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Combinational Approaches to Solve Scheduling Problems

Permanent Link: http://ufdc.ufl.edu/UFE0041090/00001

Material Information

Title: Combinational Approaches to Solve Scheduling Problems
Physical Description: 1 online resource (158 p.)
Language: english
Creator: Nemani, Ashish
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: column, combinatorial, curfew, decomposition, intermodal, load, location, negative, neighborhood, optimization, railroad, scheduling, vlsn
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this dissertation, we discuss classical scheduling problems especially in transportation industries. All of these problems come from real life applications and are very critical from the academic as well as the financial point of view. Being NP-hard problems there doesn t exist any well-defined algorithm which can solve them efficiently with good running time, and thus rules of thumbs are still being followed in practice, with very myopic use of optimization procedures. These decisions are usually worth billions of dollars per year and even a slight improvement will have a significant economic impact. Through efforts described in this document, we try to develop some holistic approaches along with heuristics, to get efficient and effective results for these decision problems. This proposed work has the potential of implementation in commercial grade software. We first suggest some hybrid approaches to solve the intermodal load planning problem, which generate very effective solutions within minutes. In our second problem, subset-disjoint minimum cost cycle problem, we suggest several exact and heuristic approaches to find the minimum cost cycles which contains at most one node from any subset. These problems occur very often as a subproblem of other other combinatorial problems. In the third problem, location routing problem, we suggest column generation algorithm and show its effectiveness by doing experiments with benchmark problems. In our last problem, which is very critical in all railway industries, we propose several models based on based on mixed integer programming, heuristics, and other hybrid approaches. These algorithms show a significant improvement over the current practices.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ashish Nemani.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Ahuja, Ravindra K.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041090:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041090/00001

Material Information

Title: Combinational Approaches to Solve Scheduling Problems
Physical Description: 1 online resource (158 p.)
Language: english
Creator: Nemani, Ashish
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: column, combinatorial, curfew, decomposition, intermodal, load, location, negative, neighborhood, optimization, railroad, scheduling, vlsn
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this dissertation, we discuss classical scheduling problems especially in transportation industries. All of these problems come from real life applications and are very critical from the academic as well as the financial point of view. Being NP-hard problems there doesn t exist any well-defined algorithm which can solve them efficiently with good running time, and thus rules of thumbs are still being followed in practice, with very myopic use of optimization procedures. These decisions are usually worth billions of dollars per year and even a slight improvement will have a significant economic impact. Through efforts described in this document, we try to develop some holistic approaches along with heuristics, to get efficient and effective results for these decision problems. This proposed work has the potential of implementation in commercial grade software. We first suggest some hybrid approaches to solve the intermodal load planning problem, which generate very effective solutions within minutes. In our second problem, subset-disjoint minimum cost cycle problem, we suggest several exact and heuristic approaches to find the minimum cost cycles which contains at most one node from any subset. These problems occur very often as a subproblem of other other combinatorial problems. In the third problem, location routing problem, we suggest column generation algorithm and show its effectiveness by doing experiments with benchmark problems. In our last problem, which is very critical in all railway industries, we propose several models based on based on mixed integer programming, heuristics, and other hybrid approaches. These algorithms show a significant improvement over the current practices.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ashish Nemani.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Ahuja, Ravindra K.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041090:00001


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First,IwanttoexpressmydeepestthankstomyadvisorDr.RavindraK.Ahujawhohelpedmeineachaspectofmylifeinthelastfouryears.Hisguidanceduringmygraduatestudynotonlyhelpedmeasaresearcherbutalsoasahumanbeing.Hisuniquewaysofencouragementalwaysmotivatedustoworkharderanddevelopadisciplinedapproachtowardslife.Iwouldalsoliketothankmycommitteemembersfortheirhelpinimprovingmyresearchpapersbyaddingnewdimensions.Aspecial"thankyou"goestomydearfriends,AbhudayaMishra,ArvindKumar,KrishnaJha,AmitAgarwal,AshwinArulselwan,andAshutoshChohanwhomademylifeeasierbytheirsuggestionsindifcultmoments.IwouldspecicallyliketomentionthenameofmycolleagueSuatBog,withwhomIworkedontwochaptersofmydissertation.Finally,Iwouldliketothankmyfamilyfortheirconsistentemotionalsupport.Andlastbutnotleast,Iwouldliketoacknowledgethesupportofmybrother,AlokNemani,forhisguidancethroughoutmylifebysuggestingmetheoptimalandimplementablesolutionsforallproblemsinmypath. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 2LOADPLANNINGPROBLEMATANINTERMODALRAILROADTERMINAL 18 2.1Introduction ................................... 18 2.2DenitionsandBackground .......................... 21 2.3ProblemDescription .............................. 24 2.3.1BasicOperations ............................ 25 2.3.2GeneralRulesofLoadingUnitsonRailcars ............. 26 2.3.3HitchUtilization ............................. 27 2.3.4AerodynamicEfciency ........................ 27 2.4ModelFormulation ............................... 28 2.5Heuristics:VeryLargeScaleNeighborhoodSearchAlgorithms ...... 33 2.5.1ConstructionHeuristics ........................ 35 2.5.2TheNeighborhoodSearchStructure ................. 35 2.5.3ImprovementGraph .......................... 36 2.5.4CyclicExchanges ............................ 37 2.5.5PathExchanges ............................ 38 2.5.6IdentifyingPathExchanges ...................... 38 2.6VeryLargeScaleNeighborhoodWithTabuSearch ............. 39 2.7HybridApproach ................................ 41 2.8ComputationalResults ............................. 41 2.8.1ComparisonofExactApproachandHeuristics ........... 42 2.8.2ResultsatDifferentStagesofHybridApproach ........... 44 2.9FlexibilityoftheModel ............................. 45 2.9.1LocalCarryCost ............................ 45 2.9.2HandlingTimeataTerminal ..................... 46 2.9.3DailyorWeeklyLoadPlan ...................... 46 2.10Conclusions ................................... 47 3SUBSETDISJOINTMINIMUMCOSTCYCLEDETECTION ........... 48 3.1Introduction ................................... 48 3.2NetworkReduction ............................... 52 3.3ExactAlgorithmsforSubsetDisjointMinimumCostCycles ........ 53 5

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............................ 54 3.3.2DynamicLabelingAlgorithm(DLA)forSDMCCProblems ..... 56 3.3.3AlgorithmAccelerationforDLA .................... 59 3.3.4DLAwithAccelerationforSDNCCProblems ............. 59 3.3.5FixedLabelingAlgorithm(FLA)forSDMCCProblems ....... 60 3.3.6All-PairslikePullAlgorithms(APPull) ................. 62 3.3.7All-PairslikePushAlgorithms(APPush) ............... 68 3.3.8StrategytoAvoidInefcientLabelExtensions ............ 68 3.4HeuristicAlgorithmsforSubsetDisjointMinimumCostCycles ....... 70 3.4.1LimitedUnprocessedLabelsHeuristics ............... 70 3.4.2LimitedCycleLengthHeuristics .................... 71 3.4.3LimitedCycleNumberHeuristics ................... 71 3.5ComputationalAnalysis ............................ 72 3.6Conclusions .................................. 85 4COLUMNGENERATIONAPPROACHFORLOCATIONROUTINGPROBLEMS 89 4.1Introduction ................................... 89 4.2ProblemDescription .............................. 90 4.3MathematicalModel .............................. 91 4.4ColumnGenerationAlgorithm ......................... 95 4.4.1TheMasterProblem .......................... 95 4.4.2ThePricingProblem .......................... 99 4.4.3TheAlgorithm .............................. 104 4.5ImplementationDetails ............................. 109 4.5.1TestProblems .............................. 109 4.5.2ComputationalPlatform ........................ 109 4.5.3ComputationalResults ......................... 110 4.6SummaryandConclusions .......................... 112 5SOLVINGTHECURFEWPLANNINGPROBLEM ................ 115 5.1Introduction ................................... 115 5.2ProblemDescription .............................. 118 5.3LiteratureReview ................................ 120 5.4Time-SpaceNetworkFormulation(TSNF) .................. 122 5.4.1MathematicalModel .......................... 124 5.4.2ComputationalAnalysis ........................ 127 5.5Duty-GenerationModel(DGM) ........................ 129 5.5.1Duty-GenerationPhase1:VariableReduction ............ 131 5.5.2Duty-GenerationPhase2:ProjectScheduling ............ 131 5.5.3Duty-GenerationPhase3:Crew-Scheduling ............. 133 5.5.4ComputationalAnalysis ........................ 134 5.6Column-GenerationModel(CGM) ...................... 135 5.6.1TheMasterProblem(MP) ....................... 136 5.6.2ThePricingProblem .......................... 138 6

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........................ 139 5.7Decomposition-BasedDutyGenerationModel ............... 141 5.8ComputationalResults ............................. 143 5.9Conclusions ................................... 147 6GENERALCONCLUSIONANDFUTURERESEARCH ............. 149 REFERENCES ....................................... 152 BIOGRAPHICALSKETCH ................................ 158 7

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Table page 2-1Examplesofrailcarsandcandidatepatterns .................... 24 2-2GapsatdifferenttimeintervalsbyIPAM. ...................... 43 2-3Comparativeanalysisofheuristics. ......................... 43 2-4Developmentofhybridapproach(timeinseconds). ................ 44 3-1ComparativeanalysisofexactSDMCCalgorithms:smallinstances. ...... 77 3-2ComparativeanalysisofexactSDMCCalgorithms:mediuminstances ..... 78 3-3ComparativeanalysisofexactSDMCCalgorithms:largeinstances ....... 79 3-4ComparativeanalysisofexactSDNCCalgorithms:smallinstances ....... 81 3-5ComparativeanalysisofexactSDNCCalgorithms:mediuminstances ..... 82 3-6ComparativeanalysisofexactSDNCCalgorithms:largeinstances ....... 83 3-7ComparativeanalysisofheuristicSDMCCalgorithms .............. 86 3-8ComparativeanalysisofheuristicSDNCCalgorithms .............. 87 4-1ConstraintsandassociateddualvaluesfortheMasterProblem ......... 100 4-2ResultsontheLRPbenchmarkinstances ..................... 110 4-3ComparisononPerl'sbenchmarkproblems .................... 111 4-4ComparisononbenchmarkproblemsthatarecompiledbyBarreto[ 11 ] .... 112 5-1Detailsofreal-lifeinstances ............................. 143 5-2Improvementinnumberofviolationsfor2007 ................... 145 5-3Improvementinnumberofviolationsfor2007 ................... 145 5-4Comparisonofalgorithms:optimalitygap(smallinstances) ........... 147 8

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Figure page 2-1Simpliedmodelofarailroadterminal. ....................... 22 2-2Patternsforarailcar. ................................. 25 2-3Flowofunitsatarailroadterminal. ......................... 26 2-450%hitchutilization. ................................. 27 2-5100%hitchutilization. ................................ 27 2-6NetworkforLPP. ................................... 29 2-7Nodescreationintheimprovementgraph ..................... 37 3-1Illustratingsubset-disjointminimumcostcyle. ................... 49 3-2Illustratingsubset-disjointnegativecostcyle. ................... 50 4-1ExampledatafortheLRP .............................. 91 4-2DynamicprogrammingalgorithmforESPPRC ................... 102 4-3Columngenerationalgorithm ............................ 106 5-1Space-Timenetworkforthecurfewplanningproblem. .............. 123 5-2VariablematrixD[p]forprojectp 129 5-3DynamicprogrammingalgorithmforESPPRC. .................. 140 5-4Algorithmframeworkofthecolumn-generationprocedure. ............ 141 9

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2 dealswiththeintermodalloadschedulingproblem.Chapter 4 discussesthelocationroutingproblem.Chapter 3 dealswiththesubsetdisjointminimumornegativecycleproblems.Chapter 5 dealswiththecurfewplanningalgorithm.Finally,wegiveageneralconclusionandfutureworkinthe 17

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54 ].Intermodaltrains'transportgoodsarestoredincontainersortrailersandthenloadedontherailcars.Determiningtheoptimalpositionofthesecontainersandtrailersontherailcarsisaverycriticaldecisionfactor,becauseitaffectstheeconomyofthetrains.Ourresearchisfocusedonthisaspectoftherailway-modeandisreferredtohereastheloadplanningproblem.Thedevelopmentintherailway-modehasrevolutionizedthegrowthofintermodaltransportation,anditiscurrentlyamongoneofthefastest-growingbusinessesintheU.S.railindustry.Inthelastdecade,thegrowthrateintransportthroughthismodehasbeenovertenpercent[ 32 ].In2001,thefreighttrafcinton-milesbyrailwas47%incomparisontotrucks'33%andships'20%[ 73 ].Earlierresearchhasshownthatfordistancesbeyond310miles,thismodeoftransportismoreefcientandresultsinsavingsinoperationalcostsandlabor[ 16 ].Railtransportalsodivertssomefreighttrafcfromtheroads,partiallyalleviatingcongestionandwearandtearonhighways[ 55 ].Therealsoisagreatscopeforimprovementduetomoredevelopedinfrastructure,betterplanning,andbetterstrategicdecisions[ 28 ].Thedemandtrendforrailservicehasbeenveryhigh,buttheoperationalactivityhasnotbeenabletomatchthissurgeduetoinefcienttechniquesandotherpolicies[ 73 ].Severaldecisionissuesarisewhenplanningthedeliveryofacontainerfromitsoriginpointthroughtoitsdestinationpoint,suchasthetimeatwhichitshouldbedispatched,itspositioninthestorageyardattheterminal,thetrainandtherailcaron 18

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48 ].Hence,evenaslightimprovementinthespaceorfuelefciencywillsavemillionsperyear.Theaerodynamicfactorofatrainisafunctionoftwomainentities:thegapbetweentheunits(containersortrailers)loadedonadjacentrailcars,andhowfarthegapisfromthefrontendofthetrain.Allofthesecharacteristicsmaketheproblemhighlycombinatorial,anditcanbeprovenNP-Completebyreductionfromageneralizedbinpackingproblem.Differentvariationsoftheproblemhavebeenextensivelystudiedintheliterature,andwebrieydiscussthosewhichareclosesttotheloadplanningproblem.OneoftherstmodelswasdevelopedbyFeoandVelarde[ 32 ].Theirpapersolvesarelaxedloadplanningproblem,whichconsidersonlytrailersandassumesthatarailcarcancarryatmosttwotrailers.Itgivesanintegerprogrammingformulationoftheproblembasedonthesetcoveringapproach.PowellandCarvalho[ 66 ]aretherstcontributorswhoconsiderthespaceandtimenetworkstructureofthisproblem.Theyintroducethemodelforoptimizingtheowofentitiesoverseveralterminalsinaplanninghorizon,whichrespectsthetechnicalconstraintsrelatedtoassigningtheunitsonrailcars.However,theydon'tconsidertheaerodynamicissues.Theyformulatetheproblembasedonalogisticqueuingnetwork,calledLQN[ 67 ].NewmanandYano[ 59 ]consideravariantoftheproblembycombiningthetrainscheduling 19

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24 ]proposeamulti-objectiveintegerprogrammingmodel,whichdevelopsaloadplantooptimizethetotalhandlingtimeofloadingandthebalancedmassdistributiononthetrain.Themassdistributionofthetrainshouldbebiasedtowardthefronttoreducethewearonthebrakingmechanism.BostelandDejax[ 17 ]addresstheproblemofoptimizationofcontainerloadingontrainsinarail-railtransshipmentshuntingyardtominimizethetransferwithintheyard.TheaerodynamicissueswererstcapturedandmodeledbyLaiandBarkan[ 48 ],[ 49 ].Theydiscussseveraloptionsforimprovingtheaerodynamicefciencyandfuelconsumptionofthetrains.Theirextensivestudyhelpedustoidentifythecriticalfactorsassociatedwithanintermodalfreighttrain'sefciency.ThemostrecentpaperbyLai,Barkan,andOnal[ 50 ]formulatestheproblemasanintegerprogrammingmodel,whoseobjectiveistominimizethefuelconsumption.Wegeneralizedtheirapproachandaddedotherdimensionstomakeitclosertoareal-lifescenario.Tosummarize,thealgorithmsdevelopedthusfarforsolvingtheLPPhavebeenveryhelpfultousinunderstandingallrelatedissuesandinthedevelopmentofauniquemodelthatcanaddressallrelevantfactorsandcanbeextendedtootherdimensions.WeinitiallyformulatetheproblemasanMIPandtrytosolveitusingexactalgorithms.Thismodelrstcreatesapatternset(arrangementofunitsonrailcars)andthenusesitinthenextphaseforconstructingthenetwork.Itimprovestheexibility,assomepatternsthatarenotcoveredbyrulescanbeaddedfromthehistoricaldata.Wealsodeveloptwoverylarge-scaleneighborhood(VLSN)heuristicsutilizingtabusearchthatconsidertheglobalproblemandgenerateexcellent-qualitysolutionswithin30seconds.VLSNsearchalgorithmsaregenerallyapplicabletoproblemswhichcanberepresentedaspartitionproblems[ 3 ].TheLPPisnotatruepartitionproblem(seeSection 2.5 ),sowemodifythebasicVLSNalgorithmaccordinglyandextendeditusingthetabusearch, 20

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1. Wegiveadetaileddescriptionoftheintermodalrailwayterminalandtheentities(Section 2.2 ). 2. Weexplaintherestrictionsonassigningcontainersortrailersonrailcars,theguidingcriteriafortheloadingprocess,aswellasothercriticalfactors(Section 2.3 ). 3. WeformulatetheloadplanningproblemasanMIPonanappropriatelydenednetworkwithgeneralizedowanddescribetheformulation(Section 2.4 ). 4. Weproposeaheuristic,single-unitmulti-exchangeneighborhood(SUMEN)search,whichgeneratesalocallyoptimalsolutionwithintwoseconds(Section 2.5 ). 5. Wedescribethesecondheuristic,multi-unitmulti-exchangeneighborhood(MUMEN)search,whichgeneratesnear-optimalsolutionsinaveryshorttime.Wethenimproveitbycombiningwiththetabusearch(MUMENT).WethensuggestauniquehybridprocedurethatproducessolutionsbetterthanthosegeneratedbyCPLEXinmorethanonehour(Section 2.7 ). 6. Weperformextensivecomputationalinvestigationsoftheexact,heuristics,andhybridapproachesandreporttheseresults(Section 2.8 ).Wealsodiscusstheapplicabilityofeachapproachinreal-lifeapplications. 7. Wedescribetheexibilityofourapproachesforsolvingdifferentvariationsoftheproblem(Section 2.9 ).Thisincludesdiscussingthedynamicversionoftheproblemandsuggestinghowitcanbesolvedbydirectlyextendingourapproach. 8. Finally,wepresentourconclusionsandsuggestdirectionsforfutureresearch(Section 2.10 ). 21

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2-1 .Webrieywilldiscussthemainentities,units,railcarsandhandlingequipmentthatplaysignicantrolesinallthedecisionproblemsatthissubsystem. Figure2-1. Simpliedmodelofarailroadterminal. Containersandtrailersareusedtotransportfreightintheintermodaltransportation.Thedifferencebetweenthesetwoisthatthetrailershaveasetofwheelsusedtodriveontheroad,whilethecontainersneedtobetransferredontosuitablysizedtrucks.Containersandtrailersvaryintheircharacteristicsdependingontheirtypes.Themostcommoncontainersizesusedininternationalcommerceare20ft.,28ft.,40ft.,and48ft.Othersizesare:10ft.,usedprimarilyinplaceEuropeandbythemilitaryservices;24ft.;45ft.;53ft.;and56ft.,usuallyusedatthedomesticlevel.Typicalcontainer 22

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68 ].Trailersalsohaveastandardwidth(8ft.,6inches)andheight(9ft.,6inches),andalengthvaryingfrom28to57feet.Somecontainershavespecialfeaturesandaregroupedasspecial-purposeunitsusedforcarryingtemperature-controlledcargo,liquids,gases,automobiles,andlivestock.Inrail-transport,containersandtrailersareplacedontherailcarsusinghandlingequipment.Atraingenerallyconsistsof60-85railcarsandafewlocomotives,anditscapacityisgiveneitherintermsofmaximumlengthorweight.Thereareseveraltypesofrailcarssuchasdouble-stacks,spine-and-skeletoncontainercars,skeletontrailercars,piggybacktrailers,androadrailers.Table 2-1 showssomeexamplesofrailcarsandtheircharacteristics.Railcarscomeinseveralcongurationsbasedonthenumberofhitchesonthemandtheirrelativepositionsontheplatform.Thesehitchesdenetheslotswherecontainers/trailersmaybeloaded.Thenumberofhitchesontherailcarsvariesfromone(frontrunners)toseven(spinecars).Fortrailers,therearesomespecialarrangementsfortheholdingandsafetyoftheirwheels.Roadrailersdon'trequirearailcarandattachmentsaremadedirectlytotheirbacks.Theconveyanceoftrailersandcontainersonrailcarsisreferredtoastrailer-on-atcar(TOFC)andcontainer-on-atcar(COFC).ThetransportusingCOFCisincreasingquiterapidlycomparedtoTOFC.In1996,theCOFCmarketshareaccountedforonly77percentofthetotalintermodalvolume,whilein2001,itaccountedformorethan92percent[ 63 ].Thetransportofcontainersandtrailerswithinayardandtheirloadingisperformedusinghandlingequipmentsuchasdocksidecranes,stackers,packers,frontlifttrucks,sideloaders,straddlecarriers,stackinggantrycranes,andchassis,etc.Atintermodalyards,achassisisatrailerdesignedspecicallytohaulcontainers.Forsimplicity,this 23

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55 ]. Table2-1. Examplesofrailcarsandcandidatepatterns 2-1 ,thereareseveraltrackswithrailcarsandunitsofdifferentdestinations,whicharedeliveredandstoredinyards.Asthenewunitsaredelivered,theloadplanisrevisedwiththemodiedsetofrailcarsandunits.Inarailroad,thisdynamicdecision-makingprocessiscurrentlyperformedusingrulesofthumbbyexperiencedpersonswhocoordinatethematerial-handlingoperations.Somerailroadsuseacomprehensivedecisionsupportsystemwhoseprimaryfunctionistoregulatetheinformationowbetweendifferentchannelsofthermratherthantooptimizetheloadingprocess.Theytrytoassigntheunitstotherailcarsafterconsideringseveralrestrictionsthatallowonlyasubsetofarrangements.Werefertothesearrangementsaspatterns. 24

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Patternsforarailcar. 2-2 .Next,webrieydiscusstheoperationswhichleadtotheformationandloadingofpatternsontherailcars. 2-3 ).Aloadplanshouldoptimizetheloadingandunloadingprocessestogether.Whenatrainarrivesattheplatform,generally,unloadingandloadingstartsimultaneously.Truckscarryingoutboundcontainersaredirectedtotherailcarsonwhichthecontainersaretobeloaded.Ifthereisspaceonthatrailcar,thecontainerisloadeddirectlyonit;otherwise,itisplacedonthegroundandloadedaftertherailcarisemptied.Thisprocessiscalleddouble-handling,asoneadditionalmoveofthecontainerisrequired.Loadingtheunitsisadynamicprocess,andtheloadingplanisrevisedasneweventsoccur,suchasthearrivalofnewunits.Intherevisedplan,acontainerthatalreadyhasbeenputonthegroundmaybeassignedtoadifferentrailcar.Thisreassignmentintroducesanothermovementofthecontainer. 25

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Flowofunitsatarailroadterminal. Aftertheunitsareloadedontherailcars,theyaresecuredusingfasteningdevicescalledpinsthatarelocatedontherailcars.Thecongurationofpinsinthenewarrangementofunitsontherailcarsimpactstheloadingefciency. 1 ].Someoftheconstraintsarelistedbelow: 1. Thereshouldbeaminimumclearance,about10inches,betweenthetrailersontheadjacentrailcars. 2. Thereshouldbeaminimumworkingdistanceofatleasttwoinchesbetweenanytwoadjacenttrailerstofacilitatesmoothloadingandunloadingoperations. 3. Atrailermaynotoverhangthestrikersbymorethanveinchesoneachendofarailcar. 4. Thewidthofatrailermustnotexceedthewidthoftherailcartoavoiddamagetothetires. 5. Theweightdistributiononthetrainshouldbebiasedtowardthefrontend. 6. Whilestacking,emptycontainerscannotbeputatthebottom. 7. Smallercontainers(26ft.,28ft.)generallycarryheavycommoditiesandshouldbeputatthebottomlevelindouble-stackingtoavoidwearandtearaswellasaccidents.Thereareothercommonrestrictions,suchasatrailercannotbedouble-stacked.Tosupportthedouble-stacking,containershavepinsonlyatthecorner.Thiseliminatesthepossibilityofputtingasmallcontaineronacomparativelylargeoneandvice-versa.Specialpurposeunitsmustbehandledmorecarefully.Refrigeratedcontainersandtrailersneedmorespaceforproperfunctioning.Unitscontaininghazardousmaterials 26

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32 ].Figures 2-4 and 2-5 ,respectively,giveexamplesofpoorandgoodhitchutilizationonadouble-stackrailcar.Thisisoneoftheefciencymeasurementfactorsofaloadedtrain.Higherhitchutilizationsignicantlyreducesthecostofpullinganoutboundtrain,asitreducesthenumberofrailcarsrequired.Also,thedenserarrangementofunitsentrapsalesservolumeofairthatisdraggedandhencelowersthefuelconsumption(seedetailsinSection 2.3.4 ). Figure2-4. 50%hitchutilization. Figure2-5. 100%hitchutilization. CurrenthitchutilizationintheUnitedStatesrangesbetween80and95percent.AstudyatConrail,oneoftheleadingrailcompaniesintheUnitedStates,hasshownthataonepercentimprovementinhitchutilizationcouldsave450,000dollarsperyearateachofitsfteenyards[ 32 ]. 27

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25 ].Yawangleistheanglebetweenatrain'sheadinganditsactualdirectionoftravelorcourse.Thisisanimportantfactorataparticularpoint,butintheglobalscenarioitdoesn'tplayanyactiverole.Theyawanglechangesassoonasthetrainchangesitscourseandneutralizesthecorrespondingfactor.Alargergaplengthbetweenunitsresultsinhigherentrappedairvolumeandconsequentlypooraerodynamicefciency.Theexperimentsalsosuggestthattherstlocomotiveexperiencesthehighestdragduetotheheadwindimpact;asthetrainmovesfarther,thisdragdeclinesuntilthetenthunit,andthereafteritbecomesconstant.Thissuggeststhatifunitandrailcarcombinationsaresuchthatthegapcannotbeavoided,thenthegapshouldbeattherearendofthetrain.Therelationshipbetweentheaerodynamicresistanceandtheunit'spositioninatrainisgivenasfollows[ 25 ]:CDA(ft2)=14.85824e0.29308k+9.86549e0.00007k+10.66914wherekistherailcar'spositioninthetrainandCDAisthedragarearepresentingaerodynamicresistance.Anadjustmentfactorisassociatedwitheachgapbydividingthedragareaofthatunitbythatofthe100thunit[ 48 ].Theadjustmentfactorataplaceismultipliedwiththelengthofthegapbetweentwoconsecutiveunitsatthatplacetondtheadjustedgaplengthinordertocomputethedrag.Theadjustedgaplengthisdenedasthegaplengthweightedbytheposition-in-traineffect[ 50 ]. 2.9 wediscuss 28

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2-6 .Itcontainsthreesetsofnodes,describedasi,ii,andiii.(i)Therstsetcorrespondstotheunitsgroupedintofamilies.Alltheunitsofafamilyareofthesameshape,weight,anddimensions,buttheymaydifferinrevenueearnedorpenaltyincurredifdelayed.(ii)Thesecondsetbelongstoallthefeasiblesetsofpatterns,and(iii)thethirdsetbelongstoalltherailcars.Thearcbetweenaunit-nodeandapattern-nodeshowsthattheunitcanbeassignedinthatpatternatacertaincost.Anarcexistsbetweenapattern-nodeandarailcar-nodeiftheunitscanbeassignedontherailcarinthatpattern.Duetothepresenceofthesetwosetsofarcs,theowthroughthepattern-nodesisconservedandhence,thenetworkisageneralizedownetwork[ 5 ]. Figure2-6. NetworkforLPP. Thefollowingnotationisusedtodescribethemixedintegerprogrammingformulationoftheloadplanningproblem(LPP).Theproposedmodelwillbedenotedastheintegerprogrammingassignmentmodel(IPAM).Parameters:B:setofalltheunits. 29

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58 ].Constraint 2 isthemassbalancefortheinowandoutowofcontainersandtrailersatthepatternnodes.Constraints 2 and 2 enforcetheconditionthatarailcarcanbeassignedinonlyonepatternandaunitcanbeloadedinasinglepatternandthustoasinglerailcar,respectively.Constraints 2 and 2 ensurethatthe 31

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2 and 2 maintaintheconditionthatifacontainerhasbeenleftunloaded,thennoneoftherailcarsremainunassigned.Intheexamplehere,weassumethatalloftherailcarsareexibleenoughtoaccommodateanytypeofunit.Constraints 2 and 2 givetherelationshipbetweenyandzvariables.Finally,nonnegativityandintegralityconstraintsareimposedonallofthedecisionvariables.Thelengthrestrictionoftherailcarsistakenintoconsiderationwhilebuildingthepatterns.Theweightconstraintsalsocanbeconsideredatthatstage,andithasbeenincludedintheformulationonlytomakeitmoreexible.TheIPAMmodelcapturesallrelevantissuesandgivesadetailedviewofthecompleteproblem.Inanintermodaltrain,thereareonaverage60-80railcarsand200-300units.Weconsiderthattherearefourtypesofrailcarsandthatallofthemarecapableofbeingdouble-stacked.Wegenerate200-300patternsfortheserailcars-unitscombinations.ThenumberofvariablesisO(|P|*|B|+|P|*|K|),whichonaverageis80,000,andthetotalnumberofconstraintsisaround5,000.TheIPAMmodelisveryexibleinincorporatingadditionalfeatures,becauseitconsiderspatternsasaseparateentity.Weperformthecomputationaltestingonseveralinstancescreatedrandomly(seeSection 2.8 fordetails)andsolvethemusingCPLEX11.2.Wendthatfortheinstanceswheretheavailablecapacity(railcars)islessthanthedemand(units),IPAMproducesanoptimalsolutionwithinveminutes.However,thesolutionqualitydeterioratesasthecapacityanddemandbecomecomparable.Inreal-lifeapplications,railroadplannershavetoprepareplansforseveraltrainseachday.Furthermore,onceasolutionisavailable,theyusuallyneedtomakeseveralchangeswhichcannotbecapturedinthemodel.Plannersalsomaywanttochangethepriorityofunitsandrailcarsbasedupontheoutputofrstmodel.Tomakeallofthesechanges,afastresponsetimeisrequired,andave-minutelimitispracticalbasedontheplanner'sexperience.Theseobservationsguidedindevelopingheuristicsthatcanaddressalloftheissues 32

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32 ]suggestagreedy-randomized-adaptive-search-procedure(GRASP)forapproachingthereducedloadplanningproblem.Foroptimizingtheaerodynamicefciencyofthetrain,Lai,Barkan,andOnal[ 50 ]proposeaheuristicthatrstevaluatestheaerodynamicefciencyofallpossiblepattern-carcombinationsforeachtypeofrailcarandthensortsthem.Next,itconsiderstherstslotoftherailcarandassignsthebestpossiblecombinationofunassignedunitsonit.Theprocessisrepeateduntileitherallslotsarelledorallunitsareassigned.Thisheuristicworkswellforthespine 33

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3 ]foranoverviewofVLSNsearchalgorithms.Thesealgorithmsaredesignedtosolvethepartitionproblemsasshownintheliterature.Ourproblemdoesn'tfallexactlyintothiscategoryduetotheaerodynamicfactorthatisbasedonthegapbetweenadjacentrailcars.Wehandlethisissuewhilendingthenegativecyclesbydynamicallyupdatingthearccosts.Thismodicationpreventsthebasicstructureofthealgorithmfromchanging,andletsusconsidertheLPPasaspecialcaseofthepartitionproblem,withthesetofrailcarsandunitspartitionedintomanysubsetsbasedonmanyrulesdiscussedlaterinthissection.Apartitionproblemistopartitionaset,sayS,intodifferentsubsets,sayS1,S2,S3,...,Sk,suchthatthecostofpartitionisminimum,wherethecostofpartitionisthesumofthecostofeachpart.ThepartitionproblemwassolvedusingVLSNbyThompsonandOrlin[ 71 ]andThompsonandPsaraftis[ 72 ].Ahujaetal.[ 6 ],[ 7 ]appliedthisapproachtothecapacitatedminimumspanningtreeproblemandtotheweapontargetassignmentproblem[ 4 ].WewillpresentabriefoverviewofthisapproachwhenitisappliedtotheLPP.Aneighborhoodsearchalgorithmstartswithafeasiblesolutionoftheoptimizationproblemandsuccessivelyimprovesitbyreplacingitwithanimprovedneighboruntilitreachesalocaloptimum.Thequalityofthelocallyoptimalsolutiondependsbothuponthequalityofthestartingfeasiblesolutionandthestructureoftheneighborhood; 34

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2-7 showsthedifferencebetweenthenodesoftheunit-setsoftheimprovementgraphforSUMENandMUMENstructures.ThearcsetinG1(A)(orG2(A))isdenedinamannersuchthateachcyclicorpathexchangewithrespecttoAdenesadirectedcycleinG1(A),andthecostofthecycleequalsthecostofthecorrespondingexchange.Adirectedarc(g,h)inG1(A)(orG2(A))signiesthatnodegleavesitsrootnodeC[g]andjoinstherootnodeC[h],andthatsimultaneouslynodehleavestherootnodeC[h].Toconstructtheimprovementgraph,weconsidereverypairgandhofnodesandaddarc(g,h)ifandonlyif(i)C[g]6=C[h]and(ii)S[h]S{g}\{h}isafeasiblepatternofrailcarC[j].Wedenethecostghofarc(g,h)as:gh=c({g}SS[h]\{g})-c(S[h]). 36

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BSUMENunitSet CMUMENunitSetFigure2-7. Nodescreationintheimprovementgraph LetA'denotethenewfeasiblearrangementofunitsafterthecyclicexchange.Wedenethecostofthiscyclicexchangeasc(A')c(A).c(A')c(A)=c(Prp=1(c(fgp1gSS[gp]nfgpg)c(S[gp])),whereg0isdenedasgr. 37

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2.5.1 ,whosecomplexityisO(pnm*|I|).TheimprovementgraphG1(A)containsO(n+|B|)nodesandO((n+|B|)2)arcs,andthecostofallarcscanbecomputedinO((n+|B|)2)time.Weuseadynamicprogramming-basedalgorithmbyAhujaetal.[ 4 ]toobtainsubset-disjointcycles.Thisalgorithmrstlooksforprotabletwo-exchangesinvolvingtwonodesonly;ifnoprotabletwo-exchangeisfound,itlooksforprotablethree-exchangesinvolvingthreenodes,andsoon.Thealgorithmeitherndsaprotablemulti-exchangeorterminateswhenitisunabletondamulti-exchangeinvolvingtnodes(wesett=5).Intheformercase,weimprovethecurrentsolution,andinthelattercasewedeclarethecurrentsolutiontobelocallyoptimalandstop.TherunningtimeofthedynamicprogrammingalgorithmisO((n+|B|)2*2t)periteration,anditistypicallymuchfastersincemostcyclicexchangesfoundbythealgorithmaresimpleswapping.TheVLSNsearchheuristics,particularlyMUMEN,generatedgood-qualitysolutionswithin30secondsformostoftheinstances(seeSection 2.8 ).Forsomeinstances,thegapsarehighindicatingthattheVLSNsearchgetstrappedinthelocaloptima.Toovercomethis,weperturbthesolutionandallowsomeunprotableexchanges.Weusethetabusearchconceptsforthesemodicationsasdescribedintheupcomingsection. 2.8 ).Theoptimalitygapsaregoodandcanbeimprovedfurtherifmoretimeisavailable.Thenatureoftheconstraintsrendersthefeasibleregionhighlyscattered,andthuseventheverylarge-scaleneighborhoodsearchcanconvergetoa 39

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35 ],amemory-basedpopularmeta-heuristic,alongwithsomerandomsolutionperturbations.Wehaveusedshort-termmemorytoavoidcyclingandusedperturbationtodiversifythesearch.Next,webrieydescribeourimplementationofthetabusearchforLPP.TheinitialsolutionandneighborhoodstructureforthelocalsearchiscapturedintheVLSNsearch.Oncethelocaloptimumisreached,weperturbthesolutionandallowittomovetoaninferiorpoint.DuringthesolutionanalysisofMUMEN,wenoticeintuitivelythatnorailcar-subsequencehasalternatingdouble-stackedandsingle-stackedpatterns.So,inperturbationwetrytochangethesesubsequences.Werandomlychoosetwoblocksofrailcars,onesingle-stackedandonedouble-stacked,andputallloadedunitsinanunassignedunit-set.Wereassigntheseunitsbackonemptyrailcarsbutinvertthesingle-stackedrailcarstodouble-stackedonesandvice-versa.WereapplytheMUMENsearchstartingfromtheperturbedsolutionbutprohibitthealgorithminordertoconsiderthesamemoveforacertainnumberofiterations,calledtabutenure.Forexample,ifrailcarkwasassignedinpatternpinthelastlocaloptimalsolution,wewillnotassignitinthesamepatternforapre-speciednumberofpatterns.Ifanyofthemovesinthecycleisatabu,wecallitatabucyclicexchange.Wedon'tacceptthetabumovewithinthetenure,evenifitmaygenerateimprovedresults.WeterminatethesearchifthelasttwoperturbationsfollowedbytheMUMENsearchdon'tincludethebestsolutionandiftherehavebeenatleastveperturbations.ThesolutiongeneratedbyMUMENcombinedwiththetabusearch(MUMENTsearch)improvedtheoptimalitygapssignicantly,asshowninTable 2-3 .WeobservedthatIPAMgeneratesverygoodresultsforinstanceswithaveryhighdemandandlowcapacity,whileMUMENTgeneratesconsistentsolutionsforall.Thisobservationmotivatedustocombinethesetwoapproachesintoahybrid,aswewillnowdiscuss. 40

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2.4 ,hasalimitedtypeofsuccess.Itgeneratesanoptimalsolutioninaveryreasonabletimeframeforsomeprobleminstances,whileittakeshourstoreachanacceptablesolutionpointforothers.TheMUMENTsearchgeneratesgoodsolutionsinallinstancesbutconvergestoalocaloptimum.Wetrytoutilizethegoodcharacteristicsofbothapproacheswhendevelopingahybrid.Forthehybridapproach,westartwiththeMUMENTsearchandfeeditsoutputastheinitialsolutionintotheIPAM.Itservesasagoodupperboundthathelpsthebranch-and-boundproceduretoprunethebranchesatanearlystage.TheIMAPprecededbytheMUMENTsearchresultseitherinimprovedsolutionqualityorinshorterrunningtimeinalmostallinstances.WeterminatetheIPAMphaseafterthreeminutes,andifthesolutionobtainedisnotoptimal,weapplytheMUMENTsearchonceagaintoitsoutput.TheIPAMphaseusuallygeneratesasolutiondiversefromthelastlocaloptimalsolutionandhelpsMUMENTtodiversifythesearch.Thethree-minuteterminationcriterionischosenfromacomputationalaswellasfromapracticalpointofview.IPAMimprovesthesolutionveryquicklyatthestart,buttheimprovement-slopebecomesalmostatafterward;inareal-lifesituation,veminutesisapracticaltimeboundary.Fromabusyterminal(BNSF,UPS),anaverageof15-20trainsleaveeveryday[ 38 ].Theloadplanningwillbedoneforeachtrainwiththeforecasteddata,andtheplanwillhavetoberevisedeachtimeanyunexpectedeventoccurs.Wediscussmoreaboutthisapproachwiththecomputationalresultsinthenextsection. 41

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2.5 ,weperformedsomeexperimentsontherelaxedloadplanningproblemobtainedbyignoringtheaerodynamicfactor.Weobservedthatthisrestrictionhasanegligibleeffectonthequalityofthesolution.Outofthe15instancesinvestigated,onlyonegeneratedaslightlybetterresult.Thisexperimentsuggeststhattherestrictiondoesn'treducetheneighborhoodsizesignicantlyenoughtoworsenthesolution. 2-2 .Weobservedthatthesolutionqualitycurveissharpintherstveminutes,butitsslopekeepsdecreasingastimepasses,thereforejustifyingourterminationcriterionofthreeminutesinthehybridapproach.Next,wecomparedthesolutionqualityofdifferentheuristics:SUMEN,MUMEN,andMUMENTinTable 2-3 .WeobservedthatalthoughtheSUMENsearch 42

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Table2-2. GapsatdifferenttimeintervalsbyIPAM. 803200.00%0.00%0.00%0.00%0.00%1.6mins3100.00%0.00%0.00%0.00%0.00%2.6mins3000.00%0.00%0.00%0.00%0.00%5.3mins2900.03%0.00%0.00%0.00%0.00%4.4mins2800.00%0.00%0.00%0.00%0.00%3.4mins2701.29%0.00%0.00%0.00%0.00%5.9mins2600.21%0.00%0.00%0.00%0.00%4.6mins250NoSol.9.40%9.40%9.40%0.00%31.4mins240NoSol.NoSol.29.22%29.22%0.00%23.4mins230NoSol.24.98%24.98%24.98%0.00%47.7mins220NoSol.NoSol.NoSol.5.27%0.00%27.0mins210NoSol.NoSol.NoSol.3.28%1.18%1,172.7mins200NoSol.NoSol.NoSol.NoSol.3.84%2,291.2mins195NoSol.NoSol.91.28%91.28%9.15%1,237.7mins190NoSol.NoSol.93.57%4.93%2.36%1,892.0mins18524.36%24.04%24.04%15.73%10.57%1,264.0mins18059.24%59.24%59.24%59.24%38.90%1,546.7mins Comparativeanalysisofheuristics. GapTime(Secs)GapTime(Secs)GapTime(Secs) 8032010.77%24.72%134.72%1831010.49%13.82%123.82%1530012.36%14.74%154.74%2129012.12%12.38%152.38%1928013.66%16.00%94.06%1627014.70%14.90%104.90%1426018.38%27.91%104.77%2125017.17%13.08%93.08%1424021.51%26.13%113.10%1923023.84%12.72%102.72%1722027.61%14.81%81.32%2121027.02%12.67%72.67%1420017.96%13.08%53.08%919524.61%13.31%43.31%819026.14%15.51%54.48%818510.77%26.49%45.09%1218010.49%23.65%43.65%7

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Developmentofhybridapproach(timeinseconds). GapTimeGapTimeGapTime 803200.00%970.00%510.00%1013100.00%1580.00%790.00%793000.00%1800.00%1920.00%1922900.03%1800.00%880.00%882800.00%1800.00%1110.00%1112701.29%1800.00%760.00%762600.21%1800.00%980.00%98250NoSol.1801.51%2130.08%231240NoSol.1800.00%2260.00%226230NoSol.1801.91%2120.29%222220NoSol.1801.01%2010.61%209210NoSol.1801.70%1970.50%215200NoSol.1801.88%1950.71%208195NoSol.1802.20%1980.96%206190NoSol.1804.97%1931.18%19918524.36%1804.81%2071.32%21318059.24%1801.89%1990.07%205 2-4 andderivedtheobservationsbelowfromit.(i)TheIPAMtakesmuchlesstimetondanoptimalsolutionwhenthenumberofunitsishigherthanthetotalcapacityofthetrain.Thisisbecausewhentherearelargenumbersofunitsofdifferenttypes,eachrailcarcanbeassignedinthemostoptimalway,sotherewillbefewerbranchesinthebranch-and-boundtree.However,asthenumberofunitsnearsthetotalcapacityofthetrain,thenumberofbrancheswillincreaseinordertondthebesttoftheunitstotherailcars.ThequalityofthesolutiongeneratedbyIPAMinthetimetakenbyheuristicswasvariable,andinmostcasesitwasnotbetterthanheuristics.Itimprovedwiththetimeasmorebrancheswereexamined.(ii)TheMUMENTsearchheuristicgavereasonablygoodsolutionsforallinstances,andwhenwecompareditwiththesolutionofIPAMgeneratedwithinthesametime-interval,thedifferencebecameprettysignicant.(iii)Intuitively,agoodstartingpointshouldimprovethesolutionqualityaswellastherun-timeforabranch-and-boundalgorithm,butthiswasnottrueforall 44

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2.3 .Assoonasnewinformationbecomesavailableandplannerswanttorevisetheloadingplan,theyneedtoupdatethearccostsintheimprovementgraphandndnegativecycles.Inthesectionbelow,weprovideexamplesofpossibleextensionsoftheproblem. 45

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2.3 ,thecurrentterminalmaybethedestinationforsomeoftheunitsloadedontheinboundtrainsandmayactastheintermediateterminalforothers.Thisscenariocreatesthepossibilityofdoublehandlingofoutboundunits,iftherailcarsonwhichthesehavetobeloadedarestilloccupiedwithinboundones.TooptimizetheLPPwiththisfactor,weneedtosolvethemodeleverytimeaneventoccurs.Theeventwillincludethearrivalofanoutboundunit,thearrivalofapick-uptruck,etc.Theobjectivefunctionshouldbemodiedtoincludetheprobabilitydistributionofalltheevents.Forthisscenario,theMUMENsearchalgorithmwillworkbetter,asonceanoptimalsolutionisfound,thenumberofexchangeswillbeveryfewwiththeoccurrenceofnewevents. 46

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3-1 .ThegraphinFigure 3-1 hasnode1insubset1,node2insubset2,node3insubset3,andnodes4and5insubset4.TheSDMCC,whichistheoptimalsolutionofSDMCC 48

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Figure3-1. Illustratingsubset-disjointminimumcostcyle. WeillustratethesubsetdisjointnegativecostcycleinFigure 3-2 .WekeepthenodepartitionssameasinFigure 3-1 ,andchangeonlythecostofarcstosimplifytheillustration.Inthisnetwork,wehavetwofeasiblenegativecycles:(i)(1-2-3-1)ofcost-4,and(ii)(1-2-5-3-1)ofcost-10.Therefore,theoptimalcycleofSDNCCproblemis(1-2-3-5-1).Thereexistsonemorenegativecycle(1-2-4-5-3-1)ofcost-12,butweeliminateitfromconsiderationasitnotsubset-disjoint.Itshouldbenotedthatinthisexampletheminimumissameasthenegativecycle(1-2-3-5-1).Fromthegivendenitionsandillustrations,wecaneasilyconcludefollowingtworelationsbetweenSDMCCproblemandSDNCCproblem: 1. ThefeasiblesetofnegativecycleproblemwillbeasubsetofthefeasiblesetofminimumcycleprobleminthesamegraphG. 2. Ifthereexistsafeasiblesolutionforthenegativecycleproblem,theoptimalsolutionofminimumcycleandnegativecyclewillbesame. 49

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Illustratingsubset-disjointnegativecostcyle. Thesubset-disjointminimumandnegativecostcycleproblemsareusuallythecorestructureofmostoftheroutingbasedoptimizationproblems,andmultiple-exchangebasedneighborhoodsearchalgorithms.IthasbeenprovedNP-completein[ 33 ].Wenextdiscusssomeofitsdirectandindirectapplications. 1. Prizecollectingtravellingsalesmanproblem[ 8 ]Asalesmantravelsbetweenpairsofcitiesatacostdependingonlyonthepair,andgetsaprizeineverycitythathevisits.However,heneedstopayapositivepenaltyforeverycitythathefailstovisit.Theobjectiveoftheproblemistomaximizehisearning(prizecollected-travelcosts-penalties).Thereisnoconstraintontheminimumreward.WecanreducethisproblemtotheSDMCCproblembyconsideringeachcityasnode.Wealsoassumethateachnodehasitsownsubset,i.e.eachsubsetcontainsonlyonenode.Arcs(i,j)2Aareconstructedbetweennodepair(i,j),ifthesalesmancantraveltocityjaftercityi.Thecostofeacharc(i,j)2A,isassignedasthesumofthreecostcomponents:(i)travelcostfromcityitocityj,(ii)negativeofthepenaltyofnotvisitingcityi,and(iii)prizecollectedforvisitingcityj.Aftertheconstructionofthenetwork,weapplytheSDMCCalgorithmstondtheminimumcyclewhichhasonetoonecorrespondencetotheoptimalsolutionoftheprizecollectingtravellingsalesmanproblem. 2. Verylargescaleneighborhoodexchangesearch[ 6 ]Theverylargescaleneighborhoodsearchisgenerallyusedinthecontextofsolvingpartitioningproblems,whicharehighlycombinatorialandhugesizeproblems.GivenasetSofnelements,theobjectivetondapartitionT=fT1,T2,...,TkgofS,whereTissetofsubsetsofS.Eachpairofsubsetsmustbemutuallyexclusive,andtheunion 50

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2 .AlthoughSDMCCorSDNCCproblemsareverycommonlyuseddirectlyorindirectlyinseveralalgorithms,wedon'thaveanyliteraturededicatedtothem.Generally,somedynamicprogrammingisusedwhichiscustomizedtoparticularproblemsandmaynotbeapplicabletootherclassofproblems.Inthischapter,wehavedevelopedseveralgenericalgorithmswhichcanbeappliedinallproblemswhereminimumcycleistobefoundrepetitively.Thecontributionsofthispaperinclude: 1. WesuggestfourexactalgorithmsfortheSDMCCproblemsandSDNCCproblems.Thesealgorithmscanbeseenasthegeneralizedversionofthedynamicprogrammingalgorithmsgenerallyusedfortheconstrainedshortestpathproblems. 2. Networkreduction:Thecomplexityofexactalgorithmsdependsonthesizeofnetwork(nodesandarcs).Weintroduceseveralstrategiestoreducethesizeofthenetworkandthereby,todecreasethecomplexityofthealgorithm. 51

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Algorithmacceleration:Wepresentsomenewandsomeexistingstrategiesusedinotherapplications,toimprovethepracticalrunningtimeofthealgorithms.Thesestrategiesdon'timprovetheworstcasecomplexitiesofalgorithms;however,theyprovetobeveryeffectiveinpracticalapplications. 4. Theexactalgorithmsmayperformveryefcientlyevenonlargeinstances,thesolutiontimemustbeevenlowerthanfractionofsecondsinsomealgorithmssuchasVLSNsearch.Weproposeseveralheuristicsbasedontheexactalgorithmsbycontrollingdifferentparametersoftheexactalgorithms. 5. Wenallygiveanextensivecomputationalanalysisbyrunningallalgorithmsondifferentclassesofproblems. 1. Subsetofnodei=Subsetofnodej. 2. Nodeihasnoincomingarc. 3. Nodejhasnooutgoingarc. 4. Forallk2Vsuchthat(j,k)2A^(i=k_s(k)6=s(i))^s(k)6=s(j)thereexistsaq2Ss(j),q6=jsuchthat(q,k)2A,(i,q)2Aandciq+cqk
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3.3.1 ,andremovesitfromthelist.not_dominated(l0,L(l0)):Itreturnstrueifthelabell0isnotdominatedbyanyofthelabelsintheset(L(l0)).remove_dominated(l0,L(l0),U):Ifthefunctionl0isnotdominatedbyanylabelpresentinthesetoflabelswiththesameendnode,itispossible 56

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3.3.1 ,whichensuresthatstrictly'larger'labelsarecreatedfromthecurrentlabel.Inotherwordsalabelalreadyprocessedwillneverbecreatedagain,andhence,nocyclingwilloccur,whichprovestheterminationofthealgorithm.ThecomplexityofthedynamiclabelalgorithmisO(jNjjAj2K),whichisclearlyexponentialinK.Although,evenwiththebetterdatastructuretheworstcasecomplexitycannotbeimproved,butwecandenitelyimprovetherunningtimebyacceleratingsomealgorithmicstepsasdescribedinthenextsection. 58

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3.3.4DLAwithAccelerationforSDNCCProblemsThedynamiclabelingalgorithmfortheSDNCCproblemisalmostsameasthatfortheSDMCCproblems.However,thealgorithmcanbeacceleratedatseverallevelsbyusingthepropertyspeciconlytonegativecycles.Property1:Foranynegativecostcycleq=(i1,i2,...,il,i1)oflengthlinG,thereexistsj21,2,...,lsuchthatc(pk)<08k=1,2,l,wherepk=(ij,ij+1,...,ij+k),andallsubscriptsarithmeticisperformedmodulol.Usingtheaboveproperty,wecanconcludethatforanyoptimalsolutionofSDNCCproblem,theremustexistanodeisuchthateverypathinthesubgraphofGobtained 59

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3.3.3 ,and(ii)describedusingproperty1.Unfortunately,wecannotusebothtechniquesatthesametime.Inrstpropertyweassumethatallcycleswithlowernumberednodesmusthavebeenfoundwhenlowernumberednodeswasconsideredassource.While,ifweusethesecondtechnique,somecyclemaystartfromthelargernodeandpassthroughasmallernode,whichdoesn'thaveanyoutgoingarcwithnegativecost,tocompleteanegativecostcycle.Generally,anetworkcontainsverysmallpercentageofnegativecostarcs,soifweusethesecondaccelerationtechnique,wewilleliminatemanysourcenodesandreducethenumberoflabelscreatedwhilelabelextensionprocess.However,ifthepercentageofnegativearcsisveryhigh,weshouldusetherstaccelerationtechniqueasnegativecostcriteriawouldbesatisedbyalmostallnodes.Theworst-casecomplexityofDLAforSDMCCandSDNCCproblemsremainsame.Wenextgivethedetailsofthealgorithmforcompleteness. 3.3.2 fortheDLA,however,usesdifferentstrategytostorethelabels.Inthexedlabelalgorithm,likeDLA,wemaintainasetofunprocessedlabelsU,butunlikeDLA,wemaintainthesetLiofnon-dominated,andprocessedlabelsateverynode 60

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3.3.3 isvalidforFLA,too.Weincludetheminitsalgorithmicframework.ThecorrespondingalgorithmforSDNCCproblemscanbeobtainedbymakingsamechangesaswedidwhileconvertingtheDLAofSDMCCproblemtoSDNCCproblem.Moreexplicity,wereplacetheline3byforall2Nwithci<0;i2Ndo;andline10byforall(i,j)2+((l))ANDc(l)+cij<0do.ThecomplexityofDLAandFLAforbothversionsoftheproblemremainssame.However,therearetwomaindifferencesbetweenthesetwoalgorithms:(i)numberoflabelsstoredinDLAapproachismuchhigherthanthatofFLA,and(ii)thenumberofchecksperformedforestablishingthedominanceoflabelsiscomparativelyhigherinFLA.WediscusstheeffectofthesedifferencesinthecomputationalanalysisinSection 3.5 62

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5 ].ForeachsourcenodeiinN,werstndallpathsoflength1,thenextendthemtopathoflength2,andsoontillwereachthemaximumpathlengthn.Itisnotatruegeneralizationasinouralgorithms,wegenerateallpathsoflengthkonlyfromthecurrentsourcei2Nandmovetondingpathsoflengthk+1.However,inactual"all-pairshortestpathdynamicalgorithms"all 63

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3.3.3 .TheAPPullalgorithmscanbeeasilymodiedtosolvetheSDNCCversionoftheproblem.Todifferentiatethesealgorithmswecallthem,APPull-MandAPPull-Nalgorithms,respectively.TheAPPull-Nalgorithmsusetheaccelerationstrategyspecicfornegativecyclesassumingthatthepercentageofnegativecostarcsinthenetworkisverylow.Theextend-possiblefunctioninline13,examinesiftheextendedlabel'scostremainsnegative,besidescheckingtheresourcevector.Wepresentthedetailsofthealgorithmforcompletenessofthediscussion.TheAPPullalgorithmscanbeconvertedtoapuregeneralizedversionofallpairshortestpathalgorithms,byexchangingthe"forloop"ofline3andline14.However,pureversionstoresverylargenumberoflabels,andslowsdownthecompletealgorithm.TheAPPullalgorithmreducesthememoryrequirementbystoringfewerlabels,andgeneratessolutionsinmuchlowerrunningtime.WeshowthedetailedcomputationalresultsinSection 3.5 65

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3.3.6 .Themaindifferencebetweenthesetwoalgorithmsisthewayinwhichweextendthecurrentlabelstocreatenewlabels.InAPPushalgorithms,whenconstructinglabelscorrespondingtoacertainlengthk,itrstdecidesastartnodei(startnodeanssourcenodearedifferent),andextendsallpathscorrespondingtolabelsinLk1ioriginatinginnodeifollowingitsoutgoingarcs.WhileinAPPullalgorithms,whenconstructinglabelscorrespondingtolengthk,itrstdecidesanendnodej,andextendsallpathscorrespondingtolabelsinLk1i,where(i,j)2A.Allotherfunctionsbetweenthesetwoalgorithmsremainsame.Wenextpresentthealgorithmicframework. 3.2 ,itispossibletoavoidsomelabelextensionsthatnevercanleadtoanoptimalcycle.Itwillnotonlysaveanextensionbuteliminateseverallabelswhichwouldhavegeneratedthroughthecurrentandsubsequentextensions.Assumeweareconsideringextendingalabellatnodejtonodek,toconstructanewlabell0.Iflabellhasaparentlabel,denotethenodeoftheparentlabelasi(iflabelldoesnothaveaparentlabel,thenthetechniquecannotbeapplied).Inotherwords,thelastarcofthepartialpaththatlabellrepresentsis(i,j)andweconsiderextendingthepathwitharc(j,k).IfthereexistsanodeqinSs(j)suchthat(i,q)2Aand(q,k)2Aandciq+cqk
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3.5 ),therunningtimefortheproblemclassesoflargesizewasupto25secondsevenforthemostefcientminimumcyclealgorithms.Since,minimumcycleproblemsaresolvedrepetitivelyasasubprobleminothercomplexproblems,thehighrunningtimeisusuallynotexpectable.However,mostofthetime,insteadofndingthetheoptimalcycle,agoodqualitycycleissufcient.Inthissection,wesuggestseveralheuristicsbycontrollingdifferentparametersoftheexactalgorithms.WehaveimplementedtheseheuristicsonlyforAPPullandAPPushalgorithmsastheseweremosteffectiveforlargesizeproblem-groups,butsameconceptisapplicableforDLAaswellasFLA. 3.3 ,maintainasetoflabelsoneachnode.Theselabelsmaybeanycombinationofsubsetsinwhichnodesaredivided,andhence,thetotalnumberoflabelsincreasesexponentiallywiththenumberofsubsets.Thenumberoflabelsisstoredisdirectlyproportionaltotherunningtimeofthealgorithm.Wesuggestcontrollingtheseparametersbymaintainingalimitednumberoflabelsstoredforfutureprocessingateachlabel.Morespecically,ifthelimitoneachnodeisN,andthenewlycreatedlabeltakesthetotalnumbertoN+1,wescanthelistof 70

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3.3.6 ,and 3.3.7 ,wesubstitutethe"for-loop"extendingthepathonebyonetoKbyfollowingline:forallk=2,3,...,mdoInthecyclecheckphase,wealsocheckonlyuptolengthmasthereisnothingstoredinsetsbeyondthat.Thecomputationalresultsisshownforallfourproblemgroupsoflargesizewhichhadhighrunningtimeinexactapproaches. 71

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3.5 31 ].ThesebenchmarkproblemsweregeneratedbyconsideringdifferentinstancesofthecapacitatedminimumspanningtreeproblemsinpaperbyAhujaetal.[ 6 ].Thepapersolvesthecapacitatedminimumspanningtreeproblemsusingvery-largescaleneighborhoodsearch,inwhichthesubset-disjointnegativecycleproblemissolvedasthesubproblem.Wetakeeachinstanceofthespanningtreeproblemfrom[ 6 ],andgroupallitssubproblems.Thesegroupsofsubproblemsserveastheinputforcomparingouralgorithms.Theinstancesforcapacitatedminimumspanningtreeproblemsweregeneratedintwoways: 72

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13 ].Wetestallouralgorithmsproposedonthesegroupofproblems.Wesolvetheallproblemsinagroup(subproblemsgeneratedbyaninstanceofcapacitatedminimumcycleproblem)andpresentdifferentstatisticssuchasthearcdensity,percentageofnegativearcs,averagerunningtime,averagemaximumlabelsstored.Eachgroupoftheproblemsisrepresentedbyauniqueidandthenumberofproblemssolvedinit.Forexample,aproblemswithid"44-10"standsforthegroupnumber44with10subsetdisjointminimumcostcycleinstances.Asdiscussedearlierpercentageofnegativearcshelpinselectedofaccelerationstrategyinthenegativecycleversionoftheproblem.Thenumberoflabelsstoresinanalgorithmisadirectindicationofitmemoryrequirement.Thearcdensityiscalculatedwithrespecttotheidealnumberoffullarcswhichcanbepresentinaperfectlydensenetwork.Idealarcsexcludethearcsthatarebetweenthenodesthatbelongtothesamesubsets.Themaximumnumberofarcsinaperfectlydensenetworkcanben(n1)PKk=1jSkjjSkj1.Thenegativearcaswellastotalarcdensitieshavebeenmentionedwithrespecttothemaximumnumberofidealarcs.Foeexactalgorithms,wereportthenumberofnodes,numberofsubsets(multipleentriesifnumberofsubsetsaredifferent),totalarcsdensity,negativearcsdensity,averagerunningtimeforeachgroup,andaverageofthehighestnumberofthelabelsstoredforeachgroup.Inthecaseofheuristics,wealsoreporttheaveragedeviationfromtheoptimalsolutionalongwiththevalueofparametercontrolled.Allnetworkreductionandaccelerationstrategieshavebeenusedineachalgorithm.TheseareprogrammedinC++usingefcientdatastructures,andtestedona2.39GHzPCwith1.99GBofRAM.WersttestallfourexactSDMCCalgorithmsonthreeclassesofprobleminstances,classiedasperthenumberofsubsetsinwhichallnodesaredivided. 73

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3-1 ,forthemediumsizeproblemsinTable 3-2 ,andforthelargesizeproblemsinTable 3-3 .Inallthetables"P"denotesproblem-groups,"Sets"denotesthenumberofsubsetsinthegraph,"Nodes"denotesthenumberofnodesinthegraph,"Arcs"denotesthearcdensityinpercentage,"Neg.Arcs"denotesthedensityofnegativearcsinthegraph,"AML"denotestheaveragemaximumnumberoflabelsstoredbyalgorithms,and"time"denotesthetimeconsumedinseconds.InTable 3-1 ,wecompareallalgorithmsonthesmallsizeinstances.Theproblemshavebeensortedindecreasingorderofnumberofsubsets,andforthesamenumberofsubsets,intheincreasingorderofnumberofnodes.Inalltheseinstances,fromtimeaswellasmemoryperspective,thedynamiclabelingalgorithmperformedmuchbetterthanthexedlabelingalgorithm.For17outof20instances,thetimerequiredbyDLAwasalmosthalfofthatrequiredbyFLA.TheaveragemaximumnumberoflabelsstoredbyFLAwas1.5-2timesofaveragemaintainedbyDLA.ItisnotveryintuitiveintherstglanceasintheFLA,westorelabelsforeachnode,onlyifitiscertainthatthelabelwouldneverberemovedthroughdominancetest.Inotherwords,ateachnode,wemaintainthelistofprocessedlabelswhicharexed.Duringthelabelextensionstep,weinsertallnewlycreatedlabels,onlyinthesetofunprocessedlabels.Wetakeoutlabelsonebyonefromthislistandcheckifitisdominated.Itshouldbenotedthat,whenthelabelisinsertedintotheunprocessedlabelset,itisnotdominatedanyofthelabelxedatthecorrespondingnode.However,bythetimethatlabelisremovedfromtheunprocessedlabelsetforprocessing,itmaybedominatedbythenewlabelscreatedandxedatthatnode.However,intheDLA,thelabelsstoredinsetoflabelsmaintainedateachnodemaygetdominatedbysomefuturelabels,andhence,mayberemoved.Intheremovaloperation,alllabelsfromtheunprocessedlabel-setaswellasfromthenode-specic-label-set,whicharedominatedbythenewlycreatedlabelareremoved.So,althoughwemayhavetoperformlabelremovalandlabelentrystepsmultipletime, 74

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3-2 comparesthesamesetofalgorithmsonproblemclassesofmediumsize.Weconsiderthesamesetofnodes,andincreasethenumberofsubsets(14-17)inwhichtheseareclassied.Thecomparativebehaviorofallfourexactalgorithmsisalmostsameaswehavediscussedforproblemgroupsofsmallsize.ThedifferenceinDLAandFLAwithregardtothenumberoflabelsstoredaswellassolutiontime,becomesmoreprominentastheabsolutedifferenceisveryhigh(althoughdifferenceinratioissame).Allbutfourproblemsweresolvedwithinfractionofseconds.Thesefourgroupswerelargest(199)withrespecttothenumberofnodes.ThecomparativeperformanceofAPPullalgorithmwasmuchbetterinthisproblemclassthanthesmallerclass.APPullalgorithmperformedbetterthanAPPushinveoutof20problem-groups, 75

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3-3 displaystheresultsforthelargeproblemgroups.Thenumberofnodesvaryfrom80to199,andnumberofsubsetsfrom17to21.Thearcdensityisalsohigherfromlasttwoproblemclasses.comparesthesamesetofalgorithmsonproblemclassesofmediumsize.Inthiscasetoo,weobservethesamepatternofperformanceamongfourexactalgorithms.However,thedifferenceinsolutionbetween"DLA-FLA",and"APPull-APPush"algorithmswasverysignicant.Forexample,oneoftheproblem-grouptook137.36secondsforDLA,255.88secondsforFLA,20.97secondsforAPPull,and32.77secondsforAPPush.ItclearlyshowstheefciencyofAPPullandAPPushalgorithms.AmongAPPullandAPPushalgorithms,theAPPullalgorithmsperformedbetterthanAPPushinallproblem-groups.ThisbehaviorisquiteinterestingasforsmallinstancesAPPushalgorithmshasbetterrunningtime.So,wecanconcludethatAPPullalgorithmsarebestfortheseclassesofproblems.However,eventhemostefcientalgorithmtookmorethan20secondsfortwoproblem-groups,andmorethan10secondsforonegroup.FormostofthoseproblemswhichsolveSDMCCrepetitivelyassub-problems,thisrunningtimeisunacceptable.So,wedevelopedsomeheuristics,andwediscusstheseresultsaftertheanalysisofSDNCCalgorithms.WediscusstheresultsofSDNCCproblemsforproblem-groupsofsmallsizeinTable 3-4 .Allalgorithmsperformedveryefcientlyinalltheseinstancesgeneratingoptimumresultsinlessthan0.07seconds.Thememoryrequirementwasalsoverylowstoringatmost1300labels.ThepatternbetweenDLAandFLAissameasitwasintheSDMCCproblemasshowninTable 3-1 .AmongAPPullandAPPushalgorithms,APPushperformedbetterinallinstancesfromtimeperspective.However,thepatternbetweenDLAandAPPullalgorithmwasoppositeofSDMCCresults.TheDLAperformedconsistentlybetterthanAPPullalgorithmsfromtimeperspective,andthenumberoflabelsstoredwereinthesamerange(slightlymoreinDLA).Therefore, 76

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ComparativeanalysisofexactSDMCCalgorithms:smallinstances. AMLTimeAMLTimeAMLTimeAMLTime 33-12139969.812.644,0260.257,0580.553,8200.183,8200.1635-812,139967.822.833,2740.205,0560.443,2170.163,2170.1330-15129975.013.683,6500.246,3490.553,5070.173,5070.1527-9129969.282.792,8500.184,4750.402,7450.142,7450.1239-161119962.281.415,2060.748,8081.704,8800.734,8800.5942-71119968.251.587,5801.1913,5262.807,2671.037,2670.8445-111119975.251.247,3241.0313,4872.577,0780.897,0780.7448-141119976.251.105,7910.9510,3282.305,5820.845,5820.682-1110,114085.959.597310.029870.036440.026440.0011-810,114085.969.705260.027980.025000.015000.015-6104085.719.996880.038100.026600.026600.008-11104085.7111.478140.031,0010.027430.017430.0118-14108083.725.821,2720.041,7970.091,1300.041,1300.0324-14108083.728.698640.021,3850.067810.027810.0215-12108081.925.907450.021,1960.046770.026770.0221-11108080.305.515280.018020.034830.024830.013-584081.409.892830.013520.022540.002540.016-984081.408.703750.014490.013080.003080.009-118,94081.7811.874650.015990.014090.014090.0112-77,84078.3410.091880.012440.011630.001630.00

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ComparativeanalysisofexactSDMCCalgorithms:mediuminstances AMLTimeAMLTimeAMLTimeAMLTime 14-1916,178090.387.049,1450.4718,9891.149,0460.189,0460.2023-1616,178090.397.7816,7110.5422,3161.0515,9920.2015,9920.2320-16168090.367.3912,5570.4423,4430.9212,4920.1712,4920.1832-2315,169971.113.7910,0310.4516,5331.029,5720.259,5720.2526-7159970.484.978,3700.6113,4791.238,3230.308,3230.3038-1714,1519964.842.0826,1523.8252,8658.8925,1672.0225,1672.0329-14149965.954.579,0910.5814,7891.238,9060.308,9060.2941-71419974.091.9912,3562.4425,1725.8912,1161.6412,1161.5044-101419972.791.5322,7155.5340,04011.8522,0992.7022,0992.8047-61419976.491.5915,3543.0231,8407.8015,0861.8315,0861.721-5144090.7111.902,6840.074,3470.112,6220.032,6220.034-4144090.4812.552,1240.053,1740.092,0710.032,0710.037-10144090.4817.251,9850.053,1390.091,9610.021,9610.0210-6144090.4814.771,6340.033,1620.071,5950.041,5950.02

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ComparativeanalysisofexactSDMCCalgorithms:largeinstances AMLTimeAMLTimeAMLTimeAMLTime 13-11218092.879.2415,1721.0635,7392.4815,0940.3415,0940.4019-27218092.929.1937,3512.2161,0554.1435,1260.6535,1260.8316-920,218092.849.6121,3501.5738,1343.1220,7720.4620,7720.5625-5319,219977.446.02137,25672.18219,269117.29136,5486.15136,54812.1722-11208092.6810.4650,6923.2081,2805.6750,2250.8050,2251.0331-17199969.534.4923,5352.1941,2014.4223,2170.7623,2170.9134-1818,199971.814.1318,2651.5233,4203.2617,6050.5717,6050.6528-2518,199966.755.8251,3365.8879,14010.6051,0421.4951,0421.9346-211819973.512.65140,916137.36233,972255.88139,41620.97139,41632.7743-221819976.582.1492,28147.42131,57987.2991,71110.4591,71114.6637-171819973.532.6537,5236.9087,11617.9736,4753.0036,4753.2740-5417,1819971.533.47171,963132.81283,052264.11170,94120.95170,94134.25

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3-5 ,weshowtheoutputofallalgorithmsonproblems-groupsofintermediatesize.Allthealgorithmswereveryeffectiveintheseclassesofproblemsaswellsolving12outof14groupswithin0.015secondsandtwogroupswithin0.15seconds.ThecomparativebehaviorofDLAandFLAremainssameasforallothergroups.TheAPPushalgorithmsfoundoptimalcyclesmuchquickerthan(almost1.5times)APPullalgorithmsforallinstances.However,amongDLAandAPPushalgorithms,therewasnocleardifferenceinperformance.Ineightoutof14problem-groupsDLAwasquickerthanAPPushalgorithms,whileinvegroupsAPPushalgorithmsoutperformedDLA.Therewastieinonegroup,whichshouldbecountedtowardsAPPushalgorithmsifmemoryrequirementisourslightestconcern.ThenumberoflabelsstoredwasconsistentlylowerinAPPushalgorithms.InTable 3-6 ,weanalysetheperformanceofeachexactalgorithmforSDNCCproblemsonlargesizeinstances.TherewasnochangeinthepatternofDLAandFLAfromourearlieranalysis.TheAPPushalgorithmsagainoutperformedAPPullalgorithmsinallproblemgroups.ThemostinterestingobservationwastheshiftinthebehaviorofDLAandAPPushalgorithms.Intheseproblemgroups,DLAwascompletelyovershadowedbyAPPushalgorithmsandperformedbetterinonlytwooutof14instances.Inthosetwoinstancestoo,thetimedifferencewasnotverysignicant,withAPPushalgorithmstoringmuchlessnumberoflabels.Inthemostdifcultinstanceofthisclass,DLAalgorithmtook24.5secondstondtheoptimalresults,however,theAPPushalgorithmgenerateditin5.4seconds.Itshowsthatiftheproblemsizeisnotsmalltomediumrange,DLAisabetterapproachbutifthesizeisverylarge,APPushalgorithmsperformthebest.WenowtesttheheuristicsapproachessuggestedinSection 3.4 ,whichcontrolthreeparametersoftheexactalgorithms:(i)Numberofunprocessedlables,(ii) 80

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ComparativeanalysisofexactSDNCCalgorithms:smallinstances AMLTimeAMLTimeAMLTimeAMLTime 33-12139969.812.641,0210.0171,1620.0309430.0349430.02235-812,139967.822.835480.0126320.0155260.0255260.01830-15129975.013.685460.0156160.0225120.0255120.01927-9129969.282.791860.0122040.0191650.0261650.01739-161119962.281.414770.0625440.0594070.1444070.10742-71119968.251.581,2330.0831,5390.0871,1510.1721,1510.12745-11111991991.241,3690.0771,7740.0901,2640.1511,2640.11248-141119976.251.11,0590.0741,3020.0799850.1419850.1112-1110,114085.959.592470.0002730.0042260.0012260.00111-810,114085.969.71610.0021590.0061490.0041490.0025-6104085.719.991320.0031190.0051150.0081150.0038-11104085.7111.472620.0012660.0092390.0092390.00318-14108083.725.824460.0063990.0093700.0083700.00724-14108083.728.693990.0035040.0083290.0103290.00815-12108081.925.92300.0062500.0071940.0091940.00721-11108080.35.513410.0043310.0043100.0103100.0073-584081.49.89920.003980.000840.006840.0006-984081.48.7960.000920.005770.003770.0029-118,94081.7811.872340.0042510.0071900.0061900.00112-77,84078.3410.09890.002850.000780.002780.000

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ComparativeanalysisofexactSDNCCalgorithms:mediuminstances AMLTimeAMLTimeAMLTimeAMLTime 14-1916,178090.387.042,1710.0203,3810.0392,0210.0322,0210.01923-1616,178090.397.782,1630.0222,2560.0262,1200.0262,1200.02120-16168090.367.391,5200.0181,8340.0261,4780.0331,4780.01732-2315,169971.113.791,3080.0181,4660.0271,2360.0371,2360.02326-7159970.484.975250.0185070.0224970.0404970.02238-1714,1519964.842.083,4850.1286,4210.1953,2860.2733,2860.16729-14149965.954.572,7540.0263,1620.0452,6630.0472,6630.03041-71419974.091.991,0070.0921,1500.0949150.2359150.14744-101419972.791.533,4800.1594,3000.2223,3470.2803,3470.18647-61419976.491.591,9940.0942,5580.1041,8720.2061,8720.1431-5144090.7111.96050.0036030.0095750.0095750.0064-4144090.4812.555240.0045130.0125070.0195070.0047-10144090.4817.251,0980.0131,1570.0281,0660.0091,0660.00610-6144090.4814.777030.0138590.0266840.0056840.000

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ComparativeanalysisofexactSDNCCalgorithms:largeinstances AMLTimeAMLTimeAMLTimeAMLTime 13-11218092.879.242,2990.0262,4320.0402,2640.0362,2640.02319-27218092.929.1911,1710.17711,3520.25210,7480.09410,7480.08216-920,218092.849.611,9560.0232,3750.0311,8610.0261,8610.01925-5319,219977.446.0240,0115.79159,7599.67739,6430.89339,6431.21822-11208092.6810.464,7970.0576,9830.0994,5290.0484,5290.04031-17199969.534.492,9210.0463,3180.0652,8530.0622,8530.03834-1818,199971.814.131,5950.0281,8990.0541,5330.0511,5330.03528-2518,199966.755.8221,1740.63224,7461.01720,9200.21420,9200.22046-211819973.512.6525,2901.72836,0192.89324,9290.86424,9290.78043-221819976.582.1416,2050.78618,7781.01616,0430.50916,0430.42937-171819973.532.652,4300.1343,1660.1522,3200.3202,3200.19340-5417,1819971.533.47133,62124.586185,56337.862132,9093.064132,9095.485

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3-7 ,and 3-8 ,"P"denotestheproblem-groups,"Opt."denotestheoptimalsolution,"Dev."denotesthedeviationofheuristicsolutionfromtheoptimal,and"Time"denotestherunningtimeinseconds.FortheLULheuristics,wexedthelimitofunprocessedlabelsto10,50,and1000forAPPullandAPPushalgorithms.Thesolutionqualityandrunningtime,asexpectedimprovedwithincreaseinthenumberoflabelsallowed.Asweincreasedthelabellimitfrom10to1000,thegapbetweentheheuristicsolutionandoptimalsolutionimprovedfrom30%toalmost1%whiletimeincreasedfromonesecondtothreesecond.Inthemostdifcultinstance,thetimetakentogenerateoptimalsolution,forAPPullandAPPushalgorithms,was20secondsand34secondswhichimprovedto3.74secondsand3.60secondsrespectively.Theoptimalitygapwasaround2%.Bothgeneratedoptimalsolutionforoneproblem-group.AmongAPPullandAPPushalgorithmswithLUL,APPullgeneratedsolutionswithbettergapwhileAPPushwasfasterspeciallywithlowernumberoflabels.ForLCL,wevariedthelimitonthemaximumcyclelengthto2,4,and8.BothAPPullandAPPushversionofalgorithmsgeneratedpoorsolutionswithcyclelengthtwo,whichdenotesasimpleswapping.Itisduetothefactthataverageoptimalcyclelengthsinallproblemsgroupvaryfromeightto10.Withthecyclelengthlimit8,therunningtimewasveryhighconsideringthatitisaheuristic.TheLULoutperformedLCLheuristicsinsolutionqualityaswellasrunningtime.Finally,we 84

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85

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ComparativeanalysisofheuristicSDMCCalgorithms APPullAPPushAPPullAPPushAPPullAPPush POpt.LimitDev.TimeDev.TimeLengthDev.TimeDev.TimeLimitDev.TimeDev.Time 46-21-12.90103.900.914.000.5027.760.117.760.1137.861.346.811.60501.101.001.100.5842.810.612.810.5755.522.164.522.7610000.143.670.143.6980.339.920.3313.74101.626.851.3311.5243-22-7.23102.230.942.270.5124.730.114.730.1132.411.902.502.21500.591.020.640.6142.090.622.090.5851.143.491.234.2710000.093.570.093.5680.456.370.458.24100.148.140.1410.6237-17-5.29101.060.870.820.4821.820.111.820.1131.762.001.762.17500.180.880.240.5240.240.560.240.5050.882.250.882.6010000.002.200.002.0780.002.760.003.06100.062.950.063.1540-54-18.22105.480.935.740.5028.330.118.330.11311.612.1711.873.26502.331.002.650.5945.060.595.060.5559.742.4310.203.7610000.393.740.413.6082.028.922.0212.54105.374.395.936.48

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ComparativeanalysisofheuristicSDNCCalgorithms APPullAPPushAPPullAPPushAPPullAPPush POpt.LimitDev.TimeDev.TimeLengthDev.TimeDev.TimeLimitDev.TimeDev.Time 46-21-12.90102.710.3633.100.20227.760.1117.760.11335.290.1805.290.138500.380.4440.290.23342.810.2732.810.16853.240.2713.240.20810000.000.6510.000.46380.330.5580.330.423100.710.5930.900.43343-22-7.23103.050.2912.820.18324.950.1144.950.12232.860.2022.950.152500.180.3250.320.20442.140.2462.140.17051.680.2561.680.17710000.050.4290.050.29880.450.3790.450.284100.270.3590.360.32937-17-5.29100.650.3080.710.19022.240.1112.240.11330.120.3110.120.192500.000.3250.060.19840.240.2750.240.16750.000.3770.000.21110000.000.3340.000.19380.000.3360.000.195100.000.3680.000.20640-54-18.22105.980.3655.980.21028.430.1188.430.11738.330.2088.370.144501.540.4251.440.22745.060.3005.060.18056.150.2326.090.15910000.000.6940.000.49582.020.8672.020.893102.440.4212.500.385

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88

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62 ],newspaperdistribution[ 41 ],postboxlocation[ 46 ],wastecollection[ 44 ],goodsdistribution[ 65 ]andparceldelivery[ 74 ]arejustsomeoftheexamples.Theproblemthatintegratesthefacilitylocationandvehicleroutingproblemstogetheristhecombinedlocation-routingproblem(LRP)[ 64 75 ].TheLRPconsistsofselectingasubsetoflocationsamonggivencandidatefacilitylocationsanddeterminingthevehicleroutestovisitallthecustomersfromtheselectedfacilitylocations.TheobjectiveinLRPistominimizethesumofthexedcosts,variablecostsanddeliverycostssuchthatcapacitiesofselectedfacilitylocationsandvehiclesarenotexceededwhilethedemandofeachcustomerissatised.Thexedcostsincludethecostsofestablishingdepotonacandidatelocationandthevariablecostsdependontheperunitthroughputtoacustomerfromtheselectedfacility.TheLRPreectstheinterdependencebetweenlocationcostsandroutingcosts[ 69 75 ].Hence,itisacombinationoftwoalreadydifcultproblems:FacilityLocationProblem(FLP)andVehicleRoutingProblem(VRP).BothFLPandVRPhavebeenshowntobeNP-hard[ 23 42 52 ],thustheLRPisNP-hardaswell. 89

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4.2 ,wegivetheproblemdescriptionandinsection 4.3 ,weintroducethemixedintegerlinearprogrammingformulationfortheLRP.Proposedcolumn-generationalgorithmispresentedindetailinsection 4.4 .Wediscussthemasterproblemandthepricingsubproblem.Wepresentthedynamicprogrammingalgorithmdevelopedforthepricingsubproblem.Attheendofthissection,wegivetheoverallalgorithmandexplainthestepsofthealgorithm.Insection 4.5 ,theimplementationdetailsandresultsaregiven. 90

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4-1 Xcoord. Ycoord. Capacity(T) FixedCost(FC) VariableCost(VC) 1 24 24 500 50 0.02 2 26 21 500 50 0.02 3 33 21 500 50 0.03 4 22 24 500 50 0.03 5 32 22 500 50 0.04 VehicleNumber(K) 25 VehicleCapacity(Q) 300 CostPerMile(CM) 0.75 CustNo. Xcoord. Ycoord. Demand 1 23 35 125 2 45 43 84 3 12 45 60 ... ... ... ... ... ... ... ... ... ... ... ... 34 30 16 160 35 10 40 40 36 34 60 80 ExampledatafortheLRP 91

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Theobjectivefunctionistominimizethetotalcostwhichiscomposedofthesumofxedcostsofestablishingdepots,thesumofvariablecoststhatdependoneachdepot'sthroughputandthesumofdeliverycosts.Constraints( 4 )statethatthereis 93

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4 )statethatthereisexactlyonearccomingintoeachcustomer.Constraints( 4 )ensurethatthenumberofvehiclesleavingadeparturedepotequalstothenumberofvehiclesreturningtoitscorrespondingarrivaldepot.Constraint( 4 )enforcesthatthetotalnumberofvehiclesusedshouldnotexceedtheupperboundonthenumberofvehiclesanditshouldbegreaterthanKLBwhichiscalculatedasdPi2Cdi=Qe.Inconstraintset( 4 ),thevalidinequalitythatwasproposedbyKulkarniandBhave[ 45 ]isusedinordertoeliminatecyclesthatcanoccurbetweencustomers.Constraints( 4 )givethelowerandupperboundoftheUivariables.Theloadonvehicleatdeparturefromanodeicanbeatleastequaltothedemandofthatnodeanditcanatmostbeequaltothecapacityofthevehicle.Constraints( 4 )ensurethatnodemandisassignedtoalocationifthatlocationisnotselected(thatis,yj=0)andifacandidatelocationisselected(thatis,yj=1)thenthesumofthedemandsthatareassignedtotheselectedfacilityshouldnotexceedthecapacityofthefacilitythatwillbeconstructedonthatlocation.Constraints( 4 )specifythatcustomericanbeassignedtoalocationjonlyifthatlocationisselected.Constraints( 4 )requirethateachcustomerisassignedtoexactlyonedepot.Constraints( 4 )statethatifacustomerisnotassignedtoalocationthentherecannotbeanarcthatgoesfromthatcustomertothatlocation'sarrivalnode.Similarly,constraints( 4 )statethatifacustomerisnotassignedtoalocationthentherecannotbeanarcthatgoesfromthedeparturenodeofthatlocationtothiscustomer.Constraints( 4 )and( 4 )ensurethatifcustomerjfollowscustomeriandifcustomeriisassignedtocandidatelocationkthencustomerjmustalsobeallocatedonthatsamecandidatelocation.Theremainingconstraints( 4 ),( 4 )and( 4 )specifythattheroutingvariables(xij),theallocationvariables(zik)andthelocationvariables(yj)cantakevalues0or1.ThegivenexactmodelisobservedtosolvesmallsizeLRPs(upto15customersand4depots)optimallyinthecomputationalexperimentsthatweconducted.Hence,we 94

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4.4 indetail. 4.4.1TheMasterProblemAfeasiblesolutionofaLRPiscomposedoftheselectedsubsetoflocationsandasetoffeasibleroutesoutgoingfromtheseselectedlocations.Arouteisdeclaredasfeasibleifitdoesnotviolatevehiclecapacityrestrictions,ifeachcustomerassignedtotherouteisvisitedexactlyonce(itmeansthattherouteshouldbeanelementaryroute)andiftheroutestartsandendsatthesamedepot.ThesetoffeasibleroutesthatcanexistintheoptimalsolutionofLRPisconsideredwhileconstructingtheLRPmasterproblem.Initially,ifweassumethatwehaveallfeasibleroutesthentheLRPreducestoselectingasubsetofthesefeasiblecolumnssuchthatthecostoftheroutesandxedcostofopeningthefacilitiesthattheseroutesoriginateareminimized.Eachfeasiblerouteformsabinaryvariable(thatis,acolumn)oftheMPmodel.TheMPmodelsolvestheLRPifallthefeasiblecolumns,thesetR,isreadilyavailable.GiventhesetR,thetaskoftheMPmodelreducestoassignallcustomerstotheroutessuchthateachcustomerisassignedtoexactlyonerouteandMPalsohandlesthefacilitycapacityconstraintssothatfeasibilityisnotviolated.Thatiswhy,theMPisasetpartitioningproblemwithsideconstraints.Thenaturalissuesthatcometomindarethedifcultyofndingthefeasibleroutesefcientlyandtheexponentialnumberoffeasibleroutesthatshouldbeconsidered.Theproblemofhugenumberoffeasibleroutesisaddressedbyrelaxingthemasterproblemwhichisanintegerprogrammingproblemandsolvingtheresultantlinearprogram(RMP)bycolumngeneration.Westartthecolumngenerationalgorithmbyconsideringonlyasubsetoffeasiblecolumns.Thenusingtheinsightofsolvingalinearprogramming(LP)problem,weaddonlythecolumnsthathavenegativereducedcoststothemasterproblem.Theproblemofndingthefeasibleroutesthathavenegativereducedcostsissolvedinthesubproblemofthecolumngeneration 95

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4.4.2 .Themathematicalformulationofthemasterproblemisgivenbelow:Sets:R:setoffeasibleroutesJ=f1,2,..,Mg:setofcandidatelocationsC=f1,2,..,Ng:setofcustomersModelParameters:K:Numberofavailablevehiclesir:1ifthecustomeriisassignedtorouter,0otherwise;iC,rRjr:1iftherouteroriginatesfromfacilityj,0otherwise;jJ,rRPDr:Demandofrouter;rRCr:Costofrouter;rRTj:Capacityofcandidatefacilityj;jJFCj:Fixedcostofcandidatefacilityj;jJDecisionVariables:xr:1iftherouterisselected,0otherwise;rRyj:1ifthecandidatefacilityjisselected,0otherwise;jJMinimize 96

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TheobjectivefunctionoftheMPistominimizethecostoftheselectedroutesamongthesetoffeasibleroutesRandthexedcostofselectedfacilitylocations.ObjectivefunctionvaluesoftheMPcorrespondtotheexactLRPmodelpresentedpreviously.Notethatthevariablecostsforeachselectedlocationisincorporatedtothecostofeachroutefound.Hence,thecostofanewrouter=fi1,i2,..,itgcanbefoundbytheequation( 4 ): 4 )aredenedforeachcustomerandformthesetpartitioningconstraints.TheseconstraintsstatethateachcustomeriisservedbyexactlyonerouteamongthesetoffeasibleroutesR.Constraintset( 4 )restrictsthenumberofpathsoriginatingfromacandidatefacilitylocation.Ifacandidatelocationjisselectedthenitrequiresthatatleastonerouteoriginatesfromthatlocationandthemaximumnumberofroutesthatcanoriginatefromthelocationissettotheavailablenumberofvehicles.Ifthecandidatelocationjisnotselectedthenitrequiresthatnorouteswilloriginatefromthislocation.Constraintset( 4 )isrstgivenin[ 14 ].OneclassicalconstraintthatcanbewritteninsteadofthisconstraintcouldbexrPjJjryj,8rR.Theclassical 97

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4 )istighterwhencomparedtotheclassicalone.Therefore,relaxationvaluesofthemodelwithconstraintset( 4 )willbegreaterthanorequaltorelaxationvalueswiththeclassicalmodel.Anotheradvantageisthereducednumberofconstraintssince( 4 )isdenedforeachcustomerandcandidatelocation.However,theclassicalconstraintisdenedforeachfeasiblerouteR.Notethatbyusingtheirandjrparameters,( 4 )canberephrasedastheconstraint( 4 ),thatis,theyareequivalent.Constraint( 4 )setsthelowerboundonthenumberoffacilitiesthatcanbeselectedamongthecandidateset.LBwiscalculatedasdPi2Cdi=TmaxewhereTmaxisthemaximumcandidatefacilitycapacityamongall.Constraint( 4 )statesthatthetotalnumberofroutesselected(orvehiclesused)shouldnotexceedthethenumberofavailablevehiclesanditshouldbegreaterthanLBvwhichiscalculatedasdPi2Cdi=Qe.Constraintset( 4 )requiresthatsumofthedemandsofthecustomersassignedtoafacilityshouldnotexceedthefacility'scapacityifthefacilityisselected.Theconstraints( 4 )and( 4 )specifythattherouteselectionvariables(xr)andthelocationvariables(yj)cantakevalues0or1. 98

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4-1 showstheconstraintnumberandassociateddualvaluesthatareobtainedasthemasterproblemissolved.Thus,themodiedarccost^cijintheconstructednetwork 99

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ConstraintsandassociateddualvaluesfortheMasterProblem MPconstraintnumberDualvalue 4 0q[i]ifiCandjC[N+1cij 0 29 ].Anintegermodelwhosesolutionidentiesanelementaryshortestpathonthedenedsubnetworkforalocationjwouldgiveonlythecolumnwiththemostnegativereducedcost.However,thegoalistodeterminewhetheranegativereducedcostcolumnexistsornot.Soobtainingasetofgoodroutesforeachlocationjismoreimportantthanndingthebestrouteforafastevolvingalgorithm.Thatiswhy,forthepricingsubproblemweusedynamicprogrammingtondasetofroutesthathavenegativereducedcosts.ThestructureofthealgorithmweusedforsolvingtheESPPRCispresentedinFigure 4-2 .Apathiscallednon-elementaryifanodeisvisitedmorethanoncehenceifthepathcontainsanyloop;otherwiseitiscalledanelementarypath.The 100

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51 ]thatsolvestherelaxedversionoftheESPPRCinwhichnon-elementarypathsareallowed.Inhisdynamicprogrammingalgorithm,Larsen[ 51 ]usesthebasicprinciplesofDijkstra'salgorithminordertondfeasiblerouteswithnegativereducedcosts.Normally,anoptimalsolutiontoaLRPdoesnotinvolvecyclessinceeachcustomerisvisitedonce.However,cycleconstraintisgenerallyrelaxedandashortestpathproblemwithresourceconstraints(SPPRC)issolved.SPPRCisalsoNP-Hardbutpseudo-polynomialalgorithmsexisttosolvetheSPPRC[ 27 ]hencetheycanbesolvedmoreefciently(e.g.see[ 5 ]).Usingthissolutionapproach,thesolutionoftheRMPmightcontainnon-elementarypathssincenon-elementarypathsareaddedtotheMP.However,ithasbeenshownthatanoptimalintegersolutionwillonlycontainelementarypathsbecauseofthetriangularinequalityassumption[ 20 ].ThedisadvantageofthisapproachisthatthelowerboundobtainedbythesolutionofRMPisweakenedsincenegativecycleshelptoreducecosts.Inourdynamicprogrammingalgorithmwestorethepredecessornodesforeachpartialpathandeliminatek-cycles.ThemodiedarccostsareupdatedeachtimeRMPissolved.Wecreatetheinitiallabelf0,0ginstep1ofthedynamicprogrammingalgorithm.Eachlabel(fi,AccDemig)representsapartialpath.Therstentryinalabelisthelastnodeofthepartialpathandthesecondistheaccumulateddemandforthepartialpath.CostofalabelisshownbyCost(fi,AccDemig).Thecostoftheinitiallabelissetto0.Whenalabeliscreateditisaddedtothesetofunprocessedlabels.Instep2,BestLabel()functionchoosesthelabelthathasminimumaccumulateddemandamongtheunprocessedlabels.Sincethelabelsareprocessedinorderofincreasingaccumulateddemand,alabelcanbechosenandprocessedonlyonetimeinstep2.Notethatthelabelisremovedfromthesetofunprocessedlabelsattheendofstep2afterithasbeenprocessed.Thelabelfu,AccDemugthathastheminimumaccumulateddemandisextendedtotheother 101

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DynamicprogrammingalgorithmforESPPRC 102

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IfanextensionofthesecondpathisfeasiblethenthesameextensionwillalsobefeasiblefortherstpathwithalowercostconsideringtheconditionsgiveninEqs.( 4 )and( 4 ).Thus,thelabelforthesecondpathcanbediscarded.However,fortheelementaryshortestpathproblemwithresourceconstraintsthebasicdominancerulecannotbeapplieddirectlyandamodieddominancerulehasbeenproposedrstbyDumasandDesrosier[ 30 ]andChabrier[ 20 ].ForSPPRC,therstandsecondpathcanbothcontinueonthesamenodeifvehiclecapacityisnotexceeded.However,forESPPRCthesecondpathhasthechanceofcontinuingwithanodeoftherstpathandthisextensionmightbeusefullater.Notethattherstpathcannotcontinuewithanodealreadypresentinitspartialpath.Hence,thepromisingnodecannotbeaddedtotherstpath.Athirdconditionthatisrequiredfortherstpathtodominatethesecondis 103

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4 ). 20 ],ourcomputationalexperiencealsoshowsthatifallthethreeconditionsareapplied,thenumberoflabelsdiscardedduetodominancearemuchlesscomparedtobasicdominance.Hence,ourstrategyistoapplybasicdominanceaslongasthealgorithmcouldndcolumnswithnegativecosts.Whenthealgorithmcannotidentifyacolumnwithnegativecost,thethirdconditionisalsoappliedtodecideondominance.Asthealgorithmcontinues,ifalabelfN+1,AccDemvgwithnegativereducedcostisdiscovered,thesetSfN+1,AccDemvgisformedfromthenodesofthelabel.LetSN+1denotesthesetthatkeepssetsofeachdistinctpathasitselements.IfSfN+1,AccDemvg2SN+1,thenthepathisalreadydiscoveredandweonlyupdatetheorderofthenodes.ButifSfN+1,AccDemvg=2SN+1,thenthissetisanewlydiscovereddistinctpathandweaddthepathtothesetSN+1.ThesolutionofthesubproblemforfacilitylocationjisterminatedwhentheunprocessedlabelsareonlytheonesassociatedwithN+1.ThelabelsfN+1,AccDemvgarenotprocessedsincenodeN+1isthearrivalnodeandhencetheycannotbeextendedtoafurthernode.IfBestLabel()functionmustreturnalabelofnodeN+1sincealltheotherlabelsareprocessedthenthealgorithmcontinueswithstep3.Inthisstep,allthedistinctpathsassociatedwithnodeN+1arepreparedforandaddedtothemasterproblem. 104

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4.4.1 .Columngenerationtechniqueisparticularlyappliedforthesolutionoftherelaxedmasterproblem(RMP)whichhappenstohaveanexponentialnumberoffeasiblecolumns.TheRMPistheLPrelaxationoftheMP.Asinanycolumngenerationtechnique,westartwithasubsetoffeasiblecolumnsandwetrytondnewpromisingcolumnsinthesubproblemwhichisalsocalledthepricingproblem.ForthepricingproblemwesolveelementaryshortestpathproblemwithresourceconstraintsbymodifyingthealgorithmofLarsen[ 51 ].TheobjectivefunctionvalueoftheRMPformsalowerboundtotheoriginalproblemwhennomorefeasiblerouteswithnegativereducedcostcanbefoundbythesubproblem.TheoutlineofourcolumngenerationalgorithmisgiveninFigure 4-3 .Intheinitializationstep,werstimportthedataforthegivenproblem.Afterdeterminingthenumberofcustomers(N)andthenumberofcandidatelocations(M),thesetofdeparturelocations(D)andthesetofarrivallocations(A)areestablished(Step1.2).ThearctravelcostsarecalculatedbymultiplyingtheEuclideandistancesbetweenpairsofcoordinateswiththecostpermileparameter(CM).Thedistancesarecalculatedwithvedecimalpointandtruncationwiththeformulagiveninequation( 4 ). 105

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Columngenerationalgorithm usedcouldnotbeincorporatedinthemasterproblemmodelinthiscase.Constraints( 4 )and( 4 )oftheMPmodelmustberemovedtohaveafeasiblesolutionandtherelaxationgapincreasesiftheyarenotutilized;therefore,theintegerMPissolvedinalongercomputingtime.Second,thenatureoftheLRPrequirestollthevehiclecapacityasmuchaspossibletoforcesmallernumberofdepotsandvehicles.Thatiswhy,multi-noderoutesthatutilizethevehiclecapacitybetterarelikelytobepresentinoptimalsolutions.Asexpected,usingsingle-noderoutesonlygivesworseinitialfeasiblesolutionsinthesensethattheyarefarfromoptimal.Hence,thenumberofpricing 106

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22 ]asaconstructionheuristicanditisfollowedbyaseriesofrouteimprovementheuristics,namely2-opt,1-1exchange,crossoverandsimulatedannealing.Instep1.5,theinitialsetoffeasibleroutesarepreparedforthemasterproblem.Notethateachfeasiblerouterisincorporatedtothemasterproblemasavariablexr,thatiswhy,inordertopreparearoutefortheMP,themodellingcomponentsrelatedtothevariablexrmustbeconstructed.Hence,inthepreparationstep,wecalculatethecostoftheroute(Cr),thedemandoftheroute(PDr)andarrangetheparameterirwhichholdsthecustomersassignedtorouterandparameterjrwhichholdsthedepotthatisusedinrouter.Theparameteriris1ifthecustomeriisassignedtorouter,0otherwiseandtheparameterjris1iftherouteroriginatesfromdepotj,0otherwise.Instep2,weconstructtheMPmodelwhichisanintegerprogrammingproblemandwhenwesolvethemasterproblemforthersttimewiththesepreparedcolumns,weobtainaninitialfeasiblesolutionandkeeptheinitialsolutiontoserveasastartingintegersolutionforourinstance.NotethatbyusingCPLEX11.1,MPmodelisconstructedcolumnwiseonlyonetimeandlaterthenewlyfoundcolumnsareaddedtothemodelwithoutreconstructingtheentiremodelbutbyusingtheaddColumn()routineoftheCPLEX.Weiteratestep3untilwecannotndafeasiblecolumnthathasnegativereducedcostforanycandidatelocationj.Aftertherelaxedmasterproblem(RMP)issolved(step3.1),thelowerboundisupdatedandtheedgecostsaremodiedusingthedualvalues.Atthismoment,itisverybenecialtocheckwhetherthesolutionoftheRMPisintegralornot.ComputationalexperienceshowedthatitisnotuncommontohaveintegralsolutionsfortheRMPsincethesizeoftheRMPincreasesstepbystepateachiterationofstep3.2.EspeciallyfortheinitialiterationsinwhichsmallsizeLPsare 107

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4.5.1TestProblemsWetrytosolveageneralLRPwherewehavecapacitatedmultiplefacilitiesandmultiplevehicles.TherearenocommonlyacceptedandwidelystudiedbenchmarkproblemsforthistypeofLRP.The3benchmarkproblemsofPerl[ 64 ]aretheonlyproblemsetsthatmatchwiththegeneralLRPwestudy.LRPbenchmarksofPerl[ 64 ]arestudiedin[ 65 ],[ 37 ]and[ 76 ].Theauthorsofthesepaperscomparetheircomputationalresultswitheachother.AnotherLRPdatasetwhichisclosetothegeneralLRPwestudyisprovidedbyOr[ 61 ]andisstudiedin[ 62 ].However,facilitiesareuncapacitatedintheLRPbenchmarkofOr[ 61 ].Recently,Barreto[ 11 ]triedtoformastandardizeddatabaseforthegeneralLRPsbyrespectingtheformatproposedbyPerl[ 64 ].TheinstancesthatarecompiledbyBarreto[ 11 ]aremostlyfromtheliteratureofvehicleroutingproblems.TheseedpapersorthesesfortheseinstancesareduetoPerl[ 64 ],Daskin[ 26 ],Gaskell[ 34 ],Min[ 56 ],Or[ 61 ],Srivastava[ 70 ]andChristodes[ 21 ].InordertoformastandardizeddatabaseBarretotookthedepotvariablecostsaszeroandtakethecostpermileparameteras1inalltheinstances.However,whilemakingcomputationaltestswerespectedtheoriginaldataofPerl's3benchmarks.Therefore,wecouldmakecomparisonswiththesolutionsof[ 65 ],[ 37 ]and[ 76 ].Fortheotherinstances,wecouldcompareourresultsonlywith[ 12 ]. 40 ].MixedintegerandprimalsimplexoptimizersofCPLEXareusedtosolveMIPandLPmodelsrespectively.ThealgorithmisimplementedwithJavaAPIofILOGCPLEXConcertTechnology. 109

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4-2 .Inthisdetailedtable,vehiclecapacity(Vcap),integerobjectivefunctionvalue(IP),lowerboundobtainedfromthecolumngenerationprocedure(LB),integralitygap(Gap),numberofvehicles/routesused(Vno)andthesetoflocationsselected(Depots)inthesolutionandnallytheruntimeofthecolumngenerationalgorithm(Time(s))isgivenrespectively.WereportedourrunningtimesinTable 4-2 .Wesolved11outof18instanceswithin1hourand13instanceswithin4hours.Thegenerationofpromisingcolumnscontinueduntilanelapsedtimelimitof12hoursfor5oftheinstances(Christodes-100c-10d,Min-134c-8d,Daskin-88c-8d,Daskin-150c-10d,Or-117c-14d).Fortheseinstances,thenalintegersolutionisreportedandTILIMiswrittenintheruntimecolumntoindicatethatalgorithmhasbeenterminatedduetotimelimit. Table4-2. ResultsontheLRPbenchmarkinstances

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ComparisononPerl'sbenchmarkproblems Perl[ 64 ]{1}2355.58{2,10,12}105795.62{2,4,5}117789.96Hansenetal.[ 37 ]{1}2355.58{2,7,12,13}105617.67{2,4,6}117551.61Wuetal.[ 76 ]{1}2355.58{5,10,12}105532.28{2,4,6}127781.21ProposedAlgorithm{1}2355.58{2,8,12}105465.51{2,4,6}117496.62 64 ],Hansenetal.[ 37 ]andWuetal.[ 76 ].Allthe3methodsareheuristicmethodsasdiscussedintheliterature.TheresultsaresummarizedinTable 4-3 .ItisobservedthatthesolutionfoundforthePerl-12c-2disthesameastheheuristicsolutionsandthesolutionsfoundforPerl-55c-15dandPerl-85c-7dislowerthanallthepreviousresults.InTable 4-4 ,wecomparetheresultsreportedinBarretoetal.[ 12 ]withourresults.InthetableprovidedbyBarretoetal.[ 12 ],the(IP)columnisthebestknownsolutionobtainedusingtheclusteringheuristic.Thelowerbound(LB)columnwasobtainedbyBarreto[ 11 ]witharelaxed2-indexintegerlinearprogrammingformulation.ItwasnotpossibletogetLBfortheinstanceMin-134c-8dusingtherelaxed2-indexintegerlinearprogrammingformulation.Usingthecolumngenerationalgorithm,avalidLBcouldnotbefoundin5oftheinstancessincethealgorithmterminatedduetotimelimit,meaningthatthealgorithmwascontinuingtogeneratepromisingcolumnswhenthetimelimithasbeenreached.Theoptimalityisreachedin2ofthe14instances(Gaskell-29c-5dandMin-27c-5d)byusingtheclusteringheuristicmethod.In8ofthe14instancestheproposedintegersolutionsislowerthantheBarreto'ssolutions.Thesolutionsobtainedinbothofthemethodsarethesamein2ofthe14instances.Intheremaining4instancesBarretoobtainedbetterresultscomparedtooursolutions.Inallthese4instancesouralgorithmterminatedduetotimelimit.FortheinstanceChristodes-100c-10d,althoughtheproposedalgorithmterminatedduetotimelimit,thesolutionfoundisbetterthantheBarreto'ssolution.Barretoetal.[ 12 ]reportsthattheintegralitygap(Gap%)fallsbetweenaminimumof0%andamaximumof19.01%with 111

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4-4 ,theaverageandthemedianofgapofBarretoetal.[ 12 ]is3.90%and2.24%respectively.Fortheproposedalgorithm,theintegralitygap(Gap%)fallsbetweenaminimumof0%andamaximumof4.31%withanaverageof1.24%andamedianof0.59%. Table4-4. ComparisononbenchmarkproblemsthatarecompiledbyBarreto[ 11 ] 12 ]ProposedAlgorithm ProblemNameVcapIPLBGap(%)IPLBGap(%) 112

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57 ]reportsthatonlyveLRParticles(lessthan16%)accountforhardtimewindowsandnoneconsiderssofttimewindows.IfaLRPwithtimewindowsissolved,theincompatiblepairandpathcutsthatisintroducedbyBardetal.[ 9 ]forthevehicleroutingproblemwithtimewindowcanbeincorporatedtothemasterproblemofthecolumngenerationalgorithm. 114

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32 ].In2001,thefreighttrafcinton-milesbyrailwas47%incomparisontotrucks'33%andships'20%[ 73 ].Railtransportalsodivertssomefreighttrafcfromtheroads,partiallyalleviatingcongestionandwearandtearonhighways[ 55 ].Itsgrowthhasguidedtheinfrastructuredevelopmentandgeneratedtheneedforstrategicandoptimizedmaintenanceoperationsthathonorsafetystandards[ 19 ].TheCurfewPlanningProblem(CPP)isoneofsuchoperationsencounteredbyrailroadswhenmaintainingtheirrailwaytracks.Itconsistsofoptimallyassigningcrewstotrackssothatallmaintenancetasksarecompletedwithaminimumdisruptionoftheregulartrainschedules.Therearetwoversionsofthisproblem:continuousandintermittent.InthecontinuousCPP,oncethecrewsstartworkingontracks,thetracksbecomeunavailableforoperationsuntilthejobisnished.Intheintermittentversionoftheproblem,crewsstopandresumetheoperationsasperthetrainschedules.Thecontinuousversionoftheproblemishardersinceanentireplanningperiodneedstobedividedintocontinuoustime-blocks.ItcaneasilybeprovedNP-Completebyshowingresourceallocationproblemsasitsspecialcase.Theintermittentcurfewplanningproblemisrelativelysimpleastherestrictionofndingcontinuoustime-blocksisrelaxed.Inthis 115

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36 ]explainthedifferencesinthesetwotechniqueswhilesuggestingthecost-effectivenessofrenewalmaintenance.InAustralianfreightoperations,themaintenancecostscomprisebetween25%and35%oftotaltrainoperatingcosts(1998).AstudyonthesubwayofHongKongsuggeststhecriticalityoftheproblembypresentingthereal-timedataofdailyridershipandtimeslotsavailableformaintenance[ 18 ].TheU.Srailroadsinvestaround$15billionforannualmaintenance[ 2 ]andevenmarginalimprovementwillsavemillionseveryyear.Allofthesestudiesshowtheglobalandeconomicimportanceoftheproblem.Additionally,thereisabroadscopeofimprovementduetomoredevelopedinfrastructure,betterplanning,andbetterstrategicdecisions[ 28 ].Therailwaynetworkofrailroadsusuallyspansanentirecountry,andthetracksaredividedintoseveralregionsandterminals.Atthebeginningoftheyear,theprojectmanagerevaluatesthenetworkandpreparesthelistoftrackstoberepairedorreplaced.Thesetracksarereferredtoasjobs,andtheyfallintosevencategories:(a)CapacityJobs,(b)CurvePatching,(c)ConcreteRepair,(d)Gauging,(e)Out-of-Face-New,(f)Out-of-Face-Repair,and(g)TieandSurfacing.Thesejobsaregroupedintotwomaincategories:Railjobs(atof),andTiejobs(g).Thejobsinaparticularregionarequiteclosetoeachotherandcollectivelyarecalledaproject.Allrailjobsofaregionaregroupedinarailprojectandtiejobsaregroupedinatieproject.Sometimesregionsalsoarereferredtoassubdivisions.Insideasubdivisionthereisusually(butnotnecessarily)arailprojectandatieproject.Someprojectscanbecompletedintherailroadyardsanddon'tdisrupttheregulartrainschedule,whilesomehavetobeperformedonthemainline.Themainlinecanbesingle-trackedordouble-tracked.The 116

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5.2 117

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5.3 .Inthispaperweconsidertheglobalproblem,includingallreal-lifeattributes,anddiscusssomeholisticapproaches.Theseapproachesaretherstattemptsatcapturingandsolvingallissuesinonemodel,sothattheycanbeimplementedinreal-lifescenarios.Weproposethreeexactalgorithmswhichvaryintheircapabilitiesandeffectivenessforhandlingdifferentaspectsoftheproblem.Wealsosuggestadecomposition-basedheuristicwhichgeneratespracticalresultswithinareasonabletime-frame.Ourmodelsconsiderallmajorfactorsaffectingtheefciencyofthecurfewplanandareexibleenoughtobeextendedforcapturingnewrequirements.InSection 5.4 ,weformulatetheCPPasaMixedIntegerProgramming(MIP)Problemonanappropriatelydenedtime-spacenetwork.Thismodelisusefulforunderstandingtheintricaciesandcomplexitiesoftheproblem.Thecontributionsofthispaperinclude: 1. Wepresentaunique,duty-generationmodelfortheCPP.Itsframeworkcapturesallissuesdiscussedearlier,anditisveryexibleandcompact(Section 5.5 ). 2. Weproposeacolumn-generationapproachtoimprovethesolutionsobtainedfromtheduty-generationmodel.Althoughitispartiallysuccessfulinthecurrentscopeoftheproblem,itgeneratesgoodsolutionsinthescenariowhereprojectcrashingisnotallowed(Section 5.6 ). 3. Wesuggestadecomposition-basedheuristicthatgeneratesverypracticalandexcellentqualitysolutionswhichcanbeimplementeddirectlyintoreal-lifeapplications.Thecomputationalresultsonreal-worldinstancestakenfromalargerailroadshowasignicantimprovementinthenumberofdisruptionsorviolations(Section 5.7 ). 118

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1. 2. 3. 4. 5. 6. 119

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39 ].Itproposesamodelaimedatdeterminingthebestallocationofmaintenanceactivitiesandcrewsforminimizingthedisruptioninthescheduledtrainsandreducingthecompletiontime.Heconsidersasubsetofthepreviouslydescribedconstraintsandgivesanon-linearformulationoftheproblem.Thenon-linearityandlargesizeoftheproblemforcetheuseofheuristicsandhenceatabu-searchheuristichasbeenproposedthatgeneratestheneighborsbyswappingtheorderingofjobs,crews,orboth.Higginsetal.[ 39 ]usethismodelforthecomputationaltestingonan89-km.trackcorridorandimprovethemanuallycreatedschedulesbyalmost7%.AdynamicschedulegenerationtechniquefortherollinghorizonisdiscussedbyBruceetal.[ 18 ].Theirstudyisbasedonreal-timedatafromtheHongKongsubwayrailwaysystem.Lakeetal.[ 47 ]havedonestudiesonAustralianfreightoperations.Theysaythatrailwaysoperateundertheconictingobjectiveofminimizingtheinfrastructurecostswhilemaintainingorimprovingtheservicequalities.Theseconictshaveintensiedduetoindustrialgrowth,hightrafcvolume,andotherrelatedfactors.Theydevelopthemodelfortheshort-termmaintenanceschedulingoftrack-maintenanceactivitiesafterthetrainschedulesandmaintenanceactivitieshavebeenplanned.Lakeetal.[ 47 ]solveitusingatwo-phaseheuristicstechnique,withtherststagegeneratingafeasiblesolution,andthesecondstageimplementingsimulatedannealing.Theyapplythesealgorithmsontwenty-tracksegmentsandonaseven-dayschedulingperiod.Foroccasionallyusedtracks,e.g.inAustraliaandsomeEuropeancountries,Budaietal.[ 19 ]showthatthetrack-possessionismodeledin-betweenoperations.Theyintroduceaslightlydifferentversionoftheproblemwherethemainpurposeiscompletingtheprojectwithinthetrack'sfreetime.Itcreatesadynamicschedulefor 120

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36 ]performastudytomeasurethecost-effectivenessofrailwayinfrastructurerenewalmaintenance.Theirresultsshowthatifrailroadsconstraintherenewalmaintenancetoreducetheoverallcapitalexpenditures,increasingmaintenanceexpenseswillmorethanoffsettemporaryreductionsincapitalspending.AdetailedsurveyofmaintenanceoperationsisdonebyOke[ 60 ]whichdescribesthecurrentstateoftheartinthiseld.Tosummarize,thealgorithmsdevelopedthusfartosolvetheCPPonlyconsidersapartoftheproblembuthavebeenveryhelpfulinpreparingthefoundationofourwork.Weconsidertheoptimizationproblemofassigningthecrewstoprojects,suchthatprimarilyallrestrictionsaresatisedandsecondarilythedistancetravelledbycrewsbetweenprojectsduringtheweekendsisminimized.Inthispaper,wesolvetheproblemusingatime-spacenetworkformulation(TSNF),theduty-generationmodel(DGM),thecolumn-generationmodel(CGM),anddecomposition-basedheuristics.Wealsosuggestseveralextensionsforeachmodelthatcanfurtherstrengthentheminspecialcases.Theproblemofassigningthecrewstotheprojectsisitselfachallengingproblem,becauseofitscombinatorialnature.IteasilycanbeshowntobeNP-Complete,astheresourceallocationproblemisitsspecialcaseandhasbeenproventobeahardproblem[ 33 ].Inthispaper,wesolvethecontinuousversionoftheproblemwhereprojectshavetobecompletedoncestarted,excludingthespecialcases(jamboreeprojects)discussedearlier.Wenowproposethetime-spacenetworkformulationoftheproblem. 121

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5-1 showsasimpliednetworkwherethetailnodesofthecoloredarcsarestartingnodesandtheheadnodesarenishingnodes.Thebluearccorrespondstothesmallcrewsandtheredarctothelargecrews.Everyarcisofunitcapacity.Therearesomeextranodes,calledpoolnodesandindicatedbyAt,whichhavebeenintroducedtodecreasethesizeofthenetwork.Wewilldiscussthemlaterinthissection.Thetimefactorofthenetworkisalongthey-axisandthespacefactorisalongthex-axisinFigure 5-1 .Toreducethenumberofconnectingarcs,weintroducetheconceptoftherelocationpools.Arelocationpoolisaplacewherecrewsgoafternishingaprojectandfrom 122

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Space-Timenetworkforthecurfewplanningproblem. theregetrelocatedtoeitherthestartingnodeofaprojectordirectlytothesinknode.Wecreateseveralrelocationpoolsineachweekbasedonthegeographicalstructureofthenetworkandthejob'svolume.Allstartingnodesandnishingnodesareconnectedthrougharcs,calledrelocationarcs,totherelocationpoolsofthesameweekwhicharewithinapredenedfactorofmaximumdistancelimitbasedonpreviousexperiments.Althoughtherelocationpoolsreducetheexactnessoftheproblem,theymaketheproblemtractable.Inourformulation,acrewmayrelocatefromprojectptoproject 123

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5.2 canberelaxed,theproblemcanbeexpressedasalinearprogram.Thesolutionofthislinearformulationwouldalwaysresultinanintegeroptimalsolutionduetothetotalunimodularityoftheconstraintmatrix.Unfortunately,thesideconstraintsmakethisproblemverydifcult.However,wecanreducethenumberofintegervariablesbydeningeverythinglinearexceptforthevariablescorrespondingtotheowontheprojectarcs.Theowconservationconstraintsmixedwiththeobjectivefunctionforceallothervariablestobeintegers.Toincreasethefeasibilityoftheproblemandndtheinitialsolutioneasily,werelaxthecontinuityrestrictionsandintroducesojournarcsbetweencontiguousrelocation-poolnodes(seeFigure 5-1 ).Thisallowsthecrewstoremaininthepoolandbeassignedlaterintheprojectspan.Butwediscouragetheseowsbyassigningveryhighcostsonthesearcs.Tofurtherimprovethefeasibility,wecanallowcrewstotravelfromarelocationpooltoanotheratveryhighcosts,sinceitwillintroducemanyinfeasibletravelsduringweekends.Pleasenotethatrelocation-poolnodesforeachprojectineachweekappeartwiceinthenetwork(Figure 5-1 )tomakethediagramcleaner,thoughtheyarethesameintheoriginalnetwork.Nextweshowthemathematicalmodelforthetime-spacenetwork. 124

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5 maintainthebalancedowofeachcrew-typethroughoutthenetwork.Constraints 5 keepacheckonthenumberofcrewsemployedwhileconstraints 5 statethateachprojectmustbecompletedbyexactlyonecrew.Constraints 5 controlthevalueofsubdivisionvariables.Constraints 5 enforcethemaximumabsolutecurfewsallowedperweek,andConstraints 5 tracktheprecedencerelations.Constraints 5 maintaintheservice-corridorabsolutecurfewrestrictions.Constraints 5 donotallowtwomutuallyexclusivesubdivisionstobecomeactivesimultaneously.Constraints 5 aretheintegralityconstraints,althoughmanyofthesecanberelaxedtolinearvariablesasdiscussedearlierinthesection. 127

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5.8 ).Weusethesamedatasetsfortestingeachformulation.TheorderofnumberofvariablesareO(|P|*W*|k|/2*r),whererisaveragenumberofprojectswhicharewithinthemaximumdistancelimitofeachother.TheorderofnumberofconstraintsareO(W*(1+|P|+k+2|S|+|U|)+|P|).Forthereal-lifeinstances,theaveragenumberofvariableswasaround600,000,andtheaveragenumberofconstraintswasaround65,000.WesolvethisformulationusingCPLEX11.2.Thebasicmodelwithoutusingtherelocationpooldidn'tgiveanysolutionwithin24hours.Wedidn'tallowtheprocesstorunformorethan24hours,becauseitlosesthepracticalityoftheapproach.Atimelimitofeighthoursisgoodfromthepracticalpointofview.Theintroductionofrelocationpoolssignicantlyimprovesthesolutiontimetotwohours,butthenumberoftravelsviolatingthemaximumdistancelimitincreasedtoanunacceptablelimit.Therailprojectsandtieprojectscanbedividedandsolvedindependentlyandnallycombinedtocreateaglobalsolution.Theprocessofcombiningneedssomere-optimization,assomeconstraintsgetviolatedwhentheindependentsolutionsaremerged.Wesolvethesedecomposedproblemsandareabletondthefeasiblesolutionsforthesameprobleminstancewithinveminutesforeachpart.Theprocessofcombiningtakesanother15minutes,butitresultsinseveralotherviolationswithpenalty.Thoughthismodelisnotverysuccessful,itisgoodforbetterunderstandingtheproblemandidentifyingthecriticalissues.Themaindrawbackofthemodelisitsincapabilityinhandlingprojectcrashingwithoutwhichbothreal-lifeinstancescauselargenumbersofviolations.Projectcrashinghelpsreducethenumberofabsolutecurfewsinaweek,andthusimprovesthefeasibility.Toincludeit,wewouldhavetointroduceanothersetofarcsforeachproject,andthatwouldmakethemodelevenmoreintractable.Theseresultsguideustowarddevelopingamorecompactformulationthatcanhandletheprojectcrashingandotherconstraintsmoreefciently.Wediscussthismodel,theduty-generationmodel,inthefollowingsection. 128

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CrewSet1CrewSet2CrewSet3CrewSet4 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 . . . . . 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 Figure5-2. VariablematrixD[p]forprojectp. WecreateamatrixD[p]foreachprojectpwhichisacollectionofallcandidatedutiesoftheprojectp.Thedutiesareidentiedbytheiruniqueid.Inthisformulation,wehavetheexibilityofcrashingtheprojectsbycombiningtwosmallcrewsortwolargecrews.Thisfeaturecannotbeimplementedintherstmodel,aseachnewcrew-setdoublesthesizeoftheproblem.Acrew-setisthecombinationofcrewsemployedtocompleteaproject.Wecreatefourcrew-setsforeachcrew-type(railandtie):(i)1-smallcrew,(ii)1-largecrew,(iii)2-smallcrew,and(iv)2-largecrew;wherethelasttwoare 129

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5-2 ,wheretheprojecttakesfour,three,two,oroneweektocompleteifdonebythefourcrew-setsdiscussedabove,respectively.Nextwepresentthemathematicalformulation:Index:d:ColumnofmatrixD.awd:ValueatwthrowanddthcolumnofmatrixD.c:Elementofacrew-setCS.Parameters:D:Setofdutiesforallprojects.D[p]:Setofdutiesfoeprojectp.D[p][c]:Setofdutiesfortheprojectpdonebycrew-setc.kd:Numberofcrew-typekindutyd.d:1ifdutydrequiresanabsolutecurfew,otherwise0.Ids:1ifdutydisinsubdivisions,otherwise0.du=1ifdutydisinservicecorridoru,otherwise0.fd:Startingweekofdutyd,i.e.theweekinwhichrst1comesincolumnd.ld:Finishingweekofdutyd,i.e.theweekinwhichlast1comesincolumnd.ct[d]:crewtypeinvolvedindutyd.Variables:xd=1ifdthcolumnofmatrixDistheduty-scheduleforitsproject;0otherwise.ysw:1ifsubdivisionsisunderabsolutecurfewinweekw;otherwise0.vcurfw:Violationinat-most-curfewconstraintsinweekw.vsubms1s2w:Violationinmutuallyexclusiveconstraintsinweekw.vservuw:Violationinservicecorridorconstraintsinweekwforcorridoru.vprecpq:Violationinprecedenceconstraintsforpairpq. 130

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5 ensurethecompletionofeachprojectbyexactlyonecrewset.Constraints 5 and 5 maintainthecontinuityrestrictioninallweeks,exceptforthelastoneaswerelaxtheconstraintinthelastweek.Aswediscussedintime-spacenetworkmodel, 132

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5 controlthesubdivisionvariablesassignment.Constraints 5 keepcheckonthenumberoftheabsolutecurfewsineachweek.Constraints 5 and 5 ,repsectively,donotallowmutuallyexclusiveprojectsandsubdivisionstobeactivesimultaneously.Constraints 5 areforprecedencerelationsamongtheprojects,andConstraints 5 maintaintheservicecorridorrestrictions.Wetrytopartiallyenforcethedistanceconstraintsthroughset 5 and 5 .TheDGMhasbeendevelopedatthecrew-typelevel,sothereisnodirectwaytoforcethemaximumdistancelimit.Thebasicconceptisthatforeachdutyendinginweekw,thereshouldbeatleastonedutycontainingatleastoneofthesamecrew-typeswithinthealloweddistancelimitstartinginweekw+1;andforeachpairofdutiesendinginweekw,thereshouldbeatleasttwodutiesstartinginweekw+1,andsoon.Settingalloftheseasconstraintswouldhavemadetheproblemintractable.Sowetryadifferentmethodtopartiallyforcetheseconstraints.InConstraints 5 ,westatethatforeachdutyendinginweekw,thereshouldbeatleastonedutycontainingatleastoneofthesamecrew-typeswithinthedistancelimitstartinginweekw+1.InConstraints 5 ,wesaythatforeachdutyendinginweekw,thereshouldbeatleastonedutycontainingatleastoneofthesamecrew-typesstartinginweekw1.Thesesetsofconstraintstogetherhandlethedistanceconstraintsveryeffectively,asshowninthecomputationalsection(Section 5.8 ).Thesetwosetsofconstraintsarereferredtoaspartialdistanceconstraintsthroughoutthepaper.Werelaxtheserestrictionsforlastveweeksfortworeasons:(i)toimprovethefeasibility,and(ii)becauseitmightbethelastprojectdonebythecrew.Set 5 aretheintegralityconstraints.Theoutputofthisphaseisthestartingweekandcrew-typeemployedforeachproject,andtheyareusedinthenextphasetogeneratethecrewschedules. 133

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1. 2. 3. 5 ].Asallsideconstraintsaresatisedintheprojectschedulingphase,optimizingtheowoveranetworkwillgeneratethebestcrew-schedules. 134

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5.8 .ThemainadvantageoftheDGMisitsexibilityineasilyincludingnewbusinessrequirements.Thesolutionsgeneratedareverygood,exceptforthedistanceviolations.Thisbehaviorencouragedustodevelopthecolumn-generationmodel,whichcanimprovethesesolutions. 10 ].ThedetailsofthisapproachcanbestudiedinthepaperbyLbbeckeandDesrosiers[ 53 ].ItstartswithaninitialfeasiblesolutionobtainedbytheDGMorlocalheuristicsandtriestoimproveit.ThesolutionsobtainedbytheDGMhaveseveraldistanceconstraintsviolations,whiletheinitialsolutionfortheCGMmustbefeasible.Sinceallconstraintsaresoftandweaimtominimizethepenalty,thesolutionsoftheDGMcanbeusedasthestartingsolution.Sowerelaxtheseconstraintsaswedidearlierbyusingpenaltystructures.Thecolumn-generationapproachhasthreebuildingblocks:(i)thegenerationofinitialschedules,(ii)theformulationofthemasterproblem,(iii)andthedevelopmentofsub-problemstructures.Wecanndtheinitialschedulesusingseveralsimplelocalheuristics.Singleprojectschedulesarethesimplesttostartwith,buttheseenduphavingaveryhighcost.Wegenerateroutesbyassigninganunscheduledprojecttothenextavailablecrew.SomepotentialroutesalsoaretakenfromtheoutputoftheDGM.SinceweallowprojectcrashingintheDGMbutnotintheCGMduetothecomplexitiesinvolvedasdiscussedlater,wesplittheprojectswhicharecrashed.Allthecharacteristicsofthesplitprojectsremainthesameexceptforthedurations.Thisimprovesthefeasibilityandallowsustousethesolutionswithgoodpotentialdirectly. 135

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5 ensurethateachprojectiscoveredbyexactlyonerouteandConstraints 5 keeptrackofthenumberofcrewsemployed.Constraints 5 and 5 forcethenumberofabsolutecurfewswithinthegivenlimit.Constraints 5 and 5 ensurethemutuallyexclusivityconstraintsatthesubdivisionandprojectlevels,respectively.Constraints 5 keepacheckonthenumberofservicecorridorsunderabsolutecurfew,andConstraints 5 maintaintheprecedencerelations.Constraints 5 enforcethevariables'integrality.Thevaluesinparenthesesarethedualvaluesusedinthepricingproblems.WenowdiscussthePricingProblem,whichissolvedrepetitivelytoinsertthepotentialcandidateroutesintotheRMP. 5.6.1 ).WeneedtorevisethearccostsofthepricingproblemseachtimewesolvetheRMPandseeknewcandidatestoenter.Wealsoneedtodynamicallycalculatethearccosts,becausethestart-timesofunscheduledprojectsareunknown. 138

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10 ].Forapartialpaththatendswithnodep,themodiedarccostccpqforthecrew-typekisgivenas:cpq2pPEWpw=SWpp4wsPEWpw=SWpAdjpp7(p,q)wPrecpSWp9pqifp2Pandq2P[N+1cpq3kifp20andq2PBasedonourobservationsinthecomputationalexperiments,oneofthemostcriticalissueswithcreatingatime-efcientalgorithmissolvingthesub-problemefciently.Thesub-problem,whichisanESPPRC,isstronglyNP-Hard[ 18 ].InsteadofsolvingitusinganMIP,weusedynamicprogrammingandobtainasetofcandidateroutestobesent,notonlytheoptimalone.Thisreducesthenumberofiterationsandthereforeimprovestherunningtimeforourcase.ThebasicframeworkofthedynamicalgorithmisshownbelowinFigure 5-3 .InFigure 5-4 ,wesummarizethealgorithmicowofthecolumn-generationmodel(CGM)implementedfortheCPP. 5.8 .ThecombinedresultsoftheCGMandtheDGMweregood,butthenumberofdistanceconstraintsviolationsin 139

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:Updatedarccosts,timewindowofeachnode,andprecedencematrix. Output :Asetofelementarypathsfromnode0tonodeN+1thathavenegativereducedcostsanddonotviolateanycapacityconstraints. Step1 :InitializationGettheupdatedarccostsassociatedwiththelastRMPsolved.CreateLabel({0,0}).Initializeasetofsetsthatstorestheelementsofeachdistinctpathfound.Initializethepredecessorset. Step2 :WHILEu6=N+1DO{u,tu}=BestLabel(unprocessedlabels)IFu
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:Distancebetweenprojects,numberofcrewsofeachtype,precedencematrix,andtimewindowsfortheprojects. Output :Scheduleforeachcrew. Step1 :Initialization.Gettheinputs.Createdeparturedepot(D)andarrivaldepot(A)sets.Initializeroutes:Constructmulti-noderoutesusingheuristics.PrepareinitialsetofroutesfortheMP. Step2 :ConstructtheMP. Step3 :SolvetheRMPwithcurrentsetoffeasibleroutes.Updatethelowerbound.GetthedualvaluesforeachconstraintoftheRMP.Modifythearccostsusingthedualvalues.FOReachcrewtypeDOSolvesubproblemswithmodiedcosts.IFReducedCostr<0foranynewpathr,gotoStep4.ELSEgotoStep5. Step4 :AddthenewpromisingcolumnstotheRMPandgotoStep3. Step5 :Optimalsolutionfortheoriginalrelaxedmasterproblemhasbeenfound.Figure5-4. Algorithmframeworkofthecolumn-generationprocedure. 141

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5-1 showsthesizeofinputsindetail.Generally,theprojectsareequallydividedbetweenrailandtietypes.Eachservicecorridorconsistsofanywherebetweenfourandtensubdivisions.Around500subdivisionpairsaremutuallyexclusive,andeachsubdivisionliesinthreeservicecorridorsonaverage.Aprojectiswithinthemaximumdistancelimitsofaround15-25otherprojects.Railroadsallowatthemost15absolutecurfewsineachweek.Weassumethateachviolationcausesthesamepenalty,althoughitmaydifferamongrailroadsbasedupontheirpriorities. Table5-1. Detailsofreal-lifeinstances 5-2 andTable 5-3 fortheyears2007and2008, 143

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15 ].Theydevelopedseveralheuristicstosolvetheproblemandcomparedthemwithrailroad-implementedschedules.Wehaveimprovedthoseresultsandsetnewbenchmarks.Asstatedearlier,theTSNFresultsasstatedwerenotverygood,eventhoughtheyaremarginallybetterthanXYZ'sschedules.WewereunabletogetfeasiblesolutionsbysolvingtheglobalTSNF,sowedecomposeditintorailandtieprojects.Wesolvedeachpartitionseparatelybutsequentially,startingwiththerailprojects.Oncewehadtheschedulesoftherailprojects,allparameterssuchastheremainingabsolutecurfewallowedperweekswereupdatedforthetieprojects.Foreachpartition,wesetatime-terminationcriteriaoffourhours.Solvingtheprobleminpartswasnoteffectivewithrespecttothesolutionquality.Butwecouldimproveitbysolvingthepartitionedproblemsrepetitivelybyupdatingtheparametersfromeachpreviousiteration;however,thiswasnotapracticalapproachgiventhetime-limitofeighthours.TheDGMisveryeffectiveiftherestrictiononmaximuminter-projectdistancetravelledisrelaxed.Themainattributewhichmakesthemodeleffectiveistheinclusionofprojectcrashinginthemodel.Projectcrashingimprovesthefeasibilitysignicantlybyallowingaprojecttobedonebytwodifferentcrewsandthusdecreasingthechancesofexceedingthemaximumcurfewlimit,violatingservicecorridorrestrictions,andsoon.ThemainlimitationoftheDGMisitsinabilitytoeffectivelyhandledistanceconstraints.Evenaddingpartialdistanceconstraintsinthemodelmakesthesizeofthemodeltoolargetosolvewithinthetimelimitusingthegivenhardwarecongurations.WethereforesolvethemwithoutdistanceconstraintsandpasstheseschedulesasinitialsolutionstotheCGM.WeseethattheCGMimprovesthesolutionmarginallyfortheyear2007butdoesn'timproveitfortheyear2008.TheadvantageofCGMisthatitdistributesthedistanceviolationsoftheDGMoverallotherviolationsandgeneratesasolutionwithbetteruniformity.Itmaybeamoreacceptablesolutiontotherailroads 144

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Table5-2. Improvementinnumberofviolationsfor2007 15 ]TSNFDGMCGMDecomposedDGM Improvementinnumberofviolationsfor2007 15 ]TSNFDGMCGMDecomposedDGM 145

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15 ].Theirformulationcapturesallissuesoftheproblem,butitisintractableevenformodest-sizedinstances.WesolvedtheexactmodelusingCPLEX11.2ona2.5GHzPCwith2GBofmemory.Wesettheterminationcriteriabasedonamaximumavailabletimeofeighthours,evenforsmallerinstances.Itgeneratedoptimalsolutionsforsixoutoftenve-weekinstances,whiletherewereoptimalitygapsof3%to8%fortherest[ 15 ].TheresultsobtainedbyBogetal.[ 15 ],TSNF,DGM,CGM,andDGM*(explainednext)forve-weekinstancesareshowninTable 5-4 .TheTSNFgeneratesoptimalsolutionsforve-weekinstancesinaroundsixminutes,butprojectcrashingisnotincluded.ThequalityoftheTSNFisnotacceptableevenforthesesmallinstancesasthegapvariesfrom2%to15%,whichclearlydemonstratestheimportanceofprojectcrashing.WetestedtheDGMbyputtingallpartialdistanceconstraints(equivalenttothedecomposedmodelforsmallerinstances)intheseinstances.Itgeneratedoptimalsolutionsforeightoutofteninstances.Intheremainingtwoinstances,thenumberofviolationswasonlyonemorethantheoptimalsolutions.WeusedtheseschedulesastheinitialsolutionsoftheCGM,whichimprovedthesolutionofoneofthesetwo.Wealsotestedtheeffectofusingthepair-wiseduties'distance 146

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5.5 .TheresultsareshowninTable 5-4 incolumnDGM*.Itconvergedtotheoptimalsolutionfornineoutofteninstances.Theseresultsshowthatthepartial-distanceconstraintsareveryeffectiveinthesereal-lifeinstancesthathaveprojectsscatteredoveralargegeographicalregion.Wealsotestedouralgorithmsonten-weekinstances.Theexactalgorithmscouldnotgenerateevenonefeasiblesolutionforanyoftheseinstances,againemphasizingtheveryhighcomplexityoftheproblem.Theresultsofothermodelsshowasimilarbehavior,withtheDGMgeneratingthebestresultsconsistently. Table5-4. Comparisonofalgorithms:optimalitygap(smallinstances) 15 ]TSNFDGMCGMDGM* 147

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AshishNemaniwasbornin1984,inBihar,India.Hereceivedhisbachelor'sdegreeinindustrialengineeringattheIndianInstituteofTechnology,Kharagpur,Indiain2005.Hereceivedhismaster'sdegreeinindustrialengineeringattheUniversityofFlorida,Gainesville,Florida,in2007.HehasbeenadoctoralstudentintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridasinceAugust2005. 158