along the route. Vernimmen, Dullaert,andEngelen(2007)statethatonlyabout52%ofthe ves-sels dispatched for liner services arrived at the planned sched-ule. Drewry (2016) presents a detailedreliability report by trade andreportsthattheaveragepercentageofon-timedeliveryranges from55% to89%.Notteboom(2006)reportsthat 93.6%ofthe de-lays are caused by disruptions in port and terminal operations. citeLee15statethat containerhandlingcapacityatportsbecomes insufficientforthegrowthinthecontainertransportdemandand the variability in port times is a vital problem for liner opera-tors.Toprevent thedelaysandmaintainschedulereliability, ves-selsmay increase their speeds, which in turnincreases the total fuelconsumptionandgreenhousegasemission.Insomecaseslike unforeseendelayinaport,vesselsmaynotreachthenextporton time even if they sail atmaximum speed. To avoid poorservice levelandmeettheplannedscheduleontime,uncertainporttimes should be consideredin speed decision. Despitethisexpectation, numerous studies in the literature do not consider the uncer-taintiesinlinershipping.PsaraftisandKontovas(2013) providea detailedreview of speed models in maritime transportation and reveal that only few studies consider the uncertainties in liner

http://dx.doi.org/10.1016/j.ejor.2016.10.002

Fig.1. Bunkerfuelprices(IFO380)atvariousports(source:bunkerindex.com).

shipping.In thispaper, we focus onspeed optimization problem withstochastic port times. We develop newmodels to optimize sailingspeedforlinershipping.

Anotherimportantfactorthataffectsshipliners’operatingcost isbunkerprice.Sincefuelcostconstitutesamajorportionof oper-ationalcost ofacontainership,shippingcompaniesfocuson effi-cient waysto reduce bunker cost (Ronen, 2011). Through private conversation with a major liner company in the Mediterranean, itwasrealized that lineroperators canmake long termor short termcontractswiththefuelsuppliers.Long termcontractis gen-erallyset fora year andduring this period, fuel price has been fixedforthecontractedamount.Lineroperatorswhowanttoavoid the risks of fluctuating bunker price, prefer long term contracts. Ontheother hand,short termcontractsare madeapproximately tendaysbeforethelinerservicestarts. Thecontract specifiesthe bunkeringports,priceandquantity.Shorttermcontractsare pre-ferredinshort-haulrouteslikethoseintheMediterranean.Fig.1 showsthe monthlyaverage fuel price atseveral bunkering ports inthe Mediterraneanand theBlack Sea area from2010 to 2016. As it isseen in this figure,bunker prices at differentports have significantdifferences.Forinstance,inFebruary2015,theaverage monthlypricesin Novorossiysk andAlgiers were184 dollars and 489dollarspermetricton.Bunkeringportssignificantlyaffectthe totalfuelcostofalinerservice.Therefore,bunkeringportselection isan important decision. Inthis paper,we also consider bunker-ing port selection strategy. In particular, we focus on short term bunkering contracts and study the decisions regarding where to bunkerand how much to bunker?Bunkeringdecision is directly relatedtofuel consumption andaffectsthe sailingspeed. For in-stance,toavoidhighfuelcostslinercompaniespreferslow steam-ingstrategy(Maloni,Paul,&Gligor,2013).Therefore,sailingspeed andbunkering decisions are interrelated.In thispaper, we study thejointspeedandbunkeringproblem.

We make the following research contributions in this paper. First,we presenta new dynamicprograming formulationforthe fuelconsumption problemin linershippingthat can handle ran-domporttimes.Ourmodeltakesintoaccountfuturepossibleport servicetimesin assessing the currentspeed decision. The objec-tiveisto optimizesailingspeedalong theroute tominimizefuel consumptioncostandthecostofdelaysbyconsidering porttime windows. We use the term “time window” to describe the re-servedtimeperiodthataportallocatestoservethevessel.Inour study, we assume that ports report the available time windows

and then, the vessel determines the arrival times according to thesegivenwindows(Meng,Wang,Andersson,&Thun,2014).The literatureaddressingtheuncertaintiesatportsislimited. The pro-posed studies generally utilize heuristical approaches due to the difficulty in solving thestochastic problem andtheseapproaches donot guarantee theglobaloptimality unlessthe objective func-tionhasaspecialstructure.Second,weexaminethepropertiesof theproposeddynamicmodelandprovidesometheoreticalresults regardingtheoptimalspeeddecisionandportarrivaltime. Third, we work on the deterministic model formulation and we show that thismodelprovides a lowerbound on theoptimalexpected cost. Consequently, by using the discretized dynamic model and deterministic model,we can provide an upperbound onthe op-timalitygap.Fourth,wedevelopanewmodelforbunkering prob-lem.Inparticular,weworkonwheretobunkerandhowmuchto bunker decisions inshort-term contracts. Different than the pro-posedbunkeringstudiesintheliterature,weallowlinervesselto bunkerattheportswhicharenotontheplannedrouteschedule. Finally,we performa computationalstudyby usingrealshipping datafromalinercompany.Ourexperimentsaremotivatedbythe factthatthelinershippingindustryisinterestedinwhat-if scenar-ios,andthisobservationleadsustoevaluatetheimpactsof prob-lem parameters onspeed and bunkering decision, fuel consump-tionandservicelevel.

Therestofthe paperis organizedasfollows.InSection 2,we provideanoverviewoftherelatedliterature.InSection3,we de-velopadynamicprogramingformulationofthespeedoptimization problemforlinershipping.Section 4presentsadeterministic ap-proximation to the dynamic model that provides a lower bound on the optimal total expected cost. In Section 5, we extend the dynamicprograming formulationby considering bunkering prob-lem. In Section 6, we presentour computational study. We con-cludeanddiscusssomefutureresearchdirectionsinSection7.

2. Reviewofrelatedliterature

Ronen (2011) points out the importance of reducing vessels’ speed on operating cost. He addresses the speed optimization problem by considering the service frequency and the required number of vessels. Fagerholt, Laporte, and Norstad (2010) and Hvattum,Norstad,Fagerholt,andLaporte(2013)workonthespeed optimization problemin liner shipping withport time windows. Theyrestrictvesseltoarrivewithinthetimewindowofeachport toachieve100%servicelevel.Fagerholtetal.(2010)discretizethe arrival times andsolve the problemby usingshortest path algo-rithm. Hvattumetal. (2013) develop an exactsolution algorithm for the deterministic problem. Wang and Meng (2012c) work on thespeedoptimizationproblemwithtransshipmentandcontainer routing. They formulate the problem as a mixed-integer nonlin-ear model and propose outer-approximation algorithm to obtain approximate solutions. Norstad, Fagerholt andLaporte (2011) in-corporate speed decisionin the trampship routing and schedul-ing problem and propose a local search method. They first de-velop a solution algorithm for speed optimization problem with fixed route. Then, they utilize this algorithm to generate an ini-tial solution for the proposed local search method. Zhang, Teo, and Wang(2014) extend the work ofFagerholt etal. (2010) and Norstad, Fagerholt and Laporte (2011), and study the optimality properties.

Later studies in this field focus on the schedule reliability in liner shipping. Mansouri, Lee, and Aluko (2015) provide a re-viewonmulti-objectivemodelsinmaritimeshippingandexamine the relation between servicelevel and fuel costs. Li, Qi, and Lee (2015)andBrouer,Dirksen,Pisinger,Plum,andVaaben(2013) an-alyze the delaysin planned arrival time due to the port disrup-tion and examinespeeding up, port omittingand port swapping options to catch up with the planned schedule. Both of these studies assume deterministic port service time. While Li et al. (2015) discuss singlevessel problem,Brouer et al.(2013) propose ageneralsolutionmethodformultiplevesselsonalinernetwork. Notteboom(2006) highlightstheeffectsofwaitingtimesand de-lays on schedule reliability and discusses the trade-off between schedulereliabilityandoperatingcosts.Healsopointsoutthatdue to thefastgrowthin thevolume ofseatransportation, port con-gestionhasbecomethemainreasonofportdelays.Duetotheport congestion,vesselsmayhavetowaitlonghoursforservice,which mayresultindelaysinthefollowingports.Incaseofsuchdelays, ship managersoflinershippingcompanies cankeep upwiththe planned scheduleby increasing vesselspeeds. However, thismay resultinhighfuelcost.Thisisthemaindilemmaexperienced by ship managers.According toSeaIntelgloballinerperformance re-port, although Maersk Line is one of the top reliable carriers in termsofontime performance, theiroperationalcosts are consid-erablyhighercomparedwiththelow-costcarriers(SeaIntel,2015). Portdelaysandresultanthighcostshavedirectedshipping compa-niestofindingwaysofanticipatingunexpecteddelays(Notteboom, 2006).

The maritime literature on fuel emission with uncertainty is scarce.WangandMeng(2012a)studytheshiprouteschedule de-sign problem by considering sea contingency and uncertain port times.Theydonotallowlatearrivalsandformulatetheproblemas nonlinear mixed integer stochastic programing model. Wang and Meng(2012b)workontherobustdesignoflinershippingschedule byallowinglatearrivals.Theypenalizethevesselifitarriveslater than thepublishedtime. Torecoverthedelays,theyforce vessels tosailathighspeedwheneveritisnecessary.Inotherwords,they decidevesselspeedwithoutconsideringthetrade-off betweenfuel anddelaycosts.Lee,Lee,andZhang(2015)workontheeffectsof slow steaming on servicelevel and fuel consumption by consid-eringuncertainporttimes.Theyconsider faststeaming and flexi-bleslowsteaming strategiestoexaminetherelationbetween de-layandfuelconsumptioncosts.Infaststeamingstrategy, avessel

sailsat itshighestspeed duringthe entirejourney.On theother hand,inflexibleslowsteamingstrategy,avesselusuallysailsatits lowestspeedanditcanswitchtothehighestspeedwhenthereis adelay.

QiandSong(2012)proposeavesselschedulingmodelby con-sidering uncertain port times. They formulate a liner shipping problemtofindtheoptimaltransittimebetweenportsand conse-quently,obtainthe plannedarrivalschedule formultiple voyages. Intheirproblemformulation,portsdonothavespecifictime win-dows forservice andhence, vessels are allowed to arrive atany time.Inotherwords,theyassumethatportsarealwaysreadyfor service.However,thisassumptionmaynotbeconsistentwiththe practice(Wang&Meng,2014).Toachievehighservicelevels dur-ing the journey, Qi and Song (2012) assume that ports penalize thevessel ifitarrives later than theplannedarrival time ofthat port.Inotherwords,theyaimtominimizethedeviationfromthe plannedschedule.Theyuselinearcostfunctiontopenalizethe de-lays and propose a stochastic approximation algorithm to tackle theproblem.

Li, Qi, andSong (2015) study thereal-time schedulerecovery problem in a liner shipping service. They propose a multi-stage stochasticmodel bydividingthe sealegsintomultiplesegments. Theirobjectiveis tofind optimaltravel time ateach segmentby considering the uncertainties in sailing times and at ports (due to port disruption). As in the work of Qi and Song (2012), they assume that ports do not haveany time windows for their con-tracted service. Instead, the vessel aims to arrive at the planned arrivaltime. Moreover, differentthanour work,they donot con-siderwaitingcost forearlyarrivals,whichis contradictoryto the currentpracticeinlinershipping.Tofacilitatetheimplementation ofthemodel,theyassume that plannedarrivaltime toeach seg-mentin a leg is predefined.Nonetheless, dueto the difficulty of solving this problem, they focus on uncertainties in sailing time andconsiderthecasewithonlyonedisruptionevent.Inour pro-posedwork,westudytheuncertaintiesatallportsbyconsidering bothearlyandlatearrivals.

The literature on bunkering management is limited. Recently, Ronen(2011)analyzestheeffectsofbunkerpricechangeonliners’ operationalcosts. Heworksonthe tradeoff between slow steam-ing and increasing the fleet size. Yao, Ng, and Lee (2012) take intoaccountthebunkerpricedifferenceacrossdifferentportsand propose a bunkering modelto findoptimal bunkering ports and bunkering amounts. However, in the proposed model they limit the number of bunkering times. They also optimize the sailing speedbetweenportsbyrestrictingthevesseltoarrivewithin pre-definedtime windows.The objectiveis to minimize bunker cost andrevenue lossdueto carryingbunker fuelweight. Sheng,Lee, andChew(2014)extendthework ofYao etal.(2012)by consid-eringthe uncertaintyinfuelconsumptionandbunkerprices.The proposedmodelaimstominimizethebunkercostandbunker in-ventory holding cost. Kim (2014) present a bunkering model to minimizethetotalcostofbunkerpurchasing,shiptime(total char-tering cost of the vessel andtime value of containers) and car-bon tax. In the model formulation, a liner vessel is assumed to arriveportsatanytime,withoutanyrestriction.Lagrangian heuris-ticisproposedtoobtainabunkeringstrategy.Ghosh,Lee,andNg (2015) focus onlong term bunker contracts. Byconsidering fluc-tuatingbunkerprices,they proposeadynamicprogramingmodel. Theyassume that vessel sails at constant speed duringthe voy-age,howeverthefuelconsumptionbetweenportsisuncertain.The objectiveistominimize totalbunkeringcost,penalty costof run-ningout offuelanddamagecost fornotfulfillingthecommitted amount.

timeuncertaintiesandport timewindowsforfixedlinershipping routehasnotbeenaddressedintheliterature.Ourstudyisclosely relatedto the work of Qi and Song (2012). We consider a liner schedule, which is repeated on regular basis (weekly, monthly etc.).DifferentfromQiandSong(2012),ourstudyfocusonspeed optimizationproblemwithtimewindowsonasinglevoyage.This enablesustoanalyzethestructure ofthedynamicmodeland in-vestigatethe properties ofoptimal speed decisions. Furthermore, the deterministic approximation that we use to obtain a lower boundon the optimalexpected cost, canbe usefulto assign the boundontheoptimalitygapforeachprobleminstance.Since sail-ingspeeddecisionisaffectedbythefuelprices,wealsostudythe relationbetweensailingspeed andbunkering.We focuson short termcontractsat the planninglevel of linershipping companies andpropose a dynamicprograming model to determine optimal vesselspeed,bunkeringportsandthebunkeringquantities. Differ-entthanthepreviousstudies,weallowalinervesseltobunkerat theportswhich are not on her plannedroute. In other words,a vesselmaydetour toselectacheaperbunkering port.Basedona realshippingdata,weanalyzetheeffectsofbunkerpricesonthe liner’sserviceschedule.

3. Dynamicspeedoptimizationmodel

Motivatedbytheshippingoperationsofamajorlinershipping company, we propose a dynamic speed optimization model. We consider a vessel which provides shipping services over a given sequence of ports-of-call denoted by set N=

{

0,1,...,n

}

. Port 0 showsthe startingnode ofthe network. We use leg i to denote the trip from port

(

i−1

)

to port i. The vessel can visit port i withinitstimewindow.Foralinershippingcompany,ontime de-liveryofthe cargois paramountasdelayed cargoresults inhigh costbycustomers.Therefore,itiscrucialtobe servedon time.If thevesselarrives earlierthantheallocatedtime,ithastowait.If thevesselmissesthereservedtime,thismaycausedeviationfrom the planned schedule.Service time of a vessel in each port is a stochastic variableand denoted by S_i forport i. Avessel has for allpracticalpurposes alower andan upperspeedlimit.We have todecidethespeedbetweenportsinordertominimize totalfuel consumptionandmaximizeservicelevel.We willusethe follow-ingnotationthroughoutthepaper.

N:setofports

Si:randomservicetimeinhouratportisuchthatli≤Si≤ui ta

i:arrivaltimeofvesselatporti td

i:departuretimeofvesselatporti [

α

_i,

β

_i]:timewindowatporti di:lengthoflegiinnauticalmile

ϕ

:fuelcostperhourduringwaitingandservicetimeataport

θ

i:delaypenaltyperhouratporti

rs:priceoffuelpertonconsumedduringsailing rp:priceoffuelpertonconsumedatports

v

i:averagespeedatlegi(nauticalmileperhour),whichis lim-itedby[

v

,

v

¯]

In the literature, quadratic function of sailing speed is gener-allyusedtocomputefuelconsumptionofavessel(Fagerholtetal., 2010).WeusetheequationproposedbyYaoetal.(2012)to calcu-latefuelconsumptionrate(tonsperday).Yaoetal.(2012)present an empirical modelto presentthe relation betweenbunker con-sumptionrateandthevesselspeedbyconsideringthesizeofthe vessels.Theproposedfuelconsumptionrateisgivenask1

v

3_i+k2, wherek1andk2areconstantsandtheirvaluesdependonthesize ofthevessel.Weassumethatthespeedofavesselisconstant be-tweentwo consecutiveports.We alsoassume that vessel’sspeed isnotaffectedby theweather conditionsduringsailing. Multipli-cationoffuelconsumptionrateby thetimerequiredtotravel the

distancebetweenportsyields the totalamount offuel consump-tion(intons).Then,thefuelconsumptionfunctionforlegiisgiven as:

g

(

v

i

)

=

(

di/24

v

i

)

k1

v

3i +k2. (1) Thefuelconsumptionfunctiong

(

v

i

)

isconvexandincreasingwith

v

ifor

v

i∈[

v

,

v

¯].Asanextension tothemodelproposedbyQiand Song(2012),wealsoconsiderthefuelconsumptionduringwaiting andservicetimeatports.Duetointernationalregulations,vessels arenot allowedto usehighsulfurfuel whileberthingatportsas opposedtoopen seacruise (EUR-Lex,2012).Low sulfurfuel con-sumed at portsis more expensive than the high sulfur fuel. We assumethat thevesselconsumesafixedamountoffuelperhour duringwaitingandservicetimeateachport.Let

κ

betheaverage amountoffuel(tons)consumedperhour.Then,

ϕ

=rp

κ

givesthe fuelcostperhour.Consequently,thetotalfuelconsumptioncostis givenby,

n

i=1

(

rsg

(

v

i

)

+

ϕ

(

tid−tia

))

.

AsinQiandSong(2012),wealsoconsidertheservicelevelin each port. Arriving later than thegiven time window will result indelayintheplannedschedule.Vesseldelayscanbeverycostly duetocargomisconnecting,reroutingthedelayedcargo,inventory handling cost and loss of customer goodwill (Qi & Song, 2012). To maximize the service level and avoid such delays, we penal-izethevesselforeachhourofthelateness.Inrealworld applica-tions,shippingcompaniesalsoquantifycostsresultingfrom sched-uledelays.Forinstance,Maerskpaysdifferentamountsofmoney to shippersdependingon thelength of thedelayin theplanned schedule(Lietal.,2015). Inthisstudy,we usea lineardelaycost function.However,ourmodelcanhandleanynondecreasingdelay costfunction.Thetotalcostisgivenby:

n

i=1

rsg

(

v

i

)

+ n

i=1

ϕ

(

td

i −tia

)

+ n

i=1

θ

i[tia−

β

i]+

where[ta

i −

β

i]+ =max

{

tia−

β

i,0

}

.Giventhespeeddecision

v

iand servicetimeS_iatport i,thestatesofthesystematthefollowing portsaredefinedbythefollowingsystemdynamicsequations:

t_ia=t_id₋₁+di/

v

i, (2)

td

i =max

{

tia,

α

i

}

+Si, i=1,...,n. (3) Sincewestartwithport0,weassumethatta

0=t0d=0.Weare in-terestedinminimizingthetotalexpectedcostoverafinitehorizon. Informulatingthisproblembyusingdynamicprograming,we di-vide the probleminto (n+1) stages and each stage corresponds to oneport. At eachstage, we haveto decidethe averagesailing speeduntilthenextport.Ourdecisiononspeedwilldesignatethe arrivaltime tothenextport.Sincesailingspeedandservicetime arebounded,the possiblearrivaltime isalsobounded.We uset¯i andt_itodenotetheseupperandlowerlimits,respectively.To cap-turethestateofthesystematstagei,weusetiasthearrivaltime atporti.Usingtiasthestatevariableatporti,theoptimalspeed policycanbefoundbycomputingthevaluefunctions{J_i(.):i_∈N} throughthefollowingequation:

Ji

(

ti

)

=E

min

vi∈ [v,v¯]

{

ϕ

(

Si+[

α

i−ti]+

)

+

θ

i[ti−

β

i]+

+rsg

(

v

i

)

+Ji+1

(

ti+1

)

}

(4)

foreveryt_i∈[t_i,t¯_i].Arrivaltimetothefollowingport(t_i+1)is com-putedbyusingrecursiveequationsgivenin(2)and(3).The bound-aryconditionforeverytn∈[tn,t¯n]isgivenby:

Sincethevesselisdeployedfromport0initially,J0(0)givesthe optimalexpectedtotalcostatthebeginningofthejourney,where 0representsthefact thatthearrival time toport0ist0=0.The firsttwocostfunctionsinEq.(4)correspondtotheporttimecost and delay penalty, respectively. These cost functions depend on thearrivaltime.Whileporttimecostfunctionisanon-increasing functionint_i,delaypenalty functionisanon-decreasingfunction. Given the realization of the service time Si, the arrival time to the next portis determinedby the speeddecision. Consequently, thestatetransitionanddecisionfunctionfortheoptimalpolicyis givenby:

(

v

i,ti+1

)

=argmin

vi∈ [v,v¯]

{

ϕ

(

Si+[

α

i−ti]+

)

+

θ

i[ti−

β

i]+

+rsg

(

v

i

)

+Ji+1

(

ti+1

)

|

Si

}

. (6) Since possible arrival times, sailing speeds and service times are continuous variables, we need to discretize time into a fi-nitenumberoftimepointstoimplementthedynamicprograming model.Thisprovidesanapproximationtothecontinuousdynamic model. Since the resulting model only considers the discretized time points,itsoptimalsolutionisfeasibleforthecontinuous dy-namicmodel.Consequently,theoptimalobjectivevalueofthe dis-crete dynamicmodelisgreater thanthecontinuousmodelandit givesanupperboundonthecontinuousmodel.Now,wewill pro-videsomeinformationabouttheoptimalsolution.

Remark 3.1. Given a realization of service times at all ports

{

S1,S2,...,Sn

}

, let

v

∗_i andt_i∗ fori ∈N be a feasible solution for speedbetweenports

(

i−1

)

andi,andcorrespondingarrivaltimes. Consider aportpwithtimewindow[

α

p,

β

p]andthefeasible so-lution

v

<

v

∗

p≤

v

¯ andt∗p<

α

p.Then, thereexists a bettersolution withsailingspeed

v

¯psuchthat

v

≤

v

¯p<

v

∗p.

Duetotheporttimewindow,avesselcannotberthassoonas itarrives totheport. Ithastowait untilport servicetime starts. Therefore,byconsidering thefuelpriceswecan saythat itisnot beneficialtoarriveataportearlierthanthetimewindowofthat port.

Lemma 3.1. Lett1

i andti2 denotethe twopossiblearrival timesfor port i suchthatt_i≤t1

i ≤ti2≤

α

i.For any givenservice time Si and arrivaltimest1

i andti2,let

v

i1∗and

v

2i∗ betheoptimalsailingspeeds atlegi,respectively.Then,wehave

v

1∗

i =

v

2i∗.

We defertheproof ofthelemma tothe supplementary docu-ment.Lemma3.1impliesthattheoptimalspeeddecisiondoesnot change when thevessel arrival time is ti ∈ [ t i,

α

i]. This obser-vation allowsustoreducethestatespaceofdynamicprograming modelateach stage.In otherwords,sailing speed

v

iatleg iwill be computed forthe arrival timesti∈[

α

i,t¯i]. The optimalsailing speed forthearrival timeperiodt_i∈[t_i,

α

_i)willbeequaltothe optimalsailingspeedwhenti=

α

i.Next,we presentsomeresults whenti∈[

α

i,t¯i].Foreveryti∈[

α

i,t¯i],theoptimalvaluefunctions oftherestrictedproblemaregivenas:

JR i

(

ti

)

=E

min

vi∈ [v,v¯]

{

ϕ

Si+

θ

i[ti−

β

i]+ +rsg

(

v

i

)

+JiR+1

(

ti+1

)

}

(7)

withtheboundarycondition,

JR

n

(

tn

)

=

ϕ

E[Sn]+

θ

i[tn−

β

n]+ . (8) Proposition3.1. JR

i

(

ti

)

isconvexandnondecreasingwithtiforevery i=0,...,n.

Theproofofthepropositionisgiveninthesupplementary doc-ument. Next, we present the relation between the sailing speed decisionandtheportarrivaltime.

Corollary3.1. Lett1

i andti2denotethetwopossiblearrivaltimesfor port i suchthat

α

i≤ti1≤ti2≤t¯i.Forany realization ofservice time

Si,let

v

1i∗and

v

i2∗betheoptimalsailingspeedsatlegigiventhatthe vessel arrivaltimesaret1

i andti2, respectively.Then, we have

v

1i∗≤

v

2∗ i .

The proof of the corollary is presented in the supplementary document.

4. Deterministicapproximation

An alternative solution method for the speed optimization problemdescribedin Section3 istosolve adeterministic model. A deterministic approximation of theabove dynamicprograming modelassumes thatall random quantities are knownin advance andthey take their expectedvalues.Hence, deterministicmodels donot capturethetemporaldynamicsoftheproblemasdynamic modelsdo(Talluri &vanRyzin,2005). However,they arepopular inpracticeduetotheircomputationalefficiency.Thedeterministic approximationisformulatedasfollows:

minimize n

i=1

rsg

(

v

i

)

+ n

i=1

ϕ

(

td

i −tia

)

+ n

i=1

θ

i[tia−

β

i]+ (9)

subjecttota i =t

d

i−1+di/

v

i, i=1,...,n, (10)

td

i =li+E[Si], i=1,...,n, (11)

li≥

α

i, i=1,...,n, (12)

li≥tia, i=1,...,n, (13)

v

≤

v

i≤

v

¯ i=1,...,n, (14) whereta

0=t0d=0.Constraints(10)and(11)correspondtothe sys-temdynamicsequations.Constraints(12)and(13)ensurethatthe vessel’sservice startswithinthe time windows atall ports. Con-straint(14) guaranteesthat the speed ofthe vesselis within the lowerandupperlimitsinalllegs.

Notethat constraint(10) is nonlinear.We define

τ

i to denote the transittime betweenports (i−1) and i and it isformulated as

τ

i=di/

v

i. Replacing di/

v

i by

τ

i and rewriting the constraints (10)and(14),we obtain a feasible region with linear constraints. Then,theobjectivefunctionin(9)isgivenas:

n

i=1

rsg

(

τ

i

)

+ n

i=1

ϕ

(

[

α

i−tia]+ +E[Si]

)

+ n

i=1

θ

i[tia−

β

i]+ . (15)

Notethatthefuelconsumption functiong(

τ

i)is convexin

τ

i.Let define h

(

ta

i

)

asthe waitingand penalty cost function for port i. Then,wehave:

h

(

ta

i

)

=

ϕ

(

[

α

i−tia]+ +E[Si]

)

+

θ

i[tia−

β

i]+ . (16) Itisclearthatthefunctionh

(

ta

i

)

isconvexintia.Consequently,the objectivefunctionin(15)isconvexin

(

τ

i,tia

)

foralli=1,...,n.

Lemma4.1. Deterministicmodel givenin (9)–(14) is a convex

pro-gram.Therefore,alocalminimumsolutionisalsotheglobalminimum solutionformodel(9)–(14).

Proposition 4.1. The optimal objective value of the deterministic modelgives a lower bound forthedynamic programingmodel, that is,Z_DM∗ ≤J0

(

0

)

.

As mentioned above, we approximate the dynamic model (4)and(5) by discretizing the time andthis approximation pro-videsan upperbound forthe continuousdynamicmodel. There-fore,thepercentagegapbetweentheobjectivefunction valuesof thedeterministicmodelandtheapproximatedynamicmodel pro-videsanupperboundfortheoptimalitygapofthecontinuous dy-namicmodel.

5. Bunkerportselection

In thissection, we studythe bunkering management strategy. Weconsiderthe planninglevelproblemforshorttermcontracts. Short termbunkering contracts are made approximately a week beforetheliner’splannedservice.Inrealpractice,priortothe ser-vice,lineroperatorsnegotiatewiththebunkersuppliersandmake bunkeringdecisions accordingto thequotes offered by suppliers. Thekeydecisionsinbunkeringproblemarewheretobunkerand howmuchtobunker?

In ourmodel formulation,we assume that thereare refueling stationsatseveralports-of-callalongthelinerservice.Inaddition, todecrease the bunkering cost a vessel may detour to the ports whicharenoton herserviceschedule.Bunkeringoperation takes approximately2–3hoursanditcanbe performedduringthe ser-vice (loading andunloading) at ports1_{. Therefore, it can be said}

that bunkering atone of theservice portsis more time efficient than detouringto another port. We use the same notationas in Section3toformulatethebunkeringproblem.Additionally,We de-fined_ij toshowthedistancebetweenportsiandj.Todenotethe bunkeringports,weintroducesetM.Thesetofallportsisgivenby N=N∪MandN_ipresentsthesetofportsthatarereachablefrom porti.Sincebunkeringisaplanninglevelproblem,weassumethat porttimetakesitsexpectedvalue.AsinSection3,weassumethat fuelconsumptionisdirectlyrelatedtothesailingspeedforwhich, we usethe empirical model of Yao etal. (2012) to compute the fuelconsumption rate. As mentioned in Section 3,vessels gener-allyconsumetwo types offuel ina voyage: low andhighsulfur fuel.Sincehighsulfurfuelismostlyusedwhilecruising,wefocus on decisions relatedto this type of fuel, which has further sim-plified theproblem. Weassume that vessel hasenough low sul-furfuelinventoryforavoyage.Theaimofthebunkeringmodelis tofindtheoptimalbunkeringports,bunkeringquantitiesand sail-ingspeedinordertominimizethetotalbunkeringanddelaycost. Theoptimalsolution ofthismodelcan be usedto decideon the bunkeringports,averagebunkering quantityrequiredtocomplete theserviceandtheaveragesailingspeed.

To include varying fuel prices, we extend the dynamic pro-gramingmodelgivenin(4)and(5).Sincebunkeringdecision de-pends on the remaining bunker on the vessel, we need to store thebunkerinventory.Wedefinewitodenotethebunkerinventory whenvesselarrivesatporti.Thestatespaceforportiisgivenby

(

i_,t_i,w_i

)

. Ateach stage, wehave todecide whethertodetour for bunkering(ifpossible),thesailingspeedtothenextportandhow muchtobunkeratthecurrentport(ifpossible)?Weuserito de-notethebunkerpricepertonatportiandx_itodenotethebunker order-up-to-levelatporti.Ifvesseldoesnotbunkeratporti,then xi=wi.LetV

(

i,ti,wi

)

denotetheminimumcostfromporti upto portngiventhatthearrivaltimetoportiist_iandtheremaining

1_{Privateconversationwithalineroperator,2016.}

bunkeriswi.Then,wehavethefollowingDPrecursion,

V

(

i,ti,wi

)

= min

vi j∈ [v,v¯] xi≤C,j∈ Ni

ϕ

(

E[Si]+[

α

i−ti]+

)

+ri

(

xi−wi

)

+

θ

i[ti−

β

i]+ +V

(

j,tj,wj

)

(17)

where tj=max

{

α

i,ti

}

+E[Si]+di j/

v

i j and wj=xi−g

(

v

i j

)

. The boundaryconditionisgivenby:

V

(

n,tn,wn

)

=

ϕ

(

E[Sn]+[

α

n−tn]+

)

+

θ

i[tn−

β

n]+ . (18) Now,wewillprovidesomestructuralresultsforthebunkering decision.ThecostfunctionV

(

i,ti,wi

)

isdecreasinginwiforevery tiandi.Inotherwords,astheinitialbunkerinventoryatporti in-creases,thetotalcostfromportitoportndecreases.Sincecarried bunkerdoesnotaffectthetotalcost,increaseintheinitialbunker inventorydoesnotincreasethetotalcost.

Lemma5.1. Letassumeavessel visitsportsi andjconsecutively in

theoptimalscheduleandthebunkerprices(dollarsperton)atports i andjaredenotedbyri andrj andri> rj.Let

v

∗i j denote the opti-malvalueofthesailingspeedatlegi−j,andx∗_i denotetheoptimal bunkerup-to-levelatport iwhen theinitialbunkerinventory iswi. Then,x∗_i=g

(

v

∗

i j

)

,ifwi<g

(

v

∗i j

)

andxi∗=wi,ifwi≥g

(

v

∗i j

)

.

The proof ofthe lemma is given in the supplementary docu-ment.Lemma5.1impliesthatwe canreducethesearchspacefor thebunkeringdecisionvariableinspecialcases.

6. Computationalresults

Wedevotethissectiontoacomputationalstudyfordiscussing different aspects of our proposed solution methods. We conduct experimentsbyusingrealdatafromamajorEuropeanliner ship-pingcompany.Inparticular,weevaluateperformancesofdynamic programing anddeterministic models andinvestigate managerial insightsobtainedfromthesemodels.Wenextexplain our simula-tionsetupindetailandthenpresentournumericalresults.

6.1. Experimentalsettings

Wedesignexperiments byusingrealdatafromthecase ship-ping companyincludingthree routeswith16,11and8ports, re-spectively.Thesedata includethedistancesbetweenports, vessel arrival and berthing times, and port service times. The detailed schedulesoftheseroutesarepresentedinTables1–3.

Table1

Anexistingscheduleoftheshippingcompanywith16ports.

Port Distance Arrival Servicestart Departure Porttime(hours) Weightofport

P0 – – – 18/09/1523:00

P1 40 19/09/1502:00 19/09/1502:00 19/09/1513:00 11.0 7 P2 125 19/09/1522:30 19/09/1523:00 20/09/1508:00 9.0 2 P3 190 20/09/1521:30 20/09/1522:30 21/09/1515:30 17.0 9 P4 1149 25/09/1502:00 25/09/1503:30 25/09/1517:30 14.0 10 P5 249 26/09/1513:00 26/09/1514:00 27/09/1500:00 10.0 7 P6 182 27/09/1513:00 27/09/1513:00 28/09/1500:00 11.0 8 P7 97 28/09/1507:00 28/09/1508:30 28/09/1517:30 9.0 4 P8 40 28/09/1520:45 28/09/1522:00 29/09/1504:30 6.5 3 P9 50 29/09/1508:30 29/09/1509:30 29/09/1516:30 7.0 7 P10 112 30/09/1501:00 30/09/1501:00 30/09/1513:30 12.5 8 P11 136 01/10/1500:00 01/10/1501:00 01/10/1511:30 10.5 2 P12 966 04/10/1517:15 04/10/1518:00 05/10/1501:00 7.0 5 P13 416 06/10/1510:00 06/10/1511:30 07/10/1501:00 13.5 3 P14 190 07/10/1514:30 07/10/1515:00 08/10/1500:30 9.5 1 P15 125 08/10/1509:30 08/10/1510:00 08/10/1516:00 6.0 9

Table2

Anexistingscheduleoftheshippingcompanywith11ports.

Port Distance Arrival Servicestart Departure Porttime(hours) Weightofport

P0 – – – 22/01/1508:30

P1 116 22/01/1516:00 22/01/1517:05 23/01/1510:30 17.5 3 P2 410 24/01/1510:00 24/01/1511:30 24/01/1519:00 7.5 7 P3 35 24/01/1521:30 24/01/1521:30 25/01/1509:30 12.0 5 P4 1546 29/01/1507:00 29/01/1508:00 29/01/1519:00 11.0 9 P5 1681 03/02/1515:30 03/02/1518:30 04/02/1518:00 23.5 3 P6 537 06/02/1507:00 06/02/1507:00 07/02/1504:00 21.0 8 P7 142 07/02/1513:00 07/02/1514:00 07/02/1523:00 9.0 6 P8 617 09/02/1507:30 09/02/1507:30 09/02/1521:30 14.0 10 P9 143 10/02/1507:00 10/02/1507:30 11/02/1501:00 17.5 4 P10 180 11/02/1513:00 11/02/1513:30 11/02/1521:30 8.0 1

Table3

Anexistingscheduleoftheshippingcompanywith8ports.

Port Distance Arrival Servicestart Departure Porttime(hours) Weightofport

P0 – – 04/01/1505:30

P1 430 05/01/1506:30 05/01/1509:30 06/01/1510:00 24.5 4 P2 593 07/02/1521:00 07/01/1521:00 08/01/1500:30 3.5 3 P3 90 08/01/1505:30 08/01/1506:00 08/01/1519:00 13.0 6 P4 484 10/01/1505:00 10/01/1508:00 10/01/1517:30 9.5 5 P5 435 11/01/1520:30 11/01/1521:30 12/01/1509:00 11.5 10 P6 100 12/01/1515:30 12/01/1515:30 12/01/1519:30 4.0 2 P7 625 14/01/1514:00 14/01/1514:00 14/01/1523:00 9.0 7

Fig.2. Comparisonofactualandestimatedfuelconsumption.

fuel)ismoreexpensivethantheoneconsumedduringsailing (in-termediate sulfur fuel).We refer readersto Bunkerworld(2016)’s websiteforthe pricerelationship betweenvariousfuel types.In our experiments, fuel prices are set by considering the average prices charged to the liner shipping company. The price of fuel used in the ports and duringsailing are set to rp= 380 dollars pertonandrs=185dollarsper ton,respectively.Tomeasurethe impactofwaiting,weapplysensitivityanalysiswithrespecttothe waitingcost.Waitingcost perhourisdirectlyrelatedtotheprice offuel consumed at the ports. According to the information ob-tainedfromtheshippingcompany,avesselconsumes2.0tonsper dayfuelon averagewhilewaitingata port whichapproximately corresponds to 31 dollars per hour. Considering thisinformation, we apply sensitivityanalysiswithrespectto thewaitingcost se-lectedas

ϕ

_∈[30dollars,50dollars].

Table4

Upperboundontheoptimalitygap.

Instances DiscreteDM Deterministicmodel Relativegap

n δ ϕ UB LB (UB– LB)/LB(%)

8 50 30 51,328 50,779 1.08

50 53,247 52,699 1.04

100 30 51,548 50,779 1.52

50 53,468 52,699 1.46

11 50 30 100,579 100,000 0.58 50 103,998 103,427 0.55 100 30 101,807 100,303 1.50 50 105,228 103,723 1.45

16 50 30 73,834 72,402 1.98

50 77,807 76,372 1.88

100 30 74,687 72,405 3.15

50 78,661 76,375 2.99

isnot available, we generate these weights randomly between 1 and10;1beingthelowestand10beingthehighestpriority.The weightsoftheportsaregivenatthelastcolumninTables1and2. Let

κ

ibe theweightofporti.Then,thedelaypenalty ofporti is givenby

θ

i=

κ

i

δ

,where

δ

canbedefinedasthedelaycoefficient. Sincewedonothavelinerdataregardingdelaypenalty,weset

δ

to[50, 100]torepresentlow andhigh penalty values.Thisleads delaycosttochangebetween50dollarsand1000dollarsperhour (Qi&Song,2012).

Another important parameter that affects the speed policy is thetime window. Theschedule ofthevessel isplanned with re-spect to the available time slots of the ports. These time win-dows are defined according to the contractual agreements with customersandports. Totest theperformancesofourmodels,we definetwovaluesforthewidthoftime windowtw_∈[3,6] corre-spondingtotightandloosewindow.Inourexperiments,time win-dowofeach port is extractedfromthe realdata ofthe shipping company.The startingtime (

α

i) ofthetime windowissetto the servicestarttimeatportiandthelatestservicestarttimeis com-putedas

β

i=

α

i+tw.Lastly,weassume thatsailingspeedranges from12.5knotsto19.5knots(Yaoetal., 2012).Toimplement dy-namicprograming,wediscretizetimebyanintervalof5minutes. Wepresentdetailedresultson theeffectofdiscretizationin sup-plementarydocument.Forour experiments,we useda computer with1.8gigahertzIntelCorei5with4processorsand8gigabytes ofRAM.ThesolutionmethodsarecodedinMATLAB8.2.0andrun underWindows8.1operatingsystem.

6.2.Resultsanalysis

As discussed before, the optimal objective values of the de-terministicmodelandthediscretizeddynamicprograming model providelowerandupperboundsforthecontinuousmodel, respec-tively.Wecomparethepercentagegapsbetweentheselower and upperboundsinTable 4.WeuseUBandLBto denotetheupper andlowerboundsprovidedbythediscretizeddynamicmodeland thedeterministicmodel,respectively.Thefirstthreecolumns indi-catethecharacteristicsofthetestinstances.Thenexttwocolumns give the optimal objective values of dynamic and deterministic models.Thelastcolumnpresentsthepercentagegapbetweenthe objectivefunctionvaluesofthesetwo boundingmodels. Ouraim istomeasurethe effectsofdelayandwaitingcosts onthe tight-nessofthesebounds.Inthisexperiment,wesettherangebetween theupperandlowerboundsontheservicetimeto6hoursandto emphasizethe effectofthe delays,we set thewidthof thetime windowto3hours.

ThefirstobservationfromtheresultsinTable4isthatthe op-timality gapis less than 4.0%.The results show that the relative differences are mostly affected by the delay penalties. As delay

penalty increases,optimal objective value of the dynamic model increases significantly. In our problem setting, we assume that delays are resulted from the uncertainty of port times. In other words,morevariationinporttimesincreasestheprobabilityof de-lays.Dynamicmodelconsidersallpossiblerealizationsoftheport timesand hence,it is sensitive tocost parameters. On the other hand,deterministicmodelcomputestheoptimalsailingspeedsby only considering theexpectedport times. Therefore,it isslightly affected by the changes in delay penalty. Consequently, the per-centage gap tends to increase with the delay penalty. Regarding theimpactoftheproblemsize,wehavenotobservedany correla-tionbetweenthenumberofportsvisitedandthepercentagegap. Weconjecture thatthepercentagegapissignificantlyaffected by theproblemstructure.When weexamine thetestproblemswith 11and16ports,weseethatthedistancebetweenportsisshorter in16porttest.Since dynamicmodelconsiderstheuncertaintyin the problem, it can estimate the effects ofdelays inshorter dis-tancesbetterthanthedeterministicmodel.

Inthenext experiment, wetest theeffectofdelaypenalty on thespeeddecision.WeusetheshippingdatagiveninTable3and setthewidthofthetimewindowto3hoursandwaitingtimeto 30dollarsperhour.Toobservethecumulativeeffectofthedelay, wesettheservicetimerealizationsofportsP1andP2totheupper bound on their servicetime. The servicetimes of the remaining portsaresetto theirexpectedvalues.Therealizationofport ser-vicetimeforthisexperimentisgivenas{30.5,9.5,16,12.5,14.5,7, 12}forportsP1toP7,respectively.Inthisexperiment,weanalyze thereactionofthedynamicmodeltothislongporttime.Table5 presents the results of this experiment. The first column corre-spondto the delaypenalty parameter. The next columnspresent theresultingdelaytime(inhours)andsailingspeedsaccordingto theporttimerealization.Thefirstobservationwemadeisthatthe speeddecisionissignificantlyaffectedbythedelaypenalty.When delaypenaltyishigh,vesselisonlylateatportP3.Thisdelayis re-sultedfromthelongporttimeinP1andP2.Whendelaypenaltyis low,vesselarrivesatportsP3andP6laterthanthetimewindows. Moreover,we observethat sailingspeed decisiondependsonthe fuelconsumption cost andthepenalty paid forbeinglate. When delaypenaltyislow,sailingatslowspeedandarrivinglatetoports canbemorecosteffectivethansailingathighspeed.Thisbehavior canbeattributedtotheimpactofportweights.Asmentioned be-fore,each porthasdifferentdelaypenalty andthesepenaltiesare computedwithrespect tothe port weights.Weight ofport P6is 2whilethe maximumweightis 10andhence,thevesselprefers to arrivelater than the time windowinstead ofsailing athigher speed.Wecandeducefromthisresultthatinunexpectedevents, beinglatetosomeportscanbelesscostly.

6.3. Simulationresults

Table5

Analysisondelaypenaltyandwaitingcost.

Instances Delaytime(hours)/ Portindex

δ speed(knots) P1 P2 P3 P4 P5 P6 P7

50 Delay – – 3.83 – – 0.38 –

vi 15.35 19.33 19.29 17.23 17.23 17.66 16.00

100 Delay – – 3.58 – – – –

vi 15.35 19.50 19.29 17.34 17.40 16.66 15.82

Table6

Simulationresultsover250runs.

Instances DDM DTM HM %gapwithDDM

n tw _{δ ϕ Mean Stddev Mean Stddev Mean Stddev DTM(%) HM(%)}

8 3 50 30 51,424 1260 52,072 1583 52,315 1528 1.26 1.73 50 50 53,356 1350 54,031 1650 54,246 1614 1.27 1.67 100 30 51,640 1382 53,005 2459 52,845 1995 2.64 2.33 100 50 53,572 1469 54,963 2521 54,776 2074 2.60 2.25 6 50 30 50,484 1079 51,088 1716 51,913 1403 1.20 2.83 50 50 52,416 1169 53,028 1791 53,845 1490 1.17 2.73 100 30 50,549 1094 52,198 3081 52,156 1715 3.26 3.17 100 50 52,481 1184 54,137 3151 54,088 1795 3.16 3.06 11 3 50 30 100,800 1799 102,666 2601 103,207 2182 1.85 2.39 50 50 104,228 1890 106,134 2663 106,636 2270 1.82 2.31 100 30 102,124 2698 105,374 4889 105,906 3866 3.18 3.70 100 50 105,553 2781 108,843 4947 109,335 3945 3.11 3.58 6 50 30 98,931 1312 101,000 2662 101,629 1734 2.09 2.73 50 50 102,361 1408 104,453 2732 105,058 1824 2.04 2.64 100 30 99,392 1531 103,184 4724 103,124 3015 3.81 3.75 100 50 102,821 1623 106,637 4790 106,552 3094 3.71 3.63 16 3 50 30 73,941 1856 77,085 4594 76,742 3712 4.25 3.79 50 50 77,916 1958 81,132 4662 80,713 3798 4.13 3.59 100 30 74,830 2915 81,386 8871 79,804 7010 8.76 6.65 100 50 78,806 3000 85,426 8991 83,775 7084 8.40 6.31 6 50 30 72,548 1204 74,992 3736 75,513 3113 3.37 4.09 50 50 76,521 1317 79,011 3808 79,484 3200 3.25 3.87 100 30 72,748 1471 77,989 7203 77,576 5826 7.20 6.64 100 50 76,722 1573 82,068 7306 81,547 5897 6.96 6.29

Table6presentsaveragecostsoverallsimulationrunsfor vary-ingfactors.Inthisexperiment,wesettherangebetweentheupper andlowerboundsontheservicetimeto6hours.Weaimto com-paretheperformancesofspeedoptimizationmethodswithrespect towidthoftimewindows,delay,waitingcostandsizeofthe prob-lem. Thetotal cost obtainedby DDM isused asa baseapproach to report therelative performancesofthe solutionmethods. Itis validated that thepercent gaps betweenthe totalexpected costs obtainedbyDDMandtheremainingsolutionsmethodsare statis-ticallysignificantat99%level.ThedetailsofANOVAandthe post-hocanalysesarepresentedinsupplementarydocument.

ComparingthepercentagegapsinTable6,weobservethatthe dynamicmodel constantly outperformsthe other solution strate-gies. Moreover, the performance gaps between DDM andthe re-maining solution methods are statistically significant across all problems instanceswith 8,11and16 ports, respectively(see Ta-blesB.6–B.8insupplementarydocument).WhenwelookintoDTM andHM,weobservethatperformanceofDTMisbetterforlow de-lay penalty. The performance of deterministic model deteriorates asthedelaypenaltyincreases.Thespeeddecisionofdeterministic modelisstaticanditdoesnotchangewiththerealizationofport time.Therefore,delayinoneportleadstodelaysinthesuccessive ports. Onthe other hand,heuristic method aims to avoiddelays andhence,itsspeedpolicyisadvantageousforhighdelaypenalty. As itisseen inTable6,DTM performsworse than HMinalmost allinstanceswhenthedelaypenaltyishigh.Wecautionthereader thattheperformancesofDTMandHMdeterioratewhenthe prob-lemsizeislarge.Aswementionedbefore,uncertaintyinthe prob-lemincreaseswiththenumberofports.Therefore,DDMperforms significantlybetterthantheothermethods.

Next,we apply sensitivityanalysis withrespect to the uncer-tainty in port times and analyze its effect on delay and sailing speed. In this experiment, we compare the speed policy of dy-namic anddeterministic modelwith respect to service time un-certainty. We set waiting cost to 30 dollars per hour and time window length to 3 hours. We test the models by setting the rangebetweenupperandlowerbound valuesofthe servicetime =ui−li as6and10.Fig.3 illustratesthe trendinaverage de-layandsailingspeed.Inthisfigure,thehorizontalaxescorrespond tothetestinstancesdenotedby(n, ,

δ

).Asdelaypenalty coeffi-cient

δ

increases,optimalpoliciesofthedynamicanddeterministic modelimposetosailathighspeedsothattheoveralldelaycostis minimized.WecanobservethisresultbycomparingFig.3(a)and (b).Theeffectofdelaypenaltyismoresignificantwhenthe varia-tionatportservicetimeishighandproblemsizeislarge.An inter-estingresultisthattheaveragesailingspeed ofthedeterministic modelisgenerallyhigherthantheoneindynamicmodel.Average sailing speedsof these models are closeras the variation in the servicetime decreases.Althoughthedeterministicmodelimposes ahighersailingspeedthanthedynamicmodel,thedelayresulted fromits speed policy is higherthan the delayresulted from the dynamicspeedpolicy.

6.4.Bunkeringproblemresults

Fig.3. Analysisondelayandsailingspeed.

Table7

Alternativebunkeringports.

Alternative Distances(nauticalmiles) Bunkeringports P0 P1 P4 P5

A1 225 100 – –

A2 – – 200 1600

Table 7. In this experiment, we assume that alternative bunker portsdonothaveanytimewindows.Wesettheaverage bunker-ingtimeto3hoursandthebunkerpricesinportB1andB2to150 dollarsand125dollars,respectively.Weassumethatbunkerprices atthescheduled portsgradually decrease andincrease alongthe serviceroute (see Table 8). In thisstudy, we look into details of theoptimalspeed andbunkering decisions.Inparticular, we ana-lyzetheeffectofporttime windows,delaycostandbunkerprice onthesedecisions.

We firststudytheeffectofporttimewindowsanddelaycost. Althoughdetour forbunkering is an effectiveway toreduce fuel cost,itmaydelaytheplannedarrivaltimeforthescheduledports. In this experiment, we analyze the bunkering decision with re-spect to different delay cost and time window parameters. We setthewaitingcost to30 dollarsper hour andinitialbunker in-ventory to 50 tons. Tables 8 and 9 show optimal bunkering and speeddecisionsfortightandwidetimewindows,respectively.As

seenfromthesetables,vesselgenerallydetourstoreducefuelcost. Whenthedelaypenaltyislow,detourforbunkeringismorecost effective even if it increases the total traveled distance. We can observe this result by comparing the total bunkering cost. It is important to note that the total bunkering amounts in low and high delaypenalty cases are different due to the additional dis-tancecovered duringdetour.Moreover, vessel increasesits speed followingthedetourto preventthedelays.Thisresultshowsthat although detour leadsto additional bunker cost, it maydecrease thetotalbunkeringcostoftheservice.BycomparingTables8and 9,weobservethatthetotal bunkeringcostislower inwide time windowcase.Wide timewindowsallowavesseltosailatslower speed and hence, total required bunker amount for the service decreases.

Wenextstudytheeffectofbunkerpricesonthesailingspeed and detour decisions. We set the width of the time window to 3 hours and delay parameter to 100 dollars per hour. We con-sidertwo scenarioscorresponding tolow andhighbunkerprices. While in the first scenario bunker prices are closer, the differ-encesare moresignificant inthehighbunkerpricescenario. Op-timal bunkering and speed decisions are presented in Table 10. When bunker prices are high, detour is more cost efficient even forhighdelaypenalty.Ontheotherhand,whenbunkerpricesare closer,detour forbunkering maynot coverthe additionalbunker costduetodetouringandcostofdelay.Therefore,vesselmay pre-fertobunker ather plannedschedule.Theseresultsindicate that

Table8

Optimalbunkeringandspeeddecisionwithrespecttolowandhighdelaycost(tw_=3).

Port Bunkerprice Lowdelaycost Highdelaycost

(dollarsperton) Bunkering Sailingspeed Bunkering Sailingspeed amount(tons) (knots) amount(tons) (knots)

P0 300 18.75 13.64

A1 150 204 19.33

P1 275 19.07 15.47

P2 250 17.50 17.50

P3 225 17.37 142 16.02

P4 200 15.38 18 15.38

A2 125 306 16.24 306 16.24

P5 175 15.13 15.13

P6 200 15.77 15.77

P7 225 17.38 17.38

P8 250 14.30 14.30

P9 375 14.40 14.40

P10 300 – –

Totalbunkering 68,850 73,800

Table9

Optimalbunkeringandspeeddecisionwithrespecttolowandhighdelaycost(tw_=6).

Port Bunkerprice Lowdelaycost Highdelaycost

(dollarsperton) Bunkering Sailingspeed Bunkering Sailingspeed amount(tons) (knots) amount(tons) (knots)

P0 300 18.75 13.64

A1 150 200 19.33

P1 275 19.07 15.47

P2 250 17.50 17.50

P3 225 16.80 140 15.61

P4 200 14.81 18 15.38

A2 125 306 16.24 305 16.09

P5 175 15.79 15.79

P6 200 15.77 15.77

P7 225 16.90 16.90

P8 250 14.30 14.30

P9 375 14.40 14.40

P10 300 – –

Totalbunkering 68,250 73,225

Cost

Table10

Optimalbunkeringandspeeddecisionatdifferentbunkerprice.

Port Lowbunkerprice Highbunkerprice

Bunker Bunkering Sailingspeed Bunker Bunkering Sailingspeed price(dollarsperton) amount(tons) (knots) price(dollarsperton) amount(tons) (knots)

P0 240 13.64 600 18.75

A1 120 300 204 19.32

P1 220 15.47 550 19.07

P2 200 17.50 500 17.50

P3 180 142 16.02 450 17.37

P4 160 18 15.38 400 15.38

A2 100 306 16.24 250 306 16.24

P5 140 15.13 350 15.12

P6 160 15.77 400 15.77

P7 180 17.38 450 17.38

P8 200 14.30 500 14.30

P9 220 14.40 550 14.40

P10 240 – 600 –

delaypenaltyshouldbedeterminedbyconsideringthevariationin fuelprices.Assigninghighdelaypenaltytoaportforcesvesselto arrivewithin thetime window,whichmayincreasethefuel con-sumptioncostsignificantly.Ontheotherhand,assigninglowdelay penaltymayresultinarrivingmuchlaterthanthetimewindowof important ports, whichleads tohigh operationalcosts and dam-agesthereliabilityoflinercompany. Moreover,bunkerdecisionis directlyrelatedtobunkerpriceswhichalsoaffecttheserviceroute andsailingspeed.

7. Conclusion

Inthispaperweaddressedthespeedoptimizationand bunker-ing problem in liner shipping. We first focus on the stochastic speed optimizationproblem.Despitethefact thatvessel schedul-ing androuting hasbeen well studiedin recentyears, stochastic aspects of the problemhasnot explicitly been considered inthe literature. This paper aimed to fill this gap. By considering the uncertain port times, we formulated the problem as a dynamic program.Theproposedmodelincorporatespossiblecostsresulted fromwaiting anddelayinports, and take theport servicetimes into consideration when making speed decisions. To implement thedynamicmodel,we discretizethestatespace, whichprovides an upper bound on the optimal expected cost. We also formu-late a deterministic model by assuming that the stochastic port timestakeontheirexpectedvaluesandweshow thatthismodel provides a lower bound on the optimal expected cost. In practi-cal implementations,theselowerandupperboundsdesignatethe

estimatedminimum andmaximumtotal cost of avessel service. We also studythe propertiesof the optimalvalue functionsand investigatethevariationofsailingspeedwithrespecttothevessel arrivaltime.

Fuelconsumptioncostisoneofthemajoroperationalcosts of linercompanies. Sailingspeed andbunker priceshave direct im-pacton thiscost. Therefore,in the second part ofthe paper,we studythe relationship betweensailing speed andbunkering. We mainlyfocusonwheretobunker,howmuchtobunkerand deter-miningthevesselspeedateachlegalongtheserviceroute.We de-velopadynamicprogramingmodelforjointbunkeringandspeed optimizationproblem.Differentthantheproposedbunkering mod-elsinthe literature,ourformulation allowsa vesselto bunkerat the portsthat are not on the planned service route. The aim of theproposed modelis to plan the bunkering operation.In other words,optimalsolutionofthismodelcanbeusedtodecideonthe bunkeringports, averagebunkeringquantity requiredto complete theserviceandthe average sailingspeed. Sincebunker contracts aresetbeforethevesseldepartsforavoyage,ourmodelis formu-latedbyconsideringtheexpectedservicetimes.However, stochas-ticdynamicmodelproposedinSection3canbeusedtoassignthe sailingspeedtothe nextportifvesseldeviatesfromtheplanned schedule.

uncertainporttimescanbringsignificantcostimprovements.This result is consistent with the real life examples. As Notteboom (2006) stated, fast growth in cargo volumes has increased the possibilityof port congestion. Planning the servicescheduleonly based on the expected port times can disrupt the whole liner serviceschedule. Therefore,sailing speeds andport arrival times should be computedby considering the uncertaintiesat portsto mitigateassociatedrisks.

Ourcomputationalstudyalsoprovidesanumberofuseful prac-ticalinsights. Manystudies in maritime literature restrict vessels toarrive within thetime window andcompute the optimal sail-ing speed by only considering the fuel consumption cost. How-ever, delays are very common in real-life cases as reported by Vernimmenetal.(2007).Ournumericalexperimentsindicatethat sailing speed is significantly affected by the delay penalty. Even waitingcost can influence on the speed decision. Due to uncer-taintiesinports,avesselcanprefertosailathighspeedstoavoid delays or sail at slow speed to arrive late at some ports in or-dertoavoidarrivingearlytofollowingports.Thus,shipoperators are likely to see clear benefits when making decisions consider-ingboth costtypes.Onthe otherhand,our numericalresultsfor bunkeringproblemshowthatbunkeringamountandportsshould bedeterminedbyconsidering theplannedscheduleofvessel, de-lay cost and bunker prices. For tight schedules, it can be more costeffectivetobunkeratportsontheplannedscheduleofa ves-sel,even ifthe fuel prices atthe alternative bunkering portsare cheaper.Moreover, delaypenalty significantly affectsthe bunker-ing port selection. Therefore, delay penalty parameter should be assignedbyconsideringthebunkerprices.Wealsoanalyzethe im-plicationsofbunkerpricesonthespeeddecision.Ourresultsshow that bunkering port and quantity decisions are directly related to bunker prices which also affect the service route and sailing speed.Whenbunkerpricesare closeratall ports,vesselmaynot prefer detour forbunkering due to theadditional bunker cost of detouring.

There areanumberofresearch directionsto pursue.Although we focus on the uncertaintiesin liner shipping,we assume that weather does not affect the speed decision. Further research is neededto develop newmodels and solution methods that avoid thisassumption.Thislineofresearchcanbenefitfromthestudies proposedforweatherrouting.Anotherfutureresearchdirectionis toincorporateportswappingandportskippingoptions.In practi-calapplications,shipoperatorscandealwithdelaysbyreshuffling theorderofportsorskipsomeports. Eventherearecasesthata vesselcanleavetheportbeforecompletingitsservicetoavoid de-layto next ports.A promisingdirectionof futureresearch would betomodelthesedecisionsasadynamicprogrambyconsidering theuncertaintiesalongthejourney.

Acknowledgment

ThisresearchissupportedinpartbyEUFP7projectMINI-CHIP (Minimising Carbon Footprintin Maritime Shipping) underGrant numberPIAP-GA-2013-611693.

Supplementarymaterial

Supplementary material associated with this paper can be found,intheonlineversion,at10.1016/j.ejor.2016.10.002.

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