Assessing and improving management practices when planning packaging waste collection systems

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ContentslistsavailableatScienceDirect

Resources,

Conservation

and

Recycling

j ou rna l h o me p ag e:w w w . e l s e v i e r . c o m / l o c a t e / r e s c o n r e c

Assessing

and

improving

management

practices

when

planning

packaging

waste

collection

systems

Tânia

Rodrigues

Pereira

Ramos

a,∗

_,

_Maria

_Isabel

_Gomes

b

_,

_Ana

_Paula

_{Barbosa-Póvoa}

c

a_Instituto_{Universitário}_de_Lisboa_(ISCTE-IUL),_Business_Research_Unit_(BRU),_Av._das_Forc¸_as_Armadas,_1649-026_Lisboa,_Portugal b_CMA,_Universidade_Nova_de_Lisboa,_Campus_da_Caparica,_2829-516_Caparica,_Portugal

c_CEG-IST,_Instituto_Superior_Técnico,_Universidade_de_Lisboa,_Av._Rovisco_Pais,_1049-001_Lisboa,_Portugal

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received14December2012

Receivedinrevisedform25October2013 Accepted18December2013 Keywords: Collectionroutes Serviceareas Routesscheduling Recycling Wastemanagement Vehiclerouting

a

b

s

t

r

a

c

t

Packagingwastecollectionsystemsareresponsibletocollect,withinageographicarea,threetypes ofpackagingmaterials(paper,glassandplastic/metal)thataredisposedbythefinalconsumerinto specialbins.Thosesystemsareoftencharacterizedbyhavinganetworkwithmultipledepotsthatactas transferandsortingstations,andwherethevehiclefleetisbased.However,eachdepotisoftenmanaged independentlyandnotasapartofauniquesystem.Inthiswork,fourcurrenttactical/operationalpractices thatcontributetotheindependentmanagementofeachdepotareanalysed.Thechangeofsuchpractices isinvestigatedandtheirimpactassessedonthetotalcollectioncost.Asolutionmethodologybasedon mathematicalformulationsisdevelopedtoplanserviceareas,vehicleroutesandvehicleschedulestaking intoaccountnewalternativesolutionsinmanagingthesystemasawhole.Suchmethodologyisappliedto arealcasestudyofacompanyresponsibleforthecollectionofthepackagingwastein7municipalitiesin mainlandPortugal.Newserviceareas,collectionroutesandvehicleschedulesaredefinedandsignificant savingsareobtainedintermsofthetotaldistancetravelledaswellasintermsofthenumberofrequired vehicles,resultinginadecreasingofthetotalsystemcost.

1. Introduction

RecyclingofpackagingmaterialswasimposedbytheEuropean Union(EU)tothe MembersStates throughtheEuropean Com-munityDirective94/62/EC,whichhassettargetsforrecoveryand recyclingofpackagingwaste.Tomeetthosetargets,alarge invest-mentinwastemanagementwasmadeinPortugalsinceuntilthen alltheproducedwastewasdumpedwithoutanykindoftreatment. Anewcollectionsystem–theselectivecollection–hadtobe devel-opedgiventhatthetraditionalroutesdefinedforundifferentiated wastedidnotfittheparticularitiesoftherecyclingmaterials: dif-ferentvehicles,differentcollectionratesanddifferentbinlocations. TheGreenDotSystemwascreatedin1996topromote selec-tivecollection,sorting,recoveryandrecyclingofpackagingwaste in Portugal in order to accomplish the targets setby EU. This systemisfunded bypackers/manufactureswho areresponsible forproductsfinal disposal(accordingtotheExtended Producer Responsibilityprinciple)andhadtransferredsuchresponsibilityto anentitydulylicensedforthisactivity(theSociedadePontoVerde (SPV)whichmanagestheGreenDotSystem).Thepackerspaya

∗ Correspondingauthor.Tel.:+351217903412;fax:+351217903904. E-mailaddresses:tania.ramos@iscte.pt(T.R.P.Ramos),mirg@fct.unl.pt (M.I.Gomes),apovoa@ist.utl.pt(A.P.Barbosa-Póvoa).

“greendotfee”foreachpackagesenttothemarketaccordingto itsweightandmaterialsused.Withtherevenuesfromthegreen dotfees,SPVsupportstheselectivecollectionandsortingcosts. Theselectivecollectionandsortingoperationsareperformedby municipalandmulti-municipalwastecollectioncompaniesthat arepaidaccordingtotheweightandtype ofmaterialdelivered toSPV.Thisfinancialsupportvalue,predeterminedbythelatter, intendstocoveronlythecostsincurredwithcollectionand sor-tingoperationsforthepackagingwastematerials,deductingthe costsavoidedwithundifferentiatedcollectionandlandfilldisposal. Therefore, it is crucial thatwaste collectioncompanies operate theircollectionandsortingsystemsinanefficientway,otherwise excessivecostsareincurredthatwillnotberefunded.Togetmore insightsontheGreenDotSystemandontheefficiencyofthewaste collectioncompanies,pleaseseeMarquesetal.(2012).

Toincreaseefﬁciencyinrecyclablewastecollectionsystems,an optimizedsolutioninterms ofserviceareas,vehicleroutesand vehicleschedules shouldbepursuit.In packagingwaste collec-tionsystemsoperatinginPortugal,collectioncostsrepresentabout 66–69%,whilesortingcostsrepresentabout11–30%ofthetotal costs(APA,2008).Thus,savingsinthecollectioncostsareofgreat impactonthetotalcosts.

Theaimofthisworkistobuildnewsolutionssoastoreduce thecollectioncostsbyoptimizingtheserviceareas,vehicleroutes and vehicle schedules under alternative network management 0921-3449/$–seefrontmatter © 2013 Elsevier B.V. All rights reserved.

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scenarios.Themajorityoftherecyclablecollectionsystemshave morethanonedepotinthenetworkanditisacurrentpractice tomanageeachdepotindependently.Therefore,serviceareasare definedbydepot,vehicles arefixedtoa depotandonly closed routesareallowed(routesstartingandendingatthesamedepot). Thisworkintendstoassessandimprovefourmanagement prac-tices,commonlyusedbythepackagingwastecollectionsystems operatinginPortugal,thatinvolvethefollowingassumptions:(1) respectthemunicipalitiesboundariestodefineserviceareas;(2) theserviceareasaredefinedbydepot;(3)thevehiclesarefixed toa depotand canonly performroutes starting and ending at thatdepot;and(4)onlyclosedroutesareallowed.Toachievethis goalthenextquestionsareraisedandconsequentlyanalysed:if thesystemsresponsibleforthecollectionandsortingofpackaging wastearemulti-municipal,whyshouldthegeographicboundaries ofthemunicipalitiesdefinetheserviceareaofeachdepot?If multi-plematerialsarecollectedinindependentroutes,whynothaving serviceareasbyrecyclablematerial?Ifthesystemhasmultiple depotsandallvehiclesbelongtothecompany,whynotsharing theresources(vehicles)amongdepots?Ifthesystemhasmultiple depots,whynotallowroutestostartatadepotandtoendata differentone?

Themaincontributionofthecurrentworkisthenonassessing theimpactof breakingupwiththe currenttactical/operational practicesthatconsiderthatdepotsaremanagedindependentlyand notinanintegratedway.Forthat,weproposedauniﬁedsolution methodologythatiscapabletoplanthecollectionsystems explor-ingmoreefﬁcientpractices.Thesolutionmethodologyisapplied toarealpackagingwaste collectionsystemandtheresultsare comparedwiththecurrentsolutionwherethefourpracticesare used.

Theremainderof thepaperisstructured asfollows.Section 2 brieﬂy reviewsthe literature onrouting problems. Section 3 describesthemain features ofthe case study.In Section 4the solutionmethodologyisdescribedandtheresultsofthe applica-tiontothecasestudyarepresentedatSection5.Finally,Section 6 concludes the paper and draws some future research direc-tions.

2. Literaturereview

Recyclinghas several environmental and economic benefits suchasmitigatingresourcescarcity,decreasingdemandfor land-fillspaceandinvolvingsavingsinenergyconsumption(Craighill andPowell,1996).However,theactivityofrecyclablewaste col-lectionhasalsoseveralenvironmentaland economiccostsasit isbasicallyatransportationactivity.Todiminishsuchcosts, col-lectionroutes shouldbedefinedinorder tominimizethetotal distancetravelledorthetotalroutingcost.Theproblemof defin-ingtheoptimalcollectionroutesisknownintheliteratureasthe VehicleRoutingProblem(VRP).Thisproblemiswidelystudiedand severalmethodswereproposedtosolveitinthelastdecades(see therecentsurveysofGoldenetal.,2008;Laporte,2009).However, theroutingproblemthatappearsinthepackagingwastecollection systemsgoesfarbeyondtheclassicalVRP.Thevehiclefleetisbased atmultipledepots,multipleproductshavetobecollectedand dif-ferentcollectionfrequenciesareobservedforeachproductandsite. Therefore,theroutingproblemarisinginthepackagingwaste col-lectionsystemsistheMulti-Product,Multi-DepotPeriodicVehicle RoutingProblem(MP-MDPVRP),whereithastobedecidedfrom whichdepotthemultipleproductsateachsiteshouldbecollected, inwhichdayoftheplanninghorizoneachsiteshouldbevisitedand whatshouldbethecollectionvisitsequenceinordertominimize thetotalroutingcost.ParthanadeeandLogendran(2006)presented amathematicalmodelfortheMP-MDPVRPandthreetabusearch heuristicsaredevelopedtosolveit.However,theauthorshadcalled

theproblem“multi-product”asseveralproductshavetobe deliv-eredtocustomers,butthesamecustomercanbevisitedbyvehicles fromdifferentdepots.Inourcase,“multi-product”referstothefact thatthethreepackagingmaterialsineachsitehavetobecollected fromthesamedepot.

Consideringonlyoneproduct,theMDPVRPhasbeenstudied byHadjiconstantinouand Baldacci(1998)andrecentlybyVidal etal.(2012).Intheformerwork,a heuristicapproachbasedon tabusearchhasbeendeveloped.Theheuristicalgorithmisapplied toarealcaseofautilitycompanythatprovidespreventive mainte-nanceservicestoasetofcustomers.Thiscompanyhas17vehicles, basedon9depots,toserve162customerswithafrequencythat canvary from oncea day toonce everyfourweeks. The large scale problemmotivatedtheauthors toapplya heuristic algo-rithminstead of an exact one. In the latter work, theauthors proposedahybridgeneticalgorithmwithadaptivediversity con-troltosolveseveralclassesof multi-depotandperiodicvehicle routingproblems,includingtheMDPVRP,andappliedittoseveral testinstances.

Consideringonlyonedepot,severalheuristicapproacheshave beendevelopedforthePVRPwhereaplanninghorizonofseveral daysisconsideredascustomershavedifferentvisitingfrequencies. BeltramiandBodin(1974),RusselandIgo(1979)andTeixeiraetal. (2004)developedheuristicalgorithmsforthePVRPwhichwere appliedtowastecollectionproblems.MourgayaandVanderbeck (2007)presenteda columngenerationprocedurefollowed bya roundingheuristictosolveaPVRPwithtwoobjectives: minimiz-ingtotaldistancetravelledandbalanceworkloadamongvehicles. OtherheuristicapplicationstoPVRPcanbefoundinChristoﬁdes andBeasley(1984),GaudiosoandPaletta(1992),Chaoetal.(1995), Cordeauetal.(1997)andAlonsoetal.(2008),tonameafew.

Consideringasingletimeunitplanninghorizonwithmultiple depots,severalworkshavebeenpublishedtotackletheMDVRP. Goldenetal.(1977),SalhiandSari(1997),Cordeauetal.(1997), ThangiahandSalhi(2001),LimandWang(2005),Lauetal.(2010) aresomeworkswhereheuristicsandmeta-heuristicsapproaches were developed. Laporte et al. (1984, 1988) and Baldacci and Mingozzi(2009)haveproposedexactmethodstosolvetheMDVRP. Among the works on multi-depot problems, we highlight one whereinter-depotroutesareconsidered.Crevieretal.(2007)study anextensionoftheMDVRPinwhichvehiclesmaybereplenishedat intermediatedepotsalongtheirroute(Multi-DepotVehicle Rout-ing Problem with Inter-DepotsRoutes). The authors propose a heuristiccombiningtheadaptativememoryprincipleandatabu searchmethodforthegenerationofasetofroutes,andaninteger programmingmodelintheexecutionofasetpartitioning algo-rithmforthedeterminationoftheleastcostfeasiblerotations(the authorsdeﬁnerotationasthesetofroutesassignedtoavehicle).

Inwastecollectionproblems,thereisanothercriticalaspectto address: theestimationof thewasteamounttocollectat each bin.The majorityofworks usesdeterministicinputdata. How-ever,recentworkslikeJohansson(2006),Faccioetal.(2011)and Anghinolfietal.(2013)haveapproacheddynamicroutinginwaste collectionproblems,wheremoderntraceabilitydevices,like volu-metricsensors,identificationRFID(RadioFrequencyIdentification) systems,GPRS(GeneralPacketRadioService)andGPS(Global Posi-tioningSystem) technology, permittoobtaindata in realtime, whichisfundamentaltoimplementanefficientroutingplan.The basicideaisthat,iftherealtimepositionandreplenishmentlevel ofeachvehicleareknown,aswellastherealtimewastelevelat eachbinandwhichbinshavebeenvisited,itispossibletodecide whichbinsshouldbeemptiedandwhichcanbeavoidedatacertain time.Thisallowsanoptimizationoftherouteplanandtominimize covereddistanceandnumberofvehiclesneeded,which,asa conse-quence,wouldminimizetraveltime,numberofload–unloadstops, exhaustemissions,noiseandtrafficcongestion(Faccioetal.,2011).

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Fig.1.Depot’slocationandcurrentserviceareas.

3. Casestudy–arealpackagingwastecollectionsystem

In Portugal there are 31 recyclable collection systems (SPV, 2011),eachoneresponsibleforacertainnumberofmunicipalities (Portugalhasatotalof308municipalities).Ourcasestudyfocuses onthecompanyresponsiblefortherecyclablecollectionsystem coveringsevenruralmunicipalitieswithatotalareaof6400km2_. Thiscompanyoperatesfivedepotswherethecollectionvehiclesare based.Oneofthedepotsoperatesalsoasasortingstation(depot 235,seeFig.1).Theremainingfourareonlytransferstationswhere thepackagingwasteisconsolidatedandafterwardstransferredto thesortingstation.Givensuchlogisticsnetworkconfiguration,two typesoftransportationflowsneedtobeconsidered,namely,the inboundflowfromthecollectionsitestothedepotsandthe out-boundflowfromthedepotstothesortingstation.Thecollectionis performedbyavehiclefleetwithnocompartments,soeach pack-agingmaterialhastobecollectedinseparatedroutes.Thereare 651Glassbins,513Paperbinsand458Plastic/Metalbinsspread over230localities(seeFig.1).Itisassumedthatacollectionsite correspondstoalocalityinsteadofanindividualcontainerinorder toreducetheproblemsize.Duetotheproximityofthe contain-erswithinalocality(anaveragedistanceof650misobserved)it ispracticabletotreatthecontainerstocollectwithinalocalityas asinglenode.Therefore,acollectionsiteaggregatesoneormore containersofoneormorerecyclablematerials,meaningthatthe distancetravelledandthetimespentwithinacollectionsitehave tobeconsidered.

Allthecollectionroutesstartatadepot,visitseverallocalities collectingasingletypeofmaterial,andreturntoadepottounload. Collectionisperformedﬁve-daysaweek,8hperday,during day-time.Itisassumedthatthecollectionsitescanbecollectedatany timeofthedaywithintheworkingperiod,i.e.,between10a.m. and19p.m.(withalunchbreakofanhour).Notimewindowsare consideredforeachsite.

Thethreerecyclablematerialshavedifferentcollection frequen-cies.Glasshastobecollectedonceamonth,Plastic/Metaleverytwo weeksandPapereveryweek.Therefore,eachroutehastobe sched-uledinafour-weekplanninghorizonthatistoberepeatedevery fourweeks.Duetovehiclevolumecapacityconstraintsandtaking intoaccounteachmaterial’sdensity,vehiclescanloadamaximum of4500kgofGlass,3400kgofPaperand600kgofPlastic/Metal. Fortheoutboundtransportation,i.e.,fromthedepotstothe sor-tingstation,largervehiclesareusedwhereweightcapacitiesare increasedto12,000kgforGlass,4000kgforPaperand2000kgfor Plastic/Metal.Furthermore,oneofthecompanypoliciesisthateach depotisresponsibleforafixedsetofcontainers,thusthereisa needofdefiningthedepotsserviceareas.Nowadays,thecompany operatesserviceareasconsideringthemunicipalities’boundaries. Moreover,allrecyclablematerialsateachcollectionsitehavetobe collectedfromthesamedepot,meaningthateachdepothasonly oneserviceareacommontoallrecyclablematerials.Thecurrent serviceareasaredepictedatFig.1,whichtogetherwiththe cur-rentvehicleroutingplanimplyanaverageof28,000kmdrivenper month(fourweeks)andavehiclefleetof9vehicles.

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Fig.2.Scenario’sdescription.

Giventhecurrentsolution’scost,thecompanyenvisagesthe restructureoftheirserviceareasandvehicleroutesplaninorderto decreasesuchvalue.Anoptimizedplanispursuedwhereservice areas,vehicleroutesandrouteschedulingcanberedeﬁnedina breakthroughscenario, i.e.,where service areas do nothave to respectmunicipalitiesboundaries,can bedeﬁned bypackaging material, resourcescan be sharedamong the depotsand open routesbetweendepotsareallowed.

4. Solutionmethodology

Accordinglytothefourpracticesusedbythepackagingwaste collectionsystemoperatinginmainlandPortugal,fourscenarios willbestudywhereineachpracticeisbrokencumulativelyuntil thebreakthroughscenariois reached(seeFig.2).Theﬁrst sce-nariobreakswiththepractice“respectmunicipalitiesboundaries” andmaintainstheremainingthree.In thiscase,theproblemto besolvedistheMulti-Product,Multi-DepotPeriodicVehicle Rout-ingProblem(MP-MDPVRP).Thesecondscenariobreakswithtwo practices,“respectmunicipalitiesboundaries”and“serviceareasby depot”,andmaintainstheremainingtwo.Inthiscaseeach pack-agingmaterialissolvedindependently.Therefore,aMulti-Depot PeriodicVehicleRoutingProblem (MDPVRP)for eachpackaging materialhastobesolved.Thethirdscenariobreakswithan addi-tionalpractice,“vehiclesareﬁxedtoadepot”whilemaintainingthe “closedroutes”practice.HereaMDPVRPwithSharedResourcesis solved,wherethevehiclebasedatonedepotcanperformcollection routesstartingandendingatadifferentdepot.Inthisscenario,only onerelocationmovementisallowedpervehicle,i.e.,avehiclebased atdepoticantraveltodepotjtoperformclosedroutesfromdepot jandthenreturnstohomedepoti.Itisnotallowedtwoormore relocationmovements,suchas:avehiclebasedatdepotitravelling todepotjandthentodepothandreturningtodepoti.Finally,in thefourthscenario,thebreakthroughscenario,thefourpractices arebrokenandaMDPRVPwithInter-DepotRoutesissolved.Atthis

Fig.3.Solutionmethodologymainmodules.

pointopenroutesbetweendepotsareallowedbut,attheendofa workingday,thevehicleshavetoreturntotheirhomedepot.

Tosolvetheabovescenariosweproposedasolution method-ologytodefineserviceareas,vehicleroutesandschedules.This approachinvolvesfourmainmodules,asitisshownatFig.3.The firstmoduledefinesserviceareasbyrecyclablematerial,thesecond moduledefinesserviceareasbydepot,thethirdmoduledefines thefinalcollectionroutesandthefourthmoduledefinesvehicle schedulesaccordingtothecharacteristicsofeach scenario.This solutionmethodologyisanextensionofthemethodproposedin Ramosetal.(2014)whereonlyserviceareasandvehicleroutesare definedfortheMP-MDVRP.Theperiodicissue(routesscheduling), sharedresourcesandopenroutesarenottackledinthementioned work.

Forscenario1, wherea MP-MDPVRPissolved,modules 1–3 definetheserviceareasandthefinalvehicleroutesforeachdepot andmodule4aschedules,withintheplanninghorizon,theroutes generatedbymodule3(seeFig.4).Forscenario2,whereaMDPVRP issolved,module1definestheserviceareasforeachpackaging material,module3definesthefinalcollectionroutesandmodule 4aschedulestheroutes.Forscenarios3and4,onlythescheduling modulediffersfromscenario2.Atscenario3,thescheduling mod-ule(module4b)takesintoaccountthatthevehiclescanbeshared amongdepots.Atscenario4,besidestakingintoaccounta shared-resourcessolution,theschedulingmodule(module4c)considers alsoallthevehicleroutes definedalong thesolutionprocedure (openandclosedroutesdefinedbymodule1)togetherwiththe vehicleroutesdefinedatmodule3.

Eachmoduleofthesolutionmethodologyinvolves mathemat-icalformulations.Eachmodulewillbedetailedinthefollowing section.

4.1. Modulesdescription

4.1.1. Module1:serviceareasbyrecyclablematerial

Todeﬁneserviceareasbyrecyclablematerial,aMDVRPhasto besolvedforeachmaterial.Asmathematicalprogrammingsolvers cannotsolvelargesizeinstancesoftheMDVRP,weadoptedthe solutionmethoddevelopedintheworkofRamosetal.(2014)as

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m=1

1.1 Single-Product, MDVRP-MCO

(with capacity and duraon Constraints)

Are all routes closed?

1.2 Single-Product, MDVRP

(for each open route)

m=|M|?

Stop

No Yes

m=m+1

Yes No

Fig.5.SolutionmethodtosolvetheMDVRP.

showninFig.5.Noticethatmrepresentsthepackagingmaterials, and|M|thetotalnumberofmaterialstobecollected.

Firstly,itissolvedaSingle-Product,MDVRPwithMixedClosed andOpenRoutes(MDVRP-MCO).Thisprobleminvolvesthe deﬁni-tionoftheoptimalroutestocollectonesingleproduct,considering anetworkwithmultipledepotsandallowsclosedandopenroutes betweendepots.Asinputdata,thismodulerequiresthedistance betweeneachnode(collectionsitesanddepots),theweighttobe collectedandtheservicetimeateachcollectionsite(considering onlyonerecyclablematerial),thevehiclecapacityandthe max-imumtimeallowedforaworkingday.Theoutputwillbeaset ofcollectionroutes,wheresomeroutesstartandendatthesame depot(closedroutes)whileothersstartandendatdifferentdepots (openroutes).Sincethisformulationallowsclosedandopenroutes betweendepots,itshouldbeguaranteethatthenumberof vehi-cleroutesstartingandendingateachdepotisthesame.Notethat thenumberofvehicleroutesisnotconstrainedbecausethe vehi-cleﬂeetsizeisoneofthemodel’soutputs.Forfurtherdetailssee Ramosetal.(2013).

Ifopenroutesareproduced,aMDVRPissolvedtoclosethem sothatserviceareasaredeﬁned.Anopenroutelinkstwo differ-entdepots,meaningthat collectionsites withinsuchrouteare assignedtotwodepotsinsteadofjustone.BysolvingaMDVRP tothecollectionsitesthatintegrateeachopenroute,closedroutes areobtained.

ConsideringthepackagingmaterialsetM,thissolutionmethod havetoberunforallmaterialsincludedinsetM.

4.1.2. Module2:serviceareasbydepot

Ifserviceareas bydepotarerequired,module2 isexecuted (this willonly beof usein scenario 1).Service areasby depot implythatallpackagingmaterialsateachcollectionsiteare col-lectedfromthesamedepot.Comparingtheserviceareasdeﬁned foreachpackagingmaterialinthepreviousmodule,somesitesmay notrespectthatrule.Suchsitesarenamedas“unclearsites”since thereisnoagreementamongthepackagingmaterialsconcerning theirdepotassignment.Forthosesites,aMulti-Product,MDVRPis solved,whereaconstraintensuresthatallmaterialsateach col-lectionsiteare allocatedtothesamedepot.Thisformulationis detailedinRamosetal.(2014).

ThemathematicalformulationfortheMulti-Product,MDVRPis onlycapableofsolvingsmallinstances,i.e.,problemswithasmall numberofunclearsites(lowerthan30unclearsites,accordingto preliminarytests).Therefore,onewaytoreducetheproblemsizeis toconsiderthenumberofunclearsitesbetweeneachpairofdepot ratherthanconsideringallunclearsitesatonce.Forexample,asite

1 2 3

...

n-1 n

Fig.6.Planninghorizonrepresentationusingadirectedcyclegraphwithnvertices.

is“unclear”ifitiscollectedfromdepot1formaterialGlassandfrom depot2formaterialsPaperandPlastic/Metal.Therefore,thissite isunclearbetweendepots1and2.Sincethislackof“clearance” concernsatleasttwodepots,suchpartitionoftheunclearsites setwouldbeanaturalwaytodoso.If,withsuchdecomposition, thenumberofunclearsitesforeachpairofdepotsisstillabove 30,aheuristicassignmentrulehastobedeﬁned.Inthiscase,the unclearsitesareplacedaccordingtotheassignmentmadebythe materialwiththehighestcollectionfrequency.Someeffectiveness testsperformedintheworkofRamosetal.(2014)supportsuch heuristicrule.

4.1.3. Module3:ﬁnalcollectionroutes

Afterdefiningtheserviceareas(bydepotorbyrecyclable mate-rial),module3isexecutedtosetthefinalvehicleroutesforeach depot and each packagingmaterial. Alongwith modules1 and 2vehicleroutesaredefined,buttheaimofthosemodulesisto establishserviceareasthroughthedefinitionofthevehicleroutes. Therefore,thedefinedroutescanbeimprovedoncethefinalservice areasareset,andthisiswhatmodule3isdesignedfor.TheCVRP formulationproposedbyBaldaccietal.(2004)isextendedtodeal withdurationconstraints.

4.1.4. Module4:routescheduling

Finally,theschedulingmoduledefinesforeachdayofthe plan-ninghorizonwhichrouteshouldbeperformed,ensuringthatthe collection frequency over the planning horizon is fulfilled and aminimumanda maximumtimeintervalbetweenconsecutive collections isobserved inorder topreventcontainers overflow. Moreover,itisdesirablethatthesamerouteisperformedbythe samevehicle/driver.

Eachroute k∈K(deﬁnedinthe previousmodule)is charac-terizedby(1)distancedk;(2)durationbk,whichincludestravel, serviceandunloadingtimes;and(3)loadLk.LetKmbetheroutes subsetformaterialm.Thecollectionsitesbelongingtoroutekare givenbyabinaryparameterikthatequals1ifcollectionsitei∈Vc belongstoroutek;and0otherwise(Vcisthenodesetofcollection sites).Thestartingandendingdepotsforroutekarealsomodelled bybinaryparametersSkiandEki,respectively:Skiequals1ifroutek startsatdepoti∈VdwhileEkiequals1ifroutekendsatdepoti∈Vd (Vdisthenodesetfordepots).

LetGbethevehicleset.Asvehiclesareﬁxedatthedepot, con-siderthebinaryparameter˛giequalsto1ifvehiclegbelongsto depoti;and0otherwise.

Thecollectionfrequencyofeachcollectionsiteiwithrecyclable material m is given by fim representinghow oftena collection sitehastobevisitedduringtheplanninghorizon.Theminimum andmaximum intervalbetweentwoconsecutivecollectionsfor packagingmaterialmaregivenbyImandAm,respectively.Given theperiodicnatureoftheproblem,theschedulesolutionistobe repeatedover the nextperiod. Therefore, theplanninghorizon deﬁnedbyadiscretesetsuchasT=_{1,...,n_}ishereconsidered tobecyclicalandmodelledasadirectedcyclegraphofsizen,Cn, basedupongraphtheory(seeFig.6).

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Theschedulingmodelhasasinglesetofdecisionvariablesxktg. Thesearebinaryvariablesthatequal1ifroutekisperformedon daytbyvehiclegandequals0otherwise.

The scheduling module involves three variants:module 4a, wheretheroutesdefinedatmodule3arescheduledandthe vehi-clearefixedateachdepot;module4b,wheretheroutesdefined atmodule3arescheduled,butthevehiclescanbesharedamong depots;andmodule4c,wherethevehiclescanbesharedbutthe setofroutestobescheduledisaugmentedwiththeroutesdefined alongmodule1.Formodule4a,theobjectivefunctionisgivenby Eq.(1),whileEq.(2)modelstheobjectivefunctionofmodules4b and4c. Min

k∈K

t∈T

g∈G dkxktg+

j∈Vs

i∈Vd

m∈M

k∈Km

t∈T

g∈G EkixktgLk Om2dij (1) Min

k∈K

t∈T

g∈G dkxktg+

j∈Vs

i∈Vd

m∈M

k∈Km

t∈T

g∈G EkixktgLk Om2dij −

j∈Vs

i∈Vd

m∈M

k∈Km

t∈T

g∈G ˛_gjS_kiE_kix_ktgL_k Om2dij +

g∈G

k∈K

t∈T

i,j∈Vd 2˛giSkj(1−Eki)xktgdij (2)

Eq.(1)considersthedistancetobetravelled(inboundand out-bounddistance)whenvehiclesareﬁxedatdepotsandcannotbe shared.Theoutbounddistance(secondterm)considersthenumber ofround-tripsbetweenthesortingstationandthedepots mod-ellingtransferofallmaterialscollectedbyeachdepot(Vsisthe sortingstationsset).Tocomputetherequirednumberof round-trips,one considersthetotal loadcollectedby adepot andthe vehiclecapacityforoutboundtransportationforeachmaterialm (Om).Notethatthenumberofround-tripsisnotroundupwarddue totheﬁnitudeofthemodelledplanninghorizon.Infact,theactual managementofsuchasystem,round-tripsarerepeatedatalltime periods.

Eq.(2)considersthatalthougheachvehiclebelongstoadepot, vehicles can be shared, meaning that they can perform routes assigned toa differentdepot. Therefore, the objective function needstoaccountfor:(i)thedistancetobetravelledbetweendepots whenavehiclebelongingtodepotiisperformingroutesofdepot j(fourthtermofEq.(2));(ii)thedistancerelatedtotheoutbound transportationwhenavehiclebasedatthesortingstationperforms routesassignedtoadifferentdepot(thirdterm).Inthelattercase, theloadcollectedwillnowbeunloadedatthesortingstationand notatthetransferdepot,sothecorrespondingoutbounddistance shouldnotbeaccountedintheobjective.

Someconstraintsarethentobeconsideredinthescheduling models:

k∈Km

t∈T

g∈G xktgik=fim

∀

i∈Vc,

∀

m (3)

Constraint(3)ensuresthatacollectionsiteiwithmaterialmhas tobecollectedfimtimesovertheplanninghorizon.

k∈K xktgbk+

k∈K

j∈Vd j /=i 2˛giSkj(1−Eki)xktg

v

ij≤H

∀

t,

∀

g,

∀

i∈Vd (4) Constraint(4) statesthat theroutetotal durationperformed byvehiclegondaytwillnotexceedthemaximumnumber of

workinghoursperday(H).Ifavehicleg,belongingtodepoti, per-formsaroutestartingatdepotj,thetraveltimebetweeniandj (

v

ij)isaccounted.Itisassumedthatsitescanbecollectedanytime duringtheworkingday,i.e.,notimewindowsarebeingconsidered concerningthecollectionsites.

g∈G

t∈Pt Im+1 xktgik≤1

∀

i∈Vc,

∀

k∈Km,

∀

m,

∀

t (5)

Constraint(5)statesthatthesamerouteformaterialmhasto beperformedwithaminimumtimeintervalofIm.Thisconstraint ensuresthatnomorethanonevisitwilloccurinanyIm+1 consecu-tivedays.ThisismodelleddeﬁningIm+1consecutivedaysasapath oflengthIm+1ofthecyclegraphCn.Indetail,Pt

Im+1⊆Cndeﬁnesthe setofIm+1consecutivedaysstartingatdayt,inarollinghorizon.

g∈G

t∈Pt_Am

xktgik≥1

∀

i∈Vc,

∀

k∈Km,

∀

m,

∀

t (6)

Constraint(6)assuresthemaximumintervalAmbetween con-secutivecollections.Thus,atleastonevisitmustoccurwithinAm consecutivedays.

Pt

Am⊆CndeﬁnesthesetofAmconsecutivedaysstartingatday t,inarollinghorizon.

Fig.7showsanillustrativeexampleonhowconstraints(5)and (6)work.Consideracycleof20days,Im=3daysandAm=8days. Constraint(5)ensuresthatatmostonecollectionmayoccurwithin fourconsecutivedays.Ifsiteiiscollectedatday1,itcannotbe col-lectedatdays2,3and4(P1

4),neitheratdays18,19and20(P418). Constraint(6)ensuresthatwithin8consecutivedays,atleastone collectionmustoccur.Ifsiteiiscollectedatday1,asecond col-lectionmustoccurbetweenday2andday9(P2

8).Assumingsitei hastobecollectedfourtimesduringtheplanninghorizon(fim=4), Fig.7showsthenonepossibleschedule.

xktg+xktg≤1

∀

k,

∀

g, g∈G, g /=g,

∀

t, t∈T, t /=t (7) Constraint(7)ensuresthatthesameroutehastobeperformed bythesamevehiclealongtheplanninghorizon.Therefore,the col-lectionsitesarealwaysvisitedbythesamevehicleanddriver. xktg∈{0,1}

∀

k∈K,

∀

t∈T,

∀

g∈G (8)

Variablesdomainisgiveninconstraint(8).

Constraints(3)–(8)areappliedinmodules4aand4b.Formodule 4c,constraints(9)–(11)shouldbeaddedbecausethereare alterna-tiveroutestobeselected,giventhatallroutesdeﬁnedalongthe solutionprocedureareconsideredintheschedulingmodule.

k∈Ko SkiEkjxktg=

k_∈Ko SkjEkixktg

∀

g,

∀

t,

∀

i, j∈Vd, i/=j (9)

Constraint(9)guaranteesthatallvehiclesreturntotheirhome depotincaseopenroutesareselectedaspartofthesolution(Kois thesubsetofopenroutes).Ifanopenroutekstartingatdepotiand endingatdepotjbelongstothesolution,oneopenroutek’starting atdepotjandendingatdepotimustalsointegratethesolution.

g∈G

t∈Pt Im+1 xktgik+

g∈G

t∈Pt Im+1 xktgik≤1

∀

i∈Vc,

∀

k, k∈Km, k /=k,

∀

m,

∀

t (10)

Incaseofmodule4c,theconsecutivecollectionscanbe per-formedbythesamerouteorbytwodifferentroutes.Constraint(10) ensurestheminimumtimeintervalincasetwodifferentroutesare

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Const

raint (5)

Im

=3 days

{

,12,3,4

}

1 4= P

{

18,19,20,1

}

18 4 = P

Constraint (6)

Am

=8 days

{

2,3,4,5,6,7,8,9

}

2 8 = P

{

19,20,,12,3,4,5,6

}

19 8 = P

Admissibl

e Sc

hedule w

ith f

im

=4

11 12 13 14 15 6 7 8 9 10 16 17 18 19 20 1 2 3 4 5

X

11 12 13 14 15 6 7 8 9 10 16 17 18 19 20 1 2 3 4 5

X

11 12 13 14 15 6 7 8 9 10 16 17 18 19 20 1 2 3 4 5 11 12 13 14 15 6 7 8 9 10 16 17 18 19 20 1 2 3 4 5 11 12 13 14 15 6 7 8 9 10 16 17 18 19 20 1 2 3 4 5

X

Fig.7. Illustrativeexampleonhowconstraints(5)and(6)operate.

collectingthesamesitei,atconsecutivecollections.

g∈G

t∈Pt_Am xktgik+

g∈G

t∈Pt_Am xktgik≥1

∀

i∈Vc,

∀

k, k∈Km, k /=k,

∀

m,

∀

t (11)

Constraint(11)guaranteesthemaximumintervalAmbetween consecutivecollectionswhentwodifferentroutescollectthesame sitei.

Theoutputoftheschedulingmodulesis aschedulefor each vehicleestablishing,ineachday,theroutestobeperformed.

5. Resultsanalysis

Thesolutionmethodologyproposedisappliedtothe packag-ingwastecollectionsystemdescribedatSection3inordertoplan newserviceareas,newcollectionroutesandnewvehicle sched-ules,whileassessing,regardingthetotalcollectioncost,theimpact ofbreakingupwiththecurrentpractices.

Themathematicalformulationsdevelopedareimplementedin GAMS23.7andsolvedthroughtheCPLEXOptimizer12.3.0,onan IntelXeonCPUX5680@3.33GHz.

Theresultsforeachscenariowillbeshown,followedupbya cost-analysisconsideringthedistancetravelledandthenumberof vehiclesrequired,assessingtheimpactofeachcurrentpractice. 5.1. Scenariosresults

5.1.1. Scenario1:municipalities’boundaries

The ﬁrst module is run for each one of the three packag-ingmaterials:Glass,PaperandPlastic/Metal.Theresultsforthe MDVRP-MCO(formulation1.1)areshownatTable1.

Thedifferenceinthetotaldistancetravelledamongthethree materialsisexplainedbythecollectionfrequencyofeach mate-rialintheplanninghorizon.Glasshastobecollectedonce,while Plastic/MetaltwiceandPaperhastobecollectedfourtimes.Onthe otherhand,Plastic/Metalisthematerialwiththelowestdensity amongthematerials,andthusthevehicleweightcapacityforsuch materialissmallerforthesamevehiclevolumecapacity. There-fore,moreroutesarerequired,andconsequently,moredistanceis travelled.

Table1

Resultsforformulation1.1.

Packagingmaterial Total distance(km) No.closed routes No.open routes Totalno. routes Glass 3381 18 13 31 Paper 11,595 21 0 21 Plastic/metal 8047 38 11 49 Total 23,023 77 24 101

RegardingPaper,onlyclosedroutesareproposedinthefinal solution, thus there is no need to solve the MDVRP (formula-tion1.2).Thiscanbeexplainedbythefactthatthere isalittle differencebetween theinbound and outboundvehicle’s capac-ityforPaper.Paperhasthesmallestincreaseinvehiclecapacity fortheoutboundcapacity(anincreaseofabout17%–4000kgvs. 3400kg–againstanincreaseof166%forGlassand233%for Plas-tic/Metal)whatfavourstheassignmentofmoresitestothesorting stationtoavoidtheoutboundtransportation(asit willimplya greaterdistancetobetravelledasthevehiclecapacityissmall).For thosesitesassignedtothesortingstation(about70%ofallsites), closedroutesaredefined inordertoavoid theoutbound trans-portation.Fortheremainingsites,ifopenroutesweredefined,the increaseintheoutboundtransportationwouldnotbecompensated bythedecreaseoftheinboundtransportationbydefiningopen routes.

Fortheremainingtwomaterials,theopenroutesaretobe rede-ﬁnedintoclosedonesbyformulation1.2inFig.5.Asﬁnalresult, Glasshas17routes,Paper21andPlastic/Metal12routes,allclosed ones.

Theserviceareasproducedforthethreematerialsareshownin Fig.8.

Asdifferentserviceareasaredefined,andScenario1requires equalserviceareasamongthethreematerials,module2isrunfor theunclearsites.Whenthethreeserviceareasareoverlapped,the unclearsitesarehighlighted(seeFig.9(a)).101unclearsitesare identified:8sitesareunclearbetweendepot231anddepot233, 20sitesbetweendepots232and235,28sitesbetweendepots234 and235andtheremaining45sitesareunclearbetweendepots233 and235.TheMulti-Product,MDVRPformulationwasabletosolve thesub-problemswith8,20and28unclearsites.However,the sub-problemwith45unclearsiteshastobesolvedbytheheuristicrule. ThefinalserviceareasforScenario1arebuiltasshowninFig.9(b).

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Fig.8.Serviceareasobtainedforeachpackagingmaterial. Ramosetal.(2014).

Havingtheserviceareasdeﬁned,module3isappliedto estab-lishtheﬁnalcollectionroutesforeachdepotandeachpackaging material.Asaresult,oneobtains32routestocollectGlass,21to collectPaperand49tocollectPlastic/Metal.Thetotaldistance trav-elledintheplanninghorizonis24,405km(seeRamosetal.(2014) forthecomputationalresults).

Module4isthenappliedtoschedulethe102collectionroutes. Itisassumedaminimaltimeinterval(Im)tocollectPaperof4days, forPlastic/Metal9 daysandforGlass20 days,anda maximum interval(Am)of5,10and20days,respectively.Giventheservice areasobtainedandconsideringthatvehiclesareﬁxedatdepots,

9vehiclesarerequired,withthefollowingdistributionbydepot (Fig.10).

TheninevehicleschedulesareshowninFig.11,wheretheusage time(inminutes)isidentiﬁedforeachdayoftheplanning hori-zon.It canbe seenthat vehicle4, basedat depot 234,hasthe lowestusageratesincethisdepotis responsibletocollectonly threecollectionsites(seeFig.9(b)).Twovehiclesareassignedto depot233:vehicle2operates3950minpermonth,withausage rateof41%(3950min/(20days×480min));whilevehicle3 opera-tes5632min,withausagerateof59%.Giventhesumoftheusage rates,amacroanalysiscouldconcludethattheworkloadofdepot

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Depot 231 Depot 232 Depot 233 Depot 234 Depot 235

1

0

2

1

5

Fig.10.Vehicleﬂeetdistributionamongdepotsinscenario1.

Table2

Computationalresultsformodule4ainscenario1.

Numberof variables Numberof constraints Running time(s) Gap(%) Module4a 36,541 1,382,831 126 0

233requiresonlyonevehicle.However,giventheroutesduration andthemaximumdurationofaworkingday,itisnotpossibleto performallrouteswithasinglevehicle.Fordepot235,whichis responsibletocollect156sites,ﬁvevehiclesareneeded.These vehi-clesareusedeverydayoftheplanninghorizonandtheminimum usagerateamongthemis74%(vehicle8)andthemaximumis86% (vehicle9).NotethatDepot235actalsoasthesortingstationand, therefore,alargernumberofcollectionsiteswereassignedtoit.

Theresultspresentedbymodule4aareoptimalonesgiventhe setofroutesdeﬁnedbythepreviousmodule.Theschedulingmodel runin126sandhasanoptimalvalueof24,405km.SeeTable2for thecomputationalresults.

5.1.2. Scenario2:serviceareasbyrecyclablematerial

Inthisscenario,serviceareasaredeﬁnedbypackagingmaterial andareobtainedaftermodule1,asitisshowninFig.8.Theservice

1

2

1

4

1

Fig.12.Vehicleﬂeetdistributionamongdepotsinscenario2.

areasarequitedifferentbetweenrecyclablematerials.Forinstance, depot234isresponsibletocollect12sitesformaterialGlass,but only3sitesforPaperand31sitesforPlastic/Metal;depot232does nothaveanysiteassignedwherePapershouldbecollected,while forGlassandPlastic/Metal,atotalof7and18sitesareassigned, respectively.

Module3isappliedtoeachdepotandeachpackaging mate-rialtodeﬁnetheﬁnalcollectionroutes.Asolutionwithatotalof 23,294kmisobtained,where21routesarecreatedtocollectPaper, 50tocollectPlastic/Metaland32tocollectGlass.The computa-tionalresultscanbeseenatRamosetal.(2014).

Routeschedulingisdonebymodule4a,wheretheroutesare assignedtoadayintheplanninghorizonandtoavehicle,assuming thatsharingresourcesarenotallowed.Inthisscenario,9vehicles arealsorequired,butwithadifferentdistributionbydepot(see Fig.12).

SchedulingresultsforeachvehicleareshowninFig.13.Vehicle 1,basedatdepot231,hasausagerateof42%;vehicle2,basedat depot232,hasthelowestusagerateof18%sinceitisagainthe depotwithlowestnumberofcollectionsitesassigned;depot233 needstwovehicles(vehicle3and4)withausagerateof61%and 44%,respectively;vehicle5,basedatdepot234,hasausagerate of34%;ﬁnally,depot235hasfourvehicles,allofthemwithahigh usagerate(83%,78%,87%and85%).

1 2 3 4 5 188m 188m 399m 370m 399m 152m 360m 399m 11 12 13 14 15 6 7 8 9 10 370m 188m 399m 16 17 18 19 20 55m 55m 356m 381m Depot 231 (Veh.1) 1 2 3 4 5 355m 361m 370m 479m 152m 171m 11 12 13 14 15 6 7 8 9 10 16 17 18 19 20 55m 140m 349m 280m Depot 233 (Veh.2) 474m 416m 122m 360m 77m 110m Depot 233 (Veh.3) 215m 211m 227m _74m 349m _74m 361m 399m 1 2 3 4 5 110m 399m 474m 6 7 8 9 10 463m 399m 474m 11 12 13 14 15 211m 294m 337m 399m 474m 16 17 18 19 20 122m 211m 1 2 3 4 5 149m 133m 360m 399m 11 12 13 14 15 6 7 8 9 10 370m 16 17 18 19 20 Depot 234 (Veh.4) 476m 294m 370m 375m 163m Depot 235 (Veh.5) Depot 235 (Veh.6) 396m 477m 133m 133m 149m 282m 1 2 3 4 5 6 7 8 9 10 16 17 18 19 20 11 12 13 14 15 375m 377m 396m 391m 434m 396m 163m 360m 375m 386m 383m 375m 396m 344m 468m 263m 370m 448m 315m 419m 439m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 413m 452m 419m 444m 315m 439m 456m 314m 472m 452m 444m 461m 406m 326m 338m 314m 370m 435m 403m Depot 235 (Veh.8) Depot 235 (Veh.9) 145m 314m 1 3 4 5 6 8 9 10 11 12 13 14 15 363m 403m 383m 418m 403m 392m 418m 363m 403m 426m 422m 190m 367m 314m 459m 370m 369m 376m 407m 314m 1 3 4 5 6 8 9 10 11 12 13 14 15 472m 476m 407m 433m 478m 407m 433m 314m 471m 467m 433m 341m 407m 433m 16 17 18 19 20 16 18 19 20 Depot 235 (Veh.7) 206m 281m 370m 471m 312m 392m 278m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 449m 312m 392m 389m 312m 429m 389m 129m 413m 312m 389m 463m 392m 389m 16 17 18 19 20 2 7 16 17 18 19 20 17 2 7 168m

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268m 471m 479m 431m 471m 11 12 13 14 15 6 7 8 9 10 335m 479m 16 17 18 19 20 80m 55m 392m Depot 231 (Veh.1) 1 2 3 4 5 277m 370m 277m 11 12 13 14 15 6 7 8 9 10 16 17 18 19 20 306m Depot 232 (Veh.2) 413m 122m 360m 381m 381m Depot 233 (Veh.3) 201m 85m 85m 474m 1 2 4 5 474m 413m 6 7 8 9 10 122m 474m 413m 211m 458m 474m 413m 16 17 18 19 20 303m 383m Depot 233 (Veh.4) Depot 234 (Veh.5) Depot 235 (Veh.6) 190m Depot 235 (Veh.8) Depot 235 (Veh.9) Depot 235 (Veh.7) 1 2 3 4 5 188m 188m 188m 80m 93m 151m 201m 11 12 13 14 15 255m 3 133m 291m 74m 2 3 4 5 421m 314m 360m 399m 12 13 14 15 6 7 8 9 10 370m 16 17 18 19 20 74m 421m 314m 1 317m 133m 148m 74m 11 317m 296m 464m 370m 392m 440m 389m 459m 1 2 3 4 5 6 7 8 9 10 16 17 18 19 20 11 12 13 14 15 392m 397m 389m 355m 440m 389m 468m 386m _392m ₃₉_7m ₃₄₁_m 392m 389m 390m 380m 380m 370m 472m 314m 341m 380m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 426m 314m 341m 477m 314m 341m 457m 380m 392m 314m 265m 293m 459m 449m 16 17 18 19 20 479m 465m 370m 358m 455m 451m 475m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 471m 455m 451m 425m 455m 314m 477m 280m 452m 455m 336m 460m 300m 16 17 18 19 20 314m 465m 457m 370m 472m 433m 455m 267m 1 3 4 5 6 8 9 10 12 13 14 15 318m 433m 455m 398m 433m 469m 398m 318m 433m 398m 318m 351m 398m 2 7 16465m17 18 19 20 11 392m 370m 250m 140m 1 3 4 5 6 8 9 10 11 12 13 14 15 341m 381m 196m 381m 392m 341m 178m 250m 244m 324m 16 17 18 19 20 2 7 417m 133m 190m 291m _74m

Fig.13.Schedulebyvehicleinscenario2.

Table3

Computationalresultsformodule4ainscenario2.

Numberof variables Numberof constraints Running time(s) Gap(%) Module4a 25,921 838,483 98 0

Theresultspresentedbymodule4aaretheoptimalonesgiven thesetofroutesdeﬁnedbythepreviousmodule.Thescheduling model run in 98sand hasan optimalvalue of 23,294km (see Table3).

5.1.3. Scenario3:sharingresources

Inthisscenario,serviceareasandvehicleroutesaretheones deﬁnedforscenario2.Themaindifferencebetweenscenarios2 and3concernstheschedulingmodule:vehiclesbasedinonedepot canperformclosedroutesofotherdepotsinscenario3.Inthiscase, itisexpectedthatthedistancetravelledwillincreasebecausethe vehicleshavetomovebetweendepots,butthenumberofvehicles willdecrease.

Wetestasolutionwitheightvehicles,distributedbydepotas showninFig.14(a),andthetotaldistancetravelledincreasesto 23,421km(more0.6%comparingwiththepreviousscenario).A solutionwithsevenvehicleswasalsotested(seeFig.14(b)),andthe

(a) 8 vehicles (b) 7 vehicles

1

4

1

4

0

Fig.14.Vehicleﬂeetdistributionamongdepotsinscenario3with(a)8and(b)7 vehicles.

totaldistanceobtainedisnow24,198km(more3.9%thanscenario 2).Nointegersolutionisobtainedifonlysixvehiclesareavailable. Thevehicleschedulesfortheseven-vehiclesolutionareshown atFig.15.Inthissolutionﬁveoutofthesevenvehicleshavehigh usagerates.Vehicle2,basedatdepot233,hasthehighestrate,95%. Comparingwiththepreviousscenario,depot233hasoneless vehi-cle,meaningthatahigherusagerateisachieved.Theremaining routesfromdepot233arenowperformedbyvehiclesbasedat depot231and235,contributingforahigherusagerateofthose vehicles.Vehiclesbasedatdepot235haveusageratesvaryingfrom 87%to91%.Besidesperformingroutesfromdepot235,these vehi-clesexecuteroutesassignedtodepot233,asmentioned,andto depot232,sinceinthissolutionnovehicleshavebeenbasedin thatdepot.Vehicle1,basedatdepot231,hasausagerateof51%, ninepercenthigherthaninthepreviousscenario,sinceitalso per-formsroutesofdepot233.Thisisalsothecasewithvehicle3,based atdepot234,whichnowincreasesitsusagerateto37%(34%inthe previousscenario)asitperformsroutesassigntodepot232.

Fig.16illustratestheroutesperformedbyvehicle1,basedat depot231,ondays4and14.Inthosedays,vehicle1hastotravel todepot233,performroute104tocollectPlastic/Metal,unloadat depot233,andthenreturntoitshomedepot.Inthosedays,the vehicle1isusedduring429min.

Intermsofcomputationalresults,module4bhasnotbeenable toproveoptimalitywithinthetimelimitof1h,butalowgapis obtained(seeTable4).

5.1.4. Scenario4:openroutesbetweendepots

Scenario4breaksupwithallfourpracticesmentioned,meaning thatopenroutesareallowedbetweendepots.Inscenario3,vehicles

Table4

Computationalresultsformodule4binscenario3withsevenvehicles.

Numberof variables Numberof constraints Running time(s) Gap(%) Module4b 33,603 1,129,857 3600 0.3

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152m 335m 152m 11 12 13 15 6 7 8 9 399m 163m 479m 55m 456m Depot 231 (Veh.1) Depot 233 (Veh.2) 184m 122m 360m 239m 207m Depot 234 (Veh.3) 323m 1 2 4 5 444m 75m 6 7 8 9 10 350m 133m 211m 310m 222m 75m 16 17 18 19 20 111m 184m Depot 235 (Veh.4) Depot 235 (Veh.5) Depot 235 (Veh.6) Depot 235 (Veh.7) 188m 456m 188m 479m 11 12 13 14 15 111m 3 475m 454m 370m 457m 397m 460m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 414m 397m 472m 433m 397m 471m 433m 475m 425m 464m 433m 464m 476m 370m 449m 463m 398m 443m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 479m 463m 398m 392m 463m 398m 392m 478m 479m 463m 392m 449m 398m 392m 16 17 18 19 20 355m 472m 370m 468m 380m 465m 446m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 477m 380m 452m 336m 470m 461m 306m 469m 477m 380m 300m 468m 358m 16 17 18 19 20 468m 457m 437m 370m 447m 389m 478m 477m 1 3 4 5 6 8 9 10 12 13 14 15 432m 432m 478m 457m 389m 341m 455m 432m 432m 457m 479m 341m 455m 2 7 16458m17 18 19 20 11 399m 1 2 3 4 5 429m 204m 10 14 429m 16 17 18 19 20 458m 478m 370m 463m 474m 413m 478m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 469m 477m 413m 474m 474m 413m 474m 383m 473m 477m 474m 463m 413m 474m 16 17 18 19 20 310m 239m 392m 16 17414m18397m19 20433m

canbesharedamongdepotsbuttheytravelemptybetweendepots. Inscenario4,themainideaistominimizethenumberofthose emptyroutes.Inordertodoso,advantagewillbetakenfromthe openroutesbetweendepotsgeneratedatmodule1whichallow thevehiclerelocation.Therefore,therelocating movementscan nowbecollectionroutesandnotonlyemptyroutes.Thescheduling module4cconsidersroutesgeneratedbymodules1and3,which areclosedandopenroutes,andselectstheonesthatshouldtake partintheﬁnalschedules.

Thetotaldistancedecreaseswhencompared withthe previ-ousscenario.Witheightvehicles,atotaldistanceof23,181kmis achieved,whileatotalof23,687kmisobtainedifsevenvehicles areused.

EachofthesevenvehiclesschedulesisshownatFig.17.Vehicle 1isnowoperating4077min,correspondingto42%ofusagerate. Vehicle2hasausagerateof92%,whilevehicle3hasincreasedits usagerateto47%(37%inthepreviousscenario)sinceitperforms openroutesbetweendepots234,232and235.Thefourvehicles basedatdepot235maintainhighusagerates(from85%to91%).

Fig.18showstheroutesperformedbyvehicle3,basedatdepot 234,ondays1and11toillustratethesharingresourcesallowed

Fig.16.Illustrationoftheroutesperformedbyvehicle1ondays4and14.

byconsideringopenroutes.Oneclosedrouteandtwoopenroutes betweendepot234anddepot232areperformedbythisvehicleon thesedays.

Concerningthecomputationalresultsformodule4cwithseven vehicles(seeTable5),oneshouldsaythatoptimalityhasnotbeen provenwithinthetimelimitof1h,butalowgapisobtained(0.3%). 5.2. Costanalysis

Asmentionedintheprevioussection,theobtainedsolutions varybetween23,181kmand24,405kmandbetween7and9 vehi-cles.Table6summarizestheresultsforthescenariosstudied.It showsthat scenario4aisthescenariowiththelowestdistance travelledpermonth(23,181km)whilescenarios3band4brequire thelowestnumberofvehiclesanddrivers(7).

Tocomparethosesolutionsacost-analysisisperformed.Three maincostswillbecomputedforeach solution:fuelcosts, vehi-clesdepreciationcostsanddriver’scosts.Thefuelcostsarealinear functionofthedistancetravelled.Ithasbeenestimated0.5D per Table5

Computationalresultsformodule4cinscenario4withsevenvehicles.

Numberof variables Numberof constraints Running time(s) Gap(%) Module4c 38,041 1,241,843 3600 0.3 Table6

Resultsforeachscenario.

Scenario Distance No.vehicles No.drivers

1 24,405 9 9 2 23,294 9 9 3a 23,421 8 8 3b 24,198 7 7 4a 23,181 8 8 4b 23,687 7 7

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362m 268m 204m 11 12 13 15 6 7 8 9 399m 268m 399m 55m 268m Depot 231 (Veh.1) Depot 233 (Veh.2) 282m 122m 270m 460m Depot 234 (Veh.3) 281m 458m 281m 211m 392m 184m 16 17 18310m19 20 Depot 235 (Veh.4) Depot 235 (Veh.5) Depot 235 (Veh.6) Depot 235 (Veh.7) 479m 11 12 13 14 15 468m 451m 370m 428m 392m 451m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 445m 392m 438m 453m 392m 328m 479m 477m 465m 446m 332m 435m 455m 370m 447m 433m 477m 455m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 436m 433m 477m 403m 433m 475m 476m 456m 436m 433m 418m 479m 314m 336m 16 17 18 19 20 389m 389m 370m 457m 380m 398m 389m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 439m 380m 398m 441m 380m 398m 391m 389m 465m 380m 399m 422m 476m 16 17 18 19 20 398m 428m 463m 370m 397m 341m 436m 455m 1 3 4 5 6 8 9 10 12 13 14 15 397m 453m 436m 455m 341m 480m 455m 397m 453m 455m 397m 480m 455m 2 7 16463m17 18 19 20 11 399m 1 2 3 4 5 80m 10 14 16 17 18 19 20 467m 381m 370m 455m 413m 474m 498m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 446m 461m 474m 398m 413m 474m 461m 361m 449m 413m 473m 455m 474m 461m 16 17 18 19 20 443m 460m 427m 16 17421m18392m19 20480m 188m 340m 188m 163m 72m 102m 310m 1 2 3 5 6 7 8 10 320m 4 9 281m

kilometerdriven. Vehicle depreciationis estimated considering vehiclesacquisitioncostandtheusefullifeassumedfortaxes pur-pose. In particular, the acquisition cost is about100,000_D and theusefullife is of 5years, which leads toanannual depreci-ation of 20,000D. Thedrivers costs are estimatedin 900D per month,paid14monthsperyear,anditisconsideredonedriverper vehicle.

Fig.19depicts thetotal costforthecurrentsolutionand for eachscenariostudied.Scenario1has12.8%lessdistancetravelled thaninthecurrentsolutionandthetotalannualcostdecreases5% ifthemunicipalities’boundariesarenotrespectedwhendeﬁning theserviceareas.Comparingscenario2(serviceareasaredeﬁned bypackagingmaterial)toscenario1(serviceareasbydepot),the distancetravelleddecreases5%,whiletherequirednumberof vehi-clesismaintained.Thetotalcostdecreasesin1.7%whencompared toscenario1and6.4%whencomparedtothecurrentsolution.The largestdecreaseinthetotalcostisobtainedinthescenariowhere resourcesaresharedamongdepots(scenarios3 and4).In sce-nario3b,thetotalcostdecreases13.3%regardingscenario2and

Fig.18.Illustrationoftheroutesperformedbyvehicle3ondays1and11.

18.9%whencomparedtothecurrentsolution.Thesegainscome fromsharingvehiclesamongdepotswhichallowsthereduction ofthenumber ofvehicles toseven.Whenopenroutes between depotsare allowed, thetotal distancetravelled decreaseseven more(23,687kmagainst24,198km,about2%)andthetotalcost decreasesabout20%whencomparingwiththecurrentsolution.

Signiﬁcantsavingsareobtainedwhensomeofthecurrent prac-tices are removed. However, from an operations management perspective,itismorecomplextomanagescenario4,wherethree differentserviceareas,sharedvehiclesand openrouteshaveto bedealt with, thanthe current solutionwhere each depot has oneservicearea,withvehiclesassignedand onlyclosedroutes, whichallowseachdepottoactindependently.However,these lat-terpracticesleadtoalargertravelleddistanceandmoreresources, as vehicles and drivers. In thebreakthrough scenario (scenario 4),alldepotsand vehiclesareintegrated andactaspartofthe samesystem.Thiskindof solutiondecreasesthedistance trav-elledand theresourcesneededbutdemandsadecision support

0 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 Current Soluon Scenario Scenario 2 Scenario 1 3a Scenario 3b Scenario 4a Scenario 4b Driver Cost Vehicle Cost Fuel Cost 475400€ 452683€ 444811€ 413037€ 385487€ 411477€ 382166€

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systemtohelpmanagingtheincreasedcomplexity.Thesolution methodologydevelopedinthisworkcanbeseenasamainpillar towardsthedevelopmentofadecisionsupportsystem,supporting theroutes deﬁnitionandvehicleschedules underaninnovative scenario.

6. Conclusions

Inthis paperfourpractices currentlyusedbyrecyclable col-lectionsystemsoperatinginPortugalhavebeenanalysedsoasto proposeimprovementsontheirsetups.Giventhatthesystemsin studyhavemultipledepotsmanagedindependently,themainidea wastoexploremanagementoperationsinanintegratedwayso thatgainsintermsofefﬁciencycouldbeachieved,bydecreasing thetotaloperationalcosts.

Toachieve this goalthis paper developed a uniﬁedsolution methodologythatiscapabletoplanthecollectionsystems explor-ingdifferent practices. Thissolution methodologyis appliedto a real packaging waste collection system. Four scenarios were analysed,developed bycumulativelyremovingeachof thefour practices.TheproblemssolvedaretheMulti-Product,Multi-Depot PeriodicVehicleRoutingProblem,theMulti-DepotPeriodic Vehi-cle Routing Problem, the Multi-Depot PeriodicVehicle Routing ProblemwithSharedResourcesandMulti-DepotPeriodicVehicle RoutingProblemwithInter-DepotRoutes.Thesolution methodol-ogydeﬁnesserviceareasandvehicleroutesinanintegratedway. Afterwardsitassignstheroutestoadayontheplanninghorizon accordingtoeachscenario.

Each scenario was evaluated in terms of travelled distance and number of required vehicles. Alsothe total annual cost of eachscenarioconsideringfuelcosts,vehicledepreciationcostsand driver’scosts,wascalculated.Itwasconcludedthatbyremoving thepracticeofnotsharingresourcesamongdepotsimpliesthe highestimpactonthetotalcost,sincesharingresourcesenables adecreaseuptotwovehicles.Moreover,deﬁningserviceareasby recyclablematerialinsteadofdepot(scenario2vs.scenario1)leads toa positiveimpactof1.7%onthetotalcost.Sharingresources insteadofhavingthevehicleﬁxedatdepots(scenario3vs. sce-nario2)conductedtoapositiveimpactof13.3%intotalcost(the highestimpact,asmentioned).Performingmixedclosedandopen routesinsteadofallowingonlyforclosedroutes(scenario4vs. sce-nario3)resultedinapositiveimpactof0.9%intotalcost.Finally, comparingthecurrentsolutionwiththeinnovativescenario (sce-nario4,whereallofthefourpracticesareremoved),adecreaseof 20%inthetotalcostwasobserved.

Itisafactthateachscenariostudiedincreasesthecomplexityof managingoperationswithintherecyclablecollectionsystems,but signiﬁcantcostssavingshavebeenattained.Moreover,thecurrent situationisdrivenbymanagement optionsadoptedbypolitical issueswhichledtotheuseofmunicipalboundarieswhen plan-ningthesystems.Suchpracticeprovedtodiminishthesystem’s efﬁciencyandconsequentlyalternativesolutionsshouldbe pur-sued.Theimprovedresultsneed,however,tobedemonstratedand discussedwiththemanagementboard.Thiswasachieved,inthis work,throughaclosecollaborationbetweentheresearchandthe companymanagementteams.

Asfutureworkandstillaimingtofurtherimprovingoperation, otheralternativeoperationalpracticescanbestudied.Suchasthe nightcollectioninoppositiontothedaytimecollection,thelatter inuseinthepackagingcollectionsystemunderstudy.Noticethat performingadaytimecollectionactivitymayleadtoadecrease intheoverallsystemefficiencygiventrafficcongestion(thatmay slow down collection). Night collection can, however, increase thecollectionefficiencyintradeoffwithanincreaseofpersonnel costs.

Thesolutionmethodologyproposeddeﬁnesﬁrstlytheservice areasandvehicleroutesinanintegratedway,whilethe sched-ulingdecisionsaretakeninasecondphase.Thissequentialmethod causesalowusagerateofsomevehicles,representingapitfallof theproposedmethodology.Atighterintegrationofthosedecisions wouldleadtohigherusagerates,andtherefore,betterresults.As futureresearch,thesolutionmethodologyshouldbeimprovedto tackle thethree decisionssimultaneously.In addition,it would also beinteresting tostudy this problemby applying different techniques,suchasmeta-heuristics,makingacomparisonofthe resultsobtained.Finally,anotherfutureworkdirectionistoapply thismethodologytootherpackagingcollectionsystemsinorderto quantifytheresultsofmorecasestudiessoastocorroboratethe conclusionsofthecurrentwork.

Acknowledgements

The insightful contributions of the anonymous referees are gratefullyacknowledged.

ThisworkwaspartiallysupportedbytheCMA/FCT/UNLunder theprojectPEstOE/MAT/UI0297/2011.

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