and delivery under uncertainty A robust optimization approach for pollution routing problem withpickup Journal of Manufacturing Systems

ContentslistsavailableatScienceDirect

Journal of Manufacturing Systems

jo u r n al ho m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / j m a n s y s

Technical Paper

A robust optimization approach for pollution routing problem with pickup and delivery under uncertainty

N. Tajik

^a,∗

, R. Tavakkoli-Moghaddam

^a

, Behnam Vahdani

^b

, S. Meysam Mousavi

^c

aSchoolofIndustrialEngineering,CollegeofEngineering,UniversityofTehran,Tehran,Iran

bFacultyofIndustrial&MechanicalEngineering,QazvinBranch,IslamicAzadUniversity,Qazvin,Iran

cIndustrialEngineeringDepartment,FacultyofEngineering,ShahedUniversity,Tehran,Iran

a r t i c l e i n f o

Articlehistory:

Received4October2012

Receivedinrevisedform11October2013 Accepted31December2013

Availableonline16February2014

Keywords:

Pollutionroutingproblem Pickupanddelivery Robustoptimization Fuelconsumptions CO2emissions

a b s t r a c t

Organizationshaverecentlybecomeinterestedinapplyingnewapproachestoreducefuelconsumptions, aimingatdecreasinggreenhousegasesemissionduetotheirharmfuleffectsonenvironmentandhuman health;however,thelargedifferencebetweenpracticalandtheoreticalexperimentsgrowstheconcern aboutsignificantchangesinthetransportationenvironment,includingfuelconsumptions,carbondioxide (CO2)emissionscostandvehiclesvelocity,thatitencouragesresearcherstodesignanear-realityand robustpollutionroutingproblem.Thispaperaddressesanewtimewindowpickup-deliverypollution routingproblem(TWPDPRP)todealwithuncertaininputdataforthefirsttimeintheliterature.Forthis purpose,anewmixedintegerlinearprogramming(MILP)approachispresentedunderuncertaintyby takinggreenhouseemissionsintoconsideration.Theobjectiveofthemodelistominimizenotonlythe traveldistanceandnumberofavailablevehiclesalongwiththecapacityandaggregatedrouteduration restrictionsbutalsotheamountoffuelconsumptionsandgreenhouseemissionsalongwiththeirtotal costs.Moreover,arobustcounterpartoftheMILPisintroducedbyapplyingtherecentrobustoptimization theory.Computationalresultsforseveraltestproblemsindicatethecapabilityandsuitabilityofthe presentedMILPmodelinsavingcostsandreducinggreenhousegasesconcurrentlyfortheTWPDPRP problem.Finally,bothdeterministicandrobustmathematicalprogrammingarecomparedandcontrasted byanumberofnominalandrealizationsunderthesetestproblemstojudgetherobustnessofthesolution achievedbythepresentedrobustoptimizationmodel.

©2014TheSocietyofManufacturingEngineers.PublishedbyElsevierLtd.Allrightsreserved.

1. Introduction

Carbon dioxide (CO2) is one of the elemental green house gasesemitted throughhumanresource activity.Increasing CO₂ becomesa majorproblemforthenaturalcycleinecosystemas thenaturealwaysmaintaintheequilibriumbetweentheamount ofCO2unleashedandtheamountofCO2refined;therefore,human resourceactivityisresponsibleforthatrisinghascomeabout.CO₂ emissionsinUnitedStatewentupbyabout12%between1990and 2010[1].In2010,itaccountedforquiet84%ofallUnitedStateso thatthemainhumanresourceactivitycausesCO₂emissionswith thecombustionoffossilfuels(e.g.,coal,naturalgasandoil)forpro- ducingenergyandtransportation.TheamountofCO2emissionof fuelcombustionintransportation,suchasgasolineanddiesel,after electricityproducersisthesecondlargestsourceofCO₂pollutants.

Itisresponsiblefor31%oftotalUnitedStateCO2 emissionsand 26%oftotalgreenhousegasemissions[1].Theroadtransportation

∗Correspondingauthor.Tel.:+982182084183.

E-mailaddress:tajik.nazanin88@ut.ac.ir(N.Tajik).

accountsfor78%ofgreenhouseemissions,includingCO₂ grow- ingtheconcernabouthazardouseffectsonecosystem[2],which makesscientistsandresearchersdevelopadaptiveapproachesand modelstoexplicitlyreducetheamountoftheabovepollutants.

Thevehicleroutingproblem(VRP)determinestheoptimalset ofroutestobeperformedbyafleetofvehiclestoserveagiven setofcustomerunderspecificconstraints.Besides,thevehiclesare startfrom,andendtoadepot[3–5].Theclassicalversionofthe VRPiscapacitateVRP(CVRP),inwhichallcustomerscorrespond todeliveriesandthedemandsaredeterministic,anditmaynotbe splitandonlycapacityrestrictionsareconsideredforthevehicles.

Fungetal.[6]representacapacitatearcroutingproblemwhich usesmemeticalgorithmtofindasetofroutesholdingtheminimum costoftransportation.Thecomputationalresultsillustratethatthe qualityofthesolutionishighinareasonabletime.

AnothercommonvariantistimewindowVRP(TWVRP)thatis anextensionofCVRP,atwhicheachcustomerisassociatedtoan interval,calledtimewindow,andeach ofthemmustbemetat itsinterval.Thesignificance oftheapplicationofthis TWVRPin realworldhasmaderesearchersdoeffortstofindappropriateand strongalgorithmtosolvethesetypesofVRPmodels.Forexample, 0278-6125/$–seefrontmatter©2014TheSocietyofManufacturingEngineers.PublishedbyElsevierLtd.Allrightsreserved.

http://dx.doi.org/10.1016/j.jmsy.2013.12.009

Nagateetal.[7]introduceamemeticalgorithmforTWVRPusing newpenaltyfunctiontoeliminateviolationoftimewindowcon- straintsaswellasoriginalconstraintusesin commonmemetic algorithm.Theexperimentalresultofthismodelassertsthatthe developedalgorithmperformswell.Zachariadisetal.[8]represent ahybridsolutionapproachasacombination oftabusearchand guidedlocalsearchwhichhasamplepowertoexploreavastspace tofindabettersolutionforVRPPDwithsimultaneouspickupand deliveryservices.Jeonetal.[9]developthemainVRPPDmodeland considerdoubletrip,multidepot,andheterogeneousvehiclesand suggestahybridgeneticalgorithminordertosolvethelargescale problems.

Anextended variantin thispaperis introducedasVRPwith pickup and delivery (VRPPD), in which some customers have demandtodeliverandsomeofothershaveloadstopickup.The typicalclassofVRPPDispickinguploadfromtheentirecustomer relatedto,andthendeliveringloadtoallremainedcustomers.Ter- sanandGen [10]consider a pickupdeliveryproblemin which vehiclecandeliverandcollectgoodssimultaneouslyinorderto modelrealworldproblemsuchareverselogisticandthenexam- inethemathematicalmodelbygeneticalgorithmandrepresentthe resultswhichapprovedtheaccuracyofthemodel.Anotherintro- ductionofVRPPDwithsimultaneouspickupdeliveryannouncedby Goksaletal.[11];Inthispaper,aheuristicsolutionbasedonpar- ticleswarmoptimizationispresentedwhichusesannealing-like strategytopreservethediversityofswarm.Fortheliteratureon theVRPanditsextensions,readerscanbereferredto[12–18].

ThereisagreatdealofeffortatextendingthetraditionalVRP objectivesandconstraintsnotjusttoaccountfortheeconomiccosts buttoconsidermorecomprehensiveenvironmentalandecological impacts.AcertainamountofworksreducestheCO₂emissionswith loweringtheamountoffuelconsumptions.Bywayofillustration, ErduganandMiller-Hooks[19]introduceamathematicalmodel bykeepingfuelconsumptionstoaminimumviaminimizingthe totaltraveleddistancewhileconsideringtheneedsforrefuelingin therouteplanssooptimizedastoavoidtheriskofrunningoutof fuel.Ubedaetal.[20]representadistance-basedmodelaffected bymultiplyinganemissionfactordependedonaverageweightof atypicalvehicleandfuelconversionfactor.Thisstudyshowsthat introducingbackhaulstoavoidemptyrunningbenefitfromboth economicandecologicalmoreefficient.

Apartfromtheabove-mentionedstudies,someofotherworks involvefactorsdependedonthefeaturesofthevehicle,thefuelthat vehicleconsumes,andtherouteittravelstodepletetheamount ofCO₂ emitted.Suzuki[21]showsthatthelessdistanceadeliv- eryvehicleisanticipatedwithheavierloadtraveledthelessfuel consumptions.Also,theyconsiderthefuelconsumptionsthatare relevanttowaitingtimeincustomerserviceplace.Figliozzi[22]

representsananalyticalmodelofCO₂emissionsinvolvedinavari- etyoftime-definitivecustomerdemandsbyusingtimedependant vehicleroutingmethodin ordertoplanvehicles routes.Bektas and Laporte [23] shed light on thetrade-off betweendifferent parameters,suchasvehicleload,speed,and aggregatecost and environmentalgreenvehicleroutes,andtheyrepresentamodel namedpollutionroutingproblem(PRP).

Thereview oftherelated literature indicatesthat there is a significantgapontheapplyingthenew economic-friendly and ecological-friendlyVRPtopickupanddeliverysystems.Moststud- ieshavefailedtopayattentiontotheeffectof someeconomic impactsonecologicalinfluenceintheVRPnetworks.Inaddition, consumingtherealizationofsomeinfluentialparametersseems tobenecessarybecauseoftheimportanceofbuildingmathemati- calmodelsinthereal-worldapplications.Thispaperistoaddress a newtime window pickup-deliverypollution routingproblem (TWPDPRP)todealwithuncertaininputdataforthefirsttimein theliterature.ThepresentedproblemcombinesconventionalVRP

andtechnicalandmechanicalapproachestoestablishanewmixed integerlinearprogramming(MILP)modelbyconsideringeconomic andenvironmentalissuesconcurrently.

Themaininnovationsofthispapertodifferentiatetheefforts fromthosealready publishedonthesubjectareasfollows: (1) introducingpenalty costsforpickup nodesin ordertocreate a sequenceofpickupnodestoreducingtheamountoftravelingtime andthentheamountoffuelconsumptionsandCO2emissions;(2) integratingtimewindowconstraintsintotheVRPPDinorderto decreasethearrivaltimetodeliverynodes.Both(1)and(2)show uptheeconomiceffectsthatenrichenvironmentalinfluencesin themodel;(3)applyingrobustoptimizationwithintheconstraints consideringthevelocityofvehicleasanuncertainparameterin ordertoachieveapracticalmodel;forinstance,thisuncertaintyis relatedtotrafficflow,stopsatfuelstationsandtrafficlights.Allof theaboveconsiderationsbringthemathematicalmodelcloserto reality,unlikethepreviousstudiesbyconsideringthevelocityas asetofnumbersthatcanbechosenfrominordertooptimizethe VRPmodel[22,23];(4)consideringthechangesoftheCO₂emis- sionscostandthefuelconsumptionscostoverthetimetothink overthealteringintherealworldapplications.Bytaking(3)and (4)intoconsiderationrobustoptimizationimportsintheTWPDPRP asanewwaytocreateatrade-offbetweeneconomicandecolog- icalissuesandcombinesthemtothenaturalcases;(5)takinginto accounttherobustconceptforservicetimeofeachcustomertodeal withdifferentsituations;and(6)thinkingoverthephysicalfeature ofeachvehicle,suchasthefrontsurfacetheweight,theslopeand thefractionfactorofroadsandbringthemintotheTWVRPPDto getthebestsequencepickupanddeliverynodessothatthefuel consumptionsandCO2emissionscostgetdownconsiderably.

Theremainderofthispaperisorganizedasfollows.Sections2, 3and4provideaformaldescriptionoftheTWPDPRP,theproposed mathematical model and the robust counterpart mathematical model,respectively.Computationalresultsondeterministicand robustsolutionsarepresentedinSection5.Also,inthissection thesensitivityanalysisisreportedonvariousparameters ofthe proposedMILPmodel.Thefinalsectionendswiththeconclusions.

2. Problemdefinition

ThispaperaddressesanewTWPDPRPincludingtwogroupsof nodes.Thefirst onecontains customerswhose loadsshouldbe pickedup,andthesecondonecoverscustomerswhosedemands shouldbedelivered.Themodelemploysagroupofvehiclesfor servicetasks,unlikethepreviousstudieswithonlyonevehicle mentioned in Section 1.As illustrated in Fig. 1, after servicing allpickup customers,theirloads arecarried bytrucks(i.e.,the loadspickedupfromthefirstgroupnodesandtheloadsthatare neededtobepickedupfromdepotatthebeginningforsupplying totaldemandsaccurately),thevehiclesdistributetheloadsamong deliverycustomers.Hence,theamountoftheproductwhicheach vehicleloadedupatthestartnode(i.e.,depot)isthetotaldemands ofdeliverynodesinwhichtheyvisitduringtheirroutesminusthe totalloadscollectedfrompickuppointsduringtravelingtheroutes thattheyareassociatedto.Iftotalpickedupproductsareoveror equaltothetotaldeliverydemandsduringtherelevantroutethe vehiclefollows,itleavesthedepotempty.Inaddition,thephysical featureofeachvehicle,suchasthefrontsurfacetheweight,the slopeandthefractionfactorofroads,areconsideredbasedon[24].

Calculatingthevelocityvalue:Duetoassumingvehiclesmotion withconstantacceleration,themaximumvalueofvelocityduring travelingfrompointitopointjwiththedistanceamount,d_ij is achievedfromthebelowequation[24]:

v_ij=

v²

0+2adij (1)

Fig.1.AnillustrativeexamplefortheTWPDPRPwithninecustomers,includingsix pickupcustomers(P),onedepot(O)andthreedeliverycustomers(D).

But,thevelocityinconstantaccelerationmotionhasdifferent valueinvariouspartsoftravelingd_ij.Thus,themeanvalueofveloc- itycanbeapropervaluetofigureoutthetimetravelingbetween pointsiandjforthedeterministicsolutioncalculatedby:

¯ v=

v_ij−v₀

2 (2)

v_{isthemaximumvelocityand}v_{0istheminimumvelocityduring} travelingd_ij.

Tospecifythestudyscope,assumptionandfacilitationaresup- posedinthepresentedMILPmodelformulationasbelow:

•Allpickupanddeliverynodesmustbeserviced;however,extra loadsareallowedtoretrievetothedepotbyvehiclesattheend oftheirroutes.

•Locationsofthedepotandcustomersnodesarefixedandprede- fined.

•Allvehiclesendupgettingtodepotaftercompletingtheirroutes.

•Eachvehiclehasapredefinedandfinitecapacity.

Withconsideringtheaboveassumptions,thispapersetsoutto describeaprocessinordertoreducetheaverageamountofloads carriedbyeachvehicleandtodecreasethenegativeimpactsof carryingloadsonfuelconsumptions.Then,CO₂ emissionsbeside loweringthetraveldistance,traveltimeandarrivaltimestopoints foreachvehiclebyconsideringtheroadelements,suchasslope andfraction,andthephysicalfeaturesofeachvehicle,suchasfront surface,netweightandcapacity.Italsolinksthepenaltycostsfor pickuppointstotimewindowintervalsofdeliverypoints.

3. Modelformulation

Thefollowingnotationsareusedintheformulationofthepro- posedTWPDPRP.

3.1. Sets

P Setofpickupcustomersp=

1,2,...,P

D Setofdeliverycustomersd={n+1,n+2,...,n+n^′} K Setofvehiclek={1,2,...,K}

{0} Depotatthebeginningoftheroute(allvehiclesstartsfrom) {n+n^′+1} Depotattheendoftheroute(allvehiclesendto) V Setofallvertices{0,1,2,...,n,n+1,...,n+n^′+1}

A SetofedgesA={(i,j)|i,j∈Vj=/0,i=/n+n^′+1,i=/j}

3.2. Indices

i Indexofthepickupanddeliverycustomersandthedepotatthe startandendoftheroute

i={0,1,....,n,n+1,...,n+n^′+n+n^′+1}

j Indexofpickupanddeliverycustomersplusdepotatthestartand endoftheroutej={0,1,...,n,n+1,....,n+n^′+n+n^′+1}>

k Indexofvehiclesk={1,2,...,K}

3.3. Parameters

qi Demandforcustomeri

_{qi>0 ifi∈P}

qi<0 ifi∈D q0=qn+n^′+1=0 Ck Capacityofvehiclek

di Servicetimeatvertexi

ei Earlieststartofservicetimeatvertexi li Latestservicetimeatvertexi

Cijk Costoftravelingbetweenvertexi,jbyvehiclek tijk Traveltimebetweenvertexi,jbyvehiclek Tk Maximumtraveltimeofvehiclek Di Duedateofvertexi

CTi Costoftardinessinvertexi CEi Costofearlinessinvertexi C_f Costoffuelconsumption e CostofCO2emission

Wk Weightofvehiclekwithoutloads P Costofdriverperhours ij Slopeofedge(i,j) Ak Frontspaceofvehiclek Cd Frictionrateofair Cr Rollingresistance

sin(i,j) Sineoftheanglebetweentheroadconnectingvertexiandj andhorizonline(angleiscalculatedbasedongradient) cos(i,j) Cosineoftheanglebetweentheroadconnectingvertexiandj

andhorizonline(angleiscalculatedbasedongradient)

Airdensity

g Gravity

ϑijk Speedofvehiclektravelingbetweenvertexiandj ak Averageaccelerationofvehiclek

3.4. Decisionvariables Xijk

_{1 ifvehiclektravelsbetweenvertexiandj}

0 otherwise,

Qijk Loadsofvehiclekbeforearrivingtojafterpassingisubsequently yik Timeofstartingserviceatvertexibyvehiclek

Sjk Timeofcompletingaroutebyvehiclekwhenvertexjisthelast nodebeforevertexn+n^′+1

TTik Tardinesstimeforvehiclekinvertexi ETik Earlinesstimeforvehiclekinvertexi

Intermsoftheabovenotations,theproposedMILPmodelfor theTWPDPRPispresentedasfollows:

MinZ=

K

k=1 n+n^′+1

j=0,i=/j n+n^′+1

i=0

(Cf+e)(ak+(sin(i,j)

+Crcos(i,j))g)dijWkxijk

+

K

k=1 n+n^′+1

j=0,i=/j n+n^′+1

i=0

(C_f+e)(a_k+(sin(i,j)+Crcos(i,j))g)d_ijQ_ijk

+

K

k=1 n+n^′+1

j=0,i=/j n+n^′+1

i=0

(Cf+e)(1/2CdAk)dijtijxijk

+

K

k=1 n+n^′+1

i=0

Psik+

K

k=1 n+n^′+1

i=0

CEiETik+

K

k=1 n+n^′+1

i=0

CTiTTik (3)

s.t.

n+n^′+1

j=1

x0jk=1 ∀k∈K (4)

n+n^′

i=0

xin+n^′+1k=1 ∀k∈K (5)

K

k=1 n+n^′+1

j=1j=/i

xijk=1 ∀i∈P∪D (6)

K

k=1 n+n^′

i=0j=/i

xijk=1 ∀j∈P∪D (7)

K

k=1 n+n^′

i=0j=/i

xijk−

K

k=1 n+n^′+1

i=1j=/i

xjik=0 ∀j∈P∪D (8)

yik+t(i,j)+di−yjk≤M(1−xijk)

∀i∈P∪D∪{0},

∀jεP∪D∪

n+n^′+1

, i=/j,k∈K

(9)

yjk+t(i,n+n^′+1)+dj−sjk≤M(1−x_in+n^′_+1k) j∈P∪D∪

0

,kεK (10)

K

k=1 n+n^′

i=0j=/i

Qijk−

K

k=1 n+n^′+1

i=1j=/i

Qjik=−qj ∀j∈P∪D (11)

max{0,−q_j}≤Q_ijk≤min{C_k,C_k−q_j} i,j∈P∪D∪

0

,k∈K (12)

yik−yjk≤M

⎛

⎝

2−

⎛

⎝

n+n^′

l=0,l=/j

xljk−

n+n^′

l=0,l=/i

xlik

⎞

⎠

⎞

⎠

i∈P,j∈D,k∈K (13)

K

k=1 n

j=0 n+n^′+1

i=n+1

xijk=0 (14)

ei n+n^′

j=0,i=/j

xijk≤yik≤li n+n^′

j=0,i=/j

xijk ∀k∈K,∀i∈D (15)

max{0,yik−Di}≤TTik+M

⎛

⎝

¹⁻

n+n^′

j=0,i=/j

xjik

⎞

⎠

∀i∈P,∀k∈K (16)

max{0,D_i−y_ik}≤ET_ik+M

⎛

⎝

¹⁻

n+n^′

j=0,i=/j

x_jik

⎞

⎠

∀i∈P,∀k∈K (17)

Q0ik+

n+n^′+1

j=1 n+n^′

l=0

qixljk≥−M(1−x0ik) ∀i∈V/

0

(18)

xijk∈

0,1

∀i∈V,∀j∈V,∀k∈K (19)

yik≥0,Qijk≥0,sjk≥0,ETik≥0,TTik≥0 ∀i∈V,∀j∈V,∀k∈K (20) Theobjectivefunction(3)minimizestotalcostsincludingsix components:thefirstthreemeasurethefuelconsumptionscost andCO₂ emissionscostwhich considertheroadsphysical con- ditionselected byvehicles totravel(i.e.,the roadfraction, and roadslop),theweightof thevehicles,and theloadstheycarry, inphysicalequationsthatpresenttheamountofenergyconsumed bytheirindependentlyandinterdependentlyimpacts.Surfaceand air fraction impacts on fuel consumption are even involved in thoseequations.Accelerationandvelocityimpactsontimetrav- elingandvehiclefuelconsumptionsareconsideredaswell.The sixthcomponentcalculatesthedrivercostoneachroute.Thelast twocomponentsdescribethepenaltycostfortardinessandearli- nessinarrivaltimeofpickupcustomers.Constraints(4)ensurethat allvehiclesstarttheirroutefromthedepot.Constraints(5)assure thatallvehiclesendtheirroutestothedepot.Constraints(6)and (7)considerthateachvertexmustbevisitedjustonetime.Con- straints(8)defineaflowbywhichthenumberofarrivalsisequal tothenumberofdeparturesforallvertices.Constraints(9)aresub- tourseliminationconstraints.Thetotaltimeoftravelingforeach vehicleisrelatedtothelastnodethatitpassesbeforereachingto thedepot.Thus,byconstraints(10)theentiretravelingtimeinthe correspondingtourcalculatedbyalinearformulationderivedfrom thebelownonlinearequation:

Sjk=

yjktj+ d_j0 v_j0k

xj0k (21)

Theequilibrium ofproductsflowis describedbyconstraints (11).Theyclearthatnoloadappearsordisappearsbyitselfandthe loadsleveleachvehiclecarriesonlychangebycustomersorder.

Constraints (12) are employed to restrict total loads that each vehiclecarries by itscapacity. Constraints (13)and (14)define restrictionstoserviceallpickupcustomers beforedeliverycus- tomers. Time window limitation is shown by constraints (15).

Constraints(16)and(17)describethetardinessandearlinessof arrivingtimeforeachvertex.Constraints(18)dispersethenum- berofloadsneededtopickupfromthedepotatthebeginningof theroute.Finally,constraints(19)and(20)enforcethebinaryand non-negativityrestrictionsondecisionvariables.

4. Robustcounterpartmathematicalmodel

Recallthatthelinearoptimizationproblemisindicatedasfol- lows:

minc^Tx (22)

s.t.Ax≥b (23)

where c∈Rⁿ isvector formingobjectivefunction coefficientfor decisionvariablesandx∈Rⁿisthevectorofdecisionvariables,and finallyA∈R^m×nandb∈R^mareconstraintmatrixandrighthandside vector.

Uncertaintypartoftheproposedmodelinthispaperisconsid- eredtobeoptimizedbytherobustcounterpartlinearoptimization solution which is done based on Ben-tal and Nemirofski [25].

Beforerepresentingtheproposedrobustmodel,somedefinitions arereviewedfortherobustoptimizationtheory.

Definition1. Afeasiblesolutionforrobustlinearoptimizationisa setofvaluesrelatedtodecisionvariablesthatrepresentameaning fullobjectivefunctionvaluebesidessatisfyingalltheconstraintsas representedbelow:

{min{c^Tx+dAx≤b}}_∀(c,d,A,B)∈u (24)

Thedifferencebetween(23)and(24)isthedata(c^′d,A,b)vary- ing in a given detailed closed bounded box u, which is called

uncertaintysetaswell[25],whereu⊆R(m+1)×(n+1)anditsgeneral formisindicatedasfollows:

={ϑ∈Rⁿ:|ϑt−ϑˆt|≤Gt,t=1,...,n}, (25) inwhich ˆϑtisnormalvalueofϑt,>0isthelevelofuncertainty, andGTdefinesasuncertaintyscale.

Definition2. By consideringapotentialsolutionx,therobust objectivefunctionvalue ˆc(x)isthesupermomvalueofthat‘true’

objectiveC^Tc+dsetoverwholerealizationsofthedatafromu.

cˆ(x)=sup[c^′Tx+d](c,d,A,b)∈u (26)

Afterknowingthemeaningful solutionsfor uncertainmodel definition,theuncertainmodelmustsuggestthebestrobustvalue ofobjectivefunctionamonggivenfeasiblesolutions.Thisprocess introducesthe robustcounterpart ofan uncertainoptimization problem.

Definition3. Therobustcounterpartofanuncertainlinearisopti- mizingthebelowproblem:

minx {ˆc(x)=sup[c^′Tx+d](c,d,A,b):Ax≤b∀(c,d,A,b)∈u} (27) By solving (27), the robust optimal solution of (26)can be obtained.ThevalueofobjectivefunctioninDefinition3cansatisfy allconstraintsandgivethebestrobustoptimizedvalueforlinear models.

Forapplyingtherobustcounterpartofthepresentedmodel,fuel consumptionscost(C_f),CO₂emissionscost(e),theservicetimefor eachcustomerservicedbyacertainvehicle(d_ik),thetimetravel- ingbetweentwocertaincustomers(t_ij)areassumedasuncertain parameters.Hence,accordingtoabove-mentioneddefinitions,the robustcounterpartoftherecommendedTWPDPRPdesignsamodel foruncertainservicetime,travelingtime,fuelconsumptionsand CO₂emissionscostprovidedbyuncertaintysetu.

MinZ s.t.

K

k=1 n+n^′+1

j=0,i=/j n+n^′+1

i=0

( ¯Cf+e)(a¯ k+(sin(i,j)+Crcos(i,j))g)dijWkxijk

+

K

k=1 n+n^′+1

j=0,i=/j n+n^′+1

i=0

( ¯Cf+e)(a¯ k+(sin(i,j)+Crcos(i,j))g)dijQijk

+

K

k=1 n+n^′+1

j=0,i=/j n+n^′+1

i=0

(Cf+e)(1/2CdAk)dijtijxijk+^t_ij)+f+e

+

K

k=1 n+n^′+1

i=0

Ps_ik+

K

k=1 n+n^′+1

i=0

CE_iET_ik+

K

k=1 n+n^′+1

i=0

CT_iTT_ik≤Z

eGexijk≤e ∀i,j∈V,∀k∈K

eGexijk≥−e ∀i,j∈V,∀k∈K

eGeQ_ijk≤e ∀i,j∈V,∀k∈K

eGeQijk≥−e ∀i,j∈V,∀k∈K

cfGcfxijk≤cf ∀i,j∈V,∀k∈K

_cfG_cfx_ijk≥−_cf ∀i,j∈V,∀k∈K

cfGcfQijk≤cf ∀i,j∈V,∀k∈K

cfGcfQijk≥−cf ∀i,j∈V,∀k∈K

tG^t_ijxijk≤^t_ij ∀i,j∈V,∀k∈K

tG^t_ijxijk≥−^t_ij ∀i,j∈V,∀k∈K

n+n^′+1

j=1

x0jk=1 ∀k∈K

n+n^′

i=0

xin+n^′+1k=1 ∀k∈K

K

k=1 n+n^′+1

j=1j=/i

xijk=1 ∀i∈P∪D

K

k=1 n+n^′

i=0j=/i

xijk=1 ∀j∈P∪D

K

k=1 n+n^′

i=0j=/i

xijk−

K

k=1 n+n^′+1

i=1j=/i

xjik=0 ∀j∈P∪D

yjk−yik≥−M(1−xijk)+t(i,j)+di+dG^d_i+tG^t_ij

∀i∈P∪D∪

0

∀j∈P∪D∪

n+n^′+1

,i=/j,k∈K

sjk−yjk≥−M(1−xin+n^′+1k)+t

i,n+n^′+1

+dj+dG^d_j +tG^t_in+n′+1 j∈P∪D∪

0

,k∈K

K

k=1 n+n^′

i=0j=/i

Q_ijk−

K

k=1 n+n^′+1

i=1j=/i

Q_jik=−q_j ∀j∈P∪D

max

0,−qj

≤Qijk≤min{Ck,Ck−qj}

∀i∈P∪D∪

0

∀j∈P∪D∪i=/j,k∈K

yik−yjk≤M

⎛

⎝

2−

⎛

⎝

n+n^′

l=0,l=/j

xljk−

n+n^′

l=0,l=/i

xlik

⎞

⎠

⎞

⎠

i∈P,j∈D,k∈K

K

k=1 n

j=0 n+n^′+1

i=n+1

xijk=0

e_i

n+n^′

j=0,i=/j

x_ijk≤y_ik≤l_i

n+n^′

j=0,i=/j

x_ijk ∀k∈K,∀i∈D

max{0,y_ik−D_i}≤TT_ik+M

⎛

⎝

¹⁻

n+n^′

j=0,i=/j

x_jik

⎞

⎠

∀i∈P,∀k∈K

max{0,D_i−y_ik}≤ET_ik+M

⎛

⎝

¹⁻

n+n^′

j=0,i=/j

x_jik

⎞

⎠

∀i∈P,∀k∈K

Q0ik+

n+n^′+1

j=1 n+n^′

l=0

qixljk≥−M(1−x0ik) ∀i∈V/{0}

xijk∈

0,1

∀i∈V,∀j∈V,∀k∈K

y_ik≥0,Q_ijk≥0,s_jk≥0,ET_ik≥0,TT_ik≥0 ∀i∈V,∀j∈V,∀k∈K

5. Experimentalresults

Toshowthevalidityandreliabilityoftherepresentedmodel, severalnumericalexperiments are executedand relevant solu- tionresultsareprovidedinthissection.Fornominalexperiments, nine test problems with different sizes, as shown in Table 2, aredoneundernominaldatachosenfromuniformboundsrep- resented in Table 1 and for each size the experiments are performedbyconsideringsixdifferentuncertaintylevels(i.e.,= 0.1,0.3,0.5,0.7,0.9,1).Then,fiverandomrealizationsthathave gottenfromuniformuncertaintyset,[nominalvalue−*G*,nom- inalvalue+*G*],areexaminedforsix testproblems with(i.e., =0.1,0.5,1)toevaluatetheperformanceofthesolutionsrep- resentedbytheproposeddeterministicandrobustmodels.The secondresultsaregiveninTable3.

In the analysis of the random realizations, the mean and standard deviationare used asperformance measures for both deterministicand robust models. The uncertainty levels for all parametersintherobustmodelarethesame(i.e.e=cf=t= d)andfordeterministicmodels(=0).Allresultsrelatedtothe robustanddeterministicrealizationsaregiveninTable3.

Itisobviousthattheresultrelatedtotheproposedrobustopti- mizationmodelisworsethantheonesobtainedfromtheproposed deterministicoptimizationmodel.Itisbecausethatintherobust optimizationalluncertainparametersareregardedasworstcases inpracticetolowertheriskandlackuntilitis possibletoper- form.Moreover,astheuncertaintylevel isincreasedtherobust objectivefunctiondegenerates.Hence,byincreasingtheamount oftheuncertaintyoftheproposedmodel,thesolutionhastoadapt itselftotheuncertainconditionsandcreateworsevalues.There- fore,increasingtheuncertaintylevelthedecisionbecomesmuch moresensitiveanddifficult.Thissensitivityanddifficultyaffects ontherobustoptimizationmodeltoadjusttheconditioninorder toproducethesolutions thatcanreduce thepossibilityof fail- urein itswork rather thanthe deterministicmodel.According toTable3,therobustmodelrepresentsthesolutionwithhigher valuesandlowerstandarddeviationratherthanthedeterministic model.

Table1

Sourceofrandomgenerationsforthenominaldata.

Parameters Correspondingrandomdistribution

e Uniform(3.15,4.95)

Cf Uniform(0.35,0.55)

di Uniform(0.25,0.75)

v _{Uniform(30,80)}

Table2

Summaryoftestresultsundernominaldata.

Problemsize

|V|*|P|*|D|

Uncertainty rate()

Objectivefunctionvalue undernominaldata Deterministic Robust

4*2*2 0.1 77,300.619 77,424.144

0.3 77,671.194

0.5 77,918.244

0.7 78,165.293

0.9 78,412.343

1 78,535.868

5*2*3 0.1 114,042.620 114,203.143

0.3 114,524.187

0.5 104,845.231

0.7 115,166.275

0.9 115,487.319

1 115,647.841

6*3*3 0.1 129,302.698 129,634.650

0.3 130,298.553

0.5 130,962.457

0.7 131,626.361

0.9 132,290.265

1 132,622.217

7*5*2 0.1 147,014.565 147,467.174

0.3 148,372.393

0.5 149,277.611

0.7 150,182.830

0.9 151,088.048

1 151,540.658

8*5*3 0.1 192,356.047 193,164.917

0.3 194,782.658

0.5 196,400.398

0.7 198,018.139

0.9 199,635.880

1 200,444.750

9*5*4 0.1 197,788.229 209,088.737

0.3 202,139.392

0.5 205,040.168

0.7 207,940.944

0.9 216,296.487

1 217,197.456

10*5*5 0.1 208,017.768 209,088.737

0.3 210,890.674

0.5 212,692.612

0.7 214,494.550

0.9 210,841.719

1 212,292.107

11*5*6 0.1 265,912.423 267,040.540

0.3 269,296.773

0.5 271,553.007

0.7 273,809.241

0.9 276,065.474

1 277,193.591

12*5*7 0.1 277,232.580 278,607.945

0.3 281,358.676

0.5 284,109.406

0.7 286,860.137

0.9 289,610.868

1 290,986.233

Thesensitivityanalysisisconductedonatestproblemwith(= 0.3)andninecustomers.ItisrepresentedinTables4–6.Inaddition, allmathematicalmodelseitherdeterministicorrobustarecoded intheoptimizationsoftware(i.e.,GAMS).

In thenextsub-section, afurthersensitivity analysisis per- formed for evaluating the capability and effectiveness of the proposedMILPmodelandtheimpactsoftheparametersusedin themodel(e.g.,CO₂emissionscost,thecapacityofvehicles,andthe numberofpickup(delivery)centersinacustomerzone).Allsen- sitivityanalysesarereportedbasedontheselectedtestproblem withninecustomers.

Table3

Summaryoftestresultsunderrealization.

Problemsize|V|*|P|*|D| Uncertainty level()

Meanofobjectivefunction valuesunderrealizations

Standarddeviationofobjective functionvaluesundernominaldata

Deterministic Robust Deterministic Robust

6*4*2 0.1 132,689.092 133,010.435 5874.806 5863.886

0.5 103,816.160 105,052.459 7697.402 7243.613

1 120,900.173 127,016.307 51,434.472 44,923.780

7*3*4 0.1 147,134.402 147,587.708 4540.966 4523.098

0.5 158,813.433 160,809.728 22,819.394 22,401.967

1 187,915.287 194,173.110 37,370.501 35,899.592

8*3*5 0.1 193,851.471 194,679.441 9881.532 9826.449

0.5 170,134.691 173,692.072 20,079.781 19,419.356

1 206,627.628 214,373.962 75,911.719 73,291.846

9*6*3 0.1 197,446.889 198,973.708 7047.592 68,531.662

0.5 202,145.887 206,185.946 48,184.873 46,827.245

1 230,667.210 239,900.967 72,276.401 68,315.493

10*6*4 0.1 231,087.940 232,096.158 5687.912 5656.778

0.5 221,868.504 231,621.073 20,200.991 15,319.460

1 252,548.786 268,822.541 129,005.798 126,364.011

11*5*6 0.1 272,306.884 273,600.724 13,122.635 13,079.201

0.5 215,498.852 220,558.448 54,576.422 53,566.694

1 324,444.373 359,772.328 94,246.525 66,031.599

Table4

Resultsofsensitivityanalysisonnumberofpicksupanddeliveriesforeighttest problems.

Problemsize|V|*|P|*|D| Objectivefunctionvaluesunder nominaldata

Deterministic Robust

9*8*1 241,837.029 246,064.209

9*7*2 210,132.374 214,111.610

9*6*3 199,663.385 203,727.521

9*5*4 197,788.229 202,139.392

9*4*5 210,255.321 224,855.175

9*3*6 218,928.635 223,313.315

9*2*7 225,453.320 230,897.073

9*1*8 222,027.539 227,471.292

Table5

ResultsofsensitivityanalysisonCO2emissionscostfordeterministicandrobust testproblem.

Problemsize

|V|*|P|*|D|

CO2emission cost

Objectivefunctionvalues undernominaldata

Deterministic Robust

9*6*3 1.55 117,630.919 121,590.182

2.05 130,838.654 134,032.446

2.55 148,044.837 151,554.886

3.05 165,251.020 169,073.327

3.55 182,457.203 186,593.767

4.05 199,663.385 204,114.207

4.55 216,869.568 221,634.647

5.05 234,075.751 239,155.088

5.55 251,281.934 256,675.528

6.05 268,488.116 274,195.968

6.55 285,694.299 291,716.408

7.05 302,900.482 309,236.848

7.55 320,106.665 326,757.289

8.05 337,312.847 347,421.928

8.55 354,519.030 361,798.169

9.05 371,725.213 379,318.609

9.55 388,931.396 396,839.050

10.05 406,137.578 417,856.874

10.55 423,343.761 431,879.930

5.1. CO2emissionscost

AsensitivityanalysisontheCO₂emissionscostareperformed anddepictedinFig.2.Asitcanbeseeninthisfigure,whentheCO2

Table6

Resultsofsensitivityanalysisonthevehiclecapacityfordeterministicandrobust testproblem.

Problemsize

|V|*|P|*|D|

Capacity Objectivefunction undernominaldata

Numberoftruckused inmodel

9*6*3 25 Infeasible Infeasible

26 Infeasible Infeasible

27 Infeasible Infeasible

28 199,663.385 4

29 199,654.385 4

30 199,645.385 4

31 199,636.385 4

32 199,627.385 4

33 199,618.385 4

34 199,609.385 4

35 199,600.384 4

36 199,591.385 4

45 199,510.385 4

47 199,492.380 4

50 199,465.385 4

55 183,650.569 4

57 183,632.569 4

60 183,605.569 4

50 212,419.715 3

55 183,650.567 3

57 183,632.569 3

60 183,605.569 3

emissionscostincreases,theobjectivefunctionvaluesincreaseas well.

5.2. Numberofpickup(delivery)nodes

AccordingtoFigs.3 and4, thenumber ofpickup anddeliv- erycustomersaffectontheobjectivefunctionvalue.Forexample, whenthecustomerzonehastheleastpickupcustomer,theobjec- tivefunctionvaluestandsatthemaximumamountthemodelis able torepresent basedonthe test problem. It occurs because the average amount of the load carried during the route and thetimeofcarryingtheloadsincrease,andsotheysignificantly impactonfuelconsumptions;however,asthenumberofpickup nodesgoesuptheobjectivefunction valuedecreaseasaresult of reducing theaverage amount of loads carried in the routes andthetimetheycarriedduringthoseroutes.Ithappensbecause vehiclegets thechoice thatloads up theproductdeliverycus- tomersneedfromthepickupcustomersduringtheirrouteinstead

Fig.2.ObjectivefunctionandCO2emissionscost.

Fig.3.Objectivefunctionandnumberofpickupnodes.

Fig.4.Objectivefunctionandnumberofdeliverynodes.

Fig.5.Objectivefunctionandvehiclescapacity.

Fig.6.Objectivefunctionandnumberofvehicles.

ofloadingupatdepot.It cancontinueuntiltheobjectivefunc- tion gets the minimum value and the optimized numbers of pickupanddeliverycustomerisachieved;however,ifthenum- ber of pickups rises continuously, the objective function value willincreaseagain.It isbecauseofraisingtheamountofloads which hasto bepicked upand then carried to thedepot, and ofcoursethetimeof carryingtheseloadsthat makesfuelcon- sumedfordeliverthemtotheirdepartureseitherdeliverynode ordepot.

5.3. Capacityofvehicles

AsdepictedinFig.5,increasingthecapacitycannotaffectthe objective functionvalue impressively beforea certainsize (i.e., 55tons);however,asitreachesto55tonsthevehiclesbecomes capabletochangetheirroutesandservicingmorecustomersina moreefficientway.Itisobviousthatbyobtainingtothispointfor thecapacity,thenumberofvehiclesinvolvedinroutesdecreases.

Itispointedoutthatthenumberofvehiclesandtheircapacitiesare

interdependent.AsitisseeninFig.6,theproblemcanbesolvedby eitherthreevehiclesorfourvehiclesbeforereachingtothecapacity of55tons.Inotherwords,threevehiclesarecapabletoanswerall thecustomersdemand;however,theydothatinawayconsuming morefuel.Thus,fourvehiclesaremoreefficientforthecapacityup to55tons.Asthecapacityreachesto55tons,threevehicleschange theirroutesinthewaythatobjectivefunctionvaluesbecomesim- ilartotheobjectivefunctionvalueinwhichfourvehiclesareused forsolvingtheproblem.

6. Conclusion

Thispaperconsidersanewtimewindowpickup-deliverypol- lutionroutingproblem(TWPDPRP)andintroducesanewrobust mixedintegerlinearprogramming(MILP)modelunderuncertainty basedontherecentextensionsintherobustoptimizationtheory.

Anewvehicleroutingproblemisrepresentedthatnotonlycon- siderstheenvironmentalissuesbutalsothinksthroughthekindof vehiclerouting problem,namedpickup-deliveryvehiclerouting problemtoreducetheeconomic and ecologicmatters simulta- neously.Thepresentedmodelimpactsonpickuparrivaltimesin awaythatdecreasesthetardinessand evenearlinessofarrival timesinordertosettheminrighttimeanddependthemonthe arrivaltimesindeliverycustomersindirectlywhichareconsidered intimewindowsconstraints.Ontheotherhand,themodelconsid- erstheenvironmentalfactors,suchastheslopofroads,andfriction forces.Thenit triestodeclinetheamountofCO₂ emission and fuelconsumptionwhichmayincreasethecostdependedoneco- nomicfactors.Thegoalofthemodelistostrikeabalancebetween economicandenvironmentalfactors.Inaddition,a robustopti- mizationapproachispresentedforuncertainparameterstomodel theTWPDPRP,suchasfuelconsumptionscost,CO₂emissionscost, traveltimeandservicetime.Therobustnessofthemodelmakesit flexibletowarduncertaintyandpreventsthemodelfromproducing infeasiblesolutions.

Theseveralsensitivityanalyseshaveemphasizedthatthenum- berofpickupanddeliverycustomerscanaffectontheobjective functionvalue,andanoptimizedsetofthemisabletogivethe best(i.e.,minimum)valueofobjectivefunctionamongresultsof othersets.Also,thevehiclescapacityimpressivelytakeseffectson thecustomersetsassociatedtovehiclesandthenumberofvehi- clesassociatedtotheroutes.Asa directionfor futureresearch, morephysical conditionsrelated toroadscanbe consideredin theTWPDPRPproblem.Moreover,heterogeneousfleetandtheir mechanicalfeatures,thecost ofvehiclessetup,andtherelated analysisofeconomiccostsandbenefitsforusingvehiclescanbean importantissuetospot.

References

[1]NationalResearchCouncil(NRC).Advancingthescienceofclimatechange.

Washington,DC,USA:TheNationalAcademiesPress;2010.

[2]OhnishiH.Greenhousegasreductionstrategiesinthetransportsector:pre- liminaryreport.Tech.rep.,OECD/ITFJointTransportResearchCentreWorking

GrouponGHGreductionstrategiesinthetransportsector.Paris:OECD/ITF;

2008. http://www.internationaltransportforum.org/Pub/pdf/08GHG.pdf [accessed11.02.11].

[3]Mousavi SM, Tavakkoli-Moghaddam R, Jolai F. A possibilistic program- ming approach for a location problem of multiple cross-docks and vehicle routing scheduling under uncertainty. Engineering Optimization 2013;45(10):1223–49.

[4]MousaviSM,Tavakkoli-MoghaddamR.Ahybridsimulatedannealingalgorithm forlocationandroutingschedulingproblemswithcross-dockinginthesupply chain.JournalofManufacturingSystems2013;32:335–47.

[5]VahdaniB, Tavakkoli-MoghaddamR, ZandiehM,RazmiJ. Vehiclerouting schedulingusinganenhancedhybridoptimizationapproach.JournalofIntel- ligentManufacturing2012;23:759–74.

[6]FungRYK,LiuR,JiangZ.Amemeticalgorithmfortheopencapacitatedarc routingproblem.TransportationResearchPartE:LogisticsandTransportation Review2013;50:53–67.

[7]NagataY,BräysyO,DullaertW.Apenalty-basededgeassemblymemetic algorithmforthevehicleroutingproblemwithtimewindows.Computers&

OperationsResearch2010;37:724–37.

[8]ZachariadisEE,TarantilisCD,KiranoudisCT.Ahybridmetaheuristicalgorithm forthevehicleroutingproblemwithsimultaneousdeliveryandpick-upservice.

ExpertSystemswithApplications2009;36:1070–81.

[9]JeonH,LeepR,YoungShimJ.Avehicleroutingproblemsolvedbyusingahybrid geneticalgorithm.Computers&IndustrialEngineering2007;53:680–92.

[10]SerdarTasanA,GenM.Ageneticalgorithmbasedapproachtovehiclerout- ing