Step-wise stowage planning of roll-on roll-off ships transporting dangerous goods

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Maritime Transport Research

journalhomepage:www.elsevier.com/locate/martra

Step-wise stowage planning of roll-on roll-off ships transporting dangerous goods

Beizhen Jia

, Kjetil Fagerholt

aAalborg University, 2450 Copenhagen, Denmark

bNorwegian University of Science and Technology, 7491 Trondheim, Norway

a r t i c le i n f o

Keywords:

Stowage planning Stability and safety Dangerous goods Eco-efficiency Mix integer programming

a b s t r a c t

Planningstowagewiththepresenceofdangerousgoodsiscriticaltoensuresafetyatsea.Inthis paper,weproposeastep-wisestowageplanningapproachtogenerateoptimalstowageplansfor roll-onroll-off shipstransportingtrailers(somecontainingdangerouscargo)betweentwoports.

Theplanningapproachconsistsofthreesteps,whereStep1maximizesthenumberofdangerous cargounitstotransportwhileadheringtotheInternationalMaritimeDangerousGoodsregula- tions.Step2,whichisoptional,maximizesthesafetydistanceamongthedangerouscargounits foundinthefirststep.Finally,inStep3,theballastwaterintakeneededtoensurestabilityof theshipisminimized,asthishasasignificanteffectonthefuelconsumption.Computational resultsoninstancesgeneratedbasedonrealdatafromashippingcompanyshowthatthepro- posedplanningapproachmightbothreducetheballastwaterintake(andhencereducethefuel consumption)andincreasethesafetydistanceamongdangerouscargounits.

1. Introduction

From2015to2019,therehavebeen19,418marinecasualtiesandincidents,including496fatalities,6210personsinjuredand 21,392shipsinvolved(EuropeanMaritimeSafetyAgency,2020).Safetyatseahasbeenimprovedduringthepastyearsthrough bettershipdesignandstability,advancedmaritimetechnologiesandmorestrictinternationalregulationsdevelopedbyInternational MaritimeOrganization.Asoneofthemostinternationalanddangerousindustries,shippingisresponsibleforthetransportationof agreatamountofdangerouscargo.Whentransportingdangerousgoodsinclosedforms,theyneedtobeproperlypackagedand segregatedaccordingtotheInternationalMaritimeDangerousGoods(IMDG)Code(InternationalMaritimeOrganization,2016)in ordertobeloadedonforexample,container,roll-onroll-off (RoRo)orgeneralcargoships.TheIMDGCodeclassifiesdangerous goodsintonineclasseswithvarioussub-classeswithinandprovidesageneralsegregationruleswithadetailedexplanationwhen stowingthesecargoondifferenttypeofships.

AccordingtotheIMDGcode,therearemainlyfoursegregationrules,supplementedbyexceptionalrulesforallshippingsegments, seeFig.1.Therulesinthetablehavethefollowingmeaningingeneral:

1. “awayfrom” 2. “separatedfrom”

3. “separatedbyacompletecompartmentorholdfrom”

4. “separatedlongitudinallybyaninterveningcompletecompartmentorholdfrom”

∗Correspondingauthor.

E-mailaddresses:[email protected](B.Jia),[email protected](K.Fagerholt).

1orcid=0000-0002-4559-3693

https://doi.org/10.1016/j.martra.2021.100029

Received25March2021;Receivedinrevisedform17May2021;Accepted1June2021

2666-822X/© 2021TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Fig.1. GeneralSegregationTable(InternationalMaritimeOrganization,2016).

InthispaperweconsiderthestowageplanningproblemforRoRoshipsoperatinginshortseashipping,whicharefacinggreat challengestransportingdangerousgoods.RoRoshippingisamajortransportmodeintheworld,especiallyforcountrieswithlong coastlines,duetoitsflexibleconnectionwithroadandrailtransportation.TheseRoRoshipsarecarryinganumberoftrucktrailers andseveraloftheseareclassifiedasdangerouscargoaccordingtotheIMDGCode.Generatingastowageplanthatcanassignall dangerousgoodswithpositionsonvariousdecksonboardtheRoRoshipwhilerespectingtheirrespectivesegregationrulesand additionalconstraintsisachallengingtask,butalsocrucialforthesafetyoftheship.Furthermore,asshownbyJiaetal.(2020),a goodstowageplancanalsoreducetheneedforballastwateronboardtheship,whichagaincangivesignificantreductionsinfuel consumption,andhenceenvironmentalemissions.

Stowageplanningforshipsisacriticalpartthatlinksdifferentactivitiesofthecargooperationstogether.Thisinterestingyetchal- lengingproblemhasattractedmanyresearcherstotackleitsvariationsindifferentsectorswithinmaritimetransportation,especially thecontainersector.Mostoftheeffortshavebeenputonminimizingtheshiftingofcontainersinthecontainerstowageplanning, knownasthemasterbayplanningproblem.Afewresearchershaveinvestigatedstowingcontainershipsinthepresenceofdangerous goods.Parreñoetal.(2016)considerstacksegregationintheslotplanningproblem.AmbrosinoandSciomachen(2021)proposea novelprocedureforstowingcontainersbasedontheprincipleincludedintheIMDGCode.Wereferreaderswithinteresttoadetailed literaturereview(Voßetal.,2004)andanupdate(StahlbockandVoß,2007)onthetopicofcontainerterminaloperationincluding stowageplanning.Moreover,Hvattumetal.(2009)studythestowageprobleminbulkshippingforchemicalandproducttankers, i.e.thetankallocationproblemwiththepresenceofdangerouscargo.

StowageplanninginRoRoshippinghasnotgainedmuchattentionfromtheresearchersuntilrecently.Severalstudiesfocuson deep-seagoingcarcarriersthatusuallyoperateonrouteswithmultipleportcallsandoptionalcargo.Thereforetheproblemdealswith maximizingprofitbytakingasmanyasoptionalcargoandminimizingshiftingcostduetoblockingcargo(Hansenetal.,2020;2016;

Øvstebøetal.,2011;Puisa,2021).Otherstudieshavealsoputmorefocusonthestabilityandsafetysideofthestowageplanning.

Puisa(2021)proposesthreeimprovementstotheoptimizationofRoRostowage,namelyfinerapproachtoshipstability,firesafety, andcargohandlingefficiency.Jiaetal.(2020)proposeanintegratedstowageplanningapproachandpresentanoptimalstowage modelthatminimizesballastwaterintake.Someotherresearchersalsostudythestowageplanningofpassengerferries(Baylissetal., 2019;2021).Totheauthors’knowledge,noresearchwiththeinclusionofthedangerousgoodstransportationhasbeenconducted, whichisessentialtostowageplanningformanyRoRoships.

ThispaperaimstofillthegapbyextendingtheproblemandstudyconductedbyJiaetal.(2020),incorporatingdangerous goodssegregationandmaximizingthesafetyonboardinthestowageplanningprocess.Furthermore,incontrasttotheall-in-one deterministicstowageplanningmodel,weproposeastep-wisestowageoptimizationmethod,tobetteraccommodatetheexperts’

opinionsintothestowageplanningprocess.

Therestofthepaperisstructuredasfollows.WestartinSection2byintroducingtheproblemformulationandrelevantmathe- maticalnotationsfortheRoRostowageplanningproblemwithdangerouscargo,extendedfromthestowageproblemwithoptimal

ballastwaterintroducedbyJiaetal.(2020).Thereafter,inSection3,weproposethestep-wiseoptimizationapproachandformulate theoptimizationproblemsarisinginthedifferentstepsasbinary/mixedintegerprogrammingmodels.InSection4,thestep-wise optimizationapproachistestedonanumberofrealistictestinstances,randomlygeneratedfromhistoricaldatafromaRoRoshipping company,beforeweconcludeinSection5.

2. TheRoRoshipstowageproblemwithdangerousgoods

WeconsideragivenRoRoshipwithasetoffixeddecks.Foreachdeck𝑑,thereisasetofslots𝑑wherecargounitscan beplaced.Eachslot𝑠fitsonestandardsizedtrailer.Theshiphasintotal𝑁^𝑆slots,where𝑁^𝑆=∑

𝑑∈|_𝑑|.RoRoshipstransport primarilytrailers,butalsotrucks,cars,andotherwheeledcargounits.Thescopeofthispaperdelimitatestostandardsizedtrailers, alsocalledcargounits.

Weconsideragivendepartureorvoyagebetweentwoportsfortheshipwhereasetofbookedtrailers,hereafterreferredtoas cargounits,iswaitingtobeplannedandloadedontotheshipforitsdestinationport.Dependingonthecontentofthecargounits, itcanbefurthercategorizedasasubsetofdangerouscargounits^𝐷andasubsetofgeneralcargounits^𝐺.Usually,dangerouscargo unitshaveanearliercut-off timethangeneralcargounits,meaningthattheyarerequiredtobepresentattheterminalandready tobeloadedseveralhoursbeforeshipdeparture.Anearliercut-off timeistoensurethatstowageplannerscanhaveenoughtimeto makeagoodsegregationplanforsafetyreasons.Thedangerouscargounitsneedtobesegregatedonboardwithacertaindistance accordingtoasetofsegregationrule𝑁dependingontheirclasses.Foreachcargounit𝑐,𝑐𝑛isasubsetofcargounitsthatconflicts withcargounit𝑐accordingtosegregationrule𝑛∈𝑁._𝑑𝑠𝑛^𝑁 isasubsetofslotsondeck𝑑thatareprohibitedtoloaddangerousgoods accordingtorule𝑛.Forexample,ifacargounit𝑐isloadedataslot𝑠ondeck𝑑,thennocargounitsfrom_𝑐𝑛canbeloadedtoanyslot in_𝑑𝑠𝑛^𝑁 subjecttosegregationrule𝑛.Inaddition,thecommitmentclassofacargounitiscategorizedaseithermandatoryoroptional.

Mandatorycargounitsarerequiredtobetransportedonthegivendeparturewhereasoptionalcargounitscanwaituntilthenext departure.However,fortheoptimalutilizationofthedeckspace,itisbeneficialtoshipasmanyoptionalcargounitsaspossible.We assumedangerouscargounitsingeneralhasahighervalueandthuspriorityovergeneralcargounits.

Weintroducethefollowingnotationforvarioussubsetsofthecargounits:^𝐷,𝑀 isthesetofmandatorydangerouscargounits,

^𝐷,𝑂isthesetofoptionaldangerouscargounits,where^𝐷,𝑀∪^𝐷,𝑂=^𝐷.^𝐺,𝑀isthesetofmandatorygeneralcargounits,^𝐺,𝑂 isthesetofoptionalgeneralcargounits,where^𝐺,𝑀∪^𝐺,𝑂=^𝐺.Eachcargounit𝑐iscontainedinastandardsizedtrailerwitha specificweight𝐶_𝑐^𝑊 andeachdeck𝑑hasamaximumallowableweight𝐷_𝑑^𝑊,maxforsafetyreasons.Allcargounitsaredeliveredatthe terminalandavailabletobestowed.Loadingandunloadingoperationsareperformedbytugmastersdrivinginandoutoftheship throughtheramp.

ThegivenRoRoshiphasasetofballasttanks,includingasubsetofheelingtanks^𝐻andasubsetofregularballasttanks^𝐵. Ballasttanksarelocatedanddistributedalongsidethebottomoftheship,carryingusuallyseawaterwithadensityof𝜌tobalancethe ship.Thevolumecapacityoftank𝑖isdefinedas𝑇_𝑖^max.Heelingtanksareusedtobalancetheshiptransverselyatanytime,therefore, thetotalwatervolumestoredinheelingtanksshouldsatisfyarangebetween𝐻^max∕mintoprovidesufficientanti-heelingcapability.

Inaddition,theregularballasttankscomeintoplaceifstabilitycannotbesatisfiedbyonlyadjustingtheheelingtanks.Accordingto theAdmiraltyCoefficient(ManDiesel&Turbo,2011),foragivencargoloadandsailingspeed,themoreballastwaterashipcarries, thehigherbecomesthefuelconsumption.

Stabilityoftheshipismeasuredalongthreedimensions:vertical,transverseandlongitudinalforcesthatareinfluencedbythe distributionoftheweightofallcomponentsontheship.Duetothecomplexityofthesecalculation,weapplyagoodapproximation ofsuchmeasuresthroughthecompositeverticalcenterofgravityfromthekeel𝑉𝐶𝐺,transversecenterofgravityfrommidship𝑇𝐶𝐺 andlongitudinalcenterofgravityfromaftperpendicular𝐿𝐶𝐺,takingintoaccountnotonlytheweightofcargounitsbutalsothe weightoftheballastwaterandlightweightoftheship𝐿^𝑊.Toachieveseaworthiness,eachmeasurementshouldsatisfyitsmaximum andminimumlimitingvalues,thatis𝑉𝐶𝐺^max∕min,𝑇𝐶𝐺^max∕min,and𝐿𝐶𝐺^max∕min,respectively.Formoredetailedexplanationsofthe dimensionsandcalculations,wereferourreaderstoJiaetal.(2020)andthetextbookbyRhodes(2003).

TheaimoftheRoRoshipstowageproblemwithdangerousgoodsistominimizethefuelconsumptionbycarryingtheminimal amountofballastwaterwhileatthesametimemaximizingsafetybymaximizingthedistanceamongthedangerouscargounitson boardtheship.Weconsiderdecisionssuchasthenumberofoptionaldangerouscargounitstocarry,themassofwaterineachballast tank𝑡_𝑖,andtheplacementofeachindividualcargounitsubjecttotheIMDGsegregationrules,weightdistribution,andotherstowing requirements.Weintroducethebinarydecisionvariable𝑥_𝑐𝑑𝑠fortheplacementofthecargounits,whichisequalto1ifcargounit𝑐 isloadedatslot𝑠ondeck𝑑,and0otherwise.

3. Step-wisestowageoptimizationapproach

Asadecisionsupporttool,weproposeastep-wisestowageoptimizationapproachwiththeabilitytoincorporateexpertsinputs, thusmorerobustness,flexibilityandusabilitytothegeneratedfinalstowageplan.Thesolutionapproachconsistsofthreesteps, whereeachstepincludesanoptimizationproblemwithgivenobjectivesandconstraints.Theflowofproposedstep-wiseplanning processisillustratedinFig.2.Step1maximizesthenumberofoptionaldangerouscargounitstobecarriedonboardtheshipasit isassumedthatonealwayswantstotransportasmanydangerouscargounitsaspossibletoreducethisnumberforthefollowing departuresalongthesameroute.Itselectsalistofoptionalcargounitstobeloadedandgeneratesapreliminarystowageplanforboth mandatoryandoptionaldangerouscargounitsthatobeystheIMDGsegregationrules.Dependingonwhetherwewanttomaximize

Fig.2. Step-wisestowageoptimizationprocesswiththepresenceofdangerousgoods.

safetybymaximizingthedistanceamongthedangerouscargounitsevenbeyondtheminimumrequirementsgivenbythesegregation rules,thestep-wisesolutionapproachfollowseitheroneoftwodirections:1)fulloptimizationand2)partialoptimization.

In1)fulloptimization,weaimatmaximizingthesafety(i.e.beyondtheminimumrequirementsdefinedbythesegregationrules).

Step2isthenactivatedtomaximizethedistancebetweenslotsthatareloadedwithdangerouscargounits.Itgeneratesapreliminary planforalldangerouscargounitsselected inStep1.Nowthepreliminarystowageplanforallthedangerous goodsselectedis availableforapproval.Thestowageplannersand/orcargoofficershavetheflexibilitytomanuallyadjusttheoptimalstowageplan fordangerousgoodsifthereareanypreferencesorexceptionstobemadeduetocertaincircumstances.Intheend,Step3fixesthe approvedpreliminarystowageplanfordangerousgoodsasinputfromStep2,andstowstherestofthecargounits,namelythegeneral cargounitstominimizetheballastwaterintakeandthusreducefuelconsumption.In2)partialoptimization,Step2isskippedand Step3takesthefixedstowageplanforthedangerouscargounitsfromStep1asinput.

Therestofthesectiondescribestheoptimizationproblemofeachstep.WereferreaderstoAppendixAforacompletelistof notationsusedinthispaper.

3.1. Step1:IMDGplanning-maximizingdangerouscargounitsintake

Foragivendeparturewithaloadlistthathasmorecargounitstotransportthantheshipcapacityallows,astowageplanbecomes simplyinfeasiblewithoutselectingwhichcargounitstotransport.Thedifficultyarisesinthepresenceofdangerouscargounitssince itisnotintuitivehowmanycanbeloadedontheshipwithoutviolatingthesegregationrules.InStep1,wewanttocreateafeasible stowageplanwiththemaximumnumberofdangerousgoodstheshipcancarrywhileobeyingthesegregationrules.Thisisdone throughthefollowingoptimizationmodel:

max ∑

𝑐∈^𝐷,𝑂

∑

𝑑∈

∑

𝑠∈𝑑

𝑥_𝑐𝑑𝑠 (1)

subjectto:

∑

𝑑∈

∑

𝑠∈𝑑

𝑥_𝑐𝑑𝑠=1, 𝑐∈^𝐷,𝑀 (2)

∑

𝑐∈^𝐷

𝑥_𝑐𝑑𝑠≤1, 𝑑∈,𝑠∈_𝑑 (3)

∑

𝑐∈^𝐷,𝑂

∑

𝑑∈

∑

𝑠∈_𝑑𝑥𝑐𝑑𝑠≤𝑁^𝑆−|^𝐷,𝑀|−|^𝐺,𝑀| (4)

∑

𝑐^′∈_𝑐𝑛

∑

𝑠^′∈^𝑁_𝑑𝑠𝑛

𝑥_𝑐^′_𝑑𝑠^′≤1−𝑥_𝑐𝑑𝑠 𝑐∈^𝐷,𝑑∈,𝑠∈_𝑑,𝑛∈ (5)

𝑥_𝑐𝑑𝑠∈ {0,1} 𝑐∈^𝐷,𝑑∈,𝑠∈_𝑑 (6)

Objectivefunction(1)maximizesthenumberofoptionaldangerouscargounitsthatarearecarriedonboardtheship.Constraints (2)makesurethatallthemandatorydangerouscargounitsareloaded,whileconstraints(3)ensurethateachslotcontainsatmost one(dangerous)cargounit.Constraint(4)makessurethatthenumberofoptionaldangerouscargounitsdoesnotexceeditscapacity ontheship,whichiscalculatedbydeductingthenumberofmandatorycargounitsfromthenumberofavailableslotsontheship.

Segregationrulesarerepresentedinconstraints(5).Ifdangerouscargounit𝑐isplacedinslot𝑠ondeck𝑑,thennocargounit𝑐^′∈_𝑐𝑛 canbeloadedatanyslot𝑠^′∈_𝑑𝑠𝑛^𝑁 .Binaryrequirementsforthevariablesareimposedthroughconstraints(6).

AsaresultofsolvingtheStep1model,asetofoptionaldangerouscargounitsisselectedandapreliminarystowageplanfor themandatory andselectedoptionaldangerouscargounitsis generated.Anupdatedsetofdangerouscargounits^𝐷′including theselectedoptionaldangerouscargounitsandmandatorydangerouscargounitsisformedandusedasinputinSteps2and3.

Accordingly,sincetheshipcapacityremainsthesameandshouldbeutilizedatmost,weloadasmanygeneraloptionalcargounitsas possible.Theavailablecapacityforgeneraloptionalcargounitsistheship’scapacityminusthenumberofmandatorygeneralcargo unitsandselecteddangerouscargounits.Wedenotethenewsubsetofgeneraloptionalcargounitsthatareselectedforloadingas

^𝐺′,𝑂.Thus,byupdatingrelevantcargosets,wehavethefollowings:^𝐺′=^𝐺,𝑀∪^𝐺′,𝑂and^′=^𝐺′∪^𝐷′. 3.2. Step2:IMDGplanning-maximizingsafety

Foragivenlistofdangerouscargounitstobeloaded(obtainedfromStep1),itisimportanttoensurethatthestowagecomplies withthesegregationrules.Moreover,itisbeneficialtostowthemasfurtherapartfromeachotheraspossibletoreducetherisk ofaccidents.Step2thereforeaimstoimprovethesafetybeyondtheminimumrequirementsgiveninIMDGsegregationrules,i.e.

maximizingthedistanceamongslotsloadedwithdangerouscargounits.Weproposetwoalternativemodelsforthispurpose:an intuitivedistancemaximizationformulation(Section3.2.1)andariskminimizationformulation(Section3.2.2).

3.2.1. Distanceformulation

Wedefine𝐷𝑑𝑠𝑑^′𝑠^′asthedistancebetweenslot𝑠ondeck𝑑andslot𝑠^′ondeck𝑑^′.Ifslots𝑠and𝑠^′areonthesamedeck,thedistance iscalculatedastheminimumEuclideandistancebetweenthem.Iftheslotsareondifferentdecks,thedistanceisgivenasanumber thatisslightlylargerthanthedistancerequiredbythestrictestsegregationrule.Weassumethatiftwodangerouscargounitsare placedsofarapartfromeachotherthenitdoesnotmatteriftheyareonthesamedeck.Theobjectiveofmaximizingthedistance amongthedangerouscargounitscanthebewrittenasfollows:

max ∑

𝑐∈^𝐷′

∑

𝑑∈

∑

𝑠∈_𝑑

∑

𝑐^′∈^𝐷′

∑

𝑑^′∈

∑

𝑠^′∈_𝑑′

𝐷𝑑𝑠𝑑^′𝑠^′𝑥𝑐𝑑𝑠𝑥𝑐^′𝑑^′𝑠^′ (7)

Theobjectivefunction(7)maximizesthesumofdistancebetweenslotsloadedwithdangerouscargounits.Itcanbenotedthatit becomesquadratic.Therefore,weintroduceanewbinaryvariable𝑦𝑑𝑠𝑑^′𝑠^′,whichtakesthevalue1ifdangerouscargounitsareplaced inbothslots𝑠∈_𝑑and𝑠^′∈_𝑑^′,and0otherwise.Wecanthenobtainthefollowinglinearformulationformaximizingthedistance amongslotswithdangerouscargounits:

max ∑

𝑑∈

∑

𝑠∈𝑑

∑

𝑑^′∈

∑

𝑠^′∈_𝑑′

𝐷_𝑑𝑠𝑑^′_𝑠^′𝑦_𝑑𝑠𝑑^′_𝑠^′ (8)

subjectto:

(5)∑

𝑑∈

∑

𝑠∈𝑑

𝑥_𝑐𝑑𝑠=1, 𝑐∈^𝐷′ (9)

∑

𝑐∈^𝐷′

𝑥_𝑐𝑑𝑠≤1, 𝑑∈,𝑠∈_𝑑 (10)

𝑦𝑑𝑠𝑑^′𝑠^′≤ ∑

𝑐∈^𝐷′

𝑥𝑐𝑑𝑠, 𝑑∈,𝑠∈𝑑,𝑑^′∈,𝑠^′∈𝑑^′ (11)

𝑦_𝑑𝑠𝑑^′_𝑠^′≤ ∑

𝑐^′∈^𝐷′

𝑥_𝑐^′_𝑑^′_𝑠^′, 𝑑∈,𝑠∈_𝑑,𝑑^′∈,𝑠^′∈_𝑑^′ (12)

𝑦_𝑑𝑠𝑑^′_𝑠^′+1≥ ∑

𝑐∈^𝐷′

𝑥_𝑐𝑑𝑠+ ∑

𝑐^′∈^𝐷′

𝑥_𝑐^′_𝑑^′_𝑠^′, 𝑑∈,𝑠∈_𝑑,𝑑^′∈,𝑠^′∈_𝑑^′ (13)

𝑥_𝑐𝑑𝑠∈ {0,1} 𝑐∈^𝐷′,𝑑∈,𝑠∈_𝑑 (14)

𝑦𝑑𝑠𝑑^′𝑠^′∈ {0,1} 𝑑∈,𝑠∈𝑑,𝑑^′∈,𝑠^′∈𝑑 (15)

Themodeladditionallyrequiresthesegregationconstraints(5)introducedinSection3.1.Constraints(9)requirethatallthe dangerouscargounitsselectedinStep1areplacedinaslot.Constraints(10)ensurethateachslotcontainsatmostonecargounit.

Constraints(11),(12)and(13)linkthenewbinaryvariables𝑦withtheoriginaldecisionvariables𝑥.Constraints(11)ensurethatif slot𝑠doesnotcontaindangerouscargounit𝑐,𝑦_𝑑𝑠𝑑^′_𝑠^′isforcedtobe0,similarlywithconstraints(12).Constraints(13)force𝑦_𝑑𝑠𝑑^′_𝑠^′ valuetobe1ifandonlyifboth𝑥_𝑐𝑑𝑠 and𝑥_𝑐^′_𝑑^′_𝑠^′are1.However,sincetheobjectivefunctionmaximizesthevalueof𝑦,thissetof constraintsbecomesredundant.Finally,thebinaryrequirementsonthevariablesareimposedthroughconstraints(14)and(15). 3.2.2. Riskformulation

EventhoughthedistanceformulationinSection3.2.1isintuitive,theenumerationofcombinationofslotsondifferentdecks resultsinavastnumberof𝑦variablesandconstraints.Therefore,weproposeanotherformulationbyintroducingariskparameter 𝑅_𝑑𝑠𝑠^′torepresenttheriskmeasurementbetweenslots𝑠and𝑠^′on(thesame)deck𝑑.𝑅_𝑑𝑠𝑠^′issettoitsmaximumvalueifslots𝑠^′and 𝑠areneighboringslots,anditsvaluedecreasesasthedistancebetweenslotsincreasesuntilittakesthevalue1whenslots𝑠and𝑠^′ areasfarapartfromeachotheraspossibleonthegivendeck.Theriskparameterbetweentwoslotsondifferentdecksissettozero.

Basedonthis,wecanimplicitlymaximizethedistance betweendangerouscargounitsbyminimizingthetotalrisk withthe followingbinaryprogrammingmodel:

min ∑

𝑑∈

∑

𝑠∈_𝑑

∑

𝑠^′∈𝑑

𝑅𝑑𝑠𝑠^′𝑦𝑑𝑠𝑠^′ (16)

subjectto:

(5) (9) (10) 𝑦_𝑑𝑠𝑠^′≤ ∑

𝑐∈^𝐷′

𝑥_𝑐𝑑𝑠, 𝑑∈,𝑠∈_𝑑,𝑠^′∈_𝑑^′ (17)

𝑦_𝑑𝑠𝑠^′≤ ∑

𝑐^′∈^𝐷′

𝑥_𝑐^′_𝑑𝑠^′, 𝑑∈,𝑠∈_𝑑,𝑠^′∈_𝑑^′ (18)

𝑦_𝑑𝑠𝑠^′+1≥ ∑

𝑐∈^𝐷′

𝑥_𝑐𝑑𝑠+ ∑

𝑐^′∈^𝐷′

𝑥_𝑐^′_𝑑^′_𝑠^′, 𝑑∈,𝑠∈_𝑑,𝑠^′∈_𝑑^′ (19)

𝑥_𝑐𝑑𝑠∈ {0,1} 𝑐∈^𝐷′,𝑑∈,𝑠∈_𝑑 (20)

𝑦_𝑑𝑠𝑠^′∈ {0,1} 𝑑∈,𝑠∈_𝑑,𝑠^′∈_𝑑 (21)

ThestructureoftheconstraintsintheriskformulationresemblesthatofthedistanceformulationinSection3.2.1.Segregationis enforcedthroughconstraints(5)introducedinSection3.1.Theriskformulationsharesthesameconstraints(9)and(10)thatensure dangerouscargounitsareloadedexactlyonceandthateachslotcannotloadmorethanonecargounitrespectively.Constraints(17), (18)and(19)linkthenewlinearvariables𝑦withthedecisionvariables𝑥.Unlikethedistanceformulation,constraints(19),which force𝑦_𝑑𝑠𝑠^′tobe1ifandonlyifboth𝑥_𝑐𝑑𝑠and𝑥_𝑐^′_𝑑𝑠^′are1,arenecessaryintheriskminimizationformulationduetoitsobjectiveof minimizingtherisk.Finally,thebinaryconstraintsonthevariablesareimposedthroughconstraints(20)and(21).

Comparedtothedistance formulation,theadvantage oftheriskformulationisthatitsignificantlyreducesthenumberof𝑦 variables(andconstraints)sincewenolongerneedtoconsiderthecombinationofslotsbetweendecks.Theobjectivefunctionwill automaticallyprioritizethestowageofdangerousgoodsintoseparatedecksifpossible,wheretherisk parameterissettozero.

Therefore,wechoosetoadopttheriskformulationforStep2ofthestep-wisestowageoptimizationapproachbasedonitsbetter performance(basedalsoonpreliminarytestswithbothformulations).

ThesolutionoftheStep2riskformulationgeneratesapreliminarystowageplanforalldangerouscargounitsthatminimizesthe riskofaccidents.Theplanillustrateshowdangerousgoodscanbestowedwithmaximaldistanceinbetweenandsupportsbothcargo stowageplannersandcargoofficerstomakeasaferstowageplanthatisatleastinaccordancewiththeminimalrequirementsofthe IMDGsegregationrules.

3.3. Step3:generalcargounitsplanning-minimizingfuelconsumption

GivenapreliminarystowageplanwithfixedpositionsfordangerouscargounitseitherfromStep2(incaseoffulloptimization) orfromStep1(incaseofpartialoptimization),Step3aimstominimizethefuelconsumptionbyminimizingtheintakeofballast water.Thisstepdealswiththestowageoftherestofcargounits,i.e.thegeneralcargounitsthatdonotrequireanysegregation.By designingaplanthatoptimallyplacescargounitsintotherightslotbyusingitsweighttobalancetheshipandsatisfystabilityand safetyrequirements,wecansignificantlyreducetheamountofexcessballastwatertheshiphastocarry.

InadditiontothenotationsdescribedinSection2,weintroducethefollowingnotationsforparametersusedtocalculatestability inStep3.Inordertocalculate𝑉𝐶𝐺,𝑇𝐶𝐺, and𝐿𝐶𝐺,weintroducethevertical,transverse,andlongitudinalcenterofgravityfor cargounitsas𝐶_𝑐^𝑉𝐶𝐺,𝐶_𝑐^{𝑇 𝐶𝐺},𝐶_𝑐^𝐿𝐶𝐺,forslotsas𝑆_𝑠^𝑉𝐶𝐺,𝑆_𝑠^{𝑇 𝐶𝐺},𝑆_𝑠^𝐿𝐶𝐺andfortheshipas𝐿^𝑉𝐶𝐺,𝐿^{𝑇 𝐶𝐺},𝐿^𝐿𝐶𝐺respectively.Thevertical centerofgravityofeachballasttank𝑖∈𝑇dependsonthemassofthewater𝑡_𝑖insidethetankanditsareaofbase𝑇_𝑖^𝐴𝑜𝐵.Theobjective ofthethirdstepofstowageplanningistooptimizetheamountofwatercarriedinregularballasttanks.TheformulationofStep3is adoptedbasedonthemodelintroducedinJiaetal.(2020)andshownasbelow:

min ∑

𝑖∈^𝐵

𝑡𝑖 (22)

subjectto:

∑

𝑑∈

∑

𝑠∈𝑑

𝑥_𝑐𝑑𝑠=1, 𝑐∈^𝐺′ (23)

∑

𝑐∈^′

𝑥_𝑐𝑑𝑠≤1, 𝑑∈,𝑠∈_𝑑 (24)

∑

𝑐∈^′

∑

𝑠∈_𝑑𝐶_𝑐^𝑊𝑥_𝑐𝑑𝑠≤𝐷^max_𝑑 , 𝑑∈ (25)

𝜌𝐻^min≤ ∑

𝑖∈^𝐻

𝑡_𝑖≤𝜌𝐻^max (26)

𝑉𝐶𝐺^min≤𝑉𝐶𝐺≤𝑉𝐶𝐺^max (27)

𝑇𝐶𝐺^min≤𝑇𝐶𝐺≤𝑇𝐶𝐺^max (28)

𝐿𝐶𝐺^min≤𝐿𝐶𝐺≤𝐿𝐶𝐺^max (29)

𝑉𝐶𝐺=

∑

𝑐∈^′

∑

𝑑∈

∑

𝑠∈_𝑑(𝑆^𝑉_𝑠^𝐶𝐺+𝐶_𝑐^𝑉𝐶𝐺)𝐶_𝑐^𝑊𝑥𝑐𝑑𝑠+𝐿^𝑉𝐶𝐺𝐿^𝑊+ ∑

𝑖∈ 𝑡_𝑖 𝜌𝑇_𝑖^𝐴𝑜𝐵𝑡𝑖

∑

𝑐∈^′𝐶_𝑐^𝑊 +𝐿^𝑊 +∑

𝑖∈𝑡_𝑖 (30)

𝑇𝐶𝐺=

∑

𝑖∈𝑇_𝑖^{𝑇 𝐶𝐺}𝑡_𝑖+ ∑

𝑐∈^′

∑

𝑑∈

∑

𝑠∈_𝑑𝑆_𝑠^{𝑇 𝐶𝐺}𝐶_𝑐^𝑊𝑥_𝑐𝑑𝑠+𝐿^{𝑇 𝐶𝐺}𝐿^𝑊

∑

𝑐∈^′𝐶_𝑐^𝑊 +𝐿^𝑊 +∑

𝑖∈𝑡𝑖 (31)

𝐿𝐶𝐺=

∑

𝑖∈𝑇_𝑖^𝐿𝐶𝐺𝑡_𝑖+ ∑

𝑐∈^′

∑

𝑑∈

∑

𝑠∈𝑑

𝑆_𝑠^𝐿𝐶𝐺𝐶_𝑐^𝑊𝑥_𝑐𝑑𝑠+𝐿^𝐿𝐶𝐺𝐿^𝑊

∑

𝑐∈^′𝐶_𝑐^𝑊 +𝐿^𝑊 +∑

𝑖∈𝑡_𝑖 (32)

𝑥_𝑐𝑑𝑠∈ {0,1},𝑐∈^𝐺′,𝑑∈,𝑠∈_𝑑 (33)

𝑥_𝑐𝑑𝑠valuefromStep1orStep2,𝑐∈^𝐷′,𝑑∈,𝑠∈_𝑑 (34)

0≤𝑡_𝑖≤𝜌𝑇_𝑖^max,𝑖∈ (35)

Givenbythestowageofthedangerouscargounits(eitherfromStep1orStep2),theStep3objective(22)istominimizethetotal amountofballastwatercarriedbytheshipinordertoreducethefuelconsumptioncausedbyexcessballastwater.Fortheupdated cargolistsubjecttoshipcapacity,constraints(23)makesurethatallthegeneralcargounitswillbeassignedaslotonboard,and constraints(24)makesurethateachslotwillonlyhaveatmostonecargounitloaded.Shipsafetyandstabilityareensuredthrough limitsonmaximumdeckweight,heelingcapabilityandthreedimensionalforces.Constraints(25)limitthetotalweightofcargo unitsloadedoneachdeck.Theheelingcapabilityisguaranteedinconstraint(26)sothatthetankshavesufficientforcestoheelthe ship.Lastly,vertical,transverseandlongitudinalstabilitycalculationsarepresentedinEqs.(30),(31)and(32),andarelimitedby constraints(27),(28)and(29),respectively.Lastly,domainsfordecisionvariablesaregivenbyconstraints(33),(34)and(35).

Equation(30)showsthatverticalcenterofgravity(𝑉𝐶𝐺)ofballasttanksbecomesafunctionofthedecisionvariablesasaresult oftheinclusionofballasttanksinthedecisionvariables.Weapplytheleveldiscretizationmethodforlinearization.Wereferreaders foradetaileddescriptionofthemethodinJiaetal.(2020).Eachtank𝑖isdividedintovariousfillinglevelsdenotedbyasetofdiscrete points𝑘∈_𝑖.Asetofbinaryvariables𝑧_𝑖𝑘equalsto1ifthetank𝑖isfilledwithballastwatertoacertainlevel𝑘.Correspondingly, eachlevel𝑘isassociatedwithavolumeofwater𝑇_𝑖𝑘^𝑉𝑂𝐿anda𝑉𝐶𝐺value𝑇_𝑖𝑘^𝑉𝐶𝐺.

Thelinearizedformulationfortheamountofwaterinballasttank𝑡𝑖,itscorresponding𝑉𝐶𝐺,anditsgravitymomentcannowbe rewrittenasfollows:

𝑡_𝑖= ∑

𝑘∈𝑖

𝜌𝑇_𝑖𝑘^𝑉𝑂𝐿𝑧_𝑖𝑘 𝑖∈ (36)

𝑇_𝑖^𝑉𝐶𝐺= ∑

𝑘∈_𝑖𝑇_𝑖𝑘^𝑉𝐶𝐺𝑧𝑖𝑘 𝑖∈ (37)

𝑇^𝑉𝐶𝐺𝑡_𝑖= ∑

𝑖∈_𝑖𝑇_𝑖𝑘^𝑉𝐶𝐺𝜌𝑇_𝑖𝑘^𝑉𝑂𝐿𝑧_𝑖𝑘 𝑖∈ (38)

Correspondingly,thequadraticconstraint(27)isnowrepresentedinthefollowinglinearform:

𝑉𝐶𝐺^min(∑

𝑐∈𝐶_𝑐^𝑊 +𝐿^𝑊 +∑

𝑖∈

∑

𝑖∈_𝑖𝜌𝑇_𝑖𝑘^𝑉𝑂𝐿𝑧_𝑖𝑘)≤

∑

𝑐∈

∑

𝑑∈

∑

𝑠∈_𝑑

(𝑆_𝑠^𝑉𝐶𝐺+𝐶_𝑐^𝑉𝐶𝐺)𝐶_𝑐^𝑊𝑥𝑐𝑑𝑠+𝐿^𝑉𝐶𝐺𝐿^𝑊 +∑

𝑖∈

∑

𝑘∈_𝑖𝑇_𝑖𝑘^𝑉𝐶𝐺𝜌𝑇_𝑖𝑘^𝑉𝑂𝐿𝑧𝑖𝑘

≤𝑉𝐶𝐺^max(∑

𝑐∈

𝐶_𝑐^𝑊 +𝐿^𝑊 +∑

𝑖∈

∑

𝑖∈_𝑖

𝜌𝑇_𝑖𝑘^𝑉𝑂𝐿𝑧_𝑖𝑘) (39)

Byupdating𝑡_𝑖withconstraints(36)intheoriginalformulationandreplacingconstraint(27)withconstraint(39),weobtainalinear formulationinStep3.

TheoutputofStep3,whichisthelastpartofthestep-wisestowageplanningprocess,providesafinaloptimizedstowageplan thatmaximizesthenumberofdangerousoptionalcargounits,maximizesthesafetyinbetweendangerouscargounits(ifStep2is applied)andminimizestheexcessintakeofballastwatertoachievefuelreduction.

4. Computationalstudy

Weconductthecomputationalstudybasedondatafromtwoidenticalsistershipsdeployedontheshortsearoutebetween VlaardingenintheNetherlandsandImminghaminUK.Allshipspecificationdataandhistoricaldataonthevoyagesareprovidedby theshippingcompanythisresearchhasbeendoneincollaborationwith.Theshiptypehasatotalcapacityof262standardtrailers, distributedthroughfourfixeddecks.Theshiphasanumberofballasttanksandtwoheelingtanksalongbothsidesoftheship.The numberofdiscretizationlevelsforthetanksinStep3issettobe10,asithasbeendemonstratedwithhighaccuracyandfastrun timebyJiaetal.(2020).Duetothelargenumberofdangerouscargounitscategories,wesimplifytheclassificationaccordingto thenumberofsegregationrulesforthepurposeofdemonstrationandsimplicity.Inthispaper,dangerouscargounitsaresimplified andclassifiedintofourclasses.ThesegregationrulesappliedamongdifferentclassesareshowninTable1.Nosegregationisneeded betweendangerouscargounitsandgeneralcargounits.

Table1

IMDGsegregationrulesforfourclasses.

cargo unit general class 1 class 2 class 3 class 4

general - - - - -

class 1 - 1 1 1 1

class 2 - 1 2 2 2

class 3 - 1 2 3 3

class 4 - 1 2 3 4

Fig.3. Histogramsofhistoricaldistributionsbasedon654voyages.

Table2

Tabulardataofsamplingdistributionbasedonhistoricaldata.

Weight (ton) (0,5) [5,10) [10,15) [15,20) [20,25) [25,30) [30,35]

6.17% 9.78% 13.73% 17.94% 32.89% 18.78% 0.63%

IMDG per voyage [0,4] [5,9] [10,14] [15,19] [20,24] [25,29] [30,35]

4.28% 18.20% 32.43% 29.30% 11.47% 3.36% 1.07%

IMDG Class c1 c2 c3 c4

21.22% 77.07% 1.47% 0.24%

Inthispaper,forthepurposeofdemonstrationandsimplicity,weassumealldecksareopenandthegeneralsegregationrequire- mentsfordistanceapartonanopendeckforRoRoshipsaredefinedasfollowsforeachnumberinTable1:

1. 3m 2. 6m 3. 36m 4. 48m

WereferreaderstoSection7.5.3.2intheIMDGCodeforadetaileddescriptionofsegregationrulesfortheRoRosector.

4.1. Generationoftestinstances

Whengeneratingthetestinstances,wefixthetotalnumberofcargounitstobe280(somewhatmorethanthecapacityofthe shipsconsidered),including240mandatoryand40 optionalcargounits.Thecommitmentclassofadangerouscargounitbeing eithermandatoryoroptionalisrandomlyassignedtocargounitandregardlessofitsdangerousproperty.Inordertorepresentthe realworldinstances,wecollectedoneyearofhistoricaldatafor654voyagesonthestudiedroute.Wegenerated30instancesbased onthehistoricaldistributionsforthreekeyparameters:theweightofcargounits,thetotalnumberofdangerouscargounits,andthe compositionofdifferentclassesofdangerouscargounits,asshowninFig.3a–c,respectively.

Notethatduetothesimplificationofthedangerouscargounitsclassification,thedistributionfortheIMDGclassisaderivation oftheoriginalIMDGclassdistributionfromthehistoricaldata.Basedonthefrequencyofeachruleappearedinthehistoricaldata, weapproximateadistributionofthefourdangerouscargounitsclassessuchthatfrequenciesforeachruleresemblethehistorical ones.

Overall,thedistributionsfortheweightofcargounits,totalnumberofIMDGcargounitsandthecompositionofdifferentclasses pervoyagearesummarizedinTable2.

Wedescribeourinstancebyitsid,commitmentdistribution(“m/o”)formandatory andoptionaldangerouscargounitsand dangerousclassdistribution(“c1/c2/c3/c4”)forclass1-4cargounit,whereeachnumberrepresentsthenumberofcargounitsfor thatspecificcategory.Thetotalnumberofdangerouscargounitsmatchesthesumofmandatoryandoptionaldangerouscargounits 𝑚+𝑜,aswellasthesumofeachdangerousclasscargounits𝑐1+𝑐2+𝑐3+𝑐4.The30instancesaresortedbythetotalnumberof dangerouscargounitsfromsmalltolargeandlistedinTable3togetherwiththecomputationalresultsinSection4.2.

4.2. Computationalresults

Inorderfortheperformancetobecomparabletowhenthemodelisrunonastowageplanner’scomputer,thecomputationaltests areconductedonaWindowslaptopwithIntel(R)Core(TM)[email protected]

Table3

Computationalresults.Solutiontimesareinseconds.

Instance Full optimization Partial optimization

Step 1 Step 2 Step 3 Step 1 Step 3

id m/o c1/c2/c3/c4 Time Time Gap Δd. Δc. d. Time Obj. Time Time Obj.

1 3/0 3/0/0/0 0 0 0.0% 0.0% 0.0% 10 0 0 3 0

2 4/0 0/4/0/0 0 1 0.0% 4.3% 14.3% 12 0 0 3 0

3 5/0 0/5/0/0 0 1 0.0% 5.4% 20.5% 6 0 0 4 0

4 6/1 5/2/0/0 1 8 0.0% 12.0% 51.9% 11 0 0 3 0

5 8/0 1/7/0/0 0 13 0.0% 15.7% 116.2% 6 0 0 3 0

6 6/2 2/6/0/0 1 7 0.0% 15.7% 116.2% 8 0 0 4 0

7 7/1 1/7/0/0 1 8 0.0% 15.7% 116.2% 7 0 1 4 0

8 10/0 0/9/1/0 0 15 0.0% 1.1% 66.3% 8 0 0 3 0

9 9/2 3/8/0/0 2 3601 5.1% 5.5% -9.1% 5 0 1 3 0

10 7/4 0/11/0/0 1 3601 7.3% 1.1% 4.6% 57 34 1 15 13

11 10/1 5/6/0/0 2 3600 3.5% 5.2% 68.1% 4 0 2 3 0

12 12/0 3/8/0/1 0 3600 9.9% 3.8% 91.1% 5 0 0 5 0

13 9/3 3/9/0/0 1 3601 10.0% 2.4% 159.0% 25 46 1 5 0

14 11/2 4/9/0/0 3 3601 13.0% 4.3% 38.6% 3 0 2 3 0

15 12/1 3/10/0/0 2 3601 13.3% 0.0% 20.0% 10 9 2 6 9

16 10/4 2/12/0/0 3 3600 11.0% 5.2% 22.9% 4 0 2 3 0

17 13/1 3/11/0/0 3 3600 10.1% 7.7% 31.4% 5 0 2 3 0

18 11/3 3/11/0/0 4 2634 0.0% 1.5% 13.1% 3 0 3 3 0

19 12/2 2/12/0/0 2 3600 7.2% 1.4% 1.2% 3 0 2 3 0

20 11/4 2/13/0/0 4 3601 13.5% 1.8% 10.4% 3 0 4 4 0

21 15/1 4/10/2/0 3 3601 15.4% 3.7% 29.9% 6 0 3 4 0

22 13/4 6/11/0/0 3 3601 14.8% 5.7% 32.2% 3 0 3 3 0

23 17/1 5/12/1/0 4 3601 20.3% 6.1% 15.9% 4 0 4 3 0

24 17/1 0/17/1/0 8 173 0.0% -0.3% 3.2% 3 0 6 4 0

25 17/2 5/14/0/0 4 3601 19.1% 6.0% 11.5% 19 53 4 20 40

26 15/4 8/11/0/0 3 3601 17.0% 6.5% 7.3% 12 24 3 3 0

27 18/1 4/15/0/0 5 3600 21.2% 7.4% 4.3% 4 0 5 3 0

28 19/2 4/17/0/0 6 3601 10.7% 0.3% 6.0% 3 0 5 3 0

29 20/3 10/12/1/0 8 3601 34.7% 1.5% 22.6% 3 0 8 3 0

30 18/6 10/14/0/0 7 3601 29.7% 1.6% -9.3% 5 9 6 21 27

inJuliawithJuMPpackageandGurobioptimizer.Weconductthetestrunswithtwosetups:1)fulloptimizationwhereweoptimize theinstancewithallthreestepsandobjectivessequentiallyand2)partialoptimizationwhereweomitStep2.Theresultsforboth setupsaresummarizedinTable3forcomparison.Eachstepofoptimizationhasbeengivenatimelimitof3600s.Allsolutiontimes aremeasuredinsecondsandobjectivevaluesforStep3arepresentedintonsofballastwater.

Additionally,inordertoquantifythesignificanceofmaximizingsafety,wecomparetheaveragetotaldistancebetweendangerous cargounitsafterStep1(originaldistance)withtheaveragetotaldistancefromStep2(optimizeddistance).Theaveragedistanceis calculatedusingthetotaldistancedividedbythenumberofdangerousgoods,whereasthetotaldistanceofasolutioniscalculated accordingtotheobjectivefunction(7)ofthedistanceformulationinSection3.2.Thedistancebetweenslotsondifferentdecksisset as48mtakenfromthestrictestrule4mentionedinthebeginningofthissection.Inadditiontotheaveragetotaldistance,which iswhatweseektomaximize,wealsocomparetheaverageclosestdistancebetweenadangerouscargounitanditsclosestother dangerouscargounitafterStep1(originalclosestdistance)withtheaverageclosestdistancefromStep2(optimizedclosestdistance).

Wecalculatetheaveragedistanceimprovement(Δd.)asthe(optimizeddistance-originaldistance)/originaldistanceandaverage closestdistanceimprovement(Δc.d.)asthe(optimizedclosestdistance-originalclosestdistance)/originalclosestdistance.Positive numberssuggestanimprovement.

Whenweransomepreliminarytestswiththefulloptimization,wenoticedthattheoptimalitygaps(i.e.thegapsbetweenthe integerfeasiblesolutionsandthelowerbounds)inStep2wereverylargeevenafteronehourofrunningtime(i.e.over20%for60%of theinstances)duetosymmetryintheproblem.Attemptstoreducesymmetryhavebeenconductedbyremovinghalfofthe𝑦variables duetothesymmetrycausedbyslot𝑠and𝑠^′.Specifically,weredefinedvariables𝑦_𝑑𝑠𝑠^′where𝑑=1,2,…,||;𝑠=1,2,…,|_𝑑|−1;𝑠^′= 𝑠+1,…,|_𝑑|.However,preliminaryresultsindicatedworseperformance.Therefore,wehavechosenthetechniqueoffixingvariables toreducesymmetrytosomeextent.Thelogicoffixingvariablesisthatwefixonedangerouscargounitoneachdeckataslotthat isthefurthestawayfromtheothers.Thisreducessomeofthesymmetryandtheaveragegapisreducedsignificantlytothenumbers seeninTable3.Smallinstanceswithatotalnumberofdangerouscargounitslessthan10aresolvedtooptimalityinlessthan15s.

Notethateventhoughfixingvariablessignificantlyreducessymmetry,itmightalsoleadtosub-optimalsolutions,whichisseenfor instance24,wherethesafetydistancebecomeslargerwhenapplyingStep2.

TheresultsinTable3comparestheperformanceoffulloptimizationwithpartialoptimizationproposedinthestep-wisestowage optimizationprocess.Forthesetupofpartialoptimization,whereweoptimizethenumberofdangerousoptionalcargounits(Step

Fig.4.Stowageplanforinstance27usingpartialoptimization,aviewoftheshipfromaboveandaft.Colorgreen,yellowandredindicatethe classofcargounitsbeinggeneral,1and2.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtotheweb versionofthisarticle.)

Fig.5. Stowageplanforinstance27usingfulloptimization,aviewoftheshipfromaboveandaft.Colorgreen,yellowandredindicatetheclass ofcargounitsbeinggeneral,1and2.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversion ofthisarticle.)

1)andthenminimizetheballastwaterintake(Step3),all30instancesaresolvedtooptimalitywithinlessthan30s.InStep1, asthenumberofdangerouscargounitsgrows,thecomputationaltimealsoincreases.However,thedifferenceisalmostnegligible sincethemodelrunssofast.InStep3,thecomputationaltimedependsonprimarilytwofactors,thedistributionofthecargoweight, andtheplacementofthedangerouscargounits.Itisthereforenoclearpatternbetweennumberofdangerouscargounitsandthe computationaltime.

Inthecaseoffulloptimization,whichensuresevenmoresafetyregardingthesegregationofdangerouscargounits,theinstances withfewdangerouscargounitscanbesolvedtooptimalitywithinreasonabletime.However,ittakesasignificantamountoftimeto solvethemodelinStep2fortheinstanceswithmorethan10dangerouscargounits,evenwhenweappliedthetechniqueoffixing someofthevariables.ThegainfromincludingStep2isthatthetotaldistanceandtheclosestdistanceamongslotswithdangerous cargounitsareincreasedby5%(Δd.)and36%(Δc.d.)onaverageamongallinstances,respectively.Thisclearlyshowsthatthe safetylevelissignificantlyincreasedbysegregatingthedangerouscargounitsevenbeyondtheminimumrequirementsgivenbythe regulations,thusminimizingtheriskofaccidents.Wedemonstratethisbythesolutionsobtainedforinstance27,asshowninFig.4 forpartialoptimizationandFig.5forfulloptimization.AsstatedinTable3,thetotaldistanceamongdangerouscargounitsfor

instance27isincreasedby7.4%,whichisalsopresentedinthestowageplanoffulloptimizationwhereallthedangerouscargounits arestowedasfarawayfromeachotheraspossible.

AsforStep3,ittakeslongertimetosolveandresultsinworseobjectiveonaverageinfulloptimizationthaninpartialoptimization.

Oneexplanationcouldbethatthefixedstowageplanfordangerousgoodsismadesparsebymaximizingtheirdistanceinbetween, andittakesmorecomputationalpowertosatisfythestabilitywithamoresparselyfixedstowageplanfordangerouscargounits, thereforepotentiallymoreballastwaterisneededaswell.

Thestep-wisestowageplanningapproachhasgreatpotentialforbeingimplementedbyshippingcompaniestoimprovethesafety onboard.Firstandforemost,itensuresthattheplancomplieswiththecomplexsegregationrulesinStep1withinsecondsofcom- putationalruntime.Secondly,itenablesasignificantlybettersafetythroughStep2optimizationbymaximizingthedistanceamong dangerouscargounits.Moreover,thestep-wiseapproachprovidesexpertsthepossibilityandflexibilitytoincorporateadditionalpref- erencesandconstraintstothepreliminarygeneratedstowageplanfordangerouscargounits,beforegeneratinganoptimalstowage planforallcargounitsinStep3thatcanpotentiallyreducefuelandCO₂emissionbyaround6.7%(Jiaetal.,2020).Theapproach aimstoprovidedecisionsupporttotheplannersandcargoofficerstofacilitatetheirdailyoperationsandnottoreplaceanydecision makers.

Thechoiceofimplementingeitherfullorpartialoptimizationdependsonmanyfactors.Shippingcompaniesapplydifferentcut off timefordangerousgoods.Anearliercutoff timeensurestheavailabilityofdangerouscargounits,i.e.thosepresentattheterminal bythetimeofplanning.Thisgivesshippingcompaniesmoretimetoplanforthestowageofdangerousgoods,potentiallyusingStep 2tomaximizethesafety.Computingpowermightalsobeadeterminingfactor.Sincetheresultsaretestedonastandardlaptop tomimictheenvironmentthatisgenerallyattheterminalorontheship,thecomputationaltimecanbedecreasedsignificantly byusingmorepowerfulcomputers,e.g.onthecloud,sothatfulloptimizationbecomesrealisticallyfast.Lastbutnottheleast,the preferencebetweenbeingsaferandcomplying withminimumrequirementsguidestheadoptionoffullandpartialoptimization, respectively.

5. Conclusion

Inthispaper,wehaveaddressedtheimportantplanningproblemofgeneratingoptimalstowageplansforroll-onroll-off ships transportingtrailers(somecontainingdangerouscargo)betweentwoports.Weproposedaplanningapproachwiththeabilityto includeexperts’opinionsforgeneratingamorerobustandflexibleplan.Theplanningapproachincludesthreesteps,eachstepconsist- ingofa(mixed)integerprogrammingmodelsolvedbyacommercialsolver.Step1maximizesthenumberofdangerouscargounitsto transportwhileadheringtotheIMDGCode.Step2,whichisoptional,maximizesthesafetydistanceamongthedangerouscargounits foundinthefirststep.Finally,inStep3,theballastwaterintakeneededtoensurestabilityoftheshipisminimized,asthishasasig- nificanteffectonthefuelconsumption.Inordertotestthestep-wiseplanningapproachwegeneratedanumberoftestinstancesbased onrealdatafromashippingcompany.Thecomputationalresultsshowedgreatpotentialforindustrialimplementationconsidering improvedsafetythroughmaximizingtotaldistance(Δd.)by5%andmaximizingclosestdistance(Δc.d.)by36%;andreducedfuel consumptionandCO₂emissionbyaround6.7%.AsthefirstresearchstudyinthetopicofstowingRoRoshipstransportingdangerous goods,wehopetoprovidefundamentalinsightsandpotentialapproachfortheproblemtobothacademicresearchersandindustrial practitioners.

Futureworkmayincludeimprovedsolutionmethodstoreduce symmetryfor themodelin Step2,which istheone which experiencesthehighest computationaltimes.Itmaybe interestingtofurtherinvestigatethecauseoftheworseperformanceof removingvariablesfromtheperspectiveofsymmetrystudy.Alternatively,itwouldalsobeinterestingtodevelopaheuristicfor solvingtheintegratedprobleminonego.Thiscouldpotentiallyreducethecomputationaltimeandimprovethesolutionquality comparedtothemathematicalthree-stepoptimizationapproachandmake itanevenmoreefficientplanningtoolinapractical setting.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingfinancialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgment

TheresearchwaspartiallysupportedbyECOPRODIGIprojectfundedbyEuropeanRegionalDevelopmentFund#R070.Wewish tothankourindustrialcollaboratorDFDSfortheiropennesstoshare,innovationmindset, andvaluablesupportthroughoutthe conductofresearch.Wearegratefulforthetwoanonymousreviewersfortheirvaluablecomments.