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A Cut-and-Branch algorithm for the Multicommodity Traveling Salesman Problem

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A Cut-and-Branch algorithm for the Multicommodity Traveling Salesman Problem
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  ACut-and-BranchAlgorithmfortheMulticommodityTravelingSalesmanProblem JoaoSarubbi CEFET-MGDivin6polis-MG-BrazilEmail:joao@div.cefetmg.br GilbertoMiranda UFMGBeloHorizonte,BrazilEmail:miranda@dep.ufmg.br HenriquePaccaLuna UFALMacei6,AL,BrazilEmail:pacca@tcLufal.br GeraldoMateus UFMGBeloHorizonte,MG,BrazilEmail:mateus@dcc.ufmg.br Abstract- ThispaperpresentsaCut-and-BranchalgorithmfortheMulticommodityTravelingSalesmanProblem(MTSP),ausefulvariantoftheTravelingSalesmanProblem(TSP).TheMTSPpresentsamoregeneralcoststructure,allowingforsolutionsthatconsiderthequalityofservicetothecustomers,deliveryprioritiesanddeliveryrisk,amongotherpossibleobjectives.IntheMTSPthesalesmanpaysthetraditionalTSPfixedcostforeacharcvisited,plusavariablecostforeachofthecommoditiesbeingtransportedacrossthenetwork. We presentastrongmathematicalformulationforthisrelevantproblem. We implementaCut-and-BranchalgorithmfortheMTSPwhich is able to findoptimalsolutionsfasterthanstand-aloneCPLEXcodes. keywords:MulticommodityTravelingSalesmanProblem;MinimumLatencyProblem,Cut-and-BranchAlgorithm,BendersDecomposition I. INTRODUCTION Withtheincreaseincomputingspeedandcapacity,ithasbecomefeasibletocreateexactalgorithmsforinstances of classicaloptimizationsproblemsthatwerepreviouslysolvedusingheuristicmethods.Amongtheseproblems,theTravelingSalesmanProblem(TSP)hasbeentheobject of intenseinvestigationsincethefirstmathematicalformulationfortheproblemwasintroducedbyDantzig,FulkersonandJohnsonin1954[11].TheTSP is importantinthesolution of manyproblemsfoundintransportationandlogistics,suchasvehiclerouting,orderpicking,pickupanddeliveryproblems,forinstance.TherehasbeenmuchprogressinthederivationofspecificcombinatorialalgorithmsfortheoriginalTSPandothersimplevariants,enablingtheexactsolution of instanceswithupto85,000demandpoints(cities)[1].Inspite of thegreatnumber of algorithmsforthe TSP, problemswhichincorporateamoregeneralcoststructuretobettersuitreallifeapplicationshavenotbeenstudiedindepth.Aninterestingvariant of the TSP, whichwascalling TheMulticommodityTravelingSalesmanProblem [23] [24],isthefocus of thispaper.Inthisproblemweareinterestedinallocatingafixedcost to eacharcoftheunderlyingMTSPtourandalsoavariablecostassociatedwiththecommoditiesbeingtransportedacrosseacharc.Thecoststructure of thisnewvariantavoidsthecalculation of an optimalsolutionthatisinsensitivetothequality of serviceperceivedbythecustomerandotherfactorsthataffecttransportationwhentheproductsbeingtransportedarenothomogeneous.TheMTSPgenerallytendstopriorize,interms of deliveryorder,customersdemandinggreaterquantities of moreexpensivecommodities.Inaddition,itcanalsoconsidertheaddedcostandrisk of transportingperishableorfragileitemsortheneedtosatisfyanimportantclientregardless of theordersizeandvalue.Ourarticle is divided as follows:inthenextsectionwepresentanddiscusssomestronglyrelatedproblems.Section(III)showstheformalderivation of ourformulationforMTSPandsomevalidinequalities.Section(IV) is devotedtothedevelopment of solutiontechniquesfor MTSP. Section(V)describesthenumericalexperiments.Finally,section(VI)presentsour find remarks. II. LITERATUREBACKGROUNDANDRELATEDPROBLEMS TheclassicalTSPisaverywellknownprobleminwhichanagentmustvisitaset of citiesintheminimumpossibledistance.Forthesake of simplicitytheclassicalTSPwillnotbereviewedhere.FordescriptionandreviewsontheTSPthereadermustreferto[11][16] [9] [1]. Two TSPvariantsthatarestronglyrelatedtotheMTSParetheMinimumLatencyProblem(MLP)andtheSingleVehicleDeliveryProblem(SVDP).TheMinimumLatencyProblem(MLP) [10][19] [6][8] [26] [22],alsocalledtheTravelingRepairmanProblem,theDeliverymanProblemandtheTravelingSalesmanProblemwithCumulativeCosts,isaTSPvariantinwhichtheagentisrequiredtovisiteachnode of agraphexactlyonceinsuchawaythattheoverallwaitingtimesforallnodesisminimized.Despitetheobvioussimilaritiestotheclassical TSP, theMLPappearstobemuchlesswell-behavedfromacomputationalpoint of view[15].Withthe MLP, theobjective is tominimizethetotallatencytime of allthecustomers,whileinthestandardTSPtheobjectivefunctionisfocusedontheminimization of thetravelingtime of asingletravelingsalesman.Due to theincorporationofthecustomersconflictingobjectives,theMLP is harderthantheclassical TSP. TheMLPwasintroducedandrelatedtotheTSPin1967,byConway,MaxwellandMiller[10],atwhichtimetheMLPwasclassified as atypeofschedulingproblem.Forthistype of application,theMLPcanbeinterpreted as a978-1-4244-2013-1/08/$25.00©2008IEEE1806  if thesalesmantravelsacrossarc( i, j) , otherwise. min L (bijXij + L Cijkdkfijk (1)  i,j EE kEY where n == V I Theobjectivefunction  1) computesthecostsforallthearcsusedintheroute,withtwopartsforeacharc.Thefirstpartreferstothefixedcost of travelinginthearc.Thesecondpart (7)(8)(2)(3) (4) (5) (6) (10) Vk E V Vj E V Vk E V Vi E V Vi j kEV i=l=j (11) Vi, j E V, i   j (12) Vi j k E V Vj, k E V, j   k, j =I=(D) 1 11 111 o {O,l} n(n+l)  2 subjectto L fill == iEVli,i:1 L  fikk - fkik == iEVli=rfk L  fjik - fijk == iEVli=rfj L fijk == i,j,kEVli=rfj fijk   Xij E L Xij == iEVli=rfj L Xij == jEVli=rfj fijk セ L fIjI == jEVlj=rf1 L  fljk - fjlk == jEVlj=rf1 fij k: fraction of totaldemand of commodity k transportedacrossarc (i, j) todemandnode k. andthefollowingset of parameters: b ij : fixedcosttotravelacrossthearc (i, j Cijk: unitarycosttotakethecommodity k acrossarc (i, j). Themathematicalmodel Mis: IntheMTSP,thetotalamount of commoditiesandthevalue of each of thecommoditiesbeingcarriedacrosseacharcisanimportantfactorinthetotalcost of theroute.Thismeansthatcustomersdemandinghigherquantities of moreexpensive,important, or high-riskproductsshouldbeattendedwithhigherpriorityindeliveryorder.Anotherpossibleinterpretationisthat if traditionalTSPvariantsare cost-oriented thentheMTSPisalso customer-oriented. Forinstance,sensitivematerialscandemandspecialtransportationstructures,perishablegoodsmaypayforrefrigerationuntildelivered,importantclientscanbeattendedwithsomeprioritywhileotherkinds of commodities or clientsdonotrequiresuchahighdegree of attention.Forthisproblemwecandefineamixed-integerlinearprogrammingformulationMwiththefollowingsets of variables: III. MODELLING THE MULTICOMMODITY TRAVELING SALESMANPROBLEM TheMTSPcanbedefinedasfollows:Consideradirectedconnectedgraph G V, E , where V denotestheset of nodesand E isacollection of arcs.Supposewehaveanoriginnode1and,foreachnode k E V, ademand d k of aspecificcommodity k shouldbedeliveredduringatour.Theorigindemandisequaltooneandrepresentsthelogisticoperatorwhomustreturntoorigin.Therearefixedcostsfortravelingacrossarc (i, j E E, andalsovariablecostsforeachdifferentcommoditybeingmovedacrossthearc.Theobjectiveistodeliverallthecommoditiesminimizingthesum of thefixedandvariableflowcosts.Thefixedcostfortravelingacrossanarcisrelatedtothedistancetraveledandthetransportationinfrastructure.Thelogisticoperatorpaysthefixedcostevenwhentravelingwithanemptyvehicle.ThisfixedcostitisalsoconsideredintheoriginalTSP.Inadditiontothefixedcost,theMTSPalsoincorporatesavariablecostforeachcommoditythatcrosseseacharc.Therole of thevariablecostistoestablishthefactthatmulticommoditytransportationnetworksdealwithdifferentproducts of differentaggregatevaluesbeingshippedindifferentquantitiesoverdifferentroutes.SingleMachineSchedulingProblemwithsequence-dependantprocessingtimesinwhichoneseekstominimizethetotalflowtime of thejobs[12].The MLP isaspecialcase of theTime-DependentTravelingSalesmanProblem(TDTSP)[21][19][25].IntheTDTSP,thecost of crossinganarcbetweentwonodesmayvarywiththeposition of thisarcalongtheHamiltoniantour[19]. An extension of the MLP istheSingleVehicleDeliveryProblem(SVDP)[5].Instead of workingwithageneraltimedependentcostfunctionasintheTDTSP,theSVDPextensionconcernsadifferentamount of demand dk to be deliveredateachnode k. Manysituationsinvolvingasinglevehicleandclientswithheterogeneousdemands(persons or goods)can be modeledasaSVDP. An exampleisacarthatpicksuptouristsattheairportanddeliversthemtotheirhotels.Inthiscase,thenumber of touriststobedeliveredateachnodeisalsoimportantinordertofindthebestroute.ThefactthatintheSVDPadifferentamount of demandcan be deliveredateachnodeinducesacertainpriorityforplaceswheremoredemandwillbedelivered.NotethattheSVDPisreducedtothe MLP whenthedemandforeachnodeisasingleunit.Bianco,Mingozzi,RicciardelliandSpadoni [5] presentexactresultsforsymmetricalinstances of theSVDPwith30nodesusingLagrangeanrelaxationanddynamicprogrammingalgorithms.Otherextensions of timedependentcostfunctionshavebeenproposed,butthespecialfeature of theMTSPisthatitencompassesheterogeneousnodedemands dk andalsoincludesunitaryflowcosts Cijk thatarespecifictothetype of commodity k E V beingtransportedacrossanarc.Thetimedependenceisthenaconsequence of boththequantityandthedifferentnature of thecommoditiesflowingonthetransportationnetwork.1807  IV. PROJECTION OFTHE MTSP EXTENDED FORMULATION Givenanintegersolutionforthe x variables,isstraightforwardtoderivetheassociatedvalues of the P variables.However,thisisnottrue if oneprovidesafractionalsolution of x. Inthiscasetheaboveconstraintsareresponsibleforconsiderableimprovement of thelinearrelaxationgaps,impactingthesolutiontimesinthesamemanner.Thenumericalexperimentsinsection(V)establishtheimportance of theseadditionalconstraintsandvariablesforthesolution of largeMTSPinstances.Itisimportanttonotethatthevariables P arewritteninadifferentspacethanthevariables x. Writingtheabovecouplingconstraintswiththe x variablesmayresultininfeasibleprograms.Inthissection,anadditionalefforttosolvetheMTSPispresented.Thiseffortisacut-and-branchalgorithm,suitedtogeneratestrongvalidinequalitiesinthespace of formulation(1)-(12).Theseinequalitiesaretheresult of theprojection of thevariables of theextendedformulation of MTSPequations(13)-(18).Thismethodisarevisitation of BendersDecompositionwhichwassuggestedinseveralworks of Ballasetal.[2],[4],[3].Thismethodisalsodescribedindetailinatextbookby R. K. Martin[20].Fromtheviewpoint of mathematicalprogramming,aprojection of theproblem (1 )-(18)ontothespace of variables (x, f) canbedone.Byfixingthepair (x, f) ==  x,I), thefollowingsystem,in P, remainstobesatisfied: L: hjk Vk E V k   1(23) i,jEVli j (24)(19) (17) Vk E V, k =11(18) Vi E V Vi,j E V Vi E V 20 Vj E V o 11  21) L: hjk Vi E V  22 k,jEVli j n 2:  l - 1 Plk l=1 n I: (n - 1 + 1  Pli l=1 L fijk == i,jEVli,t:j L fijk == k,jEVli,t:j LPij= iEV LPij == 1 Vj E V (13) LPij= JEV iEV LPij == P11 = 1 Vi E V (14) n jEV L n-l+l Pli= 1 (15) l=l PII == n p > 0 Vi jEV. (16) L l-l Plk = ,)_ l=lPij   Theformulation(1)-(12)isenoughtodescribetheMTSP.Ontheotherhand,giventheassociationbetweentheQuadraticAssignmentProblem(QAP)[17]andthetravelingsalesmanproblemsincetheoriginalpaper of KoopmansandBeckman,wehavesomeinsight as tohowtostrengthentheaboveformulationwiththeaid of  QAP-like constraints.Thework of VanderWielandSahinidis[25]isanexample of howtoconnectQAPwithTSPvariants.WhenusingQAPtosolveTSP,itisimportanttorecallthattheQAPassignmentconstraintsarenotwritteninthespace of arcs.Inotherwords,theydonotmeanthatagivenarcisusedonlyonceinagivendirection.Theyjustensurethatagivennodeisvisitedatonlyoneorderpositionandthatagivenorder of visitationisusedonlyonce. To usethistype of assignmentconstraintswithmodelM,weproposeanewset of variables Pij definingtheorder j of visitationthenode i. Inspite of theobvioussolution of creatinganewset of orders of visitation,wemustrecallthatthereareasmanyvisitationorders as nodesin V. Thisobservationisthekeytowritingtheassignmentconstraints as: A. An ExtendedFormulationforMTSP refers to thetotal flow chargesforthecommodities,whichiscalculatedwiththecost of transferacrossthetraveledarcsforeachcommodity,fromthesourcenodetoitsspecificdemandnode.Constraintsets(2)and(3)aretheassignmentconstraintsintroducedin1954byDantzig,FulkersonandJohnson[11]andensurethatthereisjustonearcarrivingandonearcleavingeachnode of thetour.The x and f couplingconstraints(4)assurethatno flowis allowedonarc (i, j) unlessthefixedcost b ij ispaidtotravelacrossthearc.Constraintsets(5)-(11)assuretheconservation of flow alongthetouraspresentedin[14][18].Theredundantconstraints(10)ensurethatthenumber of commoditiesthattravelacrossallarcsisexpressedby n(n 2 +I) , where n == V I (10).Thecouplingconstraints(17)-(18)arevalidinequalitiesthatareresponsibletoimprovetheMlinearrelaxationgap.Constraints(17)statethatateachnodeagivencommodityisbeingdelivered.Constraints(18)forcethetotal flow of commodity k tobeattachedtotheorder of visitation of node k. Inordertogeneratethedeepestpossiblecutinthespace of (x, f) fromtheviolation of thissystem,wemustrewriteitasanoptimizationproblem: (25) 1808  Where e 1, e2, e3 and e4 aretheinfeasibilitiesinthespace of p resultingfromthefractionalsolutionobtainedonthe (x, f) problem.Associatingthedualvariables Vj Ui S, Z[ 2. 0, z 2. 0, Wk 2. 0and キ 2. 0respectivelytoconstraints(26),(27),(28),(29),(30),(31)and(32)itispossibletowritethelinearprogrammingdual of formulation(25)-(34)as: L h-jk Vk E V k  11) i,jEVlii=j L h-jk Vk E V k 12) i,jEVlii=j o Vi,j E V 33 o (34) L h-jk Vi E V k,jEVlii=j (48)(47) in e} + e; + e; +   subject to: LPij == iEV LPij == 1Vj E V j -11 (42) iEV LPji == 1 Vj E V j -11 (43) iEV n L n -i + I)pij == 2: hik Vj E V j -11 (44) i=1 k,iEVlii=j n L i - l)pij == L h-lj Vj E V j -11 (45) i i lEVlii=l Pij セ 0 Vi,j E V j -11. (46) determinedsinceanytourmuststartattheorigin 1. Thisisensuredatthemasterlevel(problemin (x, f), makingconstraint(21)redundantandunnecessary. It ispossibletorewritetheequations(19)-(24)as:Thisequivalentviolatedsystemallowsforthesolution of asubproblemforeach j E V, j -I 1. It isalsopossibletocomputeaviolationineachsub-systemforthemastersolution.Theproblemwhichfindsthemostviolatedcutforagivenj E V, j > 1 isstatedas: (26)(30)(28)(29)(27) Vj E V Vi E V 1 2: h-jk Vi E V k,jEVlii=jPij   el,e2,e3,e4   n L l I)Plk + e3 セ l=1 LPij == jEV   == subject to: LPij == iEV n - L l-I Plk +e4 セ l=1 n L n-l+I Pli+ el セ l=1 n -L (n - l + 1  Pli + e2 セ l=1 Sincethepolyhedrondefinedby (36)-( 40) hasafinitenumber of generators,indexingthosegeneratorsby h == 1 ... H wehavevalidinequalitiesfortheproblem (1)-(12) duetotheviolation of equations (13)-(18) intheform:Thecorrespondingdualversion of theabovelinearprogramis: •   12   12 mIn Vj + L..J Ui + L..J  jik(Zj - Zj) + L..J  ikj(Wj - Wj)  56 iEVi,kEVi,kEV (49)   hik (50) k,iEVlii=j L hik (51) k,iEVlii=j L hlj (52) i lEV lii=l L lilj (53) i lEVlii=l 0 Vi E V 54 0(55) Pij セ e},e;,e;,eJ セ LPji == iEV n L i - l)pij + e セ i n M i n L n - i + l)pij + e} セ i=1 n - L n - i + l)pij + e セ i (39)(40)(36)(35)(37)(38) subject to:8 + Ul + VI + n [zi - zi] + [wi  wn:::; 0  1,1) E V 2 Ui + Vj + (n - i + 1 [z] - zJ]+ (i-1) [w} - w;]  ; 0 V i,j) E V 2, (i,j) -I  1,1) LZ}:::; I,Lz;:::; I,Lw}:::; i lキセZZZ[ 1 iEViEViEViEV z} , z; セ 0 Vi E V キ (41) subject to :Ui + Vj + (n - i + 1) [z} - zJ] + i-1) [w} -w;]:::; 0 o  ; z} z; , w} , w;  ; Vi E  57) 1(58) It iswellknownthatwhenusingaBendersdecompositionbasedalgorithmamajordrawbackistheabsence of specialstructureintheBenderssubproblem.Observingformulation(35)-(40)onecanseethat,afteraslightmanipulation,itmaybepossibletoderiveanequivalentviolatedsystemwithspecialstructure. We startourderivationrecallingthatintheunderlyingassignmentproblem,oneassignmentisalreadyInthiscase,theexpressionforthevalidinequalitiesisstatedas: (59) 1809  Inadditiontotheclearadvantage of solving n - 1smallsubproblemsinstead of onelargesubproblem,theabovemanipulationsenabletheuse of themulticutversion of Bendersdecomposition, as proposedbyBirge, J. R. andLouveaux, F. V. [7].Thisversionisextremelyaggressive,sinceittakesintoaccountenhanceddualinformation.Themulticutversion of thevalidinequalitiesis: (60) Theresults of severalcomputationalexperimentsarediscussedinthenextsectiontocompareourCut-and-Branchschematoastandalone(monolithic)CPLEXimplementation. V. COMPUTATIONALEXPERIMENTS ThecomputationalexperimentspresentedherehavebeencarriedoutinaPentiumIVcomputerwitha3.0GHzprocessorand1Gbyte of RAMmemory.TheoperationalsystemwasLinux.ILOGCPLEX9.1.3wasusedtocomputetheoptimalintegersolution.Inourinstances,eachnode k E V hasademandbetween1andthe MaxDemand value,whichvariesfrom5to20.Theorigindemandisequalto1andallthedemandsareintegers.Thevalues of b ij (fixedcost)arerelated to thedistancesbetweenthecitiesandhavebeenrandomlygeneratedaccordingtoauniformdistributionon[1,100],towhichthetriangularinequalitywasforcedbymeans of shortestpaths[13].Thecost, Cijk (variablecost), of carryingcommodity k acrossarc (i, j) is calcutatedwiththefixedcostandsomeadditionalparameters.Each Cijk is calculatedwiththefollowingexpression: Cij k == セ .  Tj • b ij . Theparameter セ thatrepresentsapseudo-randomrelationshipbetweenthevariabletransportationcostoneacharc (i, j) andthevalueofthefixedcost b ij • Thisparameter セ isequalforallcommoditiescrossingaspecificarcbutmayvaryfromonearctoanother.Thebettertheconditions of agivenarc,thelowerthevalue of thecorresponding セ varyingfrom 0.5 and 2 of the b ij cost.Eachnodehasavariablecostparameter  Tj, chosenrandomly,between o and Q Foreachnodeaspecificcommoditymustbedelivered.Thevalue of  Tj representstherelativevalue of thecommodity,whereahigher  Tj is amorevaluablecommodity.Alltheinstancesareasymmetric.Foreachclassandvalue of n, tenpseudo-randominstanceshavebeencreated. We haveusedCPLEXwithNewton'sBarrierMethodtocomputethelinearrelaxationsolutionandtheoptimalintegersolution. Two differentformulationsweretested.Thefirstone is model M, (1)-(12).Thesecond,whichwearecallingM2,isthemodelMplusthevalidinequalities(13)-(18).Table(I)showstheaverageresults of theinstancessolvedtooptimalitybyCPLEX,forbothformulationsMandM2.In[23],SarubbiandLunapresentedexactresultsforinstancesupto 65 nodes.Thoseinstances,however,werenotbeen TABLE I TABLE OF RESULTS STANDART CPLEX CODES SizeLROptimal LR timeLROptimalLRtimeLRtimeGapTimeBarrierGapTimeBarrierDual(s)MM M s) M2 M2 M2(s)M2100.670.240.040.370.190.091.54 15 1.041.450.180.241.90.523.02202.3331.861.290.4122.562.511.34 25 2.56503.263.670.66169.016.0531.01302.5612827(1)9.350.66745.4111.58138.38355.52   17.510.742395.8627.16353.83406.04   59.280.703760.7752.42868.56457.12   101.20.9937889.4990.551919.48508.31   187.871.0786356.09( 1) 162.69   generatedfromcompletegraphs.Table(I)presentsoptimalresultsforourcompletegraphinstances of upto50nodes,whichweconsidertobemuchmoreexpressive.WithmodelM we areabletogetoptimalresults of up to30nodesandwithmodelM2weareabletogetoptimalresultsforinstances of upto50nodes.Inaddition,thelinearrelaxationgapsfrommodelM2aretighterthanthegapscomputedfrom M. We havealsotriedtosolvethelinearprogrammingrelaxation of thoseformulationsusingtheDualSimplexMethodfromCPLEX.TheseresultscanalsobeseenintheTable(I).TheNewton'sBarrierMethod,though,appearstobethebestchoice. A.Cut-and-BranchResults We haveproducedacarefulimplementation of thecut-andbranchalgorithmsuggestedinsection 5. Inordertomeasureitsefficiency,wehaverepeatedthetestbedoftheformersection.Theideahere is thatwecanmakethebranch-andboundproceduremoreefficientifwehaveamodel as tight as M2butwithnomorevariablesthan M. Sincetheenumerationstep is themosttime-consumingpart of an integerprogramsolutionandanybranch-and-boundmethod is sensitivetothenumber of variables, we areexpectingsomeimprovementincomputingtimes. We notethatwedonotneedtoaddalltheviolatedinequalities, as itissometimesbettertoaddjusta few.We onlyneedtocaptureenoughinformationabouttheviolatedsystemtoimpacttheenumerationprocedure.Intable(II)weshowacomparisonamongthreeversions of Cut-and-BranchalgorithmsandCPLEX.Inthefirstversion,whichwearecalling PI, we workwithformulation(36)(40).Thismeansthatin PI wearenottakingadvantage of specialstructure.Inthesecondversion,calledP2, we usedthespecialstructuretosolvesubproblems,butaddjustonecutperiteration,intheform(59).Thethirdversion,P3,isthemulticutversion.Thefield CPLEXTime s) givestheaveragetime,inseconds,spentbyCPLEXtosolvethetenproblemstooptimalityusingformulation(2)-(18). We havealsotabulatedtheaveragecomputingtimesconsideringthethreeversions of theCutAnd-Branchalgorithm.Theportion of thetotaltimethatwasspentgeneratingthecutsisspecifiedinthecolumn ProjTimePx s) wherexisthealgorithmversion.Inmostcases,wehavenotrepeatedcutgenerationuntilcompletion. It is 1810
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