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Network Models for Performance Analysis and Optimization

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Title:
Network Models for Performance Analysis and Optimization
Creator:
Spanton, Shantih M
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (168 p.)

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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
GEUNES,JOSEPH PATRICK
Committee Co-Chair:
SMITH,JONATHAN COLE
Committee Members:
RICHARD,JEAN-PHILIPPE P
BANERJEE,ARUNAVA
Graduation Date:
8/9/2014

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Subjects / Keywords:
Customer service ( jstor )
Customers ( jstor )
Global positioning systems ( jstor )
Optics ( jstor )
Railroad trains ( jstor )
Service time ( jstor )
Taboos ( jstor )
Time windows ( jstor )
Travel time ( jstor )
Vehicles ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
datamining -- gps -- heuristics -- mip -- networks -- routing
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:
In this document, we consider three problems that model scenarios on a graph or transportation network, each of which includes various levels of uncertainty. We consider a distinct solution approach for each of the three problems, representing a range of operations research techniques. The first problem concerns an edge routing problem in which a vehicle selects a path of length K, and gains a reward for the traversal of each edge. Each re-traversal of an edge earns a reduced reward that is a function of the number of times the edge has been previously traversed. We present a binary linear program and two mixed integer programming formulations for the problem. We solve the mixed integer programs via branch-and-cut techniques using derived feasibility cuts. The next problem considers the daily motions of a vehicle tracked by a GPS device. The vehicle is presumed to visit a prespecified sequence of locations of interests (customers) on a transportation network. We determine daily arrival times at, and travel times between, these locations using GPS data. The true geographic location of the places of interest is uncertain, as is the visit itself as the vehicle may skip a location. We use data mining clustering techniques to extract visit events, followed by optimization procedures that assign events to the set of possible locations. The final problem we consider is an uncapacitated stochastic vehicle routing problem with time windows (VRTWSC). Each customer is randomly present for any day of service, and travel and service times are also stochastic. We seek an a priori set of routes that minimizes penalties with respect to duration, and violation of customer service windows. Due to the number of stochastic elements in the problem, we employ a heuristic based on iterative tabu search. In addition to the theoretical interest of these problems, they represent important applications relevant in several industries. The second and third projects presented are a result of an ongoing industry collaboration with CSX Transportation, LLC. They solve relevant problems in the transportation industry. Much of the data utilized in these sections was received from CSX. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
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Adviser: GEUNES,JOSEPH PATRICK.
Local:
Co-adviser: SMITH,JONATHAN COLE.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-02-28
Statement of Responsibility:
by Shantih M Spanton.

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Copyright Spanton, Shantih M. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
2/28/2015
Resource Identifier:
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NETWORKMODELSFORPERFORMANCEANALYSISANDOPTIMIZATIONBySHANTIHM.SPANTONADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014ShantihM.Spanton 2

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Idedicatethistoalltheteachers,mentors,andadvisors,whodedicatedthemselvestoinspiringandeducatingme.Thankyouforbelievinginme. 3

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ACKNOWLEDGMENTS ManypeoplelledmylifeasIcompletedtheresearchthatledtothisdissertation.Thesimplewordsofthanksgivenonthepagesbelowcannotpossiblydescribethetrueimpacttheseindividualsmadeuponme,normakecompletementionofallthosewhoinuenced,orsupportedme.Mydeepestgratitudegoestomytwoco-advisors:Dr.Geunes,andDr.Smith.Afteranyconversationwitheitherofyou,IfeltlikeIcouldaccomplishgreatthings.Dr.Geunessupportedmewithconstantkindness,andgentlemotivation.HealwaysinstilledabitofextracondenceinmewhenIneededitmost.Dr.Smithgavemehisconstantenthusiasm,tirelesshelp,andadmirableperfectionism.Healsoshowedmethatonecanentertainaswellasteach.TheremainingmembersofmycommitteewereDr.RichardandDr.Banerjee.IwasveryfortunateinthatmycommitteewascomposedofthefourinstructorsIenjoyedmostduringmytimeingraduateschool.Dr.RichardwasthegreatestteacherofmathematicsIhaveeverhad;hemadeproofsatoncemesmerizingandclear.Withhishelp,IaccomplishedmoremathematicallythanIeverthoughtpossible.Dr.Banerjeeshowedmewhatanartitistoleadstudentstointuitivelyarriveatsomepieceofclevernessinaproof.Allfourmembersofmycommitteeareinspiringacademicsintheirresearchandteaching.ThankyoutothedepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida,andtoallthefacultyandstaffthatmakeitgreat.IalsothanktheinstitutionoftheUniversityofFloridaasawhole,whichhasbeenmyhomeforthepastyears.Alargeportionofthisdissertationwasmotivated,andfunded,bytheOperationsResearchgroupatCSXTransportation.Ithankthemfortheirsupport,especiallyDharmaAcharyaandJagadishJampani.Theirsupportalsoallowedmetofundmy 4

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wonderfulundergraduateresearchassistants:TroyBakerandParisFlood.Theywereagreathelptomeonnumerousdetailsofthework.Iwouldliketothankallofmyundergraduateprogrammingstudentsforgivingmetheopportunitytoteachthem.Ilearnedsomuchaboutmyselfintheclassroom,andIfeelcondentthattheylearnedsomethingaswell.IwouldalsoliketothankallthoseattheUniversityofWisconsin-EauClairewhonudgedmetowardsgraduateschool:Dr.S.Chadha,andDr.V.ChadhaoftheMathDepartment;Dr.Likkel,Dr.Thomas,andtherestofthePhysicsDepartment;andDr.P.QuinnoftheMcNairScholarshipprogram.Inparticular,IwouldliketothankmygreatresearchadvisorDr.M.M.R.Evansforbelievinginme,andalwaysmakingmefeelthatIhadahomeinacademia.Thankyoutoallthedevotedteacherswhohaveshepherdedmethroughlife.Thankyoutomyfamilywhoweremyrst(andgreatest)teachers.IalsothankmyfriendsatUFwhowentonthisgreatjourneywithme:Alexey(andKatya),Andrew,Artyom,Ayse,Behnam,Bita,ChinHon,Cinthia,Deon,Dmitriy,Ece,Emily,Emma,Gudbjort,Jorge,Jose,Lana,Liz,May,Melis,Mike,Petros,Sibel,Soheil,Steffen,Trang,Vijay.Itisamazinghowsomanyintelligent,kind,dynamic,beautifulpeoplecametogethertoliveinGainesvilleforsuchafewshortyears.IamespeciallygratefulforChrys,Matt,Ryan,andtheWednesdaywomenwhosupportedmeentirelyselesslythroughallthehardesttimes.Iamsohumbledbyyourconstantkindness.Andlastly,Kelly.Therearenowordstoexplainhowimpossiblethiswouldhavebeenwithoutyou.Duringalltheseyearstogetheryouweremyfriend,family,counselor,tutor,andallmystabilityrolledintoone.Ourfriendshipwasthenorthernstarofthesepastfewyears.Iamsothankfulforallthetimewespenttogether. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1PRELIMINARYINTRODUCTION .......................... 13 1.1TheUndirectedEdgeRoutingProblemwithReducedRewards ...... 14 1.2DataMiningServiceMetricsFromGPSData ................ 15 1.3UncapacitatedStochasticVehicleRoutingwithSoftTimeWindowsandStochasticCustomers ............................. 16 1.4Structure .................................... 18 2OPTIMALALGORITHMSFORTHEUNDIRECTEDEDGEROUTINGPROBLEMWITHREDUCEDREWARDS ............................ 19 2.1Overview .................................... 19 2.2Introduction ................................... 19 2.3RelationshipBetweenUERPRRandPriorStudies ............. 22 2.3.1TheUERPRRasanArcRoutingOrienteeringProblemVariant .. 22 2.3.2TheUERPRRasanOptimalSearcherPathProblem ........ 24 2.4FormulationsfortheUERPRR ........................ 28 2.4.1BinaryProgrammingFormulation ................... 28 2.4.2“Tourable”Algorithm1 ......................... 29 2.4.3“Tourable”Algorithm2 ......................... 34 2.5Results ..................................... 37 2.5.1ComparisonofMethodRuntimes ................... 37 2.5.2ProblemSizesandtheOSPPFormulation .............. 42 2.6Conclusion ................................... 44 3ESTIMATIONOFRAILROADSERVICEMETRICSFROMGPSDATA ..... 46 3.1Introduction ................................... 46 3.1.1Purpose ................................. 47 3.1.2LiteratureReviewandMethodology .................. 48 3.1.2.1Fixedgeofencefromcustomergeography ......... 50 3.1.2.2Fixedgeofencefromhistoricaltrainstoplocations .... 51 3.1.2.3Velocitydeterminationofstopsandslows ......... 53 3.1.2.4Concerningsmalldistancetrains .............. 57 3.2ResearchDesignandMethodology ...................... 61 6

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3.2.1PolylineAlgorithmforLongDistanceTrains ............. 62 3.2.2ClusteringAlgorithmforShortDistanceTrains ............ 67 3.2.3AssignmentofStopIntervalstoCustomers ............. 83 3.2.4DeterminingArrival/DepartureTimesfromtheAssignment ..... 92 3.3ComputationalResults ............................. 95 3.3.1DataCollection ............................. 96 3.3.2DataPreprocessing .......................... 98 3.3.3PolylineAlgorithmValidation ...................... 102 3.3.4ClusteringAlgorithmValidation .................... 104 3.3.5FunctionalEstimationofServiceTimesandTravelTimes ..... 108 3.4Conclusions ................................... 113 4ITERATIVETABUSEARCHTOSOLVEASTOCHASTICVEHICLEROUTINGPROBLMEWITHSOFTTIMEWINDOWSANDSTOCHASTICCUSTOMERS 114 4.1Overview .................................... 114 4.2Introduction ................................... 114 4.3ReviewofRelevantLiterature ......................... 117 4.4Formulation ................................... 120 4.5EstimationofArrivalTimes .......................... 125 4.5.1SimulationofArrivalTimeswithoutTimeWindows ......... 127 4.5.2SimulationofArrivalTimeswithTimeWindows ........... 133 4.6DeterminingVehicleRoutes .......................... 134 4.6.1OutlineofIteratedTabuSearchMethod ............... 134 4.6.2TabuSearch ............................... 138 4.6.2.1Neighborhoodstructure ................... 139 4.6.2.2Feasibilityofsolutions .................... 140 4.6.2.3Solutionevaluationanddiversication ........... 140 4.6.2.4Tabucriteria ......................... 142 4.6.2.5Aspirationcriteria ...................... 142 4.7ComputationalResults ............................. 143 4.7.1TestProblems .............................. 143 4.7.2PerformanceofthePermutationMethod ............... 147 4.8ProblemExtensions .............................. 147 4.8.1CapacityConstraints .......................... 154 4.8.2MinimizingtheTotalNumberofRoutes ................ 155 4.8.3MinimizingTotalTravelDistance .................... 156 4.9Conclusion ................................... 157 REFERENCES ....................................... 159 BIOGRAPHICALSKETCH ................................ 168 7

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LISTOFTABLES Table page 2-1Summaryofgeneratedtestgraphs. ........................ 38 2-2ComparisonT1modelruntimeswithtworewardreductionparameters. .... 38 2-3Worstcasenumberofvariablesandinitialconstraintsforallmodels. ...... 43 2-4Runtime(seconds)ofall4methodsfortwo10nodegraphswith=0.95. ... 44 2-5Runtimes(seconds)ofall4methodsfortwo20nodegraphswith=0.95. .. 44 3-1ReductioninadjustedR2valueofthemultivariatetmodelforeachcustomerwhenthelistedparameterisexcluded. ....................... 109 4-1Resultsfor10customerinstanceswith(i,r)=1, (i,r)=10,and(r)=100foralli2C,r2RforsolutionR. ....................... 148 4-2Resultsfor10customerinstanceswith(i,r)=10, (i,r)=10,and(r)=10foralli2C,r2RforsolutionR. ........................ 149 4-3Resultsfor15customerinstanceswith(i,r)=1, (i,r)=10,and(r)=100foralli2C,r2RforsolutionR. ....................... 150 4-4Resultsfor10customerinstanceswith(i,r)=10, (i,r)=10,and(r)=10foralli2C,r2RforsolutionR. ........................ 151 4-5Resultsfor20customerinstanceswith(i,r)=1, (i,r)=10,and(r)=100foralli2C,r2RforsolutionR. ....................... 152 4-6Resultsfor20customerinstanceswith(i,r)=10, (i,r)=10,and(r)=10foralli2C,r2RforsolutionR. ........................ 153 4-7ComparisonofrandomizedpermutationsandenhancedpermutationsforjCj=10withpenalties(i,r)=10, (i,r)=10,and(r)=10foralli2C,r2RforsolutionR. .................................... 154 8

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LISTOFFIGURES Figure page 2-1Earnedrewardsforapathofatmostthreeedges. ................ 20 2-2Possiblenextedgesfortraversalaftertheforward(reverse)traversalofanedge. ......................................... 26 2-3Subgraphinducedbyasolution,andthedegreeofthenodes. ......... 31 2-4Adisconnectedsolution,andthedegreeofthenodes. .............. 31 2-5Graphsshowingthefrequencyofsampleswithruntimelessthansomexedtimewhen=0.95. ................................. 39 2-6Graphsshowingthefrequencyofsampleswithruntimelessthansomexedtimewhen=0.85. ................................. 40 2-7AveragepercentageincreaseinruntimeofUIPcomparedwithT1ofallsamplegraphs. ........................................ 41 3-1Axedgeofencetoestimateservicetime. ..................... 51 3-2Passingthroughaxedgeofencewithoutservice. ................ 52 3-3Atoleranceleveldeningastop. .......................... 54 3-4Alongtrainmayindicateastopfurtherawayfromacustomerthanwouldashortertrain. ..................................... 56 3-5Customergeofencesmayoverlapcausingambiguitywhenassigningservicestops. ......................................... 56 3-6Serviceeventstopsmayappearonorimmediatelyoutsideofaxedgeofencemakingclassicationdifcult. ............................ 56 3-7ShortdistancetrainGPStraceanddistance-versus-timegraph. ........ 58 3-8LongdistancetrainGPStraceanddistance-versus-timegraph. ......... 63 3-9Thelinearapproximationofthedistance-versus-timegraph. ........... 64 3-10GPStraceanddistance-versus-timegraphforatrainequidistanttothereferencepoint. ......................................... 65 3-11ThedistancesfromallpointsinthedatatothenewreferencepointQarenotconstant. ....................................... 66 3-12MinimumdistancefromeachpingintheGPStracetothreecustomergeographicregions. ........................................ 71 9

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3-13Reachabilityplotforthethreecustomerexample. ................. 72 3-14PointswithlargeRDvaluesarenotconsideredlocalmaximums. ........ 75 3-15Separationpointsofcustomersmustbesignicant. ............... 79 3-16Pruningnodesoftheclusteringstructure. ..................... 81 3-17Exclusionofaninsignicantcluster. ........................ 82 3-18Solvingtheassignmentasashortestpathproblemforasinglecustomer. ... 87 3-19Incorrectformulationoftheassignmentasashortestpathproblem. ...... 89 3-20Thegraphforsolvingtheassignmentasashortestpathproblemforthreecustomerstofourpotentialserviceintervals. ................... 91 3-21DeviationsofthePolylinealgorithmfromexpected. ................ 104 3-22Overlappedcustomerserviceareas. ........................ 105 3-23MinimumdistanceofGPSpingdatatocustomerserviceregionsforfourcustomersversustime. ...................................... 106 3-24Reachabilityplotforthefourclosecustomerexample. .............. 107 3-25Minitabhistogramandtofservicedurationdatafor6customers. ....... 111 4-1Auserentersthetimewindowoverwhichservicetimeandtraveltimedistributionswillbecalculatedforagivensetofcustomers. .................. 130 4-2Ausermayviewdistributionsandotherstatisticalvaluesforeachcustomerserviceandtraveltime. ............................... 131 4-3Simulationcomparesthedurationoftwodifferentroutes. ............ 132 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyNETWORKMODELSFORPERFORMANCEANALYSISANDOPTIMIZATIONByShantihM.SpantonAugust2014Chair:JosephGeunesCochair:J.ColeSmithMajor:IndustrialandSystemsEngineeringInthisdocument,weconsiderthreeproblemsthatmodelscenariosonagraphortransportationnetwork,eachofwhichincludesvariouslevelsofuncertainty.Weconsideradistinctsolutionapproachforeachofthethreeproblems,representingarangeofoperationsresearchtechniques.TherstproblemconcernsanedgeroutingprobleminwhichavehicleselectsapathoflengthK,andgainsarewardforthetraversalofeachedge.Eachre-traversalofanedgeearnsareducedrewardthatisafunctionofthenumberoftimestheedgehasbeenpreviouslytraversed.Wepresentabinarylinearprogramandtwomixedintegerprogrammingformulationsfortheproblem.Wesolvethemixedintegerprogramsviabranch-and-cuttechniquesusingderivedfeasibilitycuts.ThenextproblemconsidersthedailymotionsofavehicletrackedbyaGPSdevice.Thevehicleispresumedtovisitaprespeciedsequenceoflocationsofinterests(customers)onatransportationnetwork.Wedeterminedailyarrivaltimesat,andtraveltimesbetween,theselocationsusingGPSdata.Thetruegeographiclocationoftheplacesofinterestisuncertain,asisthevisititselfasthevehiclemayskipalocation.Weusedataminingclusteringtechniquestoextractvisitevents,followedbyoptimizationproceduresthatassigneventstothesetofpossiblelocations.Thenalproblemweconsiderisanuncapacitatedstochasticvehicleroutingproblemwithtimewindows(VRTWSC).Eachcustomerisrandomlypresentforanyday 11

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ofservice,andtravelandservicetimesarealsostochastic.Weseekanapriorisetofroutesthatminimizespenaltieswithrespecttoduration,andviolationofcustomerservicewindows.Duetothenumberofstochasticelementsintheproblem,weemployaheuristicbasedoniterativetabusearch.Inadditiontothetheoreticalinterestoftheseproblems,theyrepresentimportantapplicationsrelevantinseveralindustries.ThesecondandthirdprojectspresentedarearesultofanongoingindustrycollaborationwithCSXTransportation,LLC.Theysolverelevantproblemsinthetransportationindustry.MuchofthedatautilizedinthesesectionswasreceivedfromCSX. 12

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CHAPTER1PRELIMINARYINTRODUCTIONTheendofthelastcenturywascharacterizedbyrapidadvancesinthedevelopmentandapplicationofexactandapproximateoperationsresearchtechniques.Inthebeginningofthiscentury,inthefaceofanexplosionofavailable(yetoftenunutilized)data,theworldhasincreasinglyturnedtodataanalystsandoperationsresearchexpertstoleveragethesetechniquestosolveincreasinglycomplexproblems.Forgoodorill,thecurrentstrategyappearstobeoneofdevelopinghighlyspecializedalgorithmsfordistinctproblems,andawayfromgeneral,uniedtheories.Thus,itisincreasinglyimportantthatresearchersfamiliarizethemselveswiththenumerouspotentialapproachestoaproblem,andusethemappropriately.Ofcourse,thisismostimportantwhentheresearchisapplieddirectlyinindustrialorcivilapplications.Withthissocialcontextinmind,theresearchinthisthesisconcernsthreeproblemssolvedbythreedistinctoperationsresearchtechniques.Whilethetechniquesusedtosolvetheproblemsareunique,theproblemsthemselvessharecommonalities.Firstly,allthreepresentedprojectsmodelproblemsconstrainedtoanetwork,suchasatransportationnetwork.Secondly,allthreeproblemswereoriginallymodeledtoaddressameasureofrealisticuncertainty(althoughtherstmaybemodeleddeterministically).ThesecondandthirdprojectspresentedaretheresultofanongoingindustrycollaborationwithCSXTransportation,LLC.Theyaredesignedtosolverelevantproblemsinthetransportationindustry,andmuchofthedatautilizedintheseprojectswasreceivedfromCSX.Themotivatingscenariofortherstprobleminvolvesasearcherwhoupdatesprobabilisticassumptionsaboutatargetlocation.Wemodelthisproblemasadeterministicoptimizationproblem,andsolveitwithintegerprogrammingtechniques.Thesecondproblemconcernstheidenticationofthearrivaltimeofservicevehiclesatcustomerswhosetruelocationsarenotknownwithcertainty,norarethecustomers 13

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themselvesservedwithcertainty.Weapplydataminingtechniquesinthiscase.Thelastproblemconsidersavariantofthevehicleroutingproblemwithtimewindowsandseveralelementsofstochasticity.Wepresentaheuristicmethodtohandlethesenumerousuncertainties.Operationsresearchtopicsrelatedtonetworksinthepresenceofuncertaintyareatopicfarlargerthanthisdissertationcancontain.Thegoalofthisthesisistopresentthreeproblems,anddiscussthevariedsolutiontechniquesweemploy.Wenowgiveabriefoverviewoftheconsideredproblems. 1.1TheUndirectedEdgeRoutingProblemwithReducedRewardsTherstproblemweconsiderisanedge-routingproblemwerefertoastheUndirectedEdgeRoutingProblemwithReducedRewards(UERPRR).Here,avehiclemustplanapathonanetworkinordertomaximizetherewardsgainedforthetraversalofeachedge.ThevehiclemaychoseapathofatmostKedges.Thevehiclemayrevisitedgesofthenetworkandreceivesarewardforeachre-traversalofanedgethatisafunctionoftheinitialrewardandthenumberoftimesthattheedgehasbeenpreviouslytraversed.Thisproblemhasseveralimmediatepracticalapplications.Weenvisionthisprobleminthecontextofthesearchforanentityonsometransportationnetwork.Inthissearchoperation,asearchermaybeginwithaknownapriorilikelihoodofidentifyingatargetatdistinctlocationsonatransportationnetwork.Inthismanner,theprobabilityofndingatargetatalocationcorrespondstotheinitial“reward”gainedforsearchingthenetworklocation.Witheachfailedsearchofanetworklocationthesearcherreducestheprobabilityassessmentofndingthetargetatthatlocation.Thesearchermustplanthesearchtovisit(andrevisit)thoseplaceswherethelikelihoodofndingthetargetisgreatest.Asecondapplicationisinthatofsalesormobileadvertising,inwhichasalespersonmaywishtorevisitcitylocationshavingthelargestnumberofpotentialnew 14

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sales.However,witheachvisittoalocation,thenumberofpotentialnewcustomersisreduced,asthesalespersonhasalreadyencounteredmuchofthepopulation.Initially,weconsiderasimplefunctionforthereducedrewards.Foreachedgetraversal,therewardismultipliedbysomereductionconstant.Thus,fortheithtraversalofanedge,theavailablerewardisequaltotheinitialrewardmultipliedbythereductionconstantraisedtothevalueofi)]TJ /F5 11.955 Tf 12.08 0 Td[(1.Iftherewardreductionsaredenedinthismanner,andtheinitialrewardsxed,wecansolvetheUERPRRdeterministically.Wepresentthreemixed-integerprogrammingmodelsforsolvingtheUERPRR.Therstmodelisacompactbinarylinearprogrammingformulation.Thisproblemmaybesolveddirectlywithanyoptimizationsolver.However,thelargenumberofvariablesrequiredintheformulationmakeitintractableforlargerprobleminstances.Thesecondandthirdformulationsusemanyfewervariablestoformulatetheproblem,butrequireanexponentialnumberofsubtoureliminationconstraints.Weimplementbranch-and-cutproceduresfortheseformulations,duringwhichthesubtoureliminationconstraintsareaddedonlyasrequired.Wecomparetheruntimeperformanceofthesethreemodels,andmakerecommendationsbasedontheresults. 1.2DataMiningServiceMetricsFromGPSDataThegoalofthesecondprojectweconsideristoestablishtwoimportantmetricsaboutcustomerserviceevents:customerservicedurationtimeandtransittimesbetweencustomers.Weworkedcloselywithourbusinesspartner,CSXTechnology,onthisprojectasitdirectlyconcernedtheirproprietarycompanydatasources.Weutilizedthreedatasourcesforthisproject,theprimarybeinglocationdatafromGPSdevicesfromalllocomotivesthatservicecustomers.Wealsoutilizedcrewrecordedworkordersthatcontainedthesequenceandidentitiesofcustomersservicedbyeachtrain.Thenalpieceofdatarequiredwasgeographicaldatarelatedtothelocationofeachcustomerserviced.Duetothelargequantityofdata,aswellastheuncertaintypresentincertaindatapieces,wereliedonthetechniquesofdatamining.Weconstructedtwo 15

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algorithms(oneforlongdistancetrains,andoneforshortdistancetrains)toextractthearrivalanddeparturetimefromeachcustomerlocation.InthelongdistancealgorithmwereliedondatavisualizationtechniquestoapproximatetheserviceintervalsonadistanceversustimegraphoftheGPSdata.Intheshortdistancealgorithmweusedaclusteringalgorithmtoidentifystopsinproximityofpotentialservicelocations.Oncepotentialserviceeventsareidentiedbyeitheralgorithm,anassignmentoftheseeventstopossiblecustomersismadebysolvingashortestpathalgorithmonamodiedacyclicgraph.Fromthisweobtainthearrivalanddepartureforallservicedcustomers.Afteranalyzingseveralmonthsofdata,weestimatedistributionsfortheservicetimeforindividualcustomers.And,likewise,weobtainthedistributionsoftraveltimesbetweenallpairsofcustomers.Thesedistributionswillbeusedinthemanagementandschedulingofcustomerserviceevents. 1.3UncapacitatedStochasticVehicleRoutingwithSoftTimeWindowsandStochasticCustomersThenalproblemweconsiderisanuncapacitatedvehicleroutingproblem(VRP)withtimewindows.InthemostbasicdenitionoftheVRP,asetofroutes(anorderedsequenceofcustomers)issoughtforaeetofvehiclessuchthatallcustomersarevisited,andthetotalcostisminimized.Clearly,therearenumerousextensionsofthisproblem,andnumerouswaysinwhichthecostoftheroutesmaybeassessed.Weconsiderseveralextensions,namely: 1. Customersrequireserviceonanydaywithsomeprobability 2. Traveltimesaredenedbyaknowndistribution 3. Servicetimesaredenedbyaknowndistribution 4. Customershaveatimewindowduringwhichtheypreferthevehicletoarrive 5. Customershavevaryinglevelsoftoleranceforservicearrivalsoutsideoftheirdesiredtimewindows 6. Eachroutehasamaximumdurationlimit 16

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7. EachroutehasatolerancelevelforthefrequencywithwhichtheroutemayexceeditsmaximumdurationToourknowledge,thisistherststudythatsimultaneouslyexaminesalloftheseelements.Weprovideastochasticmixedintegerprogrammingformulationforthisproblem.Inthisformulationthecustomer'stolerancelevelsandtheroutedurationtolerancelevelsaremodeledaschanceconstraints.Consideringthelargenumberofstochasticelementsinthisproblem,theobjectivefunctionwillbeintermsoftheexpectedbehavioroftheselectedroutes.Weutilizeaniterativetabusearchtosolvethisproblem.TheobjectivefunctionofthetraditionalVRPisoftenconsideredasaweightedfunctionofthecosttoutilizeeachvehicle(wherethenumberofvehiclesmayvary),aswellasthetotaltripdurationand/orthetotaldistancethateachvehiclemusttravel.Otherpenaltiesmaybeincludedintheobjectiveassecondaryobjectivegoals.Forexample,penaltiesmaybeimposedforfailuretomeetcustomerservicerequirements,excessivevehicleon-dutytimes,orpenaltiesforrecourseactionthatareincurredwhenthevehiclecannotcompleteserviceasexpected(vehicleisovercapacity).Forourinitialformulationoftheproblem,theobjectiveisaweightedfunctionofthepenaltyforfailuretoarrivewithinthecustomer'sservicewindow,aswellasthepenaltyforavehiclerouterunninglongerthananprespeciedtotaldurationlimit.WhiletheseareoftenconsideredassecondaryobjectivegoalsinotherVRPformulations,thecostofanadditionalvehicleoftendwarfssmallerlessquantitativecostsofmeetingcustomerexpectations,andcrewfatigue.Thus,weinitiallyxthetotalnumberofallowedroutesandoptimizeaccordingtoourprimaryobjectivesofon-timecustomerservice,andexcessiveserviceduration.Problemsintheliteraturemostsimilartoourownoftenutilizeheuristics,particularlytabusearchheuristics.Duetotheamountofstochasticityintheproblemwealsodevelopaniterativetabusearchalgorithm.Toincreasethequalityoftheheuristic 17

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results,weutilizeseveraltechniquestoencouragediversicationoftheresultingintermediatesolutions.Therearethreeoutcomesofthisproject.Therstistheintroductionandformulationofthisproblem.Thesecondisthecreationoftwosimulationproceduresthatmodeltheexpectedbehaviorofasingleroute1)withoutthepresenceofcustomertimewindowsand,2)whencustomershavedesiredservicetimewindows.Whilethesesimulationsareusefulintheirownrightforserviceplanning,thesimulationwithtimewindowsisusedasasubprocedureforouriterativetabusearchheuristic.Thisheuristicisournalcontribution.Itseekstocreateanassignmentofcustomerstoroutes,andanorderingofthecustomers,inordertoachievetheobjectivegoalsdiscussedabove. 1.4StructureBecausethethreeproblemsweconsideraredistinct,andsolvedbydifferenttechniques,wediscusseachseparatelyinitsownchapter.Eachchaptercontainsitsownintroduction,reviewofrelevantliterature,discussionofthemodelandsolution,computationalresults,andconclusion.TheUndirectedEdgeRoutingProblemwithReducedRewardsispresentedinChapter2.ThedataminingofGPSdatatoobtainservicedataispresentedinChapter3.ThetabusearchheuristicfortheUncapacitatedStochasticVehicleRoutingwithSoftTimeWindowsandStochasticCustomersispresentedinChapter4. 18

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CHAPTER2OPTIMALALGORITHMSFORTHEUNDIRECTEDEDGEROUTINGPROBLEMWITHREDUCEDREWARDS 2.1OverviewInthispaperwestudyaprobleminwhichasinglevehicletraversessomerouteonagraph,wherethenumberofedgestraversedislimitedbyagivenparameter.Arewardisassociatedwitheachedge,andthevehiclegainsarewardfromeachedgetraversedalongitsroute.Inaddition,thevehiclemayre-traverseedges,gainingarewardthatdecreasesasafunctionofthenumberofprevioustraversalsoftheedge.Hence,anoptimalroutemaytraverseedgeshavinglargerewardsseveraltimesinlieuofexploringthegraphmorebroadly.Theobjectiveofthisproblemistondamaximum-rewardroutesubjecttotherestrictiononthemaximumroutelength.Weprovidethree(mixed-)integerprogrammingmodelsforoptimizingthisroutingproblem,twoofwhichcontainanexponentialsetofsubtoureliminationconstraints(andarethereforesolvedbyabranch-and-cutapproach).Wethendemonstratetheefcacyofourapproachesonatestbedofrandomlygeneratedinstances. 2.2IntroductionThispaperexaminesanedge-routingproblemwerefertoastheUndirectedEdgeRoutingProblemwithReducedRewards(UERPRR).Inthisproblem,avehicletraversesarouteacrossanundirectedgraph.Eachedgethatistraversedintherouteyieldsareward.Furthermore,theroutecanrevisitedges,eachtimegainingareducedportionoftheinitialrewardforthatedge.Theobjectiveofthisproblemistoidentifyarouteofapre-speciedlength(givenbythenumberofedgestraversedintheroute)thatyieldsamaximumreward.ConsidertheexampleshowninFigure 2-1 .Figure 2-1 Agivestheinitialrewardgainedforthersttraversalofeachedge.Figures 2-1 Band 2-1 Cshowanoptimalthree-edgeroutecorrespondingtothecaseswhenedgerewardsarereducedforeachtraversalby50%and25%,respectively.InFigure 2-1 B,the50%reductionforces 19

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anoptimalroutetousethreedifferentedges,becausethereductionistooseveretowarrantthere-traversalofedge(1,2).Bycontrast,inFigure 2-1 C,anoptimalroutetraversesedge(1,2)threetimes,astherewardonedge(1,2)isreducedfrom0.50,to0.375,andto0.28125foreachsubsequenttraversal. 10.50 20.10 0.05 30.40 4AInitialrewards. 10.50 20.10 0.05 30.40 4B50%optimalroute. 10.50 20.10 0.05 30.40 4C25%optimalrouteFigure2-1. Figure(1a)showstherewardfortheinitialtraversalofeachedge.Figure(1b)showstheoptimaledgestraversedbyaroute(dashedlines)ifrewardsarereducedby50%aftereachtraversal.Thetotalrewardearnedis0.50+0.10+0.40=1.00.Figure(1c)showstheoptimaledgestraversedbyaroute(dashedlines)ifrewardsarereducedby25%aftereachtraversal.Thetotalrewardearnedis0.50+0.375+0.28125=1.15625. WeformallydenetheUERPRRonagraphG(V,E),whereVandEdenotethesetofnodesand(undirected)edgesofthegraph,respectively.Theroutemustbeoftheform(j0,j1))]TJ /F5 11.955 Tf 12.19 0 Td[((j1,j2))-241()]TJ /F5 11.955 Tf 41.2 0 Td[((jK)]TJ /F7 7.97 Tf 6.59 0 Td[(1,jK)whereji2V,foralli=0,...,K,and(ji)]TJ /F7 7.97 Tf 6.59 0 Td[(1,ji)2Eforalli=1,...,K.(Notethatneitherj0norjKisprespecied.Moreover,sincej0,...,jKneednotbeunique,werefertothisrouteasawalkratherthanapath.)Intheproblemweconsider,Kisagivenparameterthatlimitstheroutelength.Analternativemodelcouldspecifya“budget”forthetotalroutelength,whereeachedgeisassociatedwithatraversalcost.ThenovelchallengeoftheUERPRRarisesfromthevehicle'sabilitytoacquireareducedrewardwhenre-traversingedges.Webeginouranalysisbymodelingedgerewardswithasimplefunction.Tomodeltheserewards,deneasthesetofallpossiblevehicleroutescontainingKedgesonG,wherearoute!2isrepresentedbyanorderedsetofedges(!1,...,!K)(andwhere!1,...,!Kmaynotbeunique).Werststatetherewardgainedfromedgee2Ewhenedgeeisthekthedgevisitedonroute!. 20

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Forall!2ande2E,denene(!,k)asthenumberofvisitstoedgeeinsteps1,...,kofroute!fork2f1,...,Kg.Theoverallrewardearnedfromroute!isgivenbyKXk=1F!k(n!k(!,k)),whereFe(i)statestherewardobtainedduringtheithpassthroughedgee.InthispaperweexaminethefollowingspecialformforF.Denepe2R+astherewardobtainedwhenedgee2Eistraversedthersttime.Also,deneaparameter2(0,1)astheratiobywhichtherewardforvisitinganedgeisreducedaftereachtraversal(e.g.,=0.5intheFigure 2-1 Bexample,and=0.75intheFigure 2-1 Cexample).Therewardgainedwhentraversingedge!kisR(!,k)=p!kn!k(!,k))]TJ /F7 7.97 Tf 6.59 0 Td[(1.Observethatthereductionparameteristhesameforalledges;alternatively,theseparameterscanbeedge-dependent(onlyattheexpenseofmorecomplexnotation).TheUERPRRisthusgivenbyMaximizenPKk=1R(!,k)j!2o.TomotivatetheUERPRR,consideramobileadvertisingvehiclethatgainsa“reward”intermsofproductexposurerelatedtothenumberofindividualswhoviewtheadvertisement.Whiletheadvertisermaygainthelargestrewardbyadvertisingtoindividualsunfamiliarwiththeproduct(thus,followingaroutethatdoesnotre-traversestreets),theadvertiseralsowishestoensurethatareasofhighpopulationdensityarevisitedwithgreaterfrequency.Thevehiclemayrevisitbusystreets,atareducedbenetforeachtraversal.Thisproblemalsoarisesinthecontextofsearchingforsometargetthatmayormaynotexistinanetwork.Inthiscase,therewardscorrespondtotheprobabilityofndingthetargetwhilesearchingedgee.However,theremayexistaprobabilityofoverlookingthetarget,whichjustiessearchinganedgemultipletimes. 21

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Inthispaper,wepresentthreeinteger(ormixed-integer)programmingmodelsforsolvingtheUERPRR.TheuseofthesemodelsisjustiedbythefactthattheUERPRRisstronglyNP-hard,whichiseasytoshowbytransformationfromtheHamiltoniancycleproblem[ 37 ].Therstmodelwepresentisacompactbinarylinearprogrammingformulationfortheproblem.Thisformulationutilizesthelargestnumberofvariablesofourthreemodels,andassuch,iscapableofsolvingonlysmall-scaleinstances.Theformulationsofthesecondandthirdmodelsrequirefewervariables,butanexponentialnumberofsubtoureliminationconstraints,whichweaddviaabranch-and-cutprocess.Theremainderofthispaperisorganizedasfollows.Section 2.3 describestherelationshipoftheUERPRRtootherproblemsthathavebeenexploredintheliterature.InSection 2.4 ,wedenethreemodelsfortheUERPRR,alongwiththeirassociatedcuttingplaneswhereappropriate.WethentestourimplementationstrategiesforsolvingtheproposedmodelsinSection 2.5 ,andconcludethechapterinSection 2.6 . 2.3RelationshipBetweenUERPRRandPriorStudiesGiventhevaststudyofarc-routingandsearchproblems,thereareseveralproblemsrelatedtotheUERPRRthathavereceivedattentionintheoperationsresearchliterature.WecategorizetherelationshipbetweentheUERPRRandorienteeringproblemsinSection 2.3.1 ,andthatbetweenUERPRRandsearchproblemsinSection 2.3.2 . 2.3.1TheUERPRRasanArcRoutingOrienteeringProblemVariantTheUERPRRisanaturalextensionofvehicleroutingproblems,especiallyarcroutingvariantsoftheOrienteeringProblem(OP).InthetraditionalOP,arewardisassociatedwitheachvertexofagraph,andtraveltimesareassociatedwitheachdirectedarcinthegraph.TheOPseeksapaththatvisitsasubsetofverticesinordertomaximizethetotalcollectedrewardsubjecttosometotaltraveltimebudget.InarcvariantsoftheOP,rewardsareassociatedwitharcsinsteadofvertices.AsurveyoftheOPanditsmanyvariantscanbefoundin[ 102 ]. 22

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Incontrasttothearc-routingOP,theUERPRRallowsreducedrewardstobegainedbyrevisitinganedge.Toourknowledgenoarc-routingOPvariantshaveconsideredreducedrewardsearnedforre-traversalsonthegraph;infact,onlylimitedattentionhasbeengiventothearc-routingOPanditsvariants[ 102 ].Amongthesestudies,thePrize-CollectingRuralPostmanProblem[ 4 ]considersthecaseinwhichthevehiclegainsprotsalongacycleonwhichasingleedgemaybetraversedseveraltimes,althoughtheprotisonlygainedforthersttraversal.TheUndirectedCapacitatedArcRoutingProblemwithProts[ 5 ]solvesasimilarproblemforaeetofvehicles(resultinginseveralcycles)andalsoincludesdemandsatarcsthatmustbesatised.Anotherarc-routingOPvariant,calledtheTheProtableArcTourProblem[ 34 ],seekstoidentifyasetofcycles(allrestrictedbysomemaximumcyclelength)thatmaximizesthedifferencebetweenthetotalprotgainedacrossallcyclesandthetravelcoststotraversethem.Theentireprotofeacharcisrepeatedlyavailableuptosomemaximumnumberoftraversals,whichcanbeviewedasthearccapacity.Certainvariantsofthetraditionalnode-basedOPvarythevalueofrewardsobtainedduringthevehicle'sroute.IntheOrienteeringProblemwithTimeWindows,thevehiclecollectstherewardatanodeonlyifthenodeisvisitedwithinagiventimewindow;outsideofthiswindowtherewardisnolongeravailable[ 102 ].AvariantoftheTravelingSalesmanProblem(TSP)calledtheDiscounted-RewardTSP[ 15 ]considerstime-dependentrewards,suchthatifavehiclereachesanodebytimet,therewardgainedfromthenodeisreducedbytforsome2(0,1).Thegoalofthisproblemistomaximizethesumofthese(reduced)rewardsgainedfromeachnode.Thus,theexponentialstructureoftherewardreductionsissimilartothespecicstructureweexamineintheUERPRR,althoughtheDiscounted-RewardTSPreducesrewardsasafunctionofedgearrivaltime,insteadofthenumberofprioredgevisits.ApproximationalgorithmsfortheDiscounted-RewardTSParegivenin[ 15 ]. 23

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2.3.2TheUERPRRasanOptimalSearcherPathProblemTheUERPRRcanalsobetransformedintotheone-sidedOptimalSearcherPathProblem(OSPP),inwhichasearcheralsoseekstomaximizetheprobabilityofdetectingatarget.The“one-sided”characterizationofthisproblemstemsfromtheassumptionthatthetargetisunawareofthesearcher'sactions,anddoesnotassistorhinderthedetectionofthistarget.Stewart[ 92 ]presentssolutionmethodstotheOSPPbyextendingclassicalsearchtheorymethods[ 19 , 94 ]tosolvesearchproblemsinwhichbothsearcherandtargetmovementsareconstrainedtoastructuresuchasagraph.Thebreadthandnumberofvariationsoftheone-sidedsearchproblemareextensive,asistherealmofsearchtheoryliteratureingeneral.Wereferthereaderto[ 9 ]foracomprehensivereviewofsearchproblemsstudies.IntheOSPP,amobilesearchertraversesasearchspace(usuallyagraphoranitesetofinterconnectedcells)insearchofatarget,subjecttotheconnectivityconstraintsofthespace.Thesearcherhasanitequantityofsearchresourcesavailable,whichitallocatesatlocationsvisitedalongthesearchpathineitherdiscreteportionsorinanunlimited,innitelydivisiblemanner.Theprobabilityofdetectingatarget(giventhatthetargetexistsatthelocation),iscalledthedetectionfunction,andisanexponentialfunctionoftheallocatedsearcheffort.Thisimperfectsearcherdetectioncapabilityimpliesthatthesearchermaywishtorevisitlocationsatwhichtheprobabilityofndingthetargetisstillhigh.Ifthesearcherdetectsthetargetatanystep,thesearchends.Thetarget'sapproximatelocationisgivenasanapriorisetofprobabilitiesoverthesetofpossibletargetlocations.Thisprobabilitydistribution,whichdenesthemobiletarget'sposition,maybeupdatedthroughoutthesearch,eitherconditionally,orasimplicitlydenedbyMarkovtransitionmatrices.Foreachstepofthesearch,theprobabilityofdetectionisafunctionofthemobiletarget'sprobabilitydistributionatthatstep,andofthedetectionresourcesallocatedbythesearcheratthatstep. 24

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Stewart[ 92 ]presentsbranch-and-boundtechniquesfortheOSPPthatguaranteeoptimalityinthecaseofnon-discretesearcheffortallocation,andprovideboundsinthecasewheresearchresourcesmustbeallocateddiscretely.Seealsospecializedbranch-and-boundalgorithms[ 32 , 65 , 69 , 85 , 104 ]anddynamicprogrammingtechniques[ 31 , 99 ]forthisproblem.TheOSPPanditssolutionapproachescanalsobeextendedtoaccountformultiplesearchers[ 25 , 80 , 84 , 104 ].Toourknowledge,thefastestknownbranch-and-boundalgorithmtosolvetheOSPPisgivenin[ 65 ].Thisalgorithmtightenstheboundsandgeneralizestheworkin[ 69 ],andalsoconsidersnon-uniformtraveltimes.Here,wedemonstratethattheUERPRRisaspecialcaseoftheOSPPwithdiscretesearcheffortrestrictions.TheinterconnectedlocationsthatthesearcherinspectsintheOSPPcanbemodeledasagraph,wherethegraphedgescorrespondtothesearchedlocations.TheUERPRRvehiclemustchoosearouteofexactlyKarcsonthegraph,whichcorrespondstothediscreteeffortrestrictionontheOSPPsearcher,whereoneunitofresourcemustbeallocatedateachlocationofthesearch,andonlyKtotalunitsofresourceareavailabletothesearcher.TheOSPPobjectivefunctiontypicallyminimizestheprobabilityofnotdetectingthetargetduringthesearch,whichisequivalenttomaximizingthesumoftherewardsgainedalongthevehiclerouteintheOSPPwhenthoseawardscorrespondtodetectionprobabilities.TheUERPRRrewardfunctionisrepresentedviatheOSPPbyconsideringtheinitialtargetdistributiontoremainxed(asifthetargetwereimmobile).ThereductionstotherewardsarethenmanagedbyadjustingthedetectionfunctionatanylocationoftheOSPPtobedependentuponthenumberofprevioussearchesofthelocation.Let!=(!1,...,!K)beavehiclerouteinUERPRR.Notethatp!k,therewardinitiallyavailableateachedge!k2EoftheUERPRR,isanalogoustotheaprioriprobabilityofndingthetargetatthecorrespondinglocationintheOSPP.Byviewing 25

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n!k(!,k))]TJ /F7 7.97 Tf 6.59 0 Td[(1asthedetectionfunctionforeachlocation,R(!,k)intheUERPRRrepresentstheprobabilityofdetectingthetargetinlocation!katstepkofthesearch.MotivatedbythegeneralformulationofthediscreteeffortallocationOSPPgivenin[ 32 ],wepresentthefollowingminimizationmodelfortheUERPRR.First,denethevariablesy(i,j,t)astheamountofeffortthatisredistributedbythesearcherfromlocationiintimeperiodttolocationjintimeperiodt+1.IntheUERPRRcontext,y(i,j,t)equals1ifedgeiisthetthedgevisitedonthevehicle'srouteandedgejisthe(t+1)thedgevisited(assumingthatiandjshareacommonincidentnode).BecausetheUERPRRisanedgesearchproblem,theconnectivityconstraintsoftheformulationmustaccountforthefactthatifedgee=(n1,n2)iscrossedfromnoden1tonoden2,thenextvisitededgemustbeincidentton2.Foreveryedgee=(n1,n2)2E,deneS1(e)astheedgesincidentton1,andS2(e)astheedgesincidentton2,wherebothsetsexcludeedgeeitself;distinctvariablesmustrepresentthedirectionofanedgere-traversal.Weutilizezf(e,t)toindicateare-traversalintheforwarddirectionattimet,andzr(e,t)forthereverse. 1 i 2 k 3i 4j 5 6 i 7 k AEdgesadjacenttoedgej. 1 i" 2 3i 4j+3 5zr(j)ee k@@ k 6 i 7BForwardtraversalofedgej. 1^^ i 2 k| 3ooi 4zf(j)99 5 jks 6)]TJ /F20 9.963 Tf 0 0 Td[()]TJ ET q 1 0 0 1 314.79 -375.69 cm 0.478 w 1 J 1 j []0 d 6.503 -56.656 m 6.586 -56.56 l S Q BT /F9 7.97 Tf 324.21 -420.6 Td[(i 7 k CReversetraversalofedgej.Figure2-2. A.Theedgesadjacenttoedgej,wherealledgeslabelediareinsetS1(j)andalledgesk2S2(j).B.Ifedgei2Si(j)istraversedbeforej,thenextedgeselectedmustbeoneinthesetS2(j)[jr(showndashed).C.Ifedgek2S2(j)isselectedpriortoj,thefollowingedgechosenmustbeinS1(j)[jf(dashed). IntheexamplegiveninFigure 2-2 ,weshowhowtheadjacencylistschangewiththedirectionoftraversalforanedgej.Usingthesedenitions,wewritetheUERPRRas 26

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thefollowingnonlinearprogrambasedonthediscreteeffortallocationOSPP: MinimizeXe2Epeexp8<:)]TJ /F5 11.955 Tf 11.3 0 Td[(ln()K)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xt=0Xi2S1(e)[S2(e)y(e,i,t)+zf(e,t)+zr(e,t)9=;, (2a)subjecttoXi2S1(j)y(i,j,t)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+zf(j,t)]TJ /F5 11.955 Tf 11.96 0 Td[(1))]TJ /F12 11.955 Tf 16.76 11.36 Td[(Xl2S2(j)y(j,l,t))]TJ /F6 11.955 Tf 11.96 0 Td[(zr(j,t)=0,j=1,...,jEj,t=1,...,K)]TJ /F5 11.955 Tf 11.95 0 Td[(1, (2b)Xi2S2(j)y(i,j,t)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+zr(j,t)]TJ /F5 11.955 Tf 11.95 0 Td[(1))]TJ /F12 11.955 Tf 16.76 11.36 Td[(Xl2S1(j)y(j,l,t))]TJ /F6 11.955 Tf 11.96 0 Td[(zf(j,t)=0,j=1,...,jEj,t=1,...,K)]TJ /F5 11.955 Tf 11.95 0 Td[(1, (2c)Xi2E0@Xj2S2(i)y(i,j,0)+zr(i,0)+Xj2S1(i)y(i,j,0)+zf(i,0)1A=1, (2d)y(i,j,t)2f0,1g,i=1,...,jEj,j2S1(i)[S2(i)t=0,...,K)]TJ /F5 11.955 Tf 11.95 0 Td[(1, (2e)zr(i,t)2f0,1g,i=1,...,jEj,t=0,...,K)]TJ /F5 11.955 Tf 11.95 0 Td[(1, (2f)zf(i,t)2f0,1g,i=1,...,jEj,t=0,...,K)]TJ /F5 11.955 Tf 11.96 0 Td[(1. (2g)Theobjectivefunction( 2a )minimizestheprobabilityofnotdetectingthetarget.Here,thesummationPi2S1(e)[S2(e)y(e,i,t)+zf(e,t)+zr(e,t)willequaloneifedgeewasvisitedattimestept.Ifthetargetispresentonedgee,thentheprobabilityofdetectingthetargetinthatsearchis1)]TJ /F5 11.955 Tf 11.95 0 Td[(expn)]TJ /F5 11.955 Tf 11.29 0 Td[(ln()Pi2S1(e)[S2(e)y(e,i,t)+zf(e,t)+zr(e,t)o.Constraints( 2b )–( 2c )areowbalanceconstraintsthatforcethesearchertoutilizeaconnectedwalkontheedgesofthegraph.AstheUERPRRisanedgesearchproblem,Formulation( 2 )denesthecontinuitybyessentially“doubling”theowbalanceconstraints(oneforeachpossibledirectionoftraversal)oftheoriginalOSPPformulation.Constraint( 2d )forcesthesearchertobeginthesearchwalkatonlyoneedge.NotethatFormulation( 2 )requiresO(KjEj2+2KjEj)binaryvariables. 27

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Becauseofitsgenerality,theOSPPformulationrequiresmorevariablestoformulatetheUERPRRthannecessary.Thisfactmotivatesourresearch,whichseekstondalternativeformulationsoftheUERPRRhavingfewervariables. 2.4FormulationsfortheUERPRRWenowpresentthreemodelsforsolvingtheUERPRR.Therstisacompactbinarylinearprogram,giveninSection 2.4.1 .Weprovidetwoadditionalintegerandmixed-integerprogrammingmodelsinSections 2.4.2 and 2.4.3 .Whilerequiringfewervariablesinitially,thesecondandthirdmodelsrequiresubtoureliminationconstraintsthatwemustaddinabranch-and-cutprocess. 2.4.1BinaryProgrammingFormulationWedenebinarydecisionvariablesxijstandxjistforeachundirectededge(i,j)2E,t=1,...,K,ands=1,...,K,and.Thevariablexijstequalsoneifandonlyifthetthedgetraversedistraversedinthedirectionfromnodeitoj,andthisisthesthtimethatedge(i,j)hasbeentraversedinanydirection.Wedenotethesetofnodesreachablefromnodei2Vasadjfig.Anoptimalroutecanthenbefoundbysolvingthefollowing 28

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binarylinearprogram. UIP:MaximizeKXt=1tXs=1X(i,j)2Epijs)]TJ /F7 7.97 Tf 6.59 0 Td[(1(xijst+xjist), (2a)subjecttoKXt=1xijst+xjist1,(i,j)2E,s=1,...,K, (2b)Xi2VXj2adjfigxij11=1, (2c)tXs=1Xi2adjfjgxijst)]TJ /F9 7.97 Tf 18.68 14.94 Td[(tXs=1Xi2adjfjgxjis,t+1=0,j2V,t=1,...,K)]TJ /F5 11.955 Tf 11.96 0 Td[(1, (2d)xijst,xjist2f0,1g,(i,j)2E,s=1,...,t)]TJ /F5 11.955 Tf 11.96 0 Td[(1,t=1,...,K, (2e) (2f)Theobjectivefunction( 2a )maximizesthecollectedrewardgainedbytheroute.Foreachedge(i,j)2E,onlyKtraversalsmayoccur,representedass=1,...,K“slots”.Constraint( 2b )ensuresthateachtraversalofeachedgeusesadistinct“slot.”Notethatforagivens1,s22f1,...,Kgwiths1t. 2.4.2“Tourable”Algorithm1Asalinearbinaryprogram,theUIPformulationcanbesolvedusingcommercialmixed-integerprogramming(MIP)solvers.However,thesizeofUERPRRinstancesthatcanbesolvedwiththisformulationwithinreasonablecomputationalresourcesislimitedduetothesizeofUIP.TosolvelargerUERPRRinstances,wepresentaformulation 29

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havingfewervariables.Thisformulationwillrequiresubtoureliminationconstraints,whichweaddinabranch-and-cutprocedure.First,weconsideramodiedgraphonwhichatrail(i.e.,awalkhavingnorepeatededges)providesanoptimalvehicleroutefortheinitialgraphG=(V,E).Ontheoriginalgraph,thevehiclemaytraverseanedgee2EatmostKtimesinafeasiblewalk.Inthemodiedgraph,wecreateKparalleledgesforeache2E,onecorrespondingtoeachpossibletraversalofeintheoriginalgraph.Wealsoaddonedummynodetothemodiedgraph,andjVjedgesofzeroweightfromthedummynodetoallothernodesinthegraph.Thevehicleroutebeginsatthedummynode,andweintroducebinaryvariablesbi,whichequal1ifandonlyifthevehicle'sroutebeginsatnodei,foralli2V.ObservethatsolvingtheUERPRRontheoriginalgraphisidenticaltondingamaximumweightedtrailoflengthKbeginningatthedummynodeonthemodiedgraph.Toourknowledge,theproblemofndingaheaviestweightedK-trailonagraphhasbeengivenonlylimitedattentionpreviously,asstudiedin[ 35 ]fortheproblemofndingthelongesttrail.Todenethisformulation,weindexeachedgeinthemodiedgraphbyanorderedpair(e,s).Indexingedges(i,j)2Eintheoriginalgraphase2f1,...,jEjg,theKparalleledgesinthemodiedgraphbetweennodesiandjarelabeled(e,1),...,(e,K).Givene2Eands2f1,...,Kg,edge(e,s)hasaweightofpes)]TJ /F7 7.97 Tf 6.58 0 Td[(1inthemodiedgraph.Foreachedge(e,s)inthemodiedgraph,thebinaryvariablexesequals1ifedge(e,s)ispartofthevehicle'strailonthemodiedgraph,and0otherwise.Wealsodeneabinarydecisionvariableyidsuchthatyid=1ifdedgesincidenttonodeiaretraversedinthevehicle'sroute,and0otherwise,foralli2V,andd=0,...,K+1.ThevariablesyidtrackthenodedegreesofthesubgraphinducedbyafeasiblerouteonthemodiedgraphthatformtheedgesoftheK-trail.Sincetheroutebeginsatthedummynode,thesubgraphinducedbytheK-trailonthemodiedgraphwillalwayshaveonlyonenodeofodddegreeotherthanthedummynode,whichwillalwayshavedegree1(see 30

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3 d3=0dX=1 X b3 b2 b1 2d2=3 1 d1=4Figure2-3. Afeasiblesolution(showndashed)inducesasubgraphonamodiedgraphforK=3.Thedegreeofthenodesrelativetotheinducedsubgrapharelabeledasdi.Thedummynode,labeledasX,willalwayshavedegree1.Onlyoneothernodewillhaveodddegreeinthesubgraphofafeasiblesolution. X b3 b2 b1 b4 1 2 3 4Figure2-4. Adisconnectedsolutionoflengththree(showndashed)withasubtouronamodiedgraph.OnlythedummynodeX(dX=1)andnode1(d1=1)haveodddegree.Asubtourexistsbetweennodes2and4,whered3=2andd4=2. Figure 2-3 ).Whileasinglenode(excludingthedummynode)withodddegreeintheinducedsubgraphisanecessaryconditionfortheroutetobeatrail,itisnotsufcienttoensurethatdisconnectedsubtoursarenotpresent(seeFigure 2-4 ).Thus,werststateabaseinteger-programmingformulation,andthendescribeabranch-and-cutproceduretodynamicallyremovedisconnectedsubtours. 31

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T1:MaximizeKXs=1Xe2Epes)]TJ /F7 7.97 Tf 6.59 0 Td[(1xes, (2a)subjecttoKXs=1Xe2Exes=K, (2b)Xi2Vbi=1, (2c)Xi2VbK 2c+1Xd=1yi,2d)]TJ /F7 7.97 Tf 6.59 0 Td[(1=1, (2d)bi+KXs=1Xe2adjfigxes=K+1Xd=0dyid,i2V, (2e)K+1Xd=0yid=1,i2V, (2f)xes2f0,1g,e2E,s=1,...,K, (2g)yid2f0,1g,i2V,d=0,...,K+1, (2h)bi2f0,1g,i2V. (2i)Theobjectivefunction( 2a )maximizestherewardgainedintheexpandedgraph.Constraint( 2b )restrictsthevehicletoselectarouteofonlyKedges.Constraint( 2c )forcestheroutetostartonanedgeincidenttothedummynode.Constraint( 2d )forcesexactlyoneothernon-dummynodetobeofodddegree.Constraints( 2e )and( 2f )setanode'sdegreetoequalthenumberofedgesincidenttothenodethatareusedinthetrail.NotethatT1containsjEjK+jVj(K+3)decisionvariables.Wenowdenethesubtoureliminationconstraints.ConsideranymaximalsubtourSonthemodiedgraphthatisnotconnectedtothedummynode.Toformallydenethenotionofamaximalsubtourwithrespecttoafeasiblesolution(^x,^y,^b)toT1,letSbethe(nonempty)setofedgesinasubtour,andletVSbethesetofnodesincidenttotheedgesofS.DeneESasthesetofedgeshavingbothincidentnodesinVS,i.e.,ES=fe=(i,j)2Eji,j2VSg.Likewise,deneESasthesetofedgeshavingexactly 32

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onenodeinVS,i.e.,ES=fe=(i,j)2Eji2VS,j62VSg.WesaythatSisamaximalsubtourifthesubgraphinducedbySisconnected;jSj
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NowconsiderasolutioncontainingamaximalsubtourS,andnotethattheRHSofinequality( 2 )equalsjSj>0.TheLHSof( 2 )equalszerobytheassumptionthatSismaximal.Therefore,( 2 )cutsoffanysolutionhavingamaximalsubtourS. Addingallinequalities( 2 )totheformulationaprioriwouldrequireenumeratingallpossiblesubtoursofsizelessthanKonthemodiedgraph.Toavoidthis,wesolveT1usingabranch-and-cutprocedure.FormulationT1initiallyincludesnosubtoureliminationconstraints.Ateachincumbentintegersolutionfoundatanodeofthebranch-and-boundtree,wedetermineifthex-solutionisconnected.Ifnot,weidentifyeachmaximalsubtourinthesolution,andaddaninequality( 2 )correspondingtoeachidentiedsubtourasacuttingplanetotheformulation. 2.4.3“Tourable”Algorithm2Inthissection,weformulationtheUERPRRasanonlinearMIP.AsinSection 2.4.2 ,theinitialformulationrequiressubtoureliminationconstraintsthatweaddviabranch-and-cut.Foreachedgee2E,deneanintegervariablexe2Z+,whichrepresentsthenumberoftimesedgeehasbeentraversedintheroute.Usingthisnotation,weobtainthe 34

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followingformulation: T2:MaximizeXe2Epe xeXi=0i!, (2a)subjecttoXe2Exe=K, (2b)Xi2Vbi=1, (2c)Xi2VbK 2c+1Xd=1yi,2d)]TJ /F7 7.97 Tf 6.58 0 Td[(1=1, (2d)bi+Xe2adjfigxe=K+1Xd=0dyid,i2V, (2e)K+1Xd=0yid=1,i2V, (2f)xe2Z+,e2E, (2g)yid2f0,1g,i2V,d=0,...,K+1, (2h)bi2f0,1g,i2V. (2i)Inthisformulation,constraints( 2b )–( 2f )areanalogoustoconstraints( 2b )–( 2f )oftheformulationin( 2 ).ThesubtoureliminationconstraintsweuseinthisformulationaresimilartothosegiveninSection 2.4.2 ,andaredenedinthefollowingproposition. Proposition2.4.2. GivenamaximalsubtourS,aswellassetsVS,ES,andES(asdenedinSection 2.4.2 ),thefollowingisavalidsubtoureliminationconstraintforT2: (K)]TJ /F5 11.955 Tf 11.95 0 Td[(1)Xe=ESxe+KXi2VSbiXe2ESxe. (2) Theproofthat( 2 )isavalidcuttingplanefollowsasimilaranalysistothatgivenforProposition 2.4.1 .Althoughtheobjectivefunction( 2 )isnonlinear,itiseasilylinearizedwiththeadditionofjEjconstraintsandjEjnonnegativerealvariables.SinceKisniteandxeKforalle2E,theobjectivecanbewrittenasaconcavepiecewiselinearfunction 35

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ofx-variableswithadditionalcontinuousvariablesze2R+foreache2E.Theobjective( 2a )canbereplacedwiththefollowing: MaximizeXe2Epeze, (2a)alongwithadditionalconstraintsdeningthez-variables: ze jXi=0i!+j(xe)]TJ /F6 11.955 Tf 11.96 0 Td[(j)j=0,...,K,e2E, (2a)ze2R+. (2b) Proposition2.4.3. Constraints( 2 )areavalidlinearizationoftheobjectivefunc-tion( 2a ). Proof. Denefunctionfe(j)=Pji=0i+j(xe)]TJ /F6 11.955 Tf 12.88 0 Td[(j)foreache2E,whichistheRHSof( 2a ).Becausezeismaximizedin( 2a ),itwilltakeonthelargestpossiblevalueallowedby( 2a ),i.e.,ze=minj=0,...,Kfe(j)atoptimality.Next,observethatfe(xe)=Pxei=0i,whichisthedesiredcontributionfromedgeetotheobjectivefunctionin( 2a ).Therefore,itissufcienttoprovethatforeache2E,fe(xe)fe(j),forallj=0,...,xe)]TJ /F5 11.955 Tf 11.95 0 Td[(1,xe+1,...,K.Forj
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wheretheinequalityholdsbecauseiisadecreasingsequence.Forj>xe,wehavethat fe(j))]TJ /F6 11.955 Tf 11.96 0 Td[(fe(xe)=jXi=xe+1i)]TJ /F4 11.955 Tf 11.95 0 Td[(j(j)]TJ /F6 11.955 Tf 11.96 0 Td[(xe) (2a)=jXi=xe+1(i)]TJ /F4 11.955 Tf 11.95 0 Td[(j)0. (2b)Thiscompletestheproof. 2.5ResultsInthissectionwecomparetheefcacyofsolvingthethreemodels(UIP,T1,andT2)presentedinSection 2.4 tosolvetheUERPRR.AllimplementationwasperformedinC++usingtheCPLEX12.1callablelibrary.TheresultswererunonaDellOptiplexGX620desktopcomputerwithtwo3.4GHzprocessorsand2GBofRAM.ThissectionalsoincludesadiscussionontheimpactoftheproblemsizeoncomputationsusingFormulation( 2 )fortheUERPRR.Throughoutthissection,modelsT1andT2aresolvedusingthebranch-and-cutapproachdevelopedforthesemodels.Inparticular,wesolvethebaseformulations,withnoinitialcuttingplanes,usingCPLEX.Whenanintegersolutionisencounteredinthebranch-and-boundtree,wetesttoseeifsubtoursexistinthesolution.Ifso,wegenerateacuttingplanecorrespondingtoeachseparatemaximalsubtourthatisidentied.Allsuchcutsaredynamicallyaddedasgloballyvalidinequalities. 2.5.1ComparisonofMethodRuntimesTocomparetheefcacyofthethreemodels(UIP,T1,andT2),wegeneratedonetestinstanceeachcorrespondingtoeverycombinationofthenumberofnodesjVj2f20,30,40,50gandgraphdensity2f0.2,0.3,0.4,0.5g.Thegraphdensityvalueimpliesthatanedgeexistsbetweenanypairofnodesintheinstancegraphwithprobability.Theaprioriedgerewardsoftheinstanceswererandomizedusinga 37

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Table2-1. Summaryoftherandomlygeneratedgraphsforagivennumberofnodesandgraphdensity(). NodesDensity()Edges 200.239200.353200.477200.598300.290300.3127300.4176300.5218400.2153400.3228400.4322400.5402500.2251500.3358500.4488500.5599 Table2-2. ComparisonofT1runtimeforrewardreductionparameter2f0.95,0.85gforcomputinganoptimalvehiclerouteoflengthK=30ontherandomlygeneratedtestgraphswithdensity=0.4. Runtime(sec)NodesEdges=0.95=0.85 20771.4530.26530176115.890.81340322121.45210.55048860.1091.766 uniformdistributionontheinterval(0,1).AsummaryofthegeneratedinstancesisgiveninTable 2-1 .Next,wefoundoptimalpathsoflengthK2f5,10,15,20,30gbysolvingmodelsUIP,T1,andT2.Wepresentcomputationalresultswhen2f0.85,0.95g,inordertopromoteedgere-traversalsatoptimality.Table 2-2 comparestheruntimetondanoptimalpathoflengthK=30when=0.85versus=0.95fortheT1modelonidenticalgraphswithadensityof=0.4andanincreasingnumberofnodes.When=0.95,runtimesaregreaterasincumbentsolutionsoftencontaindisconnectedrepeatededgesthatrequirecutstoremove. 38

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Forall16testgraphinstancesgiveninTable 2-1 ,wefoundanoptimalpathforallcombinationsofK2f5,10,15,20,30gand2f0.85,0.95g.Foreachofthesetestcombinations,wesetaruntimelimitof50minutes.Figures 2-5 and 2-6 summarizetheresultsoftheseruns.Figure 2-5 containsalltrialrunsperformed A20Nodes. B30Nodes. C40Nodes. D50Nodes.Figure2-5. Graphsshowingthefrequencyofsampleswithruntimelessthansomexedtimewhen=0.95. with=0.95,andFigure 2-6 containsallexperimentswhere=0.85.Eachsubgraphcontainsallrunsforinstanceshavingaxednumberofnodes,forallvaluesofK.Thus,Figure 2-5 BreferstotherunsofallinstanceswherejVj=30 39

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A20Nodes. B30Nodes. C40Nodes. D50Nodes.Figure2-6. Graphsshowingthefrequencyofsampleswithruntimelessthansomexedtimewhen=0.85. withK=f5,10,15,20,30gand=0.95forUIP,T1,andT2.Thesubgraphsdepictthenumberofinstanceswitharuntimelessthanthegivenxedtime.Thex-axisofthesubgraphsistime(giveninseconds),andthey-axisisthecountofthenumberofsampleruntimeslessthanthetimegivenonthex-axis.ForeachmodelUIP,T1,andT2,thesamplegraphruntimesweredividedintobucketsoftime(0,10],(10,100],(100,500],(500,1000],(1000,1500],(1500,2000],(2000,2500],and[2500,3000](seconds)andthecountofruntimesineachbucketwasplotted. 40

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Figure2-7. AveragepercentageincreaseinruntimeofUIPcomparedwithT1ofallsamplegraphs. Forexample,Figure 2-5 BshowsthatforsamplegraphswhereN=30withK=f5,10,15,20,30gand=0.95,whenrunningtheUIPmethods:8ofthesampleshadruntimeslessthan10seconds,11ofthesampleshadruntimeslessthan100seconds,12sampleshadruntimeslessthan500seconds,andsoon.Accordingly,onthesegraphs,amethodwithalinethatliesbelowanotherhasinferiorruntimeperformance.Sinceweusedacutofftimelimitof50minutes,allsampleshavearuntimeoflessthan3000seconds.Inalltestinstanceswheretheruntimesweregreaterthan5seconds,CPLEXidentiedanoptimalsolutionusingT1fasterthanUIPorT2.ThecomparativeincreaseinruntimebetweenT1andtheslowerUIPisoftensignicant.Figure 2-7 showstheaveragepercentincreaseinruntime(overallsampleruns)usingUIPcomparedtoT1,plottedforeachvalueofKand.InadditiontotheconsiderablyhigherruntimesofUIP,thelargergraphinstances(intermsofboththenumberofedgesandvalueofK)givenhererepresenttheuppermostlimitintermsofmemorysizesolvableonourcomputersystemusingUIP.Largersample 41

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graphsorlargervaluesofKresultinout-of-memoryerrorsduetothelargenumberofvariablesrequiredtodenetheUIPproblemformulation.TheT2runtimesthearelowerthanthoseofUIPwhen=0.85.AlthoughtheT2runtimevalueswereoftenworsethanthosesolvedbymodelT1,bothformulationssolvednearlyallsamplesinunder10seconds.AsseeninFigure 2-6 ,allbutthreeoftheprobleminstancesaresolvedbyT2inunder100seconds,whileoneprobleminstancetooklongerthan100secondstosolveviaT1.Forprobleminstancesinwhich=0.85,thecomputationaldifferencebetweenT2andT1isnegligible.Bycontrast,when=0.95,theruntimesforT2aregreaterthanthoseofUIP.Presumably,whenislarger,thereisagreaterpotentialforoptimalsolutionstoincluderepeatededges,thusrequiringmorecutstobeaddedduringthebranch-and-cutprocedure.However,wedidnotencounteranyout-of-memoryerrorswhenrunninglargerinstancesusingT2,possiblyduetothefactthatthebaseT2formulationrequiresfewervariables. 2.5.2ProblemSizesandtheOSPPFormulationInthissection,wediscussthecomputationalcharacteristicsofFormulation( 2 ).Muchworkhasbeendonetocreatecustombranch-and-boundorheuristicprocedurestosolvetheOSPP,andwehavenotattemptedtore-createanyofthesemethods.Bywayofanintroductorycomparison,weimplementedtheformulationinitsentiretyinaglobaloptimizationsolver.Inthisway,weconsiderhowthenumberofvariablesrequiredforlargerinstancesmightimpacttheutilityoftheOSPPmethods,especiallyincaseswheretotalenumerationmayberequiredtoobtaintheoptimalsolution.Table 2-3 showsthenumberofvariablesandconstraintsrequiredtoinitiallydenetheproblemintheworstcaseusingeachmethod.SincemodelsT1andT2areinitiallyimplementedwithoutsubtoureliminationconstraints,theseformulationsmayrequireadditionalcuttingplanestobeaddedduringthebranch-and-cutprocedure.WeencodedFormulation( 2 )withtheGeneralAlgebraicModelingSystem(GAMS),solvedwiththelarge-scalenonlinearoptimizationsolverBARON.Toavoidlimitations 42

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Table2-3. WorstcasenumberofvariablesandinitialconstraintsforallformulationsgivenagraphG(V,E).Thenumberofconstraintsgivenexcludesanysubtoureliminationconstraintswhichmaybeaddedduringthesolutionapproach. VariablesConstraints Form.1KjEj2+2KjEj2jEjK+1UIPjEjK2+3jEjKjEjK+jVjK+1T1jEjK+jVj(K+3)2jVj+3T22jEj+jVj(K+3)jEj(K)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+2jVj+3 incomputationalpower,werantheinstancesforthisformulationontheInternet-basedNEOS(Network-EnabledOptimizationSystem)servers[ 24 , 27 , 46 ],usingtheglobalsolverBARON.BecausethelargenumberofvariablesinFormulation( 2 )madesolvingproblemswithonly20nodesdifcult,wecreated10-nodeproblems(having11and22edges)tobesolvedbyBARON.Table 2-4 showstheruntimeinsecondsforthe10-nodeinstanceswithreductionconstant=0.95.TheForm.(1)columnshowstheruntimeofFormulation( 2 ).ColumnsUIP,T1,andT2aretheruntimesinsecondsofthosethreemodels.Table 2-5 comparestheresultsofFormulation( 2 ),UIP,T1,andT2,fortwo20nodeproblemshaving53and98edgeswith=0.95.MethodsT1andT2solvethe10-nodeprobleminstancesforallvaluesofKinunder1second.Forthe20-nodeproblems,T1solvestheproblemforallvaluesofKinunder3seconds.FortheT2resultsofthe20-nodegraphs,theruntimesareunder4secondsforallbutoneinstance(whereK=30forthegraphwith53edges).InTables 2-4 and 2-5 ,OOMindicatesthatthesolverhasterminatedbeforeoptimalitywasreachedduetomemoryrestrictions.ThisoccurredonceforBARONonthe22-edgeproblemwhenK=30andalsofortheprobleminstanceshaving22and98edgesfortheUIPmethod(runontheDellOptiplexGX620computer)whenK=30.TheNEOSservershaveatimelimitof8hoursorlongerdependingonthesolverserverhostlocation.BARONrunsreachedthis8hourtimelimitbeforeoptimalitywasreachedwhenK=20oninstanceshaving11,53,and98edges.ThetimelimitwasalsoreachedwhenK=30forthe98edgegraph.InTables 2-4 and 2-5 theseinstanceshaverecorded“time*”insteadofanumerical 43

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Table2-4. Runtime(seconds)ofall4methodsfortwo10nodegraphswith=0.95. EdgesKForm.(1)UIPT1T2 1152<1<1<1111016<1<1<11115714<1<111201643<1<11130time*49<1<12251<1<1<1221026<1<1<1221537<1<1<1222025011275<1<12230OOM**OOM**<1<1 Table2-5. Runtimes(seconds)ofall4methodsfortwo20nodegraphswith=0.95. EdgesKForm.(1)UIPT1T2 5358<1<1<1531015932453152253113253208505145245330time*OOM**229985353<1<1<1981035783<1<19815239706<1<19820time*943<1<19830time*OOM**<12 runtime.Withouttheaidofthespecializedbranch-and-boundprocedures,theOSPPformulationoftheUERPRRsolvedwithBARONisintractableforallbutthesmallestinstances. 2.6ConclusionInthischapterwedevelopedthreeapproachestosolvetheUERPRR.Therstmodelwasabinaryprogrammingformulation(UIP)thatcanbesolveddirectlybyanyMIPsolver.Thenotionofthevehiclerouteasatrailalsomotivatedtwoadditionalmethods.Thesemethods(T1andT2)requiresubtoureliminationconstraints,whichweincorporateviabranch-and-cuttechniques.WeperformedcomputationaltestsonrandomlygeneratedinstancestoassesstherelativetnessofeachmethodforsolvingtheUERPRR.Whileallthreealgorithmswereeffectiveforndingtheoptimal 44

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solution,thelargernumberofvariablesrequiredtoformulatetheUIPmethodresultedinout-of-memoryissuesonlargerproblemsizes,inboththenumberofnodesandK.ThesampleruntimesoftheUIPmethodwerealsoslowerthanthequickestapproachforanysampleproblem.Despitethesefactors,theUIPmethodprovidesasimplewaytoimplementandsolvetheUERPRRforsmallprobleminstances.TheT2algorithmperformedwellforsmallerrewardreductionparameters();however,theruntimesofT2wereatleastaslargeasthosefortheUIPmethodforlargervaluesof.T1solvedallinstancesinlessthan500seconds,andoutperformedbothT2andUIPforallinstanceswithnon-negligibleruntimes.BecauseofthetractabilityoftheT1algorithmforlargerprobleminstances,werecommendusingourbranch-and-cutalgorithmtosolvetheUERPRR.ThisapproachcouldbeextendedtosolvemorecomplexversionsoftheUERPRR.Futureresearchcouldincludegeneralizationoftherewardreductions.Otherinterestingextensionscouldinvolvenonuniformtraversalcosts,capacitatededges,arrivaltimewindows,ormultiplevehicles.Inadditiontothechallengeoftheseproblemstheoretically,continuedfocusonsuchproblemswillgarnermanypracticalrewardsfortransportationandrelatedelds. 45

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CHAPTER3ESTIMATIONOFRAILROADSERVICEMETRICSFROMGPSDATA 3.1IntroductionThefollowingresearchwasperformedasajointcollaborationwiththeservicedesignandoperationsresearchdepartmentsatCSXTransportation,Inc.inJacksonville,Florida.Theprimaryfunctionofanyfreightrailroadcompanyistoprovidetransportationservicestoitsnumerouscustomers.Inordertoefcientlymanageandschedulethedeliveriestothesecustomers,accurateknowledgeofthecustomerservicedurationtimes,aswellastransittimesbetweencustomersisneeded.Todeterminethesemetrics,weconsiderdatafromthreesources:GPSdatafromlocomotives,thesequenceofcustomersservedonagivenday,andgeographicinformationabouteachcustomer.Usingonlythisdata,weconstructtwoalgorithmsfordeterminingboththetimespentateachcustomer,aswellasthetimespenttravelingbetweencustomers.TherstalgorithmwerefertoasthePolylineAlgorithm.Inthisalgorithm,weusetheGPSpingstoconstructadistance-versus-timegraphrepresentingatrain'smotionsrelativetoareferencepoint.Fromapolylineapproximationofthisgraph,weextracttimewindowsaccordingtowhenthetrainapproachedacustomerlocation.Thismethodworkswellwhensubsequentcustomersarefarapart.WealsocreatedasecondalgorithmbasedonthedensitybasedclusteringalgorithmOPTICS,toestimatepotentialcustomerservicetimesforshortdistancetrains,wherecustomerserviceregionsmaybesharedoroverlapping.Suchcloseproximitybetweencustomerscreatesuniquechallengeswhendeterminingdistinctcustomerservicetimes.OncecandidatetimewindowsareestablishedbyeitherthePolylineAlgorithmortheOPTICSalgorithm,wedeterminewhichtimewindow(s)mostlikelycorrespondstotheactualservicevisits.Thisassignmentisdoneusingaweightedshortestpathproblem. 46

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Lastly,byapplyingthisanalysistoseveralmonthsworthofdataweobtainnumerouspointestimatesfortheservicedurations,andthetraveltimebetweensubsequentcustomers.Wediscusshowweusethesepointestimatestoconstructprobabilitydistributionsrepresentingthesemetrics. 3.1.1PurposeTherailroadcompanyCSXTransportation,Inc.operatesasophisticatedserviceindustry,movingthefreightofseveralthousandcustomersacrossitsnetwork.Pickupanddeliveryatcustomerlocationsisperformedbywhatareknownaslocaltrains.Theserelativelysmallertrains,takethecustomer'sfreightcarsto,andfrom,largeroutingyards.Optimizedschedulingandworkallocationfortheselocaltrainsiscomplexduetolocalyardcongestion,variabilityinworkvolume,andcrewavailability.Toefcientlymanagetheserviceoftheselocaltrains,anaccuratecharacterizationofthetimeittakesatraintoserveacustomer,aswellasthetraveltimebetweencustomers,isneeded.Thearrivalanddeparturetimesforacustomeronasingleday,providesapointestimatefortheservicetimeofthatcustomer.Thedeparturetimeoftheonecustomer,followedbythearrivalatanother,provideapointestimateforthetraveltimebetweenthetwocustomers.Aftercalculatingthesearrivalanddeparturetimesforasufcientlylargenumberofdaysofhistoricaldata,thesenumerouspointestimatescanbeusedtocreateastatisticalcharacterizationofserviceandtraveltimes.Thesestatisticalestimationtechniquescanbeusedtocharacterizetherelationshipsbetweenthedependentvariablesofserviceandtraveltime,andthemanypossibleinuencingparameters(typeofwork,numberofcarshandled,crew,dayofweekandtrain).Givenfunctionallyestimatedcustomerserviceandtraveltimes,numerousindustryusesexist.Estimatesofserviceandtraveltimecanbeusedtooptimizecustomer-to-trainscheduling,dynamicworkloadbalancing,investigatingdeviationsfromexpectedserviceschedules,andrealtimeannouncementsofexpecteddeliverytimestocustomers. 47

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3.1.2LiteratureReviewandMethodologyInthecurrentCSXlocaltrainsystem,detailedservicetimeandtraveltimeinformationisnotrecorded.Whenservicingacustomer,thetraincrewrecordsasingletimestampvaluetoindicatewhenworkwasperformedatacustomer.Thistimestampisstoredonwhatiscalledaworkorder.Theworkorderalsocontainsinformationsuchasnumberofcars,movementtype(placeorpickup),customeridenticationnumber,train,andworkassignmentdate.Forexample,thecrewmayenterthattodayat14:20,15carswereplacedattherstcustomer.Sinceasingletimestampvalueisentered,theexacttimeworkbegan,andended,atacustomerisnotknown.Whileitispossibletomodifytoworkorderdataentrysystemtorequirethecrewtoenterbothastartingandendingservicetime,severalfactorsmakethisundesirable.Mostimportantly,recordingtwotimestampvaluesdoublestheamountofdataentryrequiredofthecrew.Thejobofthetraincrewinayardlocationisanextremelychallengingone.Eachdaypresentsanewsetofchallengeswhichrequirespontaneousmodicationsandconstantalertness.Itmaybeburdensometoperformsuchsecondaryadministrativedutiesatcertaintimesresultinginpostreportingofworkwhichimpliesthetimesenteredaredeterminedatthediscretionofthecrew.Relyingonanindividualperformingdetailedcomplexjobmaneuverstorememberstartingandendingservicetimesmayresultinunintendedinaccuracies.Inadditiontotheservicetimes,theworkordermustalsorecordtheservicedcustomerandworkinformation.CSXservesthousandsofcustomersonitsnetwork,severalofwhomhavemultiplefacilitiesatproximatelocationsdenotedbytheirownuniquecodes.EnsuringthattheworkorderiscompletedforthecorrectcustomerID,withthecorrecttimes,thecorrectnumberofcarsaswellasthequalityofwork(pickupordeliverorrelocate)createsanaddedburdenonatraincrew.Otherincentivebasedreasonsmaycauseunder/overormissedreportingofcustomerservice,suchasbreaktimes,dailyperformancegoalsordifcultiesinoverridingentereddata.Anidealdeterminationofcustomerservicetimeswillminimizecrewdataentry. 48

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ThewidespreaduseofGPS(GlobalPositioningSystem)devicesforvehicletrackingandroutingimpliesseveralpotentialsolutionmethodsforestimatingcustomerservicetime.GPStechnologyiswidelyavailabletodayandallowsthetrackingofthepositionofanyequippedvehicle.GivenaknownsequenceofcustomersservedincombinationwithGPSdatacapturedfromthetrain'slocomotivescanprovideameasureofhowlongthetrainwasphysicallynearacustomer'slocation.ThequantityofusefulGPSdataintheanalysisisverylargeandasubstantialamountoftimemustbedevotedtocleaning,interpreting,andlearninghowtocorrectlyutilizethenumerousrequireddatasources.Thephysicalnatureofeachcustomerandthewayinwhicheachisservedvarywidelymakingitdifculttoapplytraditionaltraveltime/stopconcepts.Thecreationofadataminingalgorithmthatcanrepeatedlyextractservicetimeinformationmustbecarefullyconstructed.WhilethepaththetrainfollowscanbeknownfromtheGPSmeasurements,severalfactorscomplicatetheanalysisandmakedeterminingcustomerservicetimesfromthisinformationalonedifcult.GPSspatialandtemporalmeasurementsareaccurateonlytowithintheerrortoleranceoftheGPSdevice.Considerasinglelatitudeandlongitudepositionmeasurementinthesetofallreadingsforatrainonagivenday.TheerrorintheGPSmeasurementsimpliesthatifseveraltracksarepresentatthelocationofthisreading,itmaynotbepossibletodetermineonwhichtrackthetrainwastrulylocated,eveniftheunderlyingtrackstructureisknown.Thusatrainonacustomertrackisoftenindistinguishablefromoneonanearbymainlinetrack.ForadiscussionontheinherentperformanceissuesofGPSwhichoccurontherailnetworksee[ 68 ].Intuitively,aninitialapproachtotheproblemofidentifyingpotentialcustomerserviceeventsistoutilizethesametechniquesandmethodologyforidentifyingcustomerserviceintervalsforcarsandtrucks.Unfortunately,theseapproachesfailtotakeintoaccounttheparticularwaysinwhichtrainsoperate.Wediscusstheseparticularitiesinthefollowingsectionasweintroducethetraditionalmethodsusedincarandtruck 49

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research.TraditionalmethodsrelyontwoassumptionsofwhatwillbeobservedintheGPSpingsdataasavehicleservicesacustomer;rstthetrainwillnearthecustomer'slocation,andthenthetrainwillslowdown,orstop,duringservice.MuchworkhasbeendoneinGPStransportationresearchtoidentifyserviceeventsbasedonthesecriterion. 3.1.2.1FixedgeofencefromcustomergeographyTodeterminetheamountoftimeaGPSequippedvehicleiswithinproximityofalocation,itiscommontoemployaxedspatialboundaryorpolygondrawnaroundthelocationofinterest(oftenreferredtoasageofence).Intuitively,therstGPSreadingthatoccursinsideofthegeofencecanbeusedasthevehicle'sarrivaltimeandthelastreadinginsideofthegeofencecanbeusedasthevehicle'sdeparturetime.TheuseofspatialgeofencesiscommoninGISsystemdevelopment.Usingthesegeofencestomonitorvehicleactivityhasbeenperformedacrossseveralindustries.Indetermininghowtimeisspentduringabordercrossingbyashippingtruck,staticgeofenceswereusedbytheshippingcompanytoreducethetimetheirvehiclesspendatthebordercrossing[ 70 ].Otherapplicationsincludelocation-basednotications[ 72 , 103 ],vehiclemonitoringforsensitivecargosuchashazardousmaterials[ 78 ],andwastevehicletaskmanagement[ 107 ].Inallthesepapers,thegeofenceisaxedbarrier.ThevehicleisconsideredtohavearrivedatthegeofenceonceanyGPSpingappearsinsideofthisregion.Andlikewise,itisconsidereddeparted,oncetheGPSpingsexitthegeofence.AnobviousmeansofconstructingageofenceforeachcustomerwouldbetouseGPSdataabouteachcustomer'sfacilitytodeneasuitableregion.SinceCSXmaintainsdetailedGPSrecordsabouteachoftheircustomersfacilitiesincludingtrackoutlinesaroundeachlocation,itispossibletocreatesuchageofence.Unfortunatelythisinformationisnotsufcientinpracticetodeneameaningfulgeofence.Customersmaynotreceiveservicewithintheirphysicalfacilitylocations.Carsaresometimesplacedalongmainlinetracknearthecustomertobemovedatalatertime.Thisissueiscompoundedfurtherbecausecustomersmaybegeographicallyclosetooneanother, 50

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Figure3-1. AxedgeofenceestimatesservicetimebyconsideringGPSreadingswhichfallinside. includingthepossibilityofparalleltracksorevensharedtracks.Becauseofthis,customergeofencesneedtobedenedinsuchawayastoclearlyindicatewhichcustomerisbeingservicedatagivenpointintime. 3.1.2.2FixedgeofencefromhistoricaltrainstoplocationsInsteadofusingpurelygeographicaldataaboutacustomer'sfacility,historicaldataaboutserviceandstoptimesandlocationscanbeusedinthedeterminationofanappropriategeofence.Usingthisdata,apilotstudywasdoneconsideringonlyafewcustomerlocations.TwoyearsworthofGPSdatawascompiledandusedtodetermineasuitablegeofenceforeachofthecustomers.BycomparingthedensityoftheGPSpings,thestudywasabletodeterminelocationswherethetrainsspentmostoftheirtime.Oftheseareas,thoseclosesttoeachcustomer'slocationscouldbeinterpretedastheserviceareaforthatcustomer.Bydrawingaxedpolygonaroundalloftheareasidentiedasbelongingtoacustomer'sserviceareaageofenceforthatcustomerwasestablished.Withthisestablished,thetimespentbythetraininsidethegeofencecouldbeconsideredastheirservicetime;seeFigure 3-1 .Ifthisprocedurecouldbegeneralizedthencustomerservicetimescouldbeextractedalgorithmicallythroughoutthecustomernetwork.Unfortunately,thereareseveralcomplicatingfactorstothismethodology.First,thismethodrequiresyearsofGPSdatainordertodenelocationsof“dense”activityaroundeachcustomer.InthisrawGPSdata,alldaysofserviceareconsidered,includingthosedaysduringwhichthecustomerswerenotserved.Eventhoughitispossibletolteroutthosedaysonwhichthecustomerwasnotserved,thedatamaystill 51

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Figure3-2. Atrainpassesthroughacustomer'sxedgeofencewithoutservicingthecustomer. containidentiablydenseareasthatdonotcorrespondtoacustomerservicearea.Forexample,ifthecustomer'sfacilityissituatedclosetoabusyrailyardoronafrequentlytraveledsectionoftrack,theGPSpingswillindicatethattrainsspendalargeamountoftimenearthecustomerbutnotnecessarilythatthecustomerisbeingservedduringthosetimes.Thiscanalsobethecasewhenseveralcustomershavefacilitiesincloseproximitywithoverlappingserviceareas.Second,customersmayhavedifferentlocationswithintheirfacilitywhereserviceisprovided,possiblydependentonthenatureoftheserviceneeded.Forexample,acustomermaydesirethatcarsbepickedupfromaparticularlocationintheirfacilityanddroppedoffinaseparatelocation.Thisissuecouldbeaddressedwithfurtherknowledgeaboutthecustomer'sneedsbutdeningageofencewiththisdatacouldalsogreatlyexaggeratethesizeofthecustomersfacility.Calculatingservicetimesbasedonthislargegeofencecouldleadtooverestimationsintheactualtimerequiredtoservicethecustomer.Third,itishardevenwithawelldenedgeofencetodeterminewhatconstitutesthebeginningandendofacustomerserviceevent.Typicallywhenusinggeofencestodetermineservicetimes,therstGPSpinginsideofthegeofenceisdesignatedasthestarttimeandthenalGPSpingastheendtime.Becausetrainsarelimitedtothetrackastheymove,itiscommonfortrainstotravelbackandforththroughacustomer'sgeofenceseveraltimesduringaday.Becauseofthis,thetypicalmethodfordeterminingservicetimeispronetogreatlyoverestimatingtheservicetimeofacustomerorfalselyidentifyingserviceeventsthatdidnotoccur;seeFigure 3-2 . 52

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Itispossibletoavoidfalsepositivesandoverestimatesifwearerstabletoidentifytimesduringwhichthetrainisslowedorstopped.Sinceweexpectthatatrainslowsorstopswhenservicingacustomerwecanthenconsideronlythesetimesaspotentialcustomerserviceevents.Withthisinmind,wenowconsidertheproblemofidentifyingintervalsduringwhichthetrainisslowedorstoppedsowecanthenidentifywhichofthemcorrespondtocustomerserviceevents. 3.1.2.3VelocitydeterminationofstopsandslowsResearchthatdetermineswhenavehicleisstoppedfromGPSdataisespeciallyimportantinpublicbustransit[ 13 , 43 ]andhouseholdtravelactivityresearch.IntrackingbusesviaGPSpingdata,axedgeofencemaynotbegivenduetothelargenumberofbusstopsonthebusroute.Someattemptsatautomaticallyextractingstoptimesofbusdatahavebeenmadeusingvelocitytolerancelevels.However,thepresenceofroadwaystopsmakesuchstopsdifculttodistinguish[ 13 , 43 ].Busesmoveathighervelocitiesthandotrains,makingtheidenticationofstopsinthismannerfeasible.Distributionsthatcharacterizestopsandaidinthepredictionofarrivaltimesatupcomingstopsmaythenbeconstructed.GPSdatahasbeenusedasatoolinresearchingthetravelhabitsofpeopleindaytodaylife(oftencalledactivityresearch)[ 6 , 73 , 95 , 100 , 108 , 109 ].Stoptimes,andarrivalanddeparturetimesfromGPSpingdataarecombinedwithadditionaldatatodetermineconsumerandtransportationtrends.PriortothewidespreaduseofGPSdevices,theonlymeansofgatheringdatawasviawrittenreportsorphoneinterviews.Whilemuchofthedatafortheseapplicationsisstillrecordedbyconsumers,thisresearchiscomingtorelyheavilyonautomaticallyrecordedGPSgeospatialdata,andattemptingtodeterminedeterministicrulesforclassication.Traditionally,thetrackingsystemworksusingaGPSdevicestoredinthevehiclewhichperiodicallyrecordspositionandtimedatatoalog.Thislogcanbetranslatedintoinformationaboutthevehicle'smovementsbycalculatingthechangeinlocation 53

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Figure3-3. AstopismeasuredsuchthatsubsequentGPSmeasurementsarewithinatolerance. withrespecttotimeprovidingatraceofthevehicle'spathduringtheday.IfGPSmeasurementsreportasimilarlocationforseveralsuccessivereadingsthenthevehiclehasmovedverylittleifatallduringthattimeandcanbeconsideredstopped.Intheaboveliterature,avehicleisidentiedasstoppedifsuccessiveGPSmeasurementsfallwithinaspecictoleranceofoneanother;seeFigure 3-3 .Insomecases,theGPSdevicealsoreportsvelocitydataaboutthevehicle,andthiscanalsobeusedindeterminingifavehicleisstopped.Insuchcasesavehiclecanbeconsideredstoppedifitsvelocityisbelowatolerancelevel.SuchisthecasewiththeGPSdeviceshousedonCSXtrains;unfortunatelythisvelocitydatadoesnotprovideareliablemeansofdeterminingstops.First,GPSdevicessufferfromloweredaccuracywhenthevehicleistravelingatlowspeedandforsafetyandlogisticalreasons,CSXtrainstravelatextremelylowspeeds.Second,eveniftheGPSdevicesweremoreaccurateatlowspeeds,thetoleranceunderwhichastopcouldbeidentiedwouldbedifculttodetermine.ForthesereasonswemustrelyonGPSlocationdataaloneinsteadofthevelocitydata.Ourdiscussionnowhasledustoanobviousalgorithmicapproachtoidentifyingcustomerservicetimes.First,weidentifyageofenceforeachcustomerbasedontheavailableinformation.Then,usingGPSreadingsweidentifystopsmadebythetrainaspotentialserviceevents.Finally,ifwecanidentifythecustomertowhicheachstopshouldbeassigned,wecansimplycalculatetheservicetimeofeachcustomerasthedifferencebetweentheearliestandlatesttimesassociateswiththecustomer'sstops. 54

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Unfortunately,therearecomplicationstodeterminingintervalsduringwhichatrainmaybeconsideredservingacustomerwhichareinherentinthewaystrainsoperate.Muchofthetimeduringaserviceevent,thetrainisnotactuallystopped.Thisisincontrastwithadeliverytruck,forexample,whichstopsforthedurationoftheserviceevent.Uponarrivalatacustomer'sfacility,atrainmayhavetonavigateaseriesofswitchesinordertoproperlypositionitselffortheservice.Onceinposition,thetrainmaythenberequiredtopullawayfromthecustomerlocationwhileadditionalcarsaremovedaroundorwhileswitchesarealigned.Itiscommonforthispracticeofcontinualrepositioningtooccurduringacustomerserviceeventascustomercarsareoftenstoredonseveralparalleltracks.Becausetrainsoperateinthismanner,atrainmaynoteveractuallystopduringacustomerserviceevent.Furtherinformationabouthowtrainsoperateduringacustomerserviceeventrevealsanothercomplicatingfactor,inthatthelengthofthetrainchangesthroughouttheday.SincetheGPSdeviceishousedinthelocomotiveofthetrain,itsmeasurementsdonotexactlyreectthelocationwhereserviceistakingplace.Typically,trainsserveacustomerbyeitherdroppingoffcarsfromthebackofthetrainorbypickingupcarsandaddingthemtotheback,andineithercasetheserviceeventdoesnotoccuratthelocationoftheGPSdevice.Sincetrainscanreachlengthsinexcessofhalfamile,wenowneedtoconsiderwhatqualiesasastopoccurringclosetoacustomer'sfacility.Ifthetrainislongenough,itispossiblethatacustomermaybeservicedinadaybutthetrainthatprovidedtheservicemayneveractuallyenterthegeofenceestablishedforthatcustomer;seeFigure 3-4 .Anobvioussolutionistoextendthecustomergeofencetoarangethatwillaccountforanysizetrainprovidingservice.Unfortunately,thissolutionfailsasextendingcustomergeofencescancauselargeoverlapsbetweennearbycustomers.Ifbothoverlappingcustomersareservicedonthesamedaythenitmaybedifculttoidentifywhatstopshouldbeassignedtoeachone;seeFigure 3-5 .Overlappingissuesaside,extendingthegeofenceseforeachcustomerstillallows 55

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Figure3-4. Alongtrainmayindicateastopfurtherawayfromacustomerthanwouldashortertrain. Figure3-5. Customergeofencesmayoverlapcausingambiguitywhenassigningservicestops. Figure3-6. Serviceeventstopsmayappearonorimmediatelyoutsideofaxedgeofencemakingclassicationdifcult. forstopstobemissedbecauseoftrainlengthinthecaseofaparticularlylongtrain,andadditionallymaycausethemisidenticationofstopsunrelatedtothecustomer'sserviceevent.Forexample,ifacustomerfacilityliesneararoadcrossingthenstopsmadeatthatcrossingmayresidewithintheextendedgeofencecausingthemtobeidentiedaspartofthecustomerserviceevent;seeFigure 3-6 .Thepilotstudyfoundallcomplicatingfactorscommonlyoccurinpractice.Althoughitispossibleinlocalcasestodenelogicalrulestoeliminatesomeoftheseissuesnouniformsetofruleswouldallowthisapproachtobeextendedtotheentirecustomernetwork.Sometrainroutesmayservicecustomerswithverylargefacilitiesspreadoutovermanymilesofruralgeographywhereasothersmayserviceadenselypackedurbanareawithnumeroussmallcustomers.AuniforminterpretationofGPSdataforalltrainsisnotpossible,andfurtherestablishingrulesforeachroute'slocalgeographyandcustomerinformationwouldbesimilartoprocessingeachofthethousandsofcustomersindividually.AgoalofautomatingthedeterminationofservicetimesusingGPSdatamotivatesthealgorithmicapproachwewilloutlineinthefollowingsection.Itusesadistance-versus-time 56

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graphgeneratedfromourGPSdata.Similarstudiesconsideringautomotivenetworkshaveusedspeed-versus-timeandspeed-versus-distancegraphsintheiranalysis[ 48 , 49 , 77 ].However,asmentionedin[ 77 ],speed-versus-timeandspeed-versus-distancegraphsaredifculttointerpretasspeedcanvarydrasticallyonthesameroad.Additionally,asmentionedabove,theGPSvelocitymeasurementsareunreliableatlowspeed,alimitationthatiscompoundedbecausetheintervalswewishtoidentifycorrespondtothosewiththelowestspeedsandthusleastaccuracy.Withtheuseofthedistance-versus-timegraphswewillbeabletoidentifyintervalsduringwhichthetrainmovesverylittle.Theseintervalscanbeconsideredaspotentialcustomerserviceevents.Oncewehaveidentiedtheintervals,wemaythenattempttoassignthemtocustomersanddetermineservicetimes.Itshouldbenotedthatthedistance-versus-timegraphconstructedin[ 77 ]isquitedifferentandisappliedtosolveadifferentproblem. 3.1.2.4ConcerningsmalldistancetrainsWhenthetotaldistancethatatraintraversesissmallrelativetothemotionsthetrainmustmakewhileservingacustomer,thePolylinealgorithmisunabletodeterminewhetherthetrainhasarrivedatacustomer.Asatrainservesacustomer,itperformsaseriesofcomplexbackandforthmotions(orshovingmotions).Forverysmalltrains,thescaleoftheseshovingmotionsmaybeonthesameorderastheoveralldistancethatthetraintravels.ConsidertheimagesshowninFigure 3-7 .Figure 3-7 Aisthedistance-versus-timegraphofatrainwhosetotaldistancetraveledislessthan2miles.Thejaggedupanddownmotionscorrespondtotheshovingmotionsoftheserviceevent.Inthiscase,notonlyarethecorrespondingintervalsofpingsfortheserviceeventsnotat,theyaredistinctlyirregular.Figure 3-7 BshowsthetruecustomerserviceintervalsasdeterminedbyindustryexpertswithknowledgeofthisparticulartrainrouteviewingGPSdata.InattemptstoapplythePolylineAlgorithmtothisdata,wetriedvariousdatasmoothingtechniques,suchasGaussianltering,withoutsuccess.However,itisnotnecessarilyanyparticularrelativespeedordistancewhichconcerns 57

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AGPStrace. BDVTgraph.Figure3-7. A.Distance-versus-timegraphforasmalldistancetrain.Shovingmotionsduringserviceoccuronthesamedistance.B.Thedesiredcustomerservicetimewindows. us,buttherelativedistancetoeachindividualcustomerthetrainisexpectedtovisit.Inparticular,foragivencustomer,weseektoidentifyacollectionoftemporallycontiguouspingsthatarespatiallyclosetothecustomerforsometime.Unfortunately,inbusyyardareas,pingswhicharespatiallyclosetoonecustomermaybeequallyclosetoanothercustomerofinterestaswell.Onceatrainhasnearedtwoclosecustomersthatareservicedsequentially,weexpectthedatatorstshowaseriesofback-and-forthshovingtypemotionsandshortstopscorrespondingtotheserviceoftherstcustomer.Alongermovementtoanew(butverynearby)locationwouldthenbeseen,followedbyasecondsetofshovingmotionsandstopsasthetrainservicedthesecondcustomer.Intuitively,thegeographicallocationsofthepingsduringtherstandsecondserviceeventswillbedissimilar,albeitslightly.Likewise,thepositionsofthepingsduringthesestopswillbedissimilarfromthepositionsofthepingsduringthearrivalanddeparturemovementsofthetrain,aswellasthepingscorrespondingtothebrieftraveltime 58

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betweenthecustomers.Duringcustomerservicetimesweexpecttoseeanincreaseinthe“similarity”ofsubsequentpings.However,thisistrueofthegeographiclocationsofthepingsanytimethetrainslowsorperformssmallermaneuvers.Serviceeventsatcustomersmustbecharacterizedbytheirproximitytoservicedcustomersaswell.Indatamining,thegroupingsimilarobjectsisoftenperformedbyafamilyoftechniquesreferredtoasclusteringalgorithms.InourcasewewanttoclustertheGPSpingsonasetofattributeswhicharethedistancestoeachcustomer,whilepreservingthetimedependencyofthedata.However,traditionalclusteringmethods,suchask-meansclustering,willnotprovidemeaningfulresultshere.Thesemethodsaresuccessfulatidentifyingaprespeciednumberofclustersofsimilardensity(i.e.,dispersionofthepings).WhenidentifyingtheactivityregionsfromGPSpingdata,theshapeofclusterscorrespondingtocustomerserviceeventsareuniquetothegeographyofthecustomerandlengthofthetrain.Also,thenumberoftotalactivityregionswemustdiscoverisnotknownapriori.Traditionalclusteringassignseachpointtoaclusterandmaybeheavilyinuencedbythepresenceofoutliersinthedata.Numerouspingsonatrainroutecorrespondtolocationsfarfromcustomers,andwillnotbeconsideredmembersofanycluster(outliers).ForthesereasonsweidentifycustomeractivityregionsusingthedensitybasedclusteringalgorithmOPTICS[ 3 ].AmodicationtotheDBSCANalgorithm[ 33 ],theOPTICSalgorithmhasbeenhugelysuccessfulinpractice,asitaddressessomeofthelimitationsoftraditionalclusteringmethods.Unliketraditionalk-meansclustering,densitybasedclusteringallowsfortheidenticationofclustersofalllevelsofdispersion/densitysimultaneouslyincludingnonconvexclusters.OPTICSdoesnotrequireaprespeciedvalueofanumberkclusterstobeidentied.OPTICSisrobustinthepresenceofoutliers.Theclusteringofeachobjectisafunctionoftheobject'srelationtoarequiredminimumnumberofneighborsandas(bydenition)outlierspossesfewneighbors,theywillnotbecountedasmembersofanyclusters. 59

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OPTICSalsoprovidesanintuitivevisualinterpretationofthehighdimensionalclusteringstructures.Extractingmeaningfulrelationshipsandpatternsfromtemporallysequenceddata(referredtoasspatio-temportaldatamining),hasbeenapopularnewareaofresearchinthepasttenyears.Numerousmethodshavebeenappliedtosuchdata,andclusteringalgorithmshaveprovenespeciallyeffective.Arecentsurveyonthecurrentstateofresearchofclusteringonspatio-temporaldatabasescanbefoundin[ 56 ].Theauthorsemphasizethenewnessoftheresearch,andpointoutthatmuchofthecurrentresearchhasbeendonetodirectlysolveanindustrydrivenapplicationratherthantodevelopgeneralanalysistechniquesforthistypeofdata.Whiledensitybasedclusteringalgorithms(OPTICSinparticular)havebeensuccessfulinmanyoftheseapplications,thedetailsofeachutilizationareuniquetotheapplication.Severalproblemsutilizingspatio-temporaldataaregivenin[ 56 ],yetweonlymentionthoseinwhichdenseregionsinspace/timeareusedtodeneimportantregions.Theregionsareunknowninitially,andthesolutionstotheseproblemsextenddirectlyfromspatialclusteringalgorithms[ 54 , 74 , 111 ].Inourproblem,whilethecustomerserviceeventsmayappearasdenseclustersinspace/time,withoutclusteringthedataontheproximatedistancetoeachcustomer,theseclusterssimplyrepresentthepointsatwhichthetrainslowedorstoppedrelativetotheothermotionsofthetrain.Forverysmalldistancetrains,therelativemotionofthetrainduringshovingmaneuverswillnotappearstationary,sowecannotapplymethodsrelyingonstops[ 1 ].Evenifthedistancescaleofthetotaltrainmovementislarge,wecouldnotextractdistinctclustersfromoverlappedregionsofinterestwiththesemethods.Wedesireamodelthatleveragestheadditionalinformationwepossessuchascustomerfacilitylocationandgeography,trainlength,andserviceorder,toselectivelyandefcientlyidentifythesespecialspace/timeregions. 60

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3.2ResearchDesignandMethodologyOncevalidworkorder,customergeography,andGPSpingdataareobtainedforasingledayofatrainroute,thedeterminationofcustomerservicedurationsproceedsinthreesteps.First,wedeterminewhethertoapplythePolylineAlgorithmorOPTICStothedata.OtherthantostatethatweusethePolylineAlgorithmforlongdistancetrainsandtheOPTICSalgorithmforshortdistancetrains,untilthispointwehavenotdescribedwhatconstitutesa“long”or“short”train.AfterGPSpingdatahasbeenreceivedintothemodel,wecalculateaquadrilateralboundingboxaroundthelatitudeandlongitudevalues.Thelargerofthetwodiagonallinesbetweentheopposingpointsinthequadrilateraldeneournotionofthedistancetraveledbythetrain.ForthesetofallpingsP,wecalculatethemaximumandminimumlatitudevalues(latmaxandlatmin,respectively),aswellasthemaximumandminimumlongitudevalues(lonmaxandlonmin,respectively).Thefourcornersofthequadrilateralarethe(latmax,lonmax),(latmin,lonmin),(latmin,lonmax)and(latmax,lonmin).IfthelargestdiagonalofthisquadrilateralislessthanTOLPvC=5miles(approximately8000meters)weusetheOPTICSalgorithmtodeterminetheservicetimesforthistrainroute.ThevalueofTOLPvCwasdeterminedexperimentallybycomparingtheoutputsfrombothalgorithmsfor10trainsthattraveledvariousdistancesover6monthsofdata.ForalltheseinstancesboththePolylineAlgorithmandOPTICSwererun,andthealgorithmthatgaveresultsmostcloselyinagreementwiththedeterminationofservicetimes(viavisuallyinspectingthepingdatainGPSsoftware)byindustrypersonnelwithknowledgeofeachtrain'sexpectedserviceroutineswaschosen.ValuesofTOLPvC2f3,4,5,6,7,8,9,10gmilesweretested.TOLPvC=5mileswasfoundmaximizethenumberofinstanceswherethealgorithmthatproducedthedesiredresultswasselected.Infact,outofthehundredsoftestinstancescompared,TOLPvC=5failedtoselectthecorrectalgorithminfewerthan5cases. 61

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Second,weapplytheselectedalgorithm(eitherPolylineAlgorithmorOPTICS)tothedatatodetermineactivityregions.Theseactivityregionsrepresentpotentialcustomerserviceeventsalongthetrainroute.Thirdly,weassignasingleactivityregiontoeachcustomer(ifpossible).Thisassignmentisdoneusingaweightedshortestpathproblemonanacyclicgraph,suchthattheoverallproximityofallcustomerstotheirassignedintervalsisminimized.Atthispoint,ifanyidentiedactivityregionsareunassigned,weattempttoassigntheseactivityregionstosimilaradjacentassignedregions.Lastly,theactualtimeofdayoftheservicestartandendtimesareextractedfromtheassignedactivityregion(s).WenowdescribethePolylineAlgorithmandOPTICS.Pseudocodeisgivenformorecomplexroutineswhereitenhancesunderstandingofthemethodology. 3.2.1PolylineAlgorithmforLongDistanceTrainsWenowsummarizethecreationofthedistance-versus-timegraphusedtoidentifytimeintervalsduringwhichthetrainwasstoppedormoving.Onagivenday,atrain'sGPSmeasurementsprovideatimeorderedsequenceoflatitudeandlongitudecoordinates,seeFigure 3-8 .WewillrefertothissequenceasS.WebeginbyselectingareferencepointPoutsidetherangeoflatitudeandlongitudevaluesinS.Forexample,wehavearbitrarilyselectedapointPwithlatitudeequaltotheminimumlatitudeandlongitudeequaltothemaximumlongitudeobservedinalltheGPSpointsinS.PthenwillalwaysresidebelowandtotherightofallofourGPSmeasurements.Thedistance-versus-timegraphisgeneratedbyrstcalculatingtheHaversinedistance(theshortestdistancebetweentwopointsonthesurfaceofasphere)betweenallpointsx2SandthereferencepointPandthenplottingthesevaluesagainstthetimestampofeachmeasurement.Figure 3-8 Bshowsthedistance-versus-timegraphfortheGPSmeasurementsofanimaginarytrainrouteshowninFigure 3-8 A.ObservetheclusterofpointsinFigure 3-8 Ainthemiddleofthetrain'srouteandagainattheend.Therelativedistancebetweenpointsinthese 62

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AGPStrace. BDVTgraph.Figure3-8. A.GPSlatitudeandlongitudemeasurementsofatrainandtheirdistancefromareferencepointP.B.Distance-versus-timegraph(arbitraryunits)relativetothereferencepointP. clustersandthereferencepointPchangesverylittle.Thisbehaviorshowsuponthedistance-versus-timegraphasavisuallyatspot.Inordertoidentifytheseatregionsofthegraph,wetthegraphwithalinearcurveapproximationknownasasplit-and-mergeorpolylinettingapproachincomputervisualization.Inthisscheme,thecurveisinitiallyapproximatedbyasinglelineconnectingtheendpoints.Theapproximationisthenrenedbyrstidentifyingwhichpointonthecurveliesfarthestfromthecurrentapproximation.Thetisthenupdatedbyreplacingtheinitialapproximationwithtwolinesconnectingtheendpointstothepointlyingfarthestfromtheapproximation.Ifthisprocedurewasrepeatedinnitely,theapproximationwouldexactlytthecurvethroughalloftheGPSmeasurements.Afteranumberofthesesplits,areaswhichmayhavebeenoverttedcanbecorrectedusingvariousmergingalgorithms.WehaveusedtheDouglas-Peuckeralgorithm[ 28 , 51 ]andavertexreductionalgorithm.Suchtechniquesareeasytoimplementandcanbefound 63

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Figure3-9. Thelinearapproximationofthedistance-versus-timegraph. inanyintroductorytextoncomputervisualization;forasurveyofpolygonalsimplicationalgorithmssee[ 50 ].Thelinearapproximationofthedistance-versus-timegraphisalistoforderedlinesegmentsL=fl1,l2,...,ljLjgwhosecontiguousendpointsaredatapointsinthedistance-versus-timegraph.Oftheselinesegments,wewishtoidentifythosesegmentswherethetrainhasmovedslowlyorstopped.Suchsegmentsappearvisuallyatonthedistance-versus-timegraph.Becausethetrainsweconsideroperateonverydifferentgeographicalscalesasinglemetricfor“atness”ishardtodevelop.Sometrainstraversedistancesofnearlyonehundredmilesduringaroutewhileothersdonotventuremorethanonemilefromtheiroriginatingyard.Wescalethedistance-versus-timegraphstovaluesoftimeanddistanceontherangebetween0and10.Thisrescalingallowsustodevelopaconsistentinterpretationof“atness”acrossallofthetrains.Experimentallyitwasdeterminedthatalinesegmentshouldbeconsideredatifitsslopewaslessthan0.8.Howeverifthelinehadadistanceoflonger 64

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then0.5itwasconsideredatiftheslopewaslessthan0.2.Thesevalueswerelargerthanrequiredasthelinesegmentsonthedistance-versus-timegraphwhichcorrespondtothetrainmovinghavequitesteepslopesinpracticeanditisunlikelythatthelinearapproximationofthecurvewouldhavetinsuchawaythataatlinesegmentwouldbemisidentied.Unfortunately,itispossibleforavisuallyatspotonthedistance-versus-timegraphtocorrespondtoatimeintervalduringwhichthetrainismoving.Considerthecasewherethetrainismovinginanarcaroundthereferencepoint;insuchacasethedistancetothereferencepointremainsconstantbutthetrainisclearlynotstoppedorslowed,seeFigure 3-10 . AGPStrace. BDVTgraph.Figure3-10. A.GPSlatitudeandlongitudemeasurementsofatrainlieataconstantdistancefromareferencepointP.B.Distance-versus-timegraph(arbitraryunits)relativetothereferencepointPappearsveryat. Inordertoruleoutsuchcases,selectasecondreferencepointQwithwhichwewillconrmifatregionsinthedistance-versus-timegraphinfactcorrespondtoperiodswherethetrainisslowedorstopped.ForconsistencyweselectQasapointoppositeP 65

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Figure3-11. ThedistancesfromallpointsinthedatatothenewreferencepointQarenotconstant. relativetothedatapoints.Inessence,weusedthetworeferencepointstotriangulatethepointsx2S.DeneDPasthesetofdistancevaluescalculatedfromalltheGPSmeasurementstotheoriginalreferencepointP.Similarly,deneDQasthesetofdistancevaluescalculatedfromalltheGPSmeasurementstoasecondreferencepointQwhichliesoppositeP.OncethepolylinethasbeencompletedonthegraphgeneratedwithDPwehaveasetofcandidatelinesegmentsforwhichweneedtoconrmtheir“atness”.Consideralinesegmentl2Lthathasbeenidentiedasatandwhoseendpointscorrespondtodatapointsisanditrespectively.Iflinesegmentlinfactcorrespondstoatimewhenthetrainstoppedorslowed,thelinesegmentbetweenisandit,callitlQontheseconddistance-versus-timegraphformedusingDQ,shouldbeataswell.IflQisnotalsoatthenweshouldremovelfromourconsideration.Further,ifthelinesegmentslandlQareatonbothgraphswewouldexpectthecovariancebetweenbothsetsofdistancevaluestobeequal.WeusethePitman-Morgan[ 71 , 76 ]testforequality 66

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ofvariancebetweentwodatasetsXandY.ThistestworksbytestingthecorrelationbetweenX)]TJ /F6 11.955 Tf 12.14 0 Td[(YandX+Y.TodetermineifbothlinesegmentsareatweusearobustvariantofthismodiedfromtheBox-Scheffe[ 16 , 86 ]techniquepresentedbyWilcox[ 106 ]oncomparisonsofmethodsfortestingequivalenceofvariancesfordependentdatasets(forourpurposesDPandDQ).Anysegmentlthatwasidentiedinitiallyasatbutthenfailsthistestisnolongerconsideredat.Allremainingsegmentslwhichhavebeendeterminedandconrmedasatmaynowbeconsideredaspotentialserviceintervalsforcustomers.Toestablishwhichintervalscorrespondtowhichcustomers(ifsuchacorrespondencecanbemade)wenowsolveanassignmentproblem. 3.2.2ClusteringAlgorithmforShortDistanceTrainsHereweprovideanoutlineoftheOPTICSalgorithmaspresentedinAnkerst,etal.[ 3 ].Wealsodiscussthechangeswemadewhendeterminingtheneighborhoods,andtheparticulardistancemetricsusedtoidentifypotentialcustomeractivityregions.WebeginwithabriefoverviewoftheOPTICSclusteringalgorithmrstpresentedbyAnkerst,etal.in[ 3 ].Wethendiscusstheparticularchangeswemadetoaddresstheproblemofidentifyingpotentialcustomeractivityregions.Densitybasedclusteringoperatesbyensuringthatforeachobjecttobeconsideredamemberofacluster,theobjectmusthavesomeminimumnumberofneighborsTOLmaxwithinaradiusofsomevalue.Webeginwithtwokeydenitions.Foragivendistancemeasured(o,p)betweentwopointsinthedatasetpanbdowemakethefollowingdenitions.Apointpinthedatasetisacoreobjectifthecardinalityofthe-neighborhoodofpisgreaterthanorequaltoparameterTOLmax.Denepnasthenthpointnearesttop.Denethecoredistanceofpointp,equaltoCD(p)=undenedifphasfewerthanTOLmaxpointswithinanneighborhoodofp,andCD(p)=dist(p,pTOLmax)otherwise.Givenapointpandacoreobjectointhedataset,thereachabilitydistanceofpwithrespecttooisdenedasRD(p,o)= 67

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maxfd(p,o),CD(p)g.Intuitively,RD(p,o)isthedistancebetweenpando,unlessthesepointsareveryclose,inwhichcase,thedistancetotherstpointthatensuresoisacoreobjectisused.TheOPTICSalgorithmcreatesanorderingofalldatapointssuchthatobjectsinthesameclusterareplacedneareachotherintheordering.Ateachiterationthenextunorderedpointwiththesmallestreachabilitydistancevaluewithrespecttothecurrentlycalculatedcoreobjectsisaddedtotheordering.TheOPTICSalgorithm(showninAlgorithm( 1 ))proceedsbyselectinganinitialrandomobjectofthedatasetD.Ifthispointo2Disacoreobject,itbecomestheseedpointforacluster.Ifthe Algorithm1OPTICSalgorithm 1: procedureOPTICS(D,,TOLmax) 2: whilenotallpointsinDare'processed'do 3: Selectanypointo2Dthatisnot'processed' 4: Setstatusofoto'processed.' 5: RD(o) undened 6: ifCD(o)isdenedthen 7: No get-neighborsofo. 8: foreachpinNodo 9: UPDATEVALUES(o,Heap) 10: whileHeapnotemptydo 11: q pointinHeapwithsmallestRD 12: Setqas'processed' 13: AddqtoORDERLIST 14: ifCD(q)isdenedthen 15: UPDATEVALUES(q,Heap) 16: endif 17: endwhile 18: endfor 19: endif 20: endwhile] 21: ReturnORDERLISTandRDvalues 22: endprocedure pointoisacoreobject,OPTICSupdatestheRDvaluesofallitsunprocessedneighborsusingAlgorithm( 2 )UPDATEVALUES.TheseareaddedtotheHeapandthememberofHeapwiththesmallestRDvalue(q)isaddedtotheOPTICSalgorithmorderedlist 68

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ORDERLIST.AnyunprocessedneighborsofqarethenalsoaddedtotheheapandtheirRDvaluesupdatedbyUPDATEVALUES.OnceHeapisempty,andallthe-neighborsofpointohavebeenfathomed,allpointshavebeenaddedtothecurrentcluster.WenowselectanyunprocessedpointinD,untiloneisfoundthatisacoreobject.Thispointnowbeginsthenextcluster,anditsneighboringpointsareaddedproximately. Algorithm2UpdateUnprocessedValues 1: procedureUPDATEVALUES(Pointo,Heap) 2: foreachp2Nothatisnot'processed'do 3: Z maxfCD(o),dist(p,o)g 4: ifRD(p)isundenedthen 5: AddptoHeap 6: endif 7: ifRD(p)>Zthen 8: RD(p) Z 9: endif 10: endfor 11: endprocedure TheoutlineofthisalgorithmisintuitivelysuccessfulwhenthedataisspatialdataandthedistancefunctionusedisametricsuchastheEuclideannorm.TheOPTICSclusteringalgorithmhasalsobeensuccessfullyusedinclusteringtimeseriesdata.Anoftenutilizedapproachtoclusteringtimeseriesdependentdataistotreattimeasafourthspatialdimensionandclusteronitaccordingly.However,asthetotaltimeatrainspendsservingallcustomers,thetotaldistancethetraintravels,andtherelativedistancesbetweencustomerseachchangewitheachprobleminstance,wecouldnotarriveataconsistentandusefuldistancemetric.Weinsteadincorporatedthetimedependencyofthedataintotheneighborhoodcalculations.Consideratrainthatpassesbyacustomertwiceonthesametrainroute:oncetoservicethecustomer,andagain(afterservicingothercustomers)thetrainpausesnearthecustomeratacrossing.Ifweonlyconsiderthespatialdata,alloftheseGPSmeasurementswillbecombinedintoasinglecluster.Intuitively,weunderstandthatthedatashouldonlybegroupedifthepingsareroughlysequential 69

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inthetimeseries.Essentially,wewishtoexcludefromtheneighborhoodofeachdatapoint,anyotherdatapointoccurringmuchearlierormuchlater.Wedenetheneighborhoodofeachpointptobe,Np=fq2Djjp.time)]TJ /F6 11.955 Tf 11.96 0 Td[(q.timejg.Weusethevalue=15minutes,whichisabouttheleastamountoftimeneededtoserviceasinglecustomer.Iftwoproximatedatapingsseparatedbyalargespanoftimeshould,infact,beconsideredpartofthesamecluster,whentheintermediarypingsareaddedtotheclusteronebyone,theneighborhoodofthelateroneswilleventuallyincludethepingseparatedfarintime.Thedistancemetricweusedcannowdependonlyonthespatialcomponentsofthedata.However,wearenotclusteringstrictlyonthespatialvaluesofthepings(latitudeandlongitude)asistraditionallythecase.Weinsteadtransformeachpingintoanorderedvector.IfpingptoccursattimetwedenethevectorofsizejCjlabeledvt=fdmin(pt,c)gc2CwhereCisthesetofallcustomers.Thefunctiondmin(p,c)istheminimumHaversinedistancebetweenthepointpandtheserviceregionforcustomerc.Soeachelementofvtisvtcforallc2C.FortwoGPSdatapointspt1andpt2,thedistancemetricusedintheOPTICSalgorithmcustomeralgorithm,isthendenedasdist(pt1,pt2)=jvt1cMvt1cMj,wherecMistheMthcustomerontheworkorder,andM=argminc2Cvt1c.Notethatifpt1isfarfromitsnearestcustomer,thevalueofthedist(pt1,pt2)islargeasvt1cMislarge.Thevalueofthedist(pt1,pt2)isalsolargewheneverpt2isfarfromthenearestcustomertopt1.Thevalueofdist(pt1,pt2)isonlysmallwhenbothGPSpingsareneartothecustomer.TheonlyresultingclustersderivedwiththisdistancemetricthathavelowRDvalueswillappropriatelybethosewherepingsarelocatednearrelevantcustomers.Thereachabilityvaluesofeachpointp2DgraphedversustheopticsorderingORDERLIST,createwhatisreferredtoasthereachabilityplot.Asthereachabilityvaluesofanypointareessentiallyameasureofthepoint'sminimumdistancetothosepointsplacedpriortoitselfinORDERLIST,denseregionsofclusteredpointswillappear 70

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Figure3-12. MinimumdistancefromeachpingintheGPStracetothreecustomergeographicregions. togetherinthereachabilityplotwithlowreachabilityvalues.Theclustersarethusvisibleasdipsinthegraph.Figure 3-12 showsthreegraphscorrespondingtotheminimumdistancefromeachGPSdatapingtothecustomerserviceareaofeachofthethreecustomersforagivendayofatrainroute.ThisisthereachabilityplotforthedistancedatagiveninFigure 3-7 .Figure 3-13 showstheresultingreachabilityplotfromthe 71

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Figure3-13. Reachabilityplotforthethreecustomerexample. applicationoftheOPTICSalgorithmtothisthreecustomerexample.Theoriginaltimeorderingofthesequencehasbeenperturbedtoplacedatapointssimilartoeachotherinthesamecluster.Thethreedipsinthegraphcorrespondtothepotentialactivityregionforeachofthethreecustomers.Inthisexample,theeventualassignmentofcustomerstoclustersisobvious,astheclustersappearinthesequencegivenonthetheworkorder.Notethatthisvisualrepresentationoftheclusteringstructureisindependentofthedimensionalityoftheclustereddata.Thus,OPTICSalwaysobtainsanintuitivetwodimensionalrepresentationregardlessofthedimensionorcomplexityofthedata.OneofthemostpowerfulattributesoftheOPTICSclusteringmethodisthereachabilityplot'srelativeinsensitivitywithregardtotheinputparametersneighborhoodgeneratingdistanceandthevalueofTOLmax.Aslongastheseparametersare“large 72

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enough,”theOPTICSalgorithmcangeneratemeaningfulresults.Theneighborhooddistanceinuencestherelativedensityofclustersthatcanbeidentiedvisually,asasmallerincreasesthenumberofpointswithundenedreachabilityvalues.LargervaluesofTOLmaxhavea“smoothingeffect”onthereachabilitygraph.Yet,aTOLmaxvaluethatistoolargecanreducethevisibilityofcertainclustereffects.TheautomaticclusteringextractionalgorithmweuseisbasedonthealgorithmpresentedbySanderetal.[ 83 ].Theextractionalgorithmcreatesahierarchicalorderingofthereachabilityplotintheformofatree(ordendogram).Thenodesofthistreerepresentclustersinthedata.Nodesfurtherdownthetreedepictthesmallestanddensestclustersinthedata,whileatthetopofthetree,therootnoderepresentstheentiredataset.Theleafnodesoftheclusteringtreerepresenttheclusterspotentiallyassociatedwithcustomeractivityregions.Oncetheclusteringtreeisconstructed,weextractthetimeintervalsforeachpotentialactivityregionfromthepointsineachleafnodecluster.Wethenattempttoassigntheseclusterstothecustomerserviceeventslistedontheworkorder.Visually,thereachabilityplotallowsustoidentifydistinctclustersasregionsofpointswithlowreachabilityvalues,separatedby“spikes”ofrelativelyhighreachabilityvalues.Thus,therststepinextractingclustersistheidenticationofthesesignicantlocalmaximawithinthereachabilitygraph.Identifyingthelargestofthelocalmaximaprovidesaroughsegmentationofthereachabilitygraphintolargeclusters.Withintheselargeclusters,morenelygrainedclustersmaybeidentiedbytheinspectionofthesmallerlocalmaxima.Weaddadditionallogictotheextractionalgorithmforourparticularproblemofdeterminingthecustomerserviceactivityregions.Toreducethenumberoflocalmaximaconsidered,weremovedlocalmaximawithlargereachabilityvaluesthatcouldnotrepresenttheboundaryofacustomerserviceactivityregion,eveniftheydidrepresenttheboundaryofaclusterinthereachabilitygraph.Weaddedenhancedlogic 73

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whendeterminingthesignicanceoftheclustersidentied.Wealsoaddadditionalpostprocessinglogictocorrectunderestimatedclustersnestedinlargerclusters.Theclusterextractionalgorithm,EXTRACT,showninAlgorithm 3 ,beginsbyndingalllocalmaxima(see,line 4 )ofthereachabilityplotvaluesRD.Tobeconsideredalocalmaximum,theRDvalueofapointmustbegreaterthantheRDvalueofthepointswithinaneighborhoodofsizetotheleftandrightofthepoint.Forthosepointswithinofthebeginning(end)ofthedataset,theRDvalueiscomparedonlywiththosepointswhichexisttothepoint'sleft(right).Thevalueofisselectedtobelargeenoughtoavoidcountingnumerousadjacentlocalmaximumsinnoisyregionsofthedata,andalsosmallenoughthatanysmall,yettrue,clustersareidentied.Therelativelycloseproximityofcustomersinthelocaltraindatameansthatcertainoftheclustersweidentifymay,infact,bequitesmall.Becauseofthisweselecttobethelargerofeithermin,or%ofthetotalnumberofpointsinthedataset.Wesetmin=2,and=0.1%.Thisisnotdissimilartothevalueof0.5%usedinSanderetal. Algorithm3AutomaticClusterExtraction 1: procedureEXTRACT(RD) 2: R sizeofRD. 3: maxfjRDj,ming.Neighborhoodsize. 4: L fp2RDjRD(p)>RD(x)8x2[p)]TJ /F4 11.955 Tf 11.95 0 Td[(,p+]g..MaximaList 5: CreateRootnode. 6: Root.start 1,Root.end R,Root.parent null 7: BUILDTREE(Root,L).Recursivelybuildtree. 8: Eliminatenestedclusterswithoutsiblings. 9: endprocedure Duringtheidenticationofthelocalmaxima,weemployadditionallogicuniquetotheproblemofndingcustomeractivityregions.Becauseweseekclustersofpingsveryneartothedenedcustomerserviceareas,anyclusterweeventuallydeterminetobeacustomeractivityregionwillhaveRDvaluesveryneartozero.However,ifanysequenceofpingsoccursinanintervaloftimehassimilarlatitudeandlongitudevalues,itwillstillbeidentiedasaclusterinthereachabilityplot,regardlessoftheirdistanceto 74

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Figure3-14. ClustersappearatlargeRDvalues,suchasbetweenpointspandqandpointototheendofthereachabilityplot.Inthisexample,allpointsgreaterthan1106(indicatedbythedashedline)otherthanp,q,ando,areexcludedfromthelistoflocalmaximums.Notethatpandomaydenetheendpointsofclusters.Likewise,pointqmaydenethebeginningofacluster. thegeographiclocationofthecustomersontheworkorder,asseeninpointspandqofFigure 3-14 .SuchaclusterwithverylargeRDvalueswouldneverbeassociatedwithanactualcustomeractivityregion,althoughitdoesappearonthereachabilitygraph.AsweknowthatanypointwithaverylargeRDvaluewillneverbeincludedinacustomeractivityregion,weexcludeanypointpwithRD(p)>RPmax,RD(p+1)>RDmax,andRD(p)]TJ /F5 11.955 Tf 12.14 0 Td[(1)>RDmaxfromconsiderationasalocalmaximum.Figure 3-14 showsthatforsubsequentpointswithRDvaluesgreaterthanRDmax,onlytherightmostpointshouldbeconsideredalocalmaximumthatmaybeginacluster,andonlytheleftmostpointshouldbeconsideredtoendacluster.Inpractice,RDmax=1106.Suchalargevalueensuresconsideringallpointswhichtrulybeginclusters,andexcludingallintervalswithRDvaluessubstantiallyfarfromzero. 75

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Oncethelistoflocalmaxima(L)hasbeenobtained,atreerepresentingthehierarchicalorderingofclustersiscreatedviarecursion.Initially,arootnodeiscreatedrepresentingallpointsofthedatastructure(see,line 5 ofAlgorithm 3 ).Thisrootnodeandtheentirelistoflocalmaximaarepassedintotheroutine,BUILDTREE,denedinAlgorithm 4 .BUILDTREEiscalledrecursively.ItselectsalocalmaximumfromL,andaddsanodeintothetreeasaleft(right)childofthecurrentnodeifthepointstotheleft(right)ofthismaximumdeneasignicantcluster.TheinputstoBUILDTREEareanodeNthatcorrespondstoanintervalofpointsonthereachabilityplot,andalistoflocalmaximaL.InBUILDTREE,ifthelistoflocalmaximaisnotempty,thelocalmaximumwiththelargestRDvalueisselected.Thismaximumissavedass,andremovedfromL.ThismaybeaccomplishedinconstanttimeifthelocalmaximumlistissortedinEXTRACTpriortocallingBUILDTREE.WeexcludelocalmaximawithRDvaluessmallerthanm=0.003(seeline 5 ofBUILDTREE)toavoidcreatingnodesthatrepresentclustersinnoisy,near-zeroregionsofthereachabilityplot.Atthispointintherecursivefunctiontwopotentialnodesarecreated:Nl,andNr.NodeNl(Nr)containsthepointsinNtotheleft(right)ofs(theselectedlocalmaximum)inthereachabilityplot.Wealsopartitionthesetoflocalmaximaintothosepointstotheleftandrightofs:Ll=fx2LjRD(x)RD(s)g,andLr=fx2LjRD(x)RD(s)g.ListLl(Lr)ispassedintoBUILDTREEifitisdeterminedthatnodeNl(Nr)shouldbeaddedtothetree,andthusrecursedupon.BeforeaddingthenewnodesNl,andNrtothetreeaschildrenofN,wemustdetermineifthesenodesrepresentsignicantclusters.Todetermineifamaximumissignicantrelativetothesurroundingpointstodeneacluster,Sanderetal.comparethepercentincreaseinRDvalueofsrelativetotheaverageRDvaluesofthepointsintheneighborhoodsurroundingit.ThecomputationsinSanderetal.enforcethattheRDvalueatsmustbeatleast75%greaterthantheaverageoftheneighboringpointstodeneacluster,andtheyrecommendtheuserselectapercentcutoffbetween70-80%. 76

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Algorithm4AutomaticClusterExtraction-BuildingtheTree 1: procedureBUILDTREE(L,N) 2: ifLisemptythenReturnendif 3: s largestpointinL.Getthenextlocalmaxima 4: L L)-222(fsg 5: ifRD(s)=2andminRDl=RD(s)>=2then 19: BUILDTREE(MList,N) 20: Return. 21: elseifminRDl=RD(s)>1orminRDr=RD(s)>1then 22: ifminRDl=RD(s)<2then 23: avgRDr avgRDonleftmost50%ofNr 24: ifavgRDr=RD(s)>1thenuser falseendif 25: endif 26: ifminRDr=RD(s)<2then 27: avgRDl avgRDonrightmost50%ofNr 28: ifavgRDl=RD(s)>1thenusel falseendif 29: endif 30: endif 31: ifSizeNl
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Thesevaluesarebasedonwhatarationalobserverofthereachabilityplotwouldperceiveastheminimumrequireddifferenceinheightstowarrantaseparation.Wehaveexpandedthistestforsignicantmaximafortheproblemofidentifyingcustomerserviceactivityregions.ByusingthetimecomponentofthepingdatatorestricttheneighborhoodsintheOPTICSalgorithm,theresulting“valleys”ofpointsonthereachabilityplotscorrespondingtoourdataappearlesssharplydenedthanthesamplereachabilityplotsshownin[ 3 ]and[ 83 ].Thesidesoftheclustersinthereachabilityplotsareoftengentlysloped.TheneteffectisthatindeterminingiftheheightoftheRDvaluesofthecurrentlocalmaximumsissignicantrelativetotheaverageofallpointstotheleftandrightontheintervaldenedbythecurrentnode,wemayincorrectlyeliminates.ConsidertheexampleshowninFigure 3-15 thatillustratesanintermediaterecursionofBUILDTREE.InFigure 3-15 ,pointsaandbarethestartingandendingpointsthatdenethecurrentnodeoftherecursion.Pointcisthelocalmaximumfortheiteration.Visually,pointcclearlyseparatestwoclustersbetweenpointsaandb.However,thehighRDvaluesinbetweenpointaandpointcmayimplythattheaverageoftheRDvaluesdividedbyRD(s)>0.75.Toavoidfailingtoidentifydistinctclustersinsuchascenario,werstndthelocalminimumtotheleftandrightofs(minRDlandminRDr).However,wedonotndthesetwolocalminimawithrespecttotheentiresetofpointsspannedbyN.WhendeterminingminRDl,wendtheminimumRDvalueoverthe80%ofpointsclosesttosinNbetweenN.startands.Likewise,minRDr,istheminimumRDvalueoverthe80%ofpointsclosesttosinNbetweensandN.end.IftheratioofeitheroftheselocalminimawithRD(s)exceedstheparameter2=0.75,weskipsandthealgorithmrecursesonthenextlargestlocalmaximuminL.Wedonotconsiderthelocalminimumontheentireintervalbetweensandtheotherintervaldeningendpoint,asthisensuresthattheheightofRD(s)appearsprominentlylargerthanitsnearestneighboringpoints.ThishelpstoavoidcountingsasalocalminimumincertaincaseswheretheRDvaluesofthepointsrisewithasteadysmallslopeacross 78

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Figure3-15. ThepointswouldnotbefoundtobeasignicantseparationpointfortwoclustersifthemeasureofsignicanceisdenedbytheratioofRD(s)andtheaverageRDvaluesbetweenbothstartands,andsandend.Toavoidexcludings,considerstobesignicantifRD(s)is75%greaterthanthelocalminimumon80%ofpointstotheleftandrightofs. theentireinterval.Ifasingleoneoftheseratiosexceedsaparameter2=0.8,wedetermineifbothnodesshouldbeaddedtothetree.WenowcalculatetheaverageRDvalueforthe50%ofpointstotheleftofs,and50%ofpointstotherightofsonN.DeneavgRDlastheaverageRDvalueoverthe50%ofpointsclosesttosinNbetweenN.startands,andavgRDr,astheaverageRDvalueoverthe50%ofpointsclosesttosinNbetweensandN.end.WethencomparetheseaverageRDvalueswiththeRDvalueats.IfminRDl=RD(s)<2,yetminRDr=RD(s)<2,noclusterisfoundtotherightofs,andwedonotaddnodeNrtothetree(seeline 24 ).WesimilarlyexcludenodeNlfromthetreeifnoclusterisfoundtotheleftofs(line 28 ).Thisenhancedlogicforcheckingthesignicanceoflocalmaximahelpstoensurethatallclustersrepresentingpossiblecustomeractivityregionsarefound. 79

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Followingthechecksforsignicance,weperformonenalcheckbeforeaddinganynodestothetree.ThealgorithmnowcheckstoensurethatanyclustersfoundbysplittingthepointsrepresentedbythecurrentnodeNatscontainatleastTOLn=5points(seeline 31 ).IfbothnodeNlorNrwereexcludedfromthetreeforeitherfailingtobesignicantorhavingfewerthanTOLnpoints,sisskippedandwecontinuetherecursionwiththenextlargestlocalmaximuminL(line 33 ).ThenalstepoftheBUILDTREEmethodistoautomaticallyprunethecurrentnodeNifitshouldbereplacedentirelybythenewlyaddedchildnodesinthetree.InSanderetal.theRDvalueofthecurrentsplitpointsfornodeNiscomparedwiththesplitpointusedintherecursionfornodeN.parent.IftheRDvalueofsiswithinonestandarddeviationofthemeanofanormaldistributionwithameanequaltotheRDvalueoftheparent'ssplitpoint,andstandarddeviationequalto1,thenthecurrentnodeandtheparentnodearedeemedtobeapproximatelythesame.ThenewchildnodesNlandNrareaddedaschildrenofN.parent,andnodeNisremovedfromthetreeentirely.Figure 3-16 ,istheexamplegiveninSanderetal.whichillustratesthatwithoutpruning,thebinarytreethatresultsfromtheEXTRACTproceduremaynotbethedesiredrepresentationoftheunderlyingclusteringstructure.IfnodeNl(Nr)hasnotbeenexcludedfromthetree,wenowrecursivelyconsidertheclusterswhichmaybeaddedasitschildnodesinthetreebypassingNlandLl(NrandLr)toBUILDTREE.OncetheBUILDTREEhasfathomedallpossiblechildnodesofthetree,exhaustingthelistoflocalmaxima,theclusteringtreeiscompletelydened.Becauseatrainmayserviceacustomeratoneormorerelativelyproximatedistancestothecustomer'sgeographicregion,theright-handsideoftheclusterinthereachabilityplotforthiscustomeractivityregiontendstobegraduallysloped.ThosepointswithRDvaluesequaltozeroareplacedrstintheorderingofthereachabilityplot,followedbythosewithslightlylargerRDvalues,asseeninFigure 3-17 .DuringthecreationoftheclusteringtreeinBUILDTREE,whennodePissplitatlocalmaximums, 80

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AReachabilityplot. 0 4 6 3 5 1 2BNopruning. 0 3 5 6 1 2CPruning.Figure3-16. AnexamplesimilartothatgiveninSanderetal.showingtheautomaticpruningofnodes.A.Thereachabilityplotfortheexample.B.Theclusteringstructureobtainedwithoutpruning.C.Theclusteringstructureobtainedafterpruning.AnodewithasplitatanRDvaluesimilartothatoftheparent'ssplitvaluemaybereplacedwithchildnodes,creatingthecorrectrepresentationoftheclusteringstructureinthereachabilityplot. 81

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Figure3-17. Whenlocalmaximumswasconsideredduringthecreationofthetree,insignicantclusterXwasnotincludedinthetree.However,clusterAmaybeanunderestimateofthetotalactivityregiondenedbyP. thepointstotherightofsdonotestablishasignicantcluster(X).Thusnoright-handchildnodeisaddedintothetree.TheleftnoderepresentingthesignicantclusterAisaddedtothetreeasachildofP.However,itisvisuallyclearthatthelargerclusterPcorrectlydenesthiscluster.Whenweextractthetimerangespannedbythepointscontainedintheleafnodesfortheclusteringgraph,nodeAprovidesanunderestimateoftheactivityregion.InprocedureEXTRACTatline 8 ,weapplypostprocessinglogictoadjustthewidthofsuchnestedleafnodes.Thisisperformedstraightforwardlybyrecursivelytraversingtheentireclusteringtree,andexaminingallleafnodesofthetree.Ifaleafnodenistherightmost(andoften,only)childofitsparent,thenweadjustn.endtoequalton.parent.end.Thiscorrectstheunderestimateofthewidthofthecluster.Notethatitisnotappropriatetoremovetheparentnodefromthetree.Theparentmayhaveother(lefthand)children.Thus,itsinclusionprovidesthecorrectconstructionofthehierarchicalclusteringtree. 82

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3.2.3AssignmentofStopIntervalstoCustomersNowthatthestopintervalshavebeenidentiedfromthePolylineAlgorithmorclustersfromtheOPTICSalgorithm,wedeterminewhichofthosecorrespondtoactualcustomerserviceevents.Althoughtheworkordercontainsasequentiallistofcustomersservicedonthatday,theassignmentoftheseintervalsorclusterstocustomersiscomplicated.Duetodataentryissuestheworkordermaybeincorrectlysequencedormayevencontaincustomersthatwerenotserviced.Thus,itmaynotbepossibletoassignanactivityeventtoeachcustomergivenontheworkorder.Inaddition,sincetrainsmaymakesmallmovementsduringtheserviceofacustomer,theassignmentshouldallowformultiplestopintervalstocorrespondtoasinglecustomer.Tosolvethisproblem,weconstructaweightedgraphwherenodescorrespondtopotentialactivity-region-to-customerassignmentsandtheedgeweightsareafunctionoftherelativedistancebetweenthepointsthatdenetheactivityregionsandthecustomer.Weuseanassignmentofnulltoindicatethatnoactivityregionwasfoundcorrespondingtoacustomer'sservice.Weconstructthegraphweightssuchthatitisalwayspreferentialtomakeanassignmenttoeachcustomerratherthanleaveacustomerassignedtonullifpossible.Multipleassignmentsareaccountedforbyrstassigningeachcustomeronlyonestopinterval,andthenassigningsimilaradjacentunassignedintervalsduringpostprocessing.Inordertoaddresscomplicationsintheassignment,twoassumptionshavebeenmade.First,weassumethattheworkorderprovidesthecorrectorderingofthecustomersservicedthatday.Whileaseeminglyminorassumption,customerservicetimesmaybeincorrectlycalculatedifdataentryerrorsprovesuchanassumptionfalse.Insomecases,however,itiseasytoidentifywhencustomershavebeenservicedoutoforderandthecorrectservicetimeassignmentscanstillbemade.If,forexample,atrainservicesitscustomersinlongstoppedintervalsdirectlyinfrontofthecustomerlocation,andthetraindoesnotrecrosssectionsoftrackduringthedaythenerrorsinthe 83

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workorderareeasytoidentify.Forthiscase,thenumberofstopsandcustomersarethesameandanassignmentcanbemadebasedontheneareststoptoeachcustomer.Unfortunately,thisidealcaserarelyoccursinpractice.Duringeachdaythenumberofstopintervalswhichcouldrealisticallybeassignedtocustomersislarge.Astrainscommonlyadjusttheirpositionduringtheserviceofacustomer,itisoftenappropriatetoassignseveralsequentialstopstoacustomer.Inadditiontostopsassociatedwithpositioning,trainsoftenmakeotherstopsthatareunaccountedforontheworkorder,someofwhichmaybenearcustomerlocations.Theseotherstopsareparticularlycommonindenseurbanandsuburbanareaswheretrainsareoftenrequiredtostopduetoright-of-wayrestrictionsduetootherrailorautomobiletrafc.Itisalsocommonforlocaltrainstobeginandendserviceatthesameyard.Thisleadstothetraindoublingbackovertrackithadpreviouslytraversed,passing(andoftenslowingdownorstoppingat)customersithasalreadyservicedfortheday.Duetotheseissuesitisunlikelythatareliableassignmentcanbemadeifthelistofcustomersontheworkorderisnotobserved.Thenextassumptionisthatanycustomersthatwereservicedinfactappearontheworkorder.Itispossiblethatadataentryerrormayresultinacustomer'somissionfromtheworkorder.Also,assomecustomershavemultiplefacilities,itispossiblethatthewrongcustomerlocationwasrecorded.SincethenumberofCSXcustomersisverylargeitisimpossibletomakeavalidassignmenttoacustomernotlistedontheworkorder.However,sinceallstopintervalsareidentiedinourprocedureitispossibletolaterinvestigatestopintervalsthatwerenotassignedtoacustomer.Withtheseassumptions,theassignmentofstopintervalstocustomers(allowingsomecustomerstobeunassigned)canbedeterminedbysolvingashortestpathproblemonaacyclicdirectednetworkwithpositiveweights.Foragiventrainandday,considerthesetofdiscoveredactivityregionstobeI=fi1,i2,...,ijIjg2L,andtheorderedsetofcustomersrecordedontheworkorder 84

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C=fc1,c2,...,cjCjg.Inordertoquantifyhowclosecustomerc2CistotheseriesofGPSmeasurementscorrespondingtoactivityregioni2Iwedeneascorewci.ActivityregionsofGPSmeasurementsthatlieclosertoacustomerwillhavesmallerscoresthanthosefurtherfromthatcustomer.Notethattheappropriateassignmentofcustomerstointervalsisnotnecessarilytoassigncustomerctheintervalwiththesmallestwci.Whencustomersareclosetooneanother,itispossiblethatasinglestopintervalmaybetheminimalassignmentofseveralcustomers,yetthereareothernearbyintervalswhicharerelativelyclosetothecustomerlocationsaswell.Weformulatethisassignmentasashortestpathproblemwitharccostswci.Foractivityregioni2IderivedfromtheOPTICSalgorithm,thevalueofwciforc2Cisdenedby, wci=Pi2I(c,)dist(i,c) jI(c,)j,(3)wheredist(j,c)istheHaversinedistancebetweenthejthGPSpingandthecustomerserviceareaforcustomerc.And,I(c,)=fi2Ijdist(i,c)dist(i,c)gfori=djIje,orthesmallestpercentoftheinterval.Weused=0.25inpractice.IncaseswheretheOPTICSalgorithmhasbeenused,havingallpointsoftheactivityregiondeterminetheweightforthecustomerassignmentgraphmaynotbeeffective.Whenallcustomersareclosetooneanother,manyofthepointsintheactivityregionclustermayrepresentshovingandarrivaldeparturemotionsofthetrain.Intuitively,weselecttheweightsoftheassignmentproblemtodependonthosepointswhichareasclosetothecustomeraspossible.Whiletheserviceregionsofcloselyplacedcustomersareinallcasesmoredifculttoidentifythanthosefurtherspacedcustomers,theassignmentproblemisoftensimplerwhenOPTICSisused.BecauseOPTICSallowsGPSpingdatatobereordered,thealgorithmisabletoidentifyasingleclusterforeachcustomerwhenthosepingsclosesttothecustomerareplacedtogetherintheordering.Thepolylinealgorithmmayidentifynumerousintervalsforasingleserviceevent.Onlyoneoftheseintervalswillbe 85

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associatedwiththecustomerduringtheassignmentproblem.Theotherintervalsmustbeassignedinpostprocessingbasedontheirsimilaritytotheselectedcustomeractivityregion.WhentheactivityregionswereobtainedbythePolylineAlgorithm,customerswhowereservicedsequentially,andwhosegeofencesoverlapatanypoint,weregroupedintoasinglelargercustomer.WerepresentthesetofcustomersnowbythesetofgroupsG=fg1,g2,...,gjGjgwhereG=[c2Cc.Consideragroupofoverlappedcustomersofsizenbeginningatcustomercg.Theoverlappedcustomersfromgroupg=[ni=gci.Thegeographicdenitionofthisrepresentativecustomeristheunionofthegeographicregionsofallitscomponentcustomers.Inthiscase,asingleweightvalueforeachactivityregioni2I,wgi,isgivenforallcustomersofthegroupg2G.Oncetheassignmentofactivityregionshasbeenmadetoeachgroup,thetotaltimecanbeapportionedtoeachindividualasperbusinessrulesinpostprocessing.TheweightisanexponentialfunctionoftheaverageminimumHaversinedistanceofallpingsontheintervalitothetraceofallcustomersinthegroup.Or, wci=exp(1 t)]TJ /F6 11.955 Tf 11.96 0 Td[(s+1tXj=sminfdist(j,c)g),(3)wheredist(j,c)istheHaversinedistancebetweenthejthGPSpingandthecompositecustomerserviceareaforallcustomersingroupc.IntervalsofGPSmeasurementsthatlieclosertoacustomerwillhavesmallerscoresthanthosefurtherfromthatcustomer.Becausetheweightisexponentialinnature,intervalsclosetoacustomerarefavoredmuchmorehighlythanthosefartheraway.Withtheexceptionofweightsandpenalties,theshortestpathproblemusedtodeterminetheassignmentofactivityregionstocustomersisthesameregardlessofwhethertheactivityregionswereidentiedbythePolylineAlgorithmorOPTICS.Thus,toavoidconfusion,wewillhenceforthidentifygroupsofcustomersasasinglecustomer. 86

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(1,X) 0 (1,1) 0 "" S P1 EE w11 << w12 // w1jIj (1,2)0 // T...(1,jIj) 0 FF Figure3-18. Solvingtheassignmentasashortestpathproblemforasinglecustomer. Forexample,wedenoteallgraphweightsaswcifori2Iandc2c,althoughinthePolylineAlgorithmcasecisagroupc2G.Thegraphonwhichwesolvetheshortestpathproblemisconstructedinlayers.Therstlayerofnodesconsistsof(1,j)forj=1,...,jIjcorrespondingtothepossibleassignmentoftherstcustomerc1tointervalij2fi1,i2,...,ijIjgand(1,X)correspondingtothepossibilityofc1notbeingassignedtoanyinterval.WeconnecteachofthesenodestoadummystartnodeSwithdirectedweightedarcs.ThearcfromnodeStonode(1,j)willhaveweightw1jforj=1,...,jIjandtheweightofthearcfromSto(1,X)willbethepenaltyP1whichisincurredforfailingtoassignc1toacustomerserviceevent.Ifthereisonlyasinglecustomerontheworkorder,weaddadummyendnodeTandconnectnodes(1,X)and(1,j)forj=1,...,jIjtoTbydirectedarcsofweightzero;seeFigure 3-18 .Theassignmentproblemistrivialinthiscase,butcannowbesolvedbyndingtheshortestpathfromnodeStonodeTonthegraph.ThepenaltytermPcisaquantitativemeasureofthefailuretoassigncustomerctoanyinterval.Intuitively,Pcmustbelargeenoughtoensurethatifapotentiallyvalidintervalexiststhecustomerwillbeassignedtothatinterval,yetsmallenoughtoensureintervalsfarfromthecustomer'sserviceareawillnotbechosenasvalidservicetimes. 87

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WiththisinmindthevaluePcshouldrepresentthedistancebeyondwhichintervalij2fi1,i2,...,ijIjgisnolongeravalidstopintervaltobeassignedtocustomerc.WethenseethatPcshouldbedependentonthelengthofthetrain.WhenatrainisshortweexpecttoseetheclusterofGPSmeasurementswhichrepresenttheservicestoptobeclosetothecustomerservicearea.WhilealongtrainmayresultintheGPSreadingsoccurringfarfromthecustomerastheGPSdeviceishousedinthelocomotiveofatrainwhichisthefarthestpointfromtheactualservicelocation.Intuitively,regardlessofthetrainlengththepenaltyfunctionshouldnotexcludeintervalsveryneartoanycustomers.Becauseofthis,foratrainlengthlessthan20cars,thepenaltytermisxedtoalwaysconsiderintervalswithin420meters(thelengthof20railcarsand3locomotives)ofthecustomerservicearea.Additionally,thepenaltyfunctionisintuitivelyconstantforverylongtrainlengths.Inthiscase,aportionofthecarswouldlikelybeleftelsewhereonthemainlinetrackandmovementswouldbemadewithatrainconsistingoffewercars.Thepenaltyfunctionwasfoundexperimentallyandconformstotheseintuitions.Thepenaltyfunctionusedhereis: Pc=8>><>>:e0.74194derivedfromOPTICS,0.0024)]TJ /F7 7.97 Tf 6.58 0 Td[(0.475(DCSA(c)+L)f(),derivedfromPolylineAlgorithm(3)wheref()=0.104log101.35)]TJ /F5 11.955 Tf 12.49 0 Td[(()]TJ /F5 11.955 Tf 12.49 0 Td[(0.25)2)]TJ /F5 11.955 Tf 12.49 0 Td[(85andisthescalingtermthatwasusedtoscalethedistancevaluesofthedistance-versus-timegraphbetween0and10inthePolylineAlgorithm.InthePolylineAlgorithm,iftheworkordershowsthatmultiplecustomerswereservedatthesamelocation,DCSA(c)issettobethemaximumdistancebetweenanypointsoftheseoverlappingcustomerserviceareas.DCSA(c)iszeroifthecustomercdoesnotoverlapwithanyothercustomers.ThetermLis420metersifthelengthofthetrainwhenvisitingcustomercislessthan20andisequaltothetotallengthofthetrainotherwise. 88

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(1,X)P2 // w21 (( w22 w23 w24 (2,X)P3 // w31 (( w32 w33 w34 (3,X) 0 (1,1) P2 66 w22 (( w23 w24 (2,1) P3 66 w32 (( w33 w34 (3,1) 0 "" SP1 EE w11 << w12 // w13 "" w14 (1,2)P2 >> w23 (( w24 (2,2)P3 >> w33 (( w34 (3,2)0 // T(1,3) P2 CC w24 (( (2,3) P3 CC w34 (( (3,3) 0 << (1,4) P2 FF (2,4) P3 FF (3,4) 0 EE Figure3-19. Anincorrectlyformulatedgraphforsolvingtheassignmentasashortestpathproblemforthreecustomerstofourpotentialserviceintervals. Whentheworkorderindicatesmorethanonecustomerserviced,werepeatthesameconstructionofjIj+1nodesforthesecondcustomernowwithlabels(2,X)and(2,j)forj=1,...,jIj.Duetoourassumptionthattheworkorderhascustomersinthecorrectorder,iftherstcustomerhasbeenbeassignedtointervalij,thesecondcustomermustbeassignedtosomeintervalini2fij+1,ij+2,...,ijIjgifitisassignedatall.Thusthearcweighta(1,j),(2,k)fromnode(1,j)tonode(2,k)is a(1,j),(2,k)=8>><>>:w2,kifk
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theorderspeciedontheworkorder;however,inthisconstructioncustomer3couldbeassignedtointerval1insteadofinterval3ifw31><>>:wc+1,kifk
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(1,X)P2 // w21 ** w22 w23 w24 (2,X)P3 // w31 ** w32 w33 w34 (3,X) 0 (1,1) P2 ** w22 $$ w23 w24 (2,1) P3 ** w32 $$ w33 w34 (3,1) 0 (2,X)1P3 // w22 ** w23 w24 (3,X)1 0 "" S P1 II w11 EE w12 // w13 w14 (1,2) P2 ** w23 $$ w24 (2,2) P3 ** w33 $$ w34 (3,2)0 // T(2,X)2P3 // w33 ** w34 (3,X)2 0 << (1,3) P2 ** w24 $$ (2,3) P3 ** w34 $$ (3,3)0 EE (2,X)3P3 // w34 ** (3,X)3 0 HH (1,4) P2 ** (2,4) P3 ** (3,4) 0 JJ (2,X)4P3 // (3,X)40 KK Figure3-20. Thegraphforsolvingtheassignmentasashortestpathproblemforthreecustomerstofourpotentialserviceintervals. cars,tomovingwhenswitchingtracks,tostoppedasnewcarsareadded,andnallytomovingasthetrainleavesthecustomer'slocation.Duringthisservice,thetimespentstoppedbythetrainduringthedroppingoffandpickingupsegmentsoftheservicewouldlikelyshowupas2separateatintervalsonthedistance-versus-timegraph.Ofthesetwostopintervals,theassignmentproblemwillselecttheintervalwiththelowestdistancescoretothecustomerwhiletheotherpresumablyverysimilarscoredintervalwillremainunassigned.Afterthisinitialassignmentintervalswithsimilardistancescorescanbegroupedandassignedproperlyduringpostprocessing. 91

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3.2.4DeterminingArrival/DepartureTimesfromtheAssignmentOncetheassignmentofpolylineintervalsorclusterstocustomershasbeenmade,theactualarrivalanddeparturetimesassociatedwitheachcustomerserviceeventmustbeextracted.Forlongerdistancetrainswhosecustomerintervalsarefoundusingthepolylinealgorithm,thetimesforeachcustomercanbeeasilyobtainedbyndingtheminimumandmaximumtimeamongtheassigneddatapoints.Inthecaseoflongdistancetrains,customerswhowereclosetogetherandservedsequentiallyaregroupedintoasinglecustomer.So,whileseveralcustomersmayhavebeenservedatthelocation,thepolylinealgorithmproducesonlyasingletimeinterval.Thetimespentateachindividualcustomermaythenbeextractedusingbusinessrules,suchasportioningthetotaltimeintervalbytheratioofthenumbercarsworkedateachlocation.Ifamorepreciseunderstandingofthetimespentateachindividualinagroupofcustomersisavailable,theGPSdatapingsassociatedwithonlythisgroupmaybeextractedfromthedataset.Thesepointscanthenbegiventotheclusteringalgorithmwiththeportionoftheworkordercontainingonlythosecustomersinthegroup.Althoughmorecomputationallyintensive,thismethodprovedtobeverysuccessfulinpractice.Whenthecustomerserviceassignmentwasmadeusingdatafromtheclusteringalgorithm,thestraightforwardassignmentofarrivalanddeparturetimesusingtheminimumandmaximumtimesfromtheintervalmayfail.Duringthecreationofthereachabilityplot,weplacepointswithsimilarRDvaluesclosetogether.Datapointsmayhavebeenplacedinthereachabilityplotoutoftheiroriginaltimeordering.Becausetheclustersarederivedfromthereachabilityplot,twopossiblescenariosmayariseoncethemaximumandminimumpointsderivedfromeachcustomer'scluster(s)denethearrivalanddeparturetimes.ConsiderthelistofcustomersC=c1,c2,...,cjCjtobetheorderedlistofallservicedcustomersforwhichvalidservicetimeintervalswereobtained.Intherst 92

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scenarioallintervalsareseparateanddistinct.Bythiswemeanthatforallci2Cthearrivaltimeatciisgreaterthanthedeparturetimeatci)]TJ /F7 7.97 Tf 6.59 0 Td[(1,andthedeparturetimefromciislessthanthearrivaltimeatci+1.Inthesecondscenario,someoralloftheservicetimesatonecustomermayoverlap,orbenestedwithintheservicetimesofanothercustomer.Anexampleofanoverlappedcustomerissomecisuchthatthearrivaltimegivenforthenextcustomeroccursbeforethedeparturefromci,wherethedeparturefromcilessthanthedeparturefromci+1.Intherarecaseofanestedcustomerinterval,boththearrivalanddeparturetimesofcustomerci+1occurwithintheintervalofthearrivalanddeparturetimesofcustomerci.Wenowdiscussthemethodweusetosplitsuchoverlappedornestedintervalsintodistinctcustomerservicetimeintervals.Considertwocustomers,c1andc2,whosedeterminedserviceactivitytimewindowsoverlaporarenested.Denethesetofdatapointscurrentlyassignedtoc1'sclusterasS1,andthosepointsintheclusterforc2asS2.ThetotalsetofpointsassignedtobothofthesecustomersisthenS1[S2.Inordertodenedistinctintervalsoftimeforbothcustomers,thepointsinintervalS1[S2mustbereassignedsuchthatallpointsatthebeginningoftheintervalareassignedtoonecustomer,andallthepointsattheendoftheintervalareassignedtotheother.Intuitively,wearesearchingforapointp2S1[S2,suchthatallpointsU(p)=fp2S1[S2jppgareassignedtoc1,andallV(p)=fp2S1[S2jp>pgareassignedtoc2(orviceversa).Weseektominimizethesumofthepercentageofpointsoriginallyassignedtoacustomerthatwillbereassignedtotheotherusingseparationpointp.Duringthisreassignment,wedonotwishtodrasticallyalterthesizeorlocationoftheoriginalintervalselectedfromthecluster.However,whenanintervaloriginallycontainsmanypoints,thereassignmentofafewofitspointschangestheintervalrelativelylessthanwerewetoreassignthesamenumberofpointsfromanintervalwithveryfewpoints.Overlappedornestedintervalsmayoriginallybeverydifferentsizes.Byminimizingthepercentageofpointschanged 93

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foreachinterval,alargercosttotheobjectivefunctionisimposedforreassignmentsfromoriginallysmallerintervals.Wedeneabinaryvariablexi=1ifandonlyifpointiisassignedtoc1.Analogously,(1)]TJ /F6 11.955 Tf 12.55 0 Td[(xi)=1ifandonlyifpointiisassignedtocustomerc2.Wealsosetzi1=1(zi2)ifandonlyiftheseparationpointisatpointi,andcustomerc1(c2)isplacedrst.Wecanthendenetheproblemofdeterminingtheoptimalseparationpointbythefollowingquadraticmixedintegerprogrammingproblem: MinimizeXi2S1(1)]TJ /F6 11.955 Tf 11.95 0 Td[(xi) jS1j+Xi2S2(1)]TJ /F5 11.955 Tf 11.95 0 Td[((1)]TJ /F6 11.955 Tf 11.95 0 Td[(xi)) jS2j (3a)subjecttozj10@Xi2U(j)(1)]TJ /F6 11.955 Tf 11.96 0 Td[(xi))]TJ /F12 11.955 Tf 15.91 11.35 Td[(Xi2V(j)xi1A=0,j2S1[S2 (3b)zj20@Xi2U(j)xi)]TJ /F12 11.955 Tf 15.91 11.36 Td[(Xi2V(j)(1)]TJ /F6 11.955 Tf 11.96 0 Td[(xi)1A=0,j2S1[S2 (3c)Xi2S1[S2zi1+zi2=1 (3d)xi2f0,1g,i2S1[S2, (3e)zi1,zi22f0,1g,i2S1[S2. (3f)Theobjectivefunction( 3a )minimizesthepercentageofpointsoriginallyassignedtoaparticularcustomerwhicharereassigned.Inthersttermof( 3a ),Pi2S1(1)]TJ /F6 11.955 Tf 12.63 0 Td[(xi)equalsthenumberofpointspreviouslyassignedtoS1thatarenowreassignedtoS2.SotheobjectivefunctionistheratioofthenumberofpointsreassignedtoS2fromS1tothenumberoriginallyinS1,plustheratioofthenumberofpointsreassignedtoS1fromS2tothenumberoriginallyinS2.Constraints( 3b )and( 3c )enforcethatallpointsinS1andS2mustappearcontiguously.Forexampleifzj1=1,thentheseparationpointisplacedatjandcustomerc1appearsrst.Inthiscase,thepointsinU(j)mustbeassignedtocustomerc1,soPi2U(j)(1)]TJ /F6 11.955 Tf 12.16 0 Td[(xi)=0.Likewise,ifzj1=1,thenPi2V(j)xi=0 94

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asallpointsinV(j)areassignedtocustomerc2.Constraint( 3d )ensuresthatonlyonesplitpointmaybeselected.Asboththexandz-variablesarebinary,thisproblemmaybeeasilylinearized.Thisformulationcanbeusedtosolvetheproblemofdeningtheintervalseparationpointforlargeproblemsizeswherethechangetotheintervalscannotbeobtainedbyenumerationofallpossibleseparationpoints.BecausethedataweutilizecomesfromGPSdevicesthatrecordatroughly1datapointeachminute,andmosttrainsrunconsiderablyfewerthanthe24ofeachday,wedonotsolvetheabovequadraticprograminpractice.Wesolvetheplacementoftheseparationpostionbetweentwooverlappedcustomersbycheckingeachpossiblesplitpointp2S1[S2.Theobjectivefunction( 3a )isevaluatedateachpossiblepoint,andthepointofminimumplacementselected. 3.3ComputationalResultsInthissectionwerstdiscussseveraloftheuniquechallengespresentedintheGPSpingdata,aswellasourresolutionmethodsforthesechallenges.NextwecovercomputationalresultsobtainedfromourpilotdataforboththePolylineAlgorithmandOPTICSmethods.Attheendofthissection,wediscussthestatisticalmethodsweusedtoobtainfunctionalrepresentationsforthecustomerservicedurationandtraveltimemetrics.Firstanoteonthecomplexityofthebothalgorithms.WhilethetotalamountofstoragerequiredfortheGPSpingdata,aswellastheworkorderandcustomergeographicdata,isimmense,thedataneededtodeneasingledayforasingletrainisrelativelylittle.Becauseofthis,theamountoftimerequiredtoconnectthedataintotheprogramwhichrunsthemodeldwarfsthecomputationaltimeofeitheralgorithm.Despitethis,wementionthatthealgorithmiccomplexityofbothalgorithmsisquitegood.Computationally,therearetwopartsinthepolylinettingalgorithm.First,thesplittingprocedurerequiresO(n)operationstoidentifythepointsfarthestfromthet,uptoa 95

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maximumof250segments.Andthen,themergingprocedureusesO(n2)operationsintheworstcasetocorrectforoverttingcausedbythesplittingprocedure.OPTICSisespeciallycompetitiveamongstclusteringroutines.Duringthealgorithm,eachdatapointisprocessedonlyonce.Allthecomputationalexpenseofthealgorithmcomesfromtheneighborhoodsearch.SinceOPTICSoftenusesspatialdatabases,spatialindexingschemesareusedtoqueryallneighborsinonlyO(logn)time.ThisgivesatotalruntimeofO(nlogn).Theassignmentofintervalstocustomersthatfollowsrequiressolvingashortestpathproblemonadirectedacyclicgraphwiththeofnodesequaltothenumberofcustomerstimesthenumberofidentiedactivityregions.Thenumberofbothoftheseissmall.Thealgorithmcanrunforasingletrainanddayinunderonesecond.Allthefollowingresultswererunonadesktopcomputer(64bit,2IntelXeonprocessors(2.27GHz/2.26GHz),4GBRAM).Thealgorithm,aswellasthestatisticaltesting,wasimplementedinC#toavoidintegrationwithamorerobuststatisticalprogram.Whileindevelopment,datawasaccessedfromadevelopmentdatabaseviaMicrosoftSQLratherthanaccessingCSX'smassiveproductionservicemetricsandGPSdatabasesdirectly.WealsoaddthatduetocondentialityagreementswithCSXwereareunabletodisplayanydatathatmayallowforthedirectidenticationofanycustomer,physicallocationontherailnetwork,orprivatecompanyinformation.Thishassomewhatlimitedtheresultsweareabletodisplay. 3.3.1DataCollectionThedeterminationofcustomerservicetimesrequiresthreedatacomponents:GPSpositionmeasurementsfromthelocomotives,dailyworkordersrecordedbytraincrews,andGPSmeasurementsdeningtheoutlinesofcustomer'sgeographicserviceregions.AllthedatausedwasprovidedbyCSXTransportation. 96

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GPSdevicesareinstalledon85-90%ofallCSXlocomotives(operationalorotherwise).CSXutilizesseveraldevicesincludingTrimbleCrosschecks,thenewTrimbleTVG660,GEATS,andPinpointIandII.Whenalocomotiveisturnedon,theGPSinitiatesandbeginsrecordingitslocation.Normalfrequencyofrecordingisonemeasurementperminute.Numerousfactors,suchaspurposeandproximitytoindustrydenedareasofinterestdetermineshowoftenthedevicetransmitslocationinformation.Inanareathatrequiresreal-timereporting,thepositionmeasurementsmaybetransmittedatonemeasurementperminute.Forothertransmissions,thedevicemayrecorditspositionseveraltimesbeforetransmittingthedata.DependingonthetypeandageoftheGPSunitandthefunctionofthelocomotive,10datarecordingsmaybesentper10minutes,20per20minutesor60per60minutes.AlllocomotiveshaveWi-Fi,cellularandsatellitecommunicationdevicesonboard.Dependingontheavailabilityofasignal,theGPSmeasurementswillbetransferredviaoneofthesemethods.WhileWi-Fitransmissionispreferred,itmaynotbeavailableoutsideofyardsorurbanareas.ThedataiscommunicateddirectlytoCSXwiththeexceptionoftheTrimbledeviceswhichpassthroughathirdpartypreprocessingatTrimble.Thequantityofdatareceivedandstoredonasingledayisroughly3millionrecordsperday.Workorderinformationisrecordedbycrewsduringdailyoperationsviaon-boardworkorderdeviceshousedinlocomotives.ThisdataisthentransmittedfromthedeviceviaWi-Fi,andisstoredindatabases.Workordersarecompletedbythecrewforallworkperformedbythetrain,includingcustomerwork,movementsofcarsinstorageatyards,beginningandendofshiftcarmovements,andtransferscarsbetweenotherrailproviders.Thetracksknowntobeusedduringtheserviceofacustomerdenethegeographiccustomerserviceregion.Eachtrackcanbedenedbyanoverlayingsequenceof 97

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discretelatitudeandlongitudemeasurements.Foreachcustomeronthenetwork,wewereprovidedwithasetofsuchasequencestakenat20footintervals. 3.3.2DataPreprocessingBoththePolylineAlgorithmandtheOPTICSalgorithmutilizedatathatcomesfromthreesources:workorderdata,geographicknowledgeofcustomerserviceareas,andtheGPSpingmeasurements.Asisthecaseforanyapplicationdealingwithlargequantitiesofnumericalandcategoricaldata,corruptionofthedataorinvaliddataisaconcern.Foratrainonagivenday,workorderinformationmaynotbepresentintheworkorderdatasource.Withoutaworkorder,wedonotpossessalistofcustomersworked,andthusnoalgorithmcanberun.Likewise,thegeographicalcustomerserviceareainformationmaynotbepresentforallcustomersontheworkorder.Ifcustomerserviceregiondataismissingforallcustomersontheroute,thealgorithmcannotberun.However,ifthecustomerserviceregionofatleastonecustomerontheworkorderisgiven,wecandeterminethetimeatcustomerinformationforthissinglecustomerincaseswherethePolylineAlgorithmisused.ThisisnotthecasewhenutilizingtheOPTICSalgorithm.Ascustomersonshortrangetrainsarequiteneareachother,itislikelythatallservicetimeintervalsforcustomerswithmissingcustomerserviceregionswillbecountedtowardsthosewithvalidserviceregioninformationduringtheassignmentphaseasallintervalsare“close”toallcustomers.Inthiscase,theservicetimesofcustomerswithserviceregioninformationwillbeoverestimated.PreprocessinglogicisalsoaddedtoavoidutilizingthealgorithmsontraininstanceswithincompleteGPSdata.WhileGPSdevicestraditionallyfunctionbytakingapositionreadingatregulartimeintervals,itispossiblethatthedatacontainsaspanoftimeduringwhichnolocationpositionswereeitherrecordedortransmitted.Suchadataissuemayoccurinseveralways.Firstly,thelocomotive'sGPSdevicemayfailtorecordortransmitdata.TheGPSsystemsinruralormountainousareasmaybeforcedto 98

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utilizelessreliablemeansofdatatransmission.Additionally,whilerare,theGPSdeviceitselfmayfailinseveralways,suchasbatteryorotherelectroniccomponentissues.Secondly,recordingandtransmissionratesvarydependingonthereportingsystemavailabletotheGPSdevice.AGPSdeviceinanurbanyardwithaccesstoWi-Fimayrecordandreportreadingsmorefrequentlythenadevicerelyingonsatellitetransmissionsforaruraltrain.WhilethisisnottrulyagapintheGPSpingdata,thisreduction(orincrease)inrecordingrateleadstoirregularrecordingtimesthroughoutthedataset.Suchirregularitiesmayhaveimpactswhenprocessingthedataofsmalldistancetrains.UtilizingthemoresensitiveOPTICSdoesnotresultinagapintheGPSpingdata.Thirdly,intheeventthatatrainisplacedinanidlestatus(whichmayoccurifthecrewisawaitingupdatedordersorclearance),therecordingrateoftheGPSdevicemaybereduced.Thisisbecause,asthepositionofanidletrainwillnotchange,detailedlocationmeasurementsarenotexpectedtobeneededuntilthetrainistakenoutofidlestatusandbeginstomove.Inthiscase,ifthetrain'sstatusissettoidle,onlyasingledatareadingistakenevery10-20minutesandallthepositionreadingsofthesedatapointswillbethesame.TheseissuesresultingapsinGPSpingdatabetweenafewminutesandseveralhours.Thesegapsmayoccuratthebeginningorendofthetrainroute,oratanypointinthemiddletimeseriesoftheGPSdata.ThiscreatesdifcultieswhenapplyingeitherthePolylineAlgorithmortheOPTICSalgorithm.Ifpingsaremissingfromthedataduringthetimeswhencustomerswereservicedwemayunderestimateservicetimes,orbeentirelyunabletocalculateaservicetime.Becauseofthisweemployvariouspreprocessingroutinestoeitherfullyremovedatasetswithgapsormitigateresultantissues.Manualpostprocessingcandetermineifthegapinthepingdataoccurssignicantlyfarfromtimesofcustomerserviceandthealgorithmcanbeapplied. 99

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ToaccountforinstancesthataremissinglargeportionsforGPSpingdata,weremovealltraininstanceswithfewerthan30pings,aswellasthosethatspanfewerthana30minutetimeframeintotal.ToavoidrunningthePolylineAlgorithmalgorithmneedlessly,wealsocheckthatatleast15pingscomewithinatleast16000meters(approximately10miles)ofanycustomerontheworkorderlist.ApreprocessingstepisperformedwhichcheckswhetherthepingdataofaparticulartrainonaparticulardaycontainsagaplongerthansomeGAPmax=20minutes.Whendatagapsaresubstantial,itislikelythatanyresultsobtainedfromthedatasetwillbeinconclusiveforcustomerstopsonornearthistimerangecontainingthegap.ForlongdistancetrainsthatutilizethePolylineAlgorithm,itmaybepossibletousedatacontaininggapslongerthan20minutesifitcanbeveriedthatthetraindidnotmovesubstantiallyduringthetimeofthegap,astheresultingpolylinetwillbeequivalent.AnadditionalpreprocessingstepisusedfortrainsusingthePolylineAlgorithmtocheckforthiscase.Forshorterdistancetrains,themeasurementerrorintheGPSdevicemakesitimpossibletodetermineifthetrainisstationaryormobile,andthussimilarlogiccannotbeappliedwhentheOPTICSalgorithmisused.WhileOPTICSdoesnotrequirethetimeintervalsbetweendatareadingstoberegular,theresultingclustersaremoredifculttointerpretifthetimeintervalsbetweendatapointsdifferdrasticallythroughouttheavailabledata.OPTICSdenesahierarchicalorderingofthedatapointsand,thus,doesnotsufferduetoirregularlyirregularly,orwidely,temporallyspaceddatapoints.However,inordertodetermineifaclustermaycorrespondtoastopatacustomerweemployapostprocessingalgorithmwhichismoresuccessfulifthecustomerclustersareprominentwithrespecttothosedatapointsintheorderingwhicharenotrelatedtoacustomerstop.Clustersappearprominentlyifnumerousdatapointscanbefoundwhicharriveatacustomeratsimilartimes.Ifthegapsintimebetweenreadingsarelarge,datapointscanappearmoredissimilar(andtheclusterlessprominent)relativetootherclusterswhosedata 100

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pointsoccurinclosertime.Thisdoesnotmeanthatalessprominentclusterwillnotbeidentied.OneofthegreatadvantagesofemployingthedensitybasedclusteringalgorithmOPTICSisitsabilitytoidentifyclustersofvarioussizesandprominencelevels.Amoreprecise,“deeper”cluster,allowsforamorecondentdeterminationofarrivalanddeparturetimes.So,althoughitisunnecessaryforthemajorityofthedataweencountered,weincludethefollowingpreprocessingsteptoenhanceclustersinregionsofdatasetswithirregularrecordingfrequenciesandgapsindata:Foranygap Algorithm5PreprocessGPSGapsforClusteringAlgorithm 1: procedureMENDGAPS(GPSdataP) 2: N sizeofP. 3: forifrom1toN)]TJ /F5 11.955 Tf 11.96 0 Td[(1do 4: =P[i+1].time)]TJ /F6 11.955 Tf 11.96 0 Td[(P[i].time 5: if>maxthen 6: fortfrom1to)]TJ /F5 11.955 Tf 11.95 0 Td[(1do 7: =t= 8: z.loc P[i].loc+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()P[i+1].loc 9: z.time P[i].time+t 10: InsertzintoPbeforeP[i+1] 11: endfor 12: endif 13: endfor 14: endprocedure betweentwopingsinthepingdatagreaterthansomemax(weuseavalueofmax=2minutes),pointsareaddedintothegapatunittimeintervals.Thesepointsarealinearcombinationofthedatapointswhichboundthegap.Notethatifthegapissignicantlylarge,thelocomotivemayhavetraveledasignicantdistanceduringthistimeperiod.Inthiscase,suchanaiveinterpolationofthemissingdatamayfailtoapproximatethedatawell.Thus,thepreprocessingstepofAlgorithm 5 isonlyappliedtodatasetswithnogapslargerthanGAPmaxwhichwouldhavebeenremoved.Thenalerrorweconsiderinpreprocessingisrelatedtotheassignmentofindividuallocomotivestotrainsonagivenday.Multiplelocomotivesmaybeassignedtoanyonetrain,andeachlocomotiveproducesitsownsetofGPSpingdata.In 101

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rarecases,asinglelocomotiveoutofthoseassignedtothetrain,isnotusedtotakecarstothecustomer.Thelocomotivemayhavebeeninitiallyassignedtoatrain,butifthecarloadwassmallerthanexpecteditwasnotused.IftheGPSdeviceintheunusedlocomotiveisleftonandstillassociatedwiththetrain,itwillproduceaseriesofimmobilepingsthatappearaggregatedwiththeothermobilepingsdataofthetrain.Wesegmentthetimeorderedpingsintogroupsof10pingseach,andcheckthatallpingswithineachgrouparewithin11000meters(approximately7miles)ofeachother.Consideringthatpingsarenormallyrecordedoneeveryminute,thiserrorcheckhelpstoidentifywhetherlocomotivesassignedtothesametrainarerecordingGPSpingsdatafarfromoneanother.NotethatwhentheOPTICSalgorithmisused,thischeckwouldbeinsufcienttoremovethiserrorastheentiretrainmoveslessthan5miles.Forsmalldistancetrains,pingrecordsofanunusedlocomotivecannotbeidentied.However,smalldistancetrainsrarelytransportthelargenumberofcarsthatwouldnecessitatetheuseofmorethanonelocomotive. 3.3.3PolylineAlgorithmValidationToensurerobustnessfornumerousuniquecustomers,themodelwasdesignedandtestedusingthedatafromseveraltrainsandmanycustomers.Duringvalidation,thePolylineAlgorithmuseddatafromseventrainsthatservicedapproximately40uniquecustomersoversixmonths.UsingtheGPSsoftwareArcGIS,measurementsfromthetrainwerecomparedtocustomerinformationheldbyCSXandworkorderinformationforthatday.Then,usingexpertinformationfromcompanypersonnelwithextensiveindustryexperience,anestimateoftheactualofservicetimeswasdetermined.Whencomparingtheoutputofthealgorithmtotheestimatedservicetimes,themodelwasconsideredtodeviateifthestartorendtimesdifferedbyover5minutes.Inmostapplications,thislevelofdeviationisacceptable.ForfurthervericationofthePolylineAlgorithmresults,aspartofacustomerservicepilot,CSXhadstaffridealongthetrainsofthisdivisionforthreedays.These 102

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staffmembersrecordeddetailednotesofallofthetrainsactivities.Thisincludedarrival/departuretimes,timespentroutingswitches,whichswitcheswerechanged,whatservicewasdoneateachcustomer,andwhatdelays/unexpectedstopshappened.Thesestaffreportsallowedforexactdeterminationsofservicetimesforthedaystheywererecorded.Whencomparedtothestaffreports,thealgorithmoutputwaswithin5minutesforallservicestartandstoptimeswiththeexceptionofasinglecustomer.Uponfurtherreview,itwasdeterminedthatCSX'sfacilitydataforthiscustomerdidnotincludetracksusedtoenterthefacility.Afterrerunningthealgorithmwithupdatestothecustomerfacilityinformation,theoutputwaswithin5minutesoftheexactstart/stoptimesdeterminedfromthestaffreports.AfterrunningthePolylineAlgorithmonallofthegathereddata,thestart/stoptimesofthealgorithmoutputdeviatedfromtheestimatedstart/stoptimesbylessthan10minutesforeveryday,trainandcustomer.Figure 3-21 showsarelativecountofthosedaysonwhichthealgorithmoutputdeviatedmore/lessthan5minutesfromtheestimatedvalues.TheOver/Undercountindicatesthenumberoftimesthatthealgorithmoutput'sstart/stoptimesdeviatedtobymorethanveminutesfromtheestimatedstart/stoptime.Theexpectedcountindicatesthenumberofexperimentsinwhichthealgorithmprovidedthesamevaluesastheestimation,whichwasdetermined(asdiscussedabove)bymanuallyexaminingtheGPS,workorderandalgorithmicdataforeachdate.Theseresultsdemonstratethealgorithm'sabilitytoaccuratelyassesscustomerservicetimes.Theapparentlackofpointsforthethirdtrain(Train21)isduetothelimitednumberofcustomerserviceeventsthatoccurredonitsrouteduringthesixmonthsofdata.Thistraintypicallyperformsdutiesotherthancustomerserviceeventsbutdoesonoccasionservecustomersonitsroute.Thisillustratesthelargequantityofdatawhichmayberequiredtoproducedetermineservicetimesaccurately.Forexample,ifacustomerisservicedveryinfrequently,severalmonthsofdatamayberequiredinordertoobtainalargeenoughsampleofserviceeventstodeterminethe 103

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Figure3-21. ComparisonofnumberoftimesthePolylinealgorithmreturnedtheestimatedresultandthenumberoftimesthealgorithm'sstartandstoptimesdeviatedbygreaterthanveminutesfromtheestimatedresultforvetrainsforsixmonthsofdata. servicetime.Thisisfurthercomplicatedbythenumberoffactorsthatcanaffectthedurationofindividualserviceevents. 3.3.4ClusteringAlgorithmValidationWhenexaminingthevalidityoftheresultsoftheOPTICSalgorithm,thesameintensivecomparisonsbyindustryexpertswithGPSsoftwarewereperformed.Additionally,atrainride-alongwasdoneonalocaltrainthathadthreeoverlappedcustomerserviceareasthatwereoftenservicedsequentially,toensurethemodelclusteringwasappropriate.WhentheOPTICSalgorithmisused,customersareinverycloseproximitytooneanother.Inthesecases,visualvalidationofthealgorithmischallenging.Evenwithintimateknowledgeofatrain'sphysicalprocess,whenexamining 104

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Figure3-22. Examplewithoverlappedcustomerserviceareas.Therstthreecustomers(labeled1-3)overlap,whilethefourthcustomerislocatedapproximatelyonemilefromtheothers. discreteGPSpingdataitcanbeunclearwhentheserviceofonecustomerbeginsandtheserviceofasecond,andoverlapping,customerbegins.DespitethedifcultiesinobtainingtheclustersmanuallyformGPSdata,theOPTICSalgorithmextractstheexpectedservicetimewindowsaccuratelyandautomatically.Thealgorithmisrobustacrossnumerouscustomerfacilitygeographies,customerdistancepairproximities,anddifferenttrains.WetestedOPTICSon5trainsoveraroughly6monthperiod.ThequalityoftheresultswassimilartotheresultsforthePolylineAlgorithmdiscussedabove,andwewillnotdescribethemfurther.Toshowthetrueeffectivenessofthisalgorithm,wegiveareal-lifeexamplethatrepresentsa“worst-case”scenariointermsofcustomerproximitiesandordering.Figure 3-22 showsthecustomerserviceregionsoffourcustomerthatwillbeservicedbyatrain.Thegeographiccustomerserviceregionsoftherstthreecustomersnotonlyoverlap,butareallquitesmall.Thedistancetothenalcustomerisroughlyonemilefromtherstthree.Thefactthatthenalcustomerisactuallyquitefarfromtheothers,makesdistinguishingthedifferencesinrelativemotionbetweentherstthreecustomersdifcult,asseeninthedistance-versus-timegraphsforthesefourcustomershowninFigure 3-23 .EachpointofthegraphrepresentstheminimumdistancefromeachGPSpingtothecustomer'sserviceregion.Notehowthevariationsinproximitytotherstthreecustomersarebarelydistinguishable,asthe 105

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Figure3-23. MinimumdistanceofGPSpingdatatocustomerserviceregionsforfourcustomersversustime. laterpingsrelativelyreducethedifferencesbetweenthedistances.Clearly,applyinganysortofautomatedextractiontothisdataisdifcult.Thedistance-versus-timegraphsforthisexamplealsoillustratewhyapplyingaxedproximitytolerancelevelfailsforthisproblem.Considerthedipinthegraphbetween16:00and17:00onthedistance-versus-timegraphforcustomerfourthatclearlycorrespondstotheservicetime 106

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Figure3-24. Reachabilityplotforthefourclosecustomerexample. ofcustomerfour.Observethatthevaluesinthisdipneverquitereachzero.Inreality,onthisday,theserviceeventforthiscustomeroccurredabitoutsideofthecustomer'sgeographicalserviceregion.Considerdrawingahorizontallineacrossthegraphofcustomerfourcorrespondingtoatolerancelevelbelowwhichthepingsareconsideredtobeatacustomer.Ifthesamelinewereappliedacrosstheothergraphsforcustomers1,2,and3,theresultinginservicetimesforcustomers1and2(andpossibly3aswell)wouldbeindistinguishable.Thusauniquetolerancelinemustbeobtainedforeachcustomerdependentupontheothercustomersonthetrainroute.Figure 3-24 showstheoutputoftheOPTICSalgorithmonthisdata.Thethreeclustersrepresentingcustomers1,2,and3,areclearlyvisible,andcanbeextractedtoprovidecustomerservicetimesforeach.They-axisofthegraphisonlyshowing 107

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aportionofthedistancevaluesinordertomakethesmallclustersfortherstthreecustomersapparent.Theclusterintheupperrightofthegraphistheclustercorrespondingtothefourthcustomer.NotethatbecausetheGPSpingsforthiscustomerneverenteredthecustomerserviceareaforcustomerfour,theRDvaluesinthisclusteraregreaterthanzero.Despitethis,theclusterofcustomerfour'sservicetimeisstillclearlyvisibleandprovidesthecorrectservicedurationvalue.ItistheOPTICSalgorithm'sinsensitivitytoclusterdensityandsize,andinputparameterswhichmakethismethodeffective.ThisexampleshowstheabilityoftheOPTICSalgorithmtodeterminecustomerserviceeventsfromsequentiallyservedcustomersevenwhentheirgeographicalserviceareasoverlap. 3.3.5FunctionalEstimationofServiceTimesandTravelTimesAfterapplyingeitherthePolylineAlgorithmortheOPTICSalgorithmtotwoyearsworthofdataforournumerouspilottrains,weobtainedmultiplepointestimatesofthetimerequiredforservice.Fromthisweinterpolatedtheapproximationsoftraveltimesbetweenallpresentcustomerpairs.Wethenbegantoapplystatisticalestimationtechniquestocharacterizetherelationshipsbetweentheservicetimeforeachcustomer,andthepotentiallyinuencingparametersoftotaltrainlength,thenumberofcarshandledpercarmovementtype(eitherpickuporplacementofcars),startingtimeofthetrainroute,theorderinginthesequenceofeachcustomer,thetotalnumberofcustomers,andthecustomerservedprior.Welaterappliedthesameanalysistothetraveltimebetweencustomers.OurgoalwastoprovideCSXwithafunctionalrepresentationoftheservicetimesintermsofthesevariables.Suchafunctioncouldbeuseddailytodeterminetheexpectedservicetimeoncetheday'sworkwasknown.AllstatisticalanalysiswasdoneinMinitabversion16.Foreachcustomer,wedeterminedthetopfour(orfewer)parametersthatmostinuencedtheservicetimeatastatisticallysignicantlevelusingbothstepwiseregressionandbest-subsetregressiononlog-transformedservicedurationdata.Subsetregressiontsthemodelwith 108

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numerouscombinationsofnumbersofvariables,providingstatisticsoneach.Stepwiseregressionproducesasinglemodeldeemedthebesttbysystematicinsertionandremovalofthepossibleinuentialvariables.Therewasverylittlediscrepancybetweentheparametersselectedbyeitherbest-subsetorstepwiseregression.Oncethemostinuentialparameterswereknown,weperformedmultivariateregressiontodeterminethefunctionalrelationbetweentheservicetimesandtheparameters.Despiteourinitialassumptions,eventhetopmoststatisticallyinuentialparameterswerefoundtohaveastatisticallysignicantimpactontheservicetimeatcustomer.Todetermineanapproximatemeasureoftheimpactofeachvariablewebeganwiththetmodelforeachcustomerusingthemostsignicantparameters.WesystematicallyremovedeachvariableandexaminedthechangeintheadjustedR2forthemodelt.ThedecreaseintheR2valuewheneachvariableisremovedisrecordedinTable 3-1 .Eachcustomer Table3-1. ReductioninadjustedR2valueofthemultivariatetmodelforeachcustomerwhenthelistedparameterisexcluded.CustomerCountAdj.R2Len#PU#PLStartSeq#CustCust1 D22230.610.45.29.4D32346.68.13.730.4C4375.45.4P14020.25.41.89.9R34910.83.33.72.8C15225.77.85.016.9R25636.76.428.114.11.9T55633.56.418.76.1L15838.23.54.916.413.8G16111.24.09.1N26713.95.91.62.2W16922.58.21.41.9I18120.11.93.010.1C28751.21.620.14.910.5B111528.32.34.418.4S119029.47.62.38.2N124027.66.64.710.4 rowcontainsthenumberofservicedurationpointestimatesusedtogeneratethetmodelfollowedbytheadjustedR2valueforthemodeltusingthemoststatistically 109

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signicantparameters.Rowentrieswithoutnumericalvaluesindicatethatthevariableofthecorrespondingcolumndidnotsignicantlyinuencetheservicetime.ThenumericalvaluesindicatethedropintheR2valuewhenthevariablewasremovedfromthemodelandthetrecalculated.ThelowR2valuesindicatethemodelmaybeapotentiallypoortforthisdata.Inaddition,thesmallchangeintheR2withtheremovalofeachindependentvariableimpliesthattheparametersarepoorpredictorsofservicetime.However,wepointoutthatthisanalysisdoesnotimplythatnoparametersareinuentialindeterminingtheservicetimeforeachcustomer.Thismerelyimpliesthatforthequantityofdatawegathered,theselectedparametersfailtobeinuential.Wewereunabletotesttheimpactofthenumerousotherpotential(andlessnumeric)indicatorssuchasweather,crewexperienceandvariability,impedingtrafcfromotherfreightorpassengertrains,etc.Duetothelimitedresultsfromthemultivariateregression,coupledwithitscomplexityandpotentiallackofvalidityforinstancesoutsideoftherangesofsimilarparametersvalues,wefailedtoobtainafunctionalcharacterizationoftheservicetimeatcustomerintermsofthepossibleinuentialparameters.Despitetheapparentlackofassociationbetweenindependentparametersandthemeasuresofserviceduration,theestimatedservicetimesprovideanexcellentunivariatecharacterizationoftheservicetimes.Figure 3-25 showsthehistogramsandtheresultingtfortheservicetimesatsixrepresentativecustomersfromthepilotstudy.Theservicedurationdataofallcustomersweretwellbyeitheranormaloralognormaldistribution.Theservicedistributiondatawillbeutilizedinservicearrivalestimates,trainrouteplanning,andloadbalancing.Thefactthatthedatatssowellwiththesesimple,andwellknowndistributions,assuresincreasedaccuracy(aswellasease)ofthesecalculationsandmodels.Becauseofthenormality(orlognormality)ofthedata,weremovedoutliersusingGrubb'stest[ 47 ]appliediterativelytoeachdataset.Thesignicancelevelwassetto 110

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ATRA1Customer1 BTRA1Customer2 CTRA1Customer3 DTRB2Customer1 ETRB2Customer2 FTRB2Customer3Figure3-25. Minitabhistogramandtofservicedurationdatafor6customers. 111

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removeoutliersquiteconservatively.Asthedataisstrictlypositive,alloutliersappearedatlargevaluesofserviceduration.Fortheservicedesignapplicationsinwhichthesedistributionswillbeused,aslightoverestimateispreferredtoanunderestimatethatcouldresultintheconstructionofunreasonableservicelevelgoals.Forseveralcustomers,thenumberofpointestimatesobtainedwasfewerthan20.Forthesecustomersweprovidedstatisticsontheservicedurationsuchasminimum,maximum,quantiles,median,mean,andstandarddeviationvalues.Thedesiredoutputofthisprojectwasthecharacterizationofthetimeittakesatraintoserveacustomer,inadditiontothetraveltimebetweensubsequentcustomers.BothalgorithmsPolylineAlgorithmandOPTICSidentifyactivityregionsnearcustomersthatarethenassignedastheestimatedservicetimesforeachcustomer.ToapproximatethetraveltimebetweentwosubsequentcustomerscA2CandcB2CinthesetofcustomersontheworkorderC,thedifferencebetweenthebeginningoftheservicetimeatcustomercBandthenalservicetimeatcustomercAisused.Likewise,thetimeittakestoreachtherstcustomeroftheroutec1fromtheoriginatingstationisestimatedbythedifferencebetweenthebeginningoftheservicetimeatc1andtherstrecordedGPSpingtimestamp.ThetraveltimefromthelastcustomeroftheroutetothedestinationstationisthedifferencebetweenthelastrecordedGPSpingtimeandtheendoftheservicetimeintervalforthenalcustomer.WeassumethattheGPSdeviceinthetrainlocomotivedoesnotrecordpingsuntilthelocomotivebeginsitsworkattheoriginatingstation(areasonableassumptioninmostcases).Whenworkorderentriesaremissing,orincorrectlysequenced,businessrulesmustbeappliedtoltertheresultsdatatoensurethatthetraveltimeobtainedbetweentwocustomersistrulyrepresentative.However,whensuchworkorderdataentryerrorsarerare,onecanexpecttheaveragetraveltimeobtainedovernumerousinstances,tobelittleinuencedbyanypotentiallyincorrectestimates. 112

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Inthismanner,wecalculatedthepointestimatesforthetraveltimebetweenallcustomerpairs.Statisticalanalysisidenticaltothatusedfortheservicedurationdistributionswasappliedtothetraveltimedata.Wenotethatwefoundlittlestatisticallyvalidassociationbetweenthetraveltimesandtheindependentparametersoftotaltrainlength,thenumberofcarshandledpercarmovementtype(eitherpickuporplacementofcars),startingtimeofthetrainroute,theorderinginthesequenceofeachcustomer,thetotalnumberofcustomers,andthecustomerservedprior.Thehistogramsofthetraveltimeswereapproximatedwellbyprimarilylognormal,andoccasionallynormal,distributions. 3.4ConclusionsThetwoabovealgorithmspresentedprovidedagoodestimateofcustomerservicetimesontherailnetworkgivenGPSdata,aworkordersequence,andminimalknowledgeofthecustomergeograph.ThealgorithmsutilizedstateoftheartdataminingprocedurestosolvewhatiscurrentlyadifcultproblemforindustriesattemptingcapturebusinessknowledgefromthemassivewealthofnewlyacquiredGPSdata.Theparticularmotionsofthelocomotivesduringservicenecessitatedtechniquesbeyondthesimilarresearchperformedinbusing,trucking,andhouseholdresearch.Thegoalofattainingafunctionalestimationwasachievedbyapplyingstatisticalanalysistotheresultingservicetimeandtraveltimedata.ThesealgorithmseliminatedtheneedformanualcomparisonofGPSdatawithcrewrecordedworktimes.TheapplicationhasbeenputintoproductionatCSXandisalreadybeingusedtocompareactualarrivaltimeswithbusinessplanexpectedtimesatcustomers.Laterresearchwillleveragetheoutputsfromthisresearchtocreatesuchproductsasdynamicschedulecreationanddynamicwork-to-trainassignments,realtimeserviceestimation,detailedprolesofservicebehaviorformanagement,orrealtimearrivaltimeupdatesforcustomersslatedtobeserved.Theautomation(andmoreimportantlyoptimization)oftheseprocesseshasahighvalueforacompanyoperatingonsuchaprohibitivelylargenetwork. 113

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CHAPTER4ITERATIVETABUSEARCHTOSOLVEASTOCHASTICVEHICLEROUTINGPROBLMEWITHSOFTTIMEWINDOWSANDSTOCHASTICCUSTOMERS 4.1OverviewInthischapterwepresentavariantofthestochasticvehicleroutingproblemwithsofttimewindowsthatprioritizesmeetingcustomerservicegoals,andlimitingvehicleovertimeworkonaxednumberofroutes.Thetraveltimesbetweencustomersaretakenfromknownprobabilitydistributions,asaretheservicetimesateachcustomer.Weincludeanotherelementofstochasticityinthateachpotentialcustomerisservicedwithsomeprobabilityoneachserviceday.Weobtainasolutiontothisproblemusinganiterativetabusearchheuristic. 4.2IntroductionVehicleroutingproblemsareamongsomeofthemoststudiedintheeldofcombinatorialoptimization.Inthetraditionalvehicleroutingproblem(VRP),aeetofvehiclesisassignedtoserveasetofcustomersonatransportationnetwork.Theeetoftenbeginsatasinglestartingterminalnode.TheVRPseekstondtheoptimalsetofvehicleroutesthatserviceallcustomers,andthatminimizesthecostofthisservice,subjecttopossibleconstraintssuchasduration,andcapacity.Theseproblemsrepresentanessentialconcernformanagementplannersacrossnumerousdisciplines.Tosolvetherealisticroutingproblemsfacedbyvariousindustries,anamountofstochasticitymustbeaddressedwhensolvingtheproblem.Uncertaintymayariseinvariousformsincludingtraveltimes,servicetimes,whichcustomersaretobeserved,andcustomerdemand.Theinclusionofsuchelementsoftenhindersthetractabilityoftheseproblems,especiallywhentheprobleminstanceislarge.Inthesecases,heuristicsareoftenutilizedtoprovideefcientsolutions.Inthischapter,wepresentastochasticvehicleroutingproblemwithsofttimewindowsandstochasticcustomers,whichwerefertoastheVRTWSC.WesolvetheVRTWSCusinganiterativetabusearch.Softtimewindowsimplythateach 114

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customerhasarangeoftimeduringwhichtheyprefertheservicevehicletoarrive.Ifthevehiclemissesthistimewindow,thedeliverystilloccurs,yetapenaltyisimposedthatmaybemonetaryorintheformofalossofgoodwill.Weconsiderthreeelementsofstochasticityinourproblems:stochastictraveltimes,servicetimes,andalsostochasticityintheparticularcustomersservicedoneachserviceinstance.Ourobjectiveminimizesexpectedcostassociatedwithvehiclearrivaloutsideofthesewindows,aswellastotalroutedurationsinexcessofapredenedlimit.Notethattheprobabilityofservingeachcustomerfactorsintothecalculationoftheseexpectations.Thus,solutionswithlowerobjectivevaluesmayfavoron-timearrivalsatfrequentlyservicedcustomerswhileallowingrarelyservedcustomerstobeconsistentlyservicedwelloutsideoftheirpreferredtimewindows.Toavoidthis,weincludeservicelevelconstraintstoensurethatavehiclewillarrivetoserviceeachcustomerbeforetheendofitstimewindowinatleastacertainpercentageofdeliveriestothatcustomer.Theinclusionofstochasticcustomersalsoimpliesthatonserviceinstanceswhenseveralmorecustomersarepresentthannormallyexpectedonagivenroute,thetotalroutedurationmayrunexcessivelylong.Toreduceresultingovertimecosts,weincludeconstraintstoensurethatthetotaldurationofallroutesisbelowsomexedlimitwithsomegivenprobability.Despitethepracticalapplicabilityofsuchaproblem,wehavenotfoundsuchavehicleroutingproblemintheliteraturethatincludesstochasticityofcustomers,aswellastimewindows,andstochastictravelandservicetimes.Duetothedifcultyoftheproblem,weinitiallyconsiderthevehiclestobeuncapacitated.While,theseconstraintsaresimpletoformulate,theirinclusionchangesthefeasibleregion,andthus,impactsthesolution.WediscusstheinclusionofcapacityconstraintsinSection 4.8 .Thestochasticityoftheproblemmakesthecalculationofthearrivaltimeatacustomernon-intuitiveanddifcult.Duetothesedifcultiesweusesimulationprocedurestodeterminethearrivaltimeatacustomerforagivenroute.Werst 115

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presentthesimulationwithoutthepresenceofsofttimewindows,andthenmodifyittoincludethem.Thesesimulationsareusefulintheirownright,andwereusedtocreateaninteractiveapplicationthatsimulatesthearrivaltimesforasingleroute.Theapplicationallowstheusertogaugetheimpactofanewcustomeronacurrentroute.Italsoallowstheusertocomparearrivaltimes,andtotalroutedurations,forvarioussequencesofcustomerscurrentlyassignedtoaroute.ThesimulationprocedureisusedwithinaniterativetabusearchtosolvethelargerproblemofdetermininganassignmentofcustomerstoroutesintheVRTWSCProblem.TheVRTWSCwasmotivatedbythereal-lifeproblemofimprovingon-timearrivalsatcustomersfortheroutesoflocaltrainsforCSXTransportation,LLC.Unfortunately,atthedateofthiswriting,acompletesetofdatawasnotyetavailablefromCSX,andthuswewereunabletotestourapproachusingreal-lifedata.TherandomlygenerateddatausedinourcomputationaltestswascreatedtomimictheportionofdatacurrentlyavailabletousforCSX'slocaltrainnetwork.Aswedohaveaccesstocertaindata,wewereabletoutilizedatafromCSXtotestthesimulationprocedure.Theremainderofthischapterisasfollows.InSection 4.3 wepresentareviewofliteraturerelatedtothestochasticvehicleroutingproblemwithtimewindows,andliteratureonvehicleroutingproblemswithstochasticcustomers.Thequantityofresearchperformedonvehicleroutingisconsiderable,andthuswepresentonlythosestudiesthataremostrelevanttoourwork.WepresentaformulationoftheVRTWSCinSection 4.4 asastochasticmixedintegerprogrammingproblemwithchanceconstraints.ThedifcultiesinprovidingadeterministicapproximationofthearrivaltimesarediscussedinSection 4.5 .Thissectionalsopresentsthesimulationproceduresandtheinteractiveapplicationthatsimulatesthearrivaltimesforasingleroute.Section 4.6 outlinestheiterativetabusearch,anddescribesthenumerousdetailsofitsimplementation.Introductorycomputationaltrialsofthismethodappliedtoseveralrandomlygeneratedtestinstances 116

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arediscussedinSection 4.7 .AdiscussionofpotentialextensionstotheproblemisgiveninSection 4.8 ,andconclusionsaregiveninSection 4.9 . 4.3ReviewofRelevantLiteratureEfcientvehicleroutingisessentialfortheoperationsofmanyindustries.Numerousextensionsofthisproblemexist,andseveralsurveyscovertheVRP[ 7 , 58 – 60 ].IntheVRTWSCweconsiderthreeextensionstotheVRP,notallofwhichhavehithertobeenconsideredinconjunction.Weconsiderstochasticcustomers,stochastictraveltimesandservicetimes,andsofttimewindows.Wenotethat,stochasticservicetimesaresometimesconsideredtobemodeledidenticallytostochasticdemandsincapacitatedVRPs.Wenowdescribetheavailableliteraturerelatedtotheseproblemextensions.Therstextensionweconsideristhatofstochasticcustomers.Formoreonthistopic,theinterestedreadershouldbeginwiththesurveyin[ 40 ].Invehicleroutingwithstochasticcustomers,aroutethatvisitsallcustomersisselectedapriori.Foranyinstanceofservice,oneormorecustomersontheroutemaynotrequireservice.Inthiscase,theauthorsof[ 40 ]ascribetheenumerationofthefollowingthreepossibleoperatingpoliciesto[ 105 ].Avehiclemayfollowtheaprioriroute,simplynotperformingserviceatanabsentcustomer.Alternatively,avehiclemayskipabsentcustomers,takingaroutebetweenonlythosecustomersneedingservice.Orlastly,avehiclemayre-optimizetheremainingroutewhenacustomerdynamicallyrevealsthatitwillnotrequireservice.IntheVRTWSCweconsiderthesecondtypeofpolicytomodelstochasticcustomers.Notethatincaseswherethevehiclefollowstheaprioriroutewithoutskippingabsentcustomers,thestochasticcustomerconsiderationscanbereducedtothethoseofthestochasticdemandcase[ 40 ].Intheabsenceofotherconstraints,whenvehiclesareuncapacitated,thevehicleroutingproblemwithstochasticcustomersmostcloselyresemblestheProbabilisticTravelingSalespersonProblem(PTSP).InthePTSPeachvertexofagraphispresentwithsomeprobability.AsingleaprioriHamiltoniantourisfoundthatcontainsall 117

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vertices.InanyrealizationofthePTSP,absentcustomersareskipped.Theaprioritourisconstructedtominimizetheexpectedtourlength[ 52 , 53 ].Anexactmethodexistsforsmallproblems[ 64 ],althoughheuristicsandapproximationtechniquesareoftenusedforlargerproblemsizes.See[ 14 ]forarecentsummaryofthePTSPliterature.Thecapacityconstraintsinvehicleroutingproblemsaddconsiderabledifcultytothealreadycomplexnotionofstochasticcustomers.Bertsimas[ 11 ]providesboundsandheuristicapproacheswhenvehiclesarecapacitated.Anumberofworksconsiderstochasticdemandsinconjunctionwithstochasticcustomers.Asthedemandsarestochastic,itispossiblethatonaparticularinstanceofservice,thedemandatacustomermaycauseavehicletoexceeditscapacity.Inthiscase,thevehiclemaytakesomeaction(suchasreturntoaterminal)toobtainthecapacityneededtoservicetheremainingcustomers.Anexactalgorithmexistsforsmallerproblemsizes[ 42 , 87 ],aswellasheuristicapproaches[ 41 ].InasimilarVRPwithstochasticcustomersanddemands,thecustomersnotrequiringservicearegraduallyrevealed,andtherandomdemandsarenotactualizeduntilthevehiclearrivesatthecustomer[ 12 ].Anotherallowsforcompletereoptimizationoftheremainingrouteasdemandsarerealized[ 10 ].Whenstochasticcustomersarenotconsideredinproblemswithstochasticdemand,xedroutesarefoundforeachvehicleoftheeet.Severaltheoreticalresultsforthisproblemareestablishedin[ 11 ].Thismaybemodeledbychanceconstraintsthatensuretheprobabilityoftherouteexceedingcapacityislessthansomegivenvalue[ 8 , 63 , 93 ].Additionally,theproblemmaybemodeledbyminimizingtheexpectedcostoftherecourseactionstakenwhencapacityisexceeded[ 8 , 22 , 29 , 30 , 62 , 93 ].Asthemajorityofpracticalroutingproblemscontainsomeuncertaintyinthetraveltimes(forexample,duetotrafc,weather,anddeliverytypes),modifyingtheVRPtoincludestochastictraveltimesaddsnecessaryrealism.ForabriefsurveyofstochastictraveltimesintheTravelingSalespersonProblemandtheMultipleTravelSalespersonProblemasitrelatestovehiclerouting,see[ 40 ].Whenbothtravelandservicetimesare 118

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stochastic,andthevehiclesareuncapacitated,theVRPoftenconstrainsthedurationofeachroutetobelessthansomexedvalue.Asinthedeterministictraveltimecase,thedurationlimitationmaybeenforcedbychanceconstraintsthatensuretherouteislessthanthexedvaluewithsomepredenedprobability,orinexpectationoftheroutedurationoverthesetofallrealizedroutes[ 57 , 61 ].Alternatively,afunctionoftheexpecteddurationofthelongestroutemaybeminimized[ 55 ].Suchproblemshavebeensolvedbybranch-and-cut[ 61 ],branch-and-cutembeddedinsimulationprocedures[ 55 ],andheuristicapproaches[ 57 , 101 ].TimewindowsforthearrivaltimeofvehiclesatcustomersintheVRPwererstconsideredinthecontextofdeterministictravelandservicetimes[ 90 ].Timewindowsmaybe“hard”or“soft.”Ahardtimewindowimpliesthatthecustomerwillnotacceptserviceeventsoutsideoftimewindows,resultinginalostserviceopportunity.WithsofttimewindowVRPs,earlyand/orlatearrivalsarepenalized,andinmostmodels,thevehiclesthatarriveearlymustwaituntilthebeginningofthetimewindowtostartservice.Thereadermayndin-depthsurveysofdeterministictraveltimeVRPswithtimewindowsin[ 17 , 18 , 26 ].Asoursolutionimplementsatabusearch,wementionthatseveralsolutionsimplementatabusearchinthedeterministictraveltimecasewithsofttimewindows;forexample,[ 36 , 67 , 98 ],andalso[ 23 ](whichalsoincludesperiodicandmulti-vehiclerouting).Wedirectthereadertothefollowingsurveysforanintroductiontothetabusearch[ 44 , 45 ],anditsparticularapplicationtotheVRP[ 39 , 41 ].InthestochasticVRPwithsofttimewindows,theobjectivefunctionisoftenamulti-criteriaobjective,whichmayminimizethenumberofroutes,totalcostofalltraversals,penaltytermsfromtimewindowviolations,and/orpenaltytermsforexceedingdurationlimitsoneachroute.TheinclusionofstochasticityinVRPswithtimewindowsincreasesthecomplexity,andmostsolutionsintheliteratureutilizeheuristicapproaches,althoughacolumngenerationsolutiontechniqueexistsaswell[ 97 ].Heuristicapproachesincludegeneticalgorithms[ 2 ],andtabusearches 119

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[ 67 , 81 , 96 , 110 , 110 ].Notethatthesestudiesconstrainvehiclestoarrivewithinthetimewindowsinoneoftwoways.Firstly,theexpectedarrivaldeviationfromthetimewindowmaybeminimized[ 2 , 67 , 81 , 96 , 97 , 110 ].Secondly,achanceconstraintmaybeusedtoensurethatthearrivalsoccurwithinthetimewindowwithsomegivenprobability[ 110 ].Chanceconstraintsaremostoftenusedinthehardtimewindowcasetoensureallcustomersreceiveaminimumlevelofservice.Asmentionedinourintroduction,weutilizebothtechniquestoencouragearrivalswithinthetimewindowsinthepresenceofstochasticcustomers.Thechanceconstraintsallowustoenforcea“hard”constraintthatensuresrarelyservicedcustomersareservicedwithsomeprobability.TheproblemformulationinZhangetal.[ 110 ]alsomanagestimewindowswithbothtechniques,andtheresultingproblemissolvedwithaniterativetabusearchheuristic.Ourformulationissimilar,andwealsoutilizeaniterativetabuheuristicapproach.WefoundlimitedresearchonVRPsthatcombinebothstochasticdemandwithtimewindows[ 66 ].Thisworksolvestheproblemusinganadaptivelargeneighborhoodsearchheuristic.Timewindowsmayalsobeconsideredinthecaseofroutinginthepresenceofstochasticcustomers.ThePTSPhasrecentlybeenmodiedtoincludedeadlineconstraintsintheProbabilisticTravelingSalespersonProblemwithDeadlines(PTSPD[ 20 , 21 ]).Thisproblemdoesnotconsiderstochastictraveltimes,totalroutedurationlimitations,orcapacities.Oursurveyoftheliteraturefailedtondavehicleroutingproblem,orTSPvariant,thatincludedalltheelementsweconsiderintheVRTWSC.Thefollowingworkrepresentsaninitialattemptatformulatingandsolvingthischallengingproblemviaaheuristicapproach. 4.4FormulationTomoreformallydenetheVRTWSC,webeginwiththefollowingnotation.ConsiderallcustomersrepresentedbythesetC=f1,...,jCjg.Eachcustomeris 120

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servicedonagivendaywithsomeprobabilitypiforalli2C.Weconsiderarouteasthexedorderinwhichasubsetofthesecustomersisservicedbyasinglevehicle,andallowexactlyKroutestoserviceallthecustomersinC.DeneKasthecollectionofallsetscontainingKroutes,suchthateachcustomerisonexactlyoneroute.WeseekasolutionR2K,whereeachrouteinRmustserveatleastonecustomer,andeachcustomermustbeassignedtoasingleroute.Ifacustomerrequiresservicemorethanonceperday,eachserviceeventmayberepresentedasaseparatecustomer.Allroutesbeginatthesamestartingterminal,andendatthesamenalterminal,whicharenotnecessarilyidentical.TheVRTWSCproblemcanbedenedonagraphG=(V,A).ThesetofverticesisofsizejCj+2,andcontainsonevertexforeachi2C,vertexv0(representingthestartingterminal),andvertexvf(representingthenalterminal).Weassumethatitisalwayspossibletondapathfromthestartingterminaltoeachcustomernode,andanotherpathfromeachcustomertotheendingterminal.Forarc(vi,vj)2Abetweenverticesviandvj,thereisanassociatedstochastictraveltime.Ateachvertexviforalli2Cthereisanassociatedstochasticservicetime.Onthisgraph,weseekKpathsthatoriginateatv0,endatvf,andvisiteachnodev2Vnfv0,vfgexactlyonceintotal.Wedenethebinaryvariablexijrtoequaloneifandonlyifnodeiwillbefollowedbynodejinrouter,foreachi,j2Vandr2f1,...,Kg,andzerootherwise.ThesevariablesareusedtodenefeasibilityconstraintsofgraphG.Foraxednumberof 121

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routesK,wedeneKbythefollowingconstraints:Xi,j2Cxijr1r2f1,...,Kg, (4)Xi2CKXr=1xijr=1,j2C, (4)Xi2FS(v0)xv0ir=1,r2f1,...,Kg, (4)Xi2RS(vf)xivfr=1,r2f1,...,Kg, (4)Xi2RS(j)xijr)]TJ /F12 11.955 Tf 17.73 11.36 Td[(Xi2FS(j)xjir=0,j2C,r2f1,...,Kg, (4)xijr2f0,1g,i,j2C[fv0,vfg,r2f1,...,Kg. (4)ThesetsFS(i)andRS(i)denetheforward-starandreverse-starofcustomeri2Crespectively.NotethatFS(i)maycontaintheterminalvf,andRS(i)maycontainv0.FS(v0)isthesetofcustomersreachablefromthestartingterminal,andRS(vf)containsthosecustomersconnectedtothenalterminalvf.Constraint( 4 )ensuresthateachroutevisitsatleastonecustomer.Constraint( 4 )ensuresthateachcustomerisassignedtoaroute.Constraints( 4 )–( 4 )areowbalanceconstraintsthatensurethateachroutebeginsatthestartingterminalv0,visitsasubsetofcustomersinacontiguouspath,andreturnstotheendterminalvf.Withthisdenition,Kisthesetofallvalidrouteassignmentswhenwedonotconsidercustomerpreferredtimewindows,servicelevel,orovertimeconstraints.Onanygivendayofservice,allcustomersassignedtoaroutemaynotrequireservice.Inthiscase,thevehiclewillproceeddirectlybetweenthosecustomersthatdorequireservicewithoutpassingthroughthosethatdonot.Iftheshortestpathbetweentwocustomernodesiandjwillpassthroughanothernodek,weassumethatthetraveltimebetweeniandjcanberepresentedaprioriusingasingleprobabilitydistribution,representingthesumofdistributionsofthetraveltimesbetweeniandk,andktoj. 122

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Thus,weseektheassignment(andordering)ofcustomerstoroutesthatminimizestheexpectedtotalcosts.Crewingfactorssuchasovertimeandfatiguemakeitundesirablethatthetotaldurationofrouter2RofsomesolutionR2Kshouldexceedaparameterlr.However,itisunderstoodthattheuncertaintiesincustomerpresence,serviceandtraveltimes,mayoccasionallycausethevehicletoruninexcessofthisroutetimelimitparameter.Toenforceasofterconstraintontherouteduration,werequiretheprobabilitythatrouterrunslongerthanlrtobelessthanorequaltor.Apenaltyof(r)isplacedonanyroutewithdurationgreaterthanlrinexpectation.Eachcustomeri2Chasatimewindowofpreferredservice[ai,bi].DenotethesubsetC(r)Casthesetofcustomersassignedtorouter2RforthesolutionR2K.Ifthevehicleonrouter2Rarrivesatcustomeri2C(r)beforetimeaianearlypenalty(i,r)isincurredperunittimethevehicleisearly.Thevehiclemustwaituntilaitobeginservingthecustomer.Ifthevehiclearrivesatthecustomeraftertimebi,alatepenalty (i,r)isincurred.Eachcustomeralsopossessesarequiredserviceleveli.Ascustomersarestochastic,thedailyprobabilitythatthecustomerisservicedearly,orlate,changesbasedontheothercustomersservicedthatday.Werequirethattheoverallprobabilitythatthevehiclearriveslateatcustomeri2Cbelessthanorequaltoi.WeemploynotationcommoninvehicleroutingliteraturetodenethearrivaltimeatcustomeriasapositiverealvariableAirforalli2C(r)onrouter2R.Thearrivaltimeofrouter2RattheendterminalvfisdenotedbyrealvariableAvfr.Dene(i,r)astheamountoftime(ifany)thevehicleonrouter2Rwasearlywhenservicingcustomeri2C(r).Likewise,(i,r)isthelengthofthetimeintervalthevehiclewaslate 123

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tocustomeri.ForaxedsolutionR,thesetermsarecalculatedas:(i,r)=maxfai)]TJ /F6 11.955 Tf 11.95 0 Td[(Air,0g,i2C(r),r2R, (4)(i,r)=maxfAir)]TJ /F6 11.955 Tf 11.96 0 Td[(bi,0g,i2C(r),r2R. (4)Denetheamountbywhichrouterexceedslras:(r)=maxflr)]TJ /F6 11.955 Tf 11.95 0 Td[(Avfr,0g,r2R. (4)Weseekthecustomersequenceoneachroutethatminimizesthepenaltiesincurredforarrivingoutsideofthetimewindow,aswellasforexceedingtheroutedurationlimit.Becausethecustomersarestochastic,asarethetravelandservicestimesforeachinstanceofcustomerservice,weminimizethesevaluesinexpectation.Namely,forallindividualroutesr2RforasolutionR2Kweminimize:Z(r)=Xi2C(r)(i,r)E((i,r))+Xi2C(r) (i,r)E((i,r))+(r)E((r)), (4)whereE(x)representstheexpectedvalueofsomex.WecannowdenethefollowingformulationfortheVRTWSC:MinimizeF(R) (4)subjecttoF(R)=Xr2RZ(r), (4)PfAirbigi,i2C(r),r2R, (4)PfAvfrlrgr,r2R, (4)R2K. (4)Constraint( 4 )denestheservicelevelsandensuresthattheprobabilityofrarrivinglatetocustomeri2C(r)iswithincustomeri'sservicetolerancei.Constraint( 4 )ensuresthatinafeasiblesolution,thetotaldurationofeachrouter2Rdoesnotexceedlrinmorethanrpercentofallroutes. 124

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4.5EstimationofArrivalTimesAkeycomponentoftheVRTWSCisthedeterminationofthearrivaltimesatindividualcustomers.Becauseweconsiderstochasticcustomers,weutilizesimulationprocedurestoestimatethesevaluesgivenknownprobabilitydistributionsforthecustomerserviceduration,andtraveltimes.ArandomvariableDi,fromaknownprobabilitydistribution,representsthetimeittakestoservicecustomeri2C.Thetraveltimebetweeneachpairofcustomersi,j2CisdenotedbyarandomvariableTij.SymmetrymaynotexistbetweentheprobabilitydistributionsfromwhichTijandTjiaredrawn.Thetraveltimefromthestartingdepottoeachcustomeri2CisgivenbyarandomvariableUi.Also,thetraveltimefromeachcustomeri2CtotheendingdepotisgivenbyrandomvariableVi.TherandomvariablesDi,Tij,Ui,andVimaybeapproximatedempirically,butthedistributionsfromwhichtheyaredrawnareassumedtobeknownandindependentofallotherfactorsrelevanttotheproblem.Whencustomersarenotstochastic,anddonotpossesspreferredservicetimewindows,thearrivaltimeatcustomers,aswellasthetotalexpectedrouteduration,canbemodeledbyasumoftheserviceandtraveltimerandomvariables.IndexthesetofcustomersC(r)ofrouterasc1,c2,...,cjC(r)j,wherejC(r)jdenotesthenumberofcustomersonrouter.Inthiscase,thearrivaltimeatthejthcustomercjonrouter,isdenotedasAcjr,andcanbedeterminedbythesumofrandomvariables:Acjr=Uc1+j)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xi=1Dci+j)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=1Tci,ci+1. (4)Likewisethetotaltimeavehicletakestoreturntothenaldepotis:Avfr=Uc1+jC(r)jXi=1Dci+jC(r)j)]TJ /F7 7.97 Tf 8.94 0 Td[(1Xi=1Tci,ci+1+VcjC(r)j. (4) 125

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Inthepresenceoftimewindows,workatcustomercibeginsatmaxfAcir,acig.Thearrivaltimeatthenextcustomerinthesequencecanbedenedas:Aci+1r=maxfAcir,acig+Dci+Tci,ci+1. (4)Thus,thearrivaltimesofallcustomersonroutercanbeconstructediteratively,wherethearrivaltimeoftherstcustomeronthesequenceisgivenasAc1r=Uc1.Inlieuofsimulatingthesevaluesateachstep,approximationsbasedonthedistributionsfromwhichthesevaluesaredrawnmaybeconstructed[ 110 ].However,inthepresenceofcustomerstochasticity,thearrivaltimeatasinglecustomerdependsonthepreviouslyservicedcustomers,whichmaychangeforeachserviceday/instance.Considerasequenceofthreeorderedcustomers:c1,c2,c3.Thearrivaltimedistributionatcustomerc3maytakefourpossiblevaluesonrouter2R.Giventhatneithercustomersc1,norc2,wereserviced:A1c3r=Uc1.Giventhatcustomerc1wasserviced,andc2wasnot:A2c3r=Uc1+maxfUc1,ac1g+Dc1+Tc1c3.Giventhatcustomerc2wasserviced,andc1wasnot:A3c3r=Uc2+maxfUc2,ac2g+Dc2+Tc2c3.Giventhatallthreecustomerswereserviced:A4c3r=Uc1+maxfUc1,ac1g+Dc1+Tc1c2+maxfAc2r,ac2g+Dc2+Tc2c3. 126

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Eachcustomerisworkedwithprobabilitypifori2C.Assumingcustomerc3isworked,thentheexpectedarrivaltimeatcustomerc3is:E(Ac3r)=(1)]TJ /F6 11.955 Tf 11.95 0 Td[(pc1)(1)]TJ /F6 11.955 Tf 11.95 0 Td[(pc2)A1c3r+pc1(1)]TJ /F6 11.955 Tf 11.95 0 Td[(pc2)A2c3r+(1)]TJ /F6 11.955 Tf 11.95 0 Td[(pc1)pc2A3c3r+pc1pc2A4c3r. (4)Ascanbeseenbythisthreecustomerexample,calculatingtheprobabilitydistributionsisintractableforlargenumbersofcustomersandroutes.Thisisespeciallytruewhenthesolutionmethodologyshouldutilizegeneralizedistributions.Inadditiontothecomplexityofthiscalculation,theresultingexpectedtimemayinnowaymimicreality.Consideratwocustomerexample.Whentherstcustomerisserviced,thearrivaltimeatthesecondcustomer(t1)maybemuchlaterthanwhentherstcustomerisnotserviced(t2).Iftheprobabilityofservingtherstcustomerisonehalf,theexpectedarrivaltimeatthesecondcustomerwilllieexactlybetweenthevaluesoft1andt2.However,thisexpectedvalueisunlikelytobethetruearrivaltimeatthesecondcustomerinanycase.Becauseofthecomplexitiesthatthestochasticcustomersintroduce,weestimatethearrivaltimesofallcustomersonaroute,aswellasthetotaldurationoftheroute,viathesimulationtechniquesoutlinednext. 4.5.1SimulationofArrivalTimeswithoutTimeWindowsThepseudocodeforthesimulationofarrivaltimesonasingleroutewithstochasticcustomerswithouttimewindowsisgivenbySIMULATENOTWinAlgorithm??.Thisalgorithmtakesasinputagivensequenceofcustomers(router),andastarttimefortheroute.Withouttimewindows,thestarttimeoftherouteisarbitrary.Withouttimewindows,thestarttimemaybesettozero,andthearrivaltimescalculatedinpostprocessingforanypossiblestarttime.AlgorithmSIMULATENOTWsimulatesanumberofMSimtheoreticalservicedaysforagivensinglerouter2RforsomeR2K.Eachcustomerc2C(r)hasaprobabilitypcofbeingworkedonanyofthesimulatedservicedays.Foreachcustomerineachiterationofthesimulation,wegeneratearandomvariablefromauniformdistributionontheinterval(0,1).Thecustomerisconsidered 127

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Algorithm6RouteSimulation-StochasticCustomerwithNoTimeWindows 1: procedureSIMULATENOTW(Router,StartTime) 2: forMsimiterationsdo 3: prev ; 4: time StartTime 5: forc2C(r)do.Orderedcustomersonroute 6: ifrand(0,1)
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calculatestheexpectedroutedurationforagivensequenceofstochasticcustomers.Theservicedurationandtraveltimedistributionsareeithernormalorlognormal,andwerederivedusingthemethodsdescribedinChapter3ofthisdissertationforthetimerangespeciedbytheuser.Theprobabilityofservingacustomeronanygivendayiscalculatedbythenumberoftimesacustomerwasservicedversusthenumberoftotalservicedaysinthetimerangespeciedbytheuser.AscreenprintofthedataloadscreenfortheapplicationisgiveninFigure 4-1 .Byselectingthe“ViewAllCustomerStatistics”buttontheusermayviewthedistributions,andotherancillarystatisticsaboutthecustomerservicedurationsandtraveltimes,asshowninFigure 4-2 .Byselectingthe“Interpolate”button,theusermayllininformationnotobtainedfromhistoricaldata.Forexample,traveltimedatamayexistbetweencustomersAandB,butnotbetweenBandA.Iftheuserdeemsthatthetraveltimesareindeedsymmetric,thedistributionforthetraveltimebetweenBandAmaybesetequaltothetraveltimedistributionbetweenAandB.Theusermaywishtoaddanewcustomertotheroute.Inthiscase,theservicetimeandtraveltimesbetweenallothercustomersmaybeestimatedusingindustryknowledge.Also,iftravelneveroccurredbetweenapairofcustomersinhistory,thisinformationmayalsobeestimatedusingindustryknowledgetoprovideastartingpointforroutecomparison.Theusermaynowenteralistofcustomerservicesequencestocompareviasimulation.InFigure 4-3 ,theuserhascomparedsequenceA,BandC,B,A.TheaveragedurationofrouteA,Bis9hoursand21minutes,whiletheaveragedurationofrouteC,B,Ais9hoursand20minutes.Althoughthesecondsequencecontainsathirdcustomer,theprobabilityofservingthisadditionalcustomerissmall.ByorderingthecustomersAandB,evenwithoutincludingthethirdcustomerontheroute,theexpecteddurationincreasesbyoneminute.Ahistogramoftheaveragetimetoarriveatthenalstationisshowninthelowerleftcornerforthehighlightedsequence(C,B,A).Theminimumandmaximumofarrivaltimeatthenalstationonthehistogramdeneawiderangeofpossibletotalroutedurations.Thisinformationis 129

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Figure4-1. Auserentersthetimewindowoverwhichservicetimeandtraveltimedistributionswillbecalculatedforagivensetofcustomers. 130

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Figure4-2. Ausermayviewdistributionsandotherstatisticalvaluesforeachcustomerserviceandtraveltime. 131

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Figure4-3. Simulationcomparesthedurationoftwodifferentroutes. usefulindeningtheexpectedbehavioroftheroute,andthusdeterminingfactorssuchasdeviationfromexpectedbehavioronagivenday,andopportunitiesfordailydynamicloadbalancing. 132

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4.5.2SimulationofArrivalTimeswithTimeWindowsWemaketwomodicationstoSIMULATENOTWpseudocodetoaccountforthecaseoftimewindows.TherstchangeconcernsthetimeofdaytherouteleavestherststationStartTime.Whentimewindowsarepresent,ifthevehiclearrivesatacustomerpriortothestartofthetimewindow,thevehiclemustwaittobeginservice.Becauseofthisthetimethevehicleleavesthestationwillhaveaneffectonthesubsequentcustomerarrivaltimes.Thevalueoftimemaybetreatedasanadditionalvariabletobedeterminedinthesimulation.However,withthepresenceofstochasticcustomers,theresulting“best”timemaybenon-intuitive.Consideraroutecontainingtwocustomers.Iftherstcustomerontherouteisrarelyserviced,the“best”timemay,toalargeextent,ignorethetimewindowoftherstcustomer.Infact,itispossiblethatthetimethevehicledepartsfromthestationmaybeaftertherstcustomer'stimewindowentirely.Onecanassumeanamountoflossofgoodwilltobegeneratedbyanycustomerwhodiscoveredthatthestarttimeoftherouteentirelyexcludedtheirdesiredservicetimes.Becauseofthisweselectthestarttimeofanyroutetobebasedontherstcustomerplacedontheroute,irrespectiveoftheprobabilityofservingthatcustomer.ThevalueofStartTimeinSIMULATENOTWissettobeStartTime ac0)]TJ /F6 11.955 Tf 12.16 0 Td[(E(Uc0),wherec0istherstcustomerinC(r).Namely,thestarttimeisthebeginningoftherstcustomer'stimewindowlesstheexpectedtraveltimeforthevehicletoarriveattherstcustomer.OursecondmodicationtoSIMULATENOTWconcernsthecalculationofthearrivaltimes.Toaccountforthetimethevehiclemustwaitbeforeservicingacustomeratwhichitarrivesearly,thewaittimemustbeaddedtothevalueoftimeasdescribedinequation( 4 )priortotheestimationoftheservicedurationforthecustomer.Theresultingpseudocodeforsimulatingthearrivaltimesinthepresenceoftimewindowsisgivenhere: 133

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Algorithm7RouteSimulation-StochasticCustomerwithTimeWindows 1: procedureSIMULATE(Router) 2: StartTime ac0)]TJ /F6 11.955 Tf 11.96 0 Td[(E(Uc0) 3: forMsimiterationsdo 4: prev ; 5: time StartTime 6: forc2C(r)do.Orderedcustomersonroute 7: ifrand(0,1)acthen.Waittoserviceifearly 15: time ac 16: endif 17: Avfr time+Dc.Addservicetime 18: prev=c 19: endif 20: endfor 21: time time+Vc.Addtraveltimebacktoterminal 22: endfor 23: returnAverageAcrforallcustomersandaverageAvfr. 24: endprocedure 4.6DeterminingVehicleRoutesNowthatwehavedenedthesimulationprocedurethatevaluatesthearrivaltimesfortheVRTWSCProblem,weoutlinetheiterativetabuprocedureweusetodeterminethecustomerassignmentsandsequences.Aniterativetabusearchmakesrepeatedcallstoatabusearch,permutingthecurrentacceptedsolutionbeforeeachcall.Wediscussthespecializedrestartpermutationmethodbelow.Wealsodescribetheimplementationofthetabusearchanditsenhancements. 4.6.1OutlineofIteratedTabuSearchMethodAniteratedtabusearchfunctionssimilarlytoaniterativelocalsearch.Tointroducediversicationintotheexploredsolutionspace,thesolutionreturnedfromatabusearchisperturbedinamannerthatencouragesexplorationintoothersearchregions.The 134

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tabusearchisthenreapplied,andthebestsolutionencounteredacrossalliterationsoftabusearchesisreturnedasthebestsolution.Algorithm 8 providesabriefoutlineoftheiteratedtabusearch.Beginningataninitialsolutions,ITERATEDTABUperformsMITS Algorithm8IteratedTabuSearch 1: procedureITERATEDTABU(0) 2: s initialsolution 3: s s.Bestfeasiblesolution 4: forMITSiterationsdo 5: c Tabu(s) 6: ifsisfeasibleandc(s)
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solutionusingasecondsetofcriteria.Suchcriteriaareheavilyproblemdependent,althoughseveralgeneralizedtechniqueshaveproveneffective.Severalapplicableremovalstrategiesinclude: Random—elementstoberemovedareselectedatrandom. Worstcase—elementstoberemovedareselectedthatcausethegreatestchangetotheobjectivevalue. Relational—asingleelementisrandomlyselected.Thefollowingelementstoberemovedarethosedeemedmostclosely“related”totheselectedelement. Clustering—elementsremovedoccurinthesamegeographiccluster. Time-oriented—elementstoberemovedwereservicedatroughlythesametime. Historical—elementstoberemovedarethosewhoseremovalhasbeensuccessfulinpriorperturbations.Methodsforreinsertingtheremovedelementsbackincludethefollowing: Greedy—elementsareinsertedsuchthattheobjectiveincreasestheleast. Regret—elementsareinsertedsuchthattheymaximizethedifferenceofthecostofinsertingintothebestandsecondbestinsertionlocations. Masterframework—elementsareinsertedbasedoncriteriaderivedfromasimulatedannealingprocedurethatacceptsaninsertionifitissomepercentileworsethanthecurrentsolution. Noise—randomnoiseisaddedtosolutionsandthebetteroftheoriginalandnoisysolutionisselected. Weightedadjustments—elementsareaddedusingoneofseveralinsertionmethods.Themethodselectedhasthebestperformancethroughthecourseofthesearch,asperaweightedscoreoftheimprovementsgainedusingseveralmethods.Notethatmostofthesemethodsrequirethecomparisonofthecurrentsolutionwithupdatedsolutionsthathavebeenmodiedbytheseinsertions/removals.Sincewedonothavedeterministicknowledgeofoursolutionquality,toutilizethesemethodseffectivelywewouldhavetoperformasimulation(orseveral)foreachpossibleinsertion 136

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andremovalweconsider.Utilizingtheexpectedvaluesfromthedurationandtraveltimedistributionsasanapproximationfailstocapturethestochasticnatureofthecustomers.Wechosetocreateaninsertionheuristicthatisbothrelationalandtime-orientedwithrespecttothecustomertimewindows.Logically,thosecustomerswhosetimewindowsbeginatsimilartimesofday,aremorelikelytobeinterchangeablebetweenroutes.WemodiedtherelationalremovalheuristicofShaw[ 88 , 89 ],aspresentedin[ 79 ]thatsolvesapickupanddrop-offdeliveryproblemwithtimewindows.Inthatpaper,therelatednessbetweenanytwoelementsisdenedasaweightedfunctionofseveralparametersrelatedtotheproblem.OurremovalmethodbeginsbyselectingarandomcustomerfromarouterinthecurrentsolutionR2K,andremovingitfromtherouter.TheremovedcustomercisaddedtoalistofremovedcustomersLR.TheremainingcustomersdenethesetSRareorderedbasedontheirrelationshiptotheelementc.Totakeadvantageofthecustomertimewindowrestrictions,wedenetherelationshiptoc,asr(j)c=jaj)]TJ /F6 11.955 Tf 12.39 0 Td[(acjforallj2SR.WethenremovethekthmostrelatedelementtocfromSR,andaddittoLR.WenowrandomlyselectanyelementofromthesetLR.Therelationshipr(j)oisnowcalculatedforallj2SR,andthekthsimilarelementisremovedfromSR,andaddedtoLR.Theprocedurerepeatsiteratively,untilQelementshavebeenremoved.InourcomputationalresultswesetQ=5,althoughthisquantityisoftenrandomized,orgraduallyincreased(ordecreased)throughoutthesearchinotherapplications.Toencouragerandomness,kisrandomizedtobeavaluebetweenzeroandjSRj.NotethatjSRj
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4.6.2TabuSearchWenowprovideabriefoutlineofthetabusearchinAlgorithm 9 .Eachiterationof Algorithm9TabuSearchOutline 1: procedureTABU(SolutionR) 2: R R.bestdiversiedsolution 3: whileunimprovediterations
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VRTWSCformulation.Todiscouragetherepeatedinsertionofthesamecustomerontothesameroute,aseconddiversiedobjectiveFD()iscalculatedthatweightsthefrequencyoftheseinsertions(Subsection 4.6.2.3 ).Animprovingsolutionisselectedwithrespecttotheseevaluationcriteria.Thissolutionisacceptedifitpassesthetabucriteriaestablishedforthesearch(Subsection 4.6.2.4 ).Ifthesolutionfailstomeetthetabucriteria,itmaystillbeacceptedifitpassestheaspirationcriteriaofthesearch(Subsection 4.6.2.5 ).ThisprocedureisiterativelyrepeateduntilMTabuiterationshaveoccurredwithoutanyimprovementtothebestacceptableRsolution.InthecomputationalresultsgiveninSection 4.7 ,MTabu=25.Thebestfeasibleacceptablesolutionfoundduringthetabusearchisreturned,unlessnofeasiblesolutionwasfound.Inthiscasethebestacceptedinfeasiblesolutionisreturned.Wenowdiscusstheconceptsoftheimplementationofthetabusearch. 4.6.2.1NeighborhoodstructureThesolutionspaceweconsideristhesetofallpossibleassignmentsandorderingsK.Ineachstepofthetabusearch,alocalsearchisperformedaroundallneighborsofthecurrentsolutionR2K,whereaneighboristheelementofthesolutionspaceresultingfromtheapplicationofatransformtoR.Wedeneastraightforwardneighborhoodstructure.ForasolutionR2K,wedenetheneighborhoodN(R)tobethesetofallpossibleassignmentsofcustomerstoroutesgeneratedbyremovingonecustomerfromarouter2R,andplacingitintoanypositionofanyotherrouter2Rnfrg.NotethatevaluatingthecontributiontotheobjectivefunctionforneighboringsolutionsofRrequiressimulatingtheroutefromwhichthesinglecustomerwasremoved,aswellastherouteintowhichthecustomerwasinserted.Thus,thelocalsearchofthetabusearchisthemostcostly.Whenthecustomersarenotstochastic,evaluatingtheseneighborsmaybeapproximated[ 110 ].However,whenstochasticcustomersareconsidered,suchanapproximationmustconsidertheprobabilitiesthatanysubsetsofcustomersarevisitedontheroute. 139

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4.6.2.2FeasibilityofsolutionsWhenevaluatingthesolutionsinthetabusearch,weareconcernedwiththefeasibilityoftheresultingroutes.However,allowinginfeasiblesolutionstobeacceptedbythetabusearchmayhelpdivertthealgorithmintounexploredanddesirablesearchregions.Thus,weallowthetabusearchtoconsiderallinfeasiblesolutionsaswell.Ifanimprovingfeasiblesolutionisfound,thissolutionisreturnedfromTABU.Otherwise,themostimprovedinfeasiblesolutionisreturned.Infeasibilitycanariseintwoways.Firstly,pathinfeasibilitymayarisebyviolatingconstraints( 4 )–( 4 )inFormulation( 4.4 ).IfatraveltimedistributionTijexistsbetweenallpairsofcustomersi,j2C,theseconstraintsareinherentlysatised.Ifnotallpossibleconstructedroutesarefeasible,andfeasibilitycanbeensuredbydisallowinginsertionsordeletionsthatleadtoinvalidroutesduringthecreationofneighborhoodN(R).Thesecondformofinfeasiblilitymayarisefromviolatingtheservicelevelanddurationlevelchanceconstraints( 4 )–( 4 ).Weevaluatetherighthandsideofconstraint( 4 )bycalculatingthenumberofiterationsobservedinthesimulationthatarrivedlatetocustomeri,anddividingthisbythenumberofinstancesinwhichiwasworked.Likewise,weevaluatetheviolationofthetotaldurationlrforeachrouter2Rinconstraint( 4 ),byconsideringthenumberofiterationsofthesimulationinwhichthetotaldurationwasinexcessoflr.Thesechanceconstraintsmustbeevaluatedforall2N(R)forsolutionR2Kineachiterationofthetabusearch.Also,whentimewindowsaresmall,orthenumberofcustomersislargerelativetothenumberofvehicles,itmaybedifculttondasolutionwheretheseconstraintsaresatised,ifoneexistsatall. 4.6.2.3SolutionevaluationanddiversicationTocomparethesequenceandassignmentofcustomerstoroutesforallsolutions2N(R),weevaluatetheobjectivefunctionplusthepenaltytermsforexceedingthecustomerservicelevelsandtotalroutedurationlevels.ForagivensetofKroutes 140

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R2K,denethesumoftheviolationofservicelevels(ascalculatedviasimulationSIMULATETW)forallcustomersas:B(R)=Xr2RXi2C(r)maxf0,PfAirbig)]TJ /F5 11.955 Tf 20.59 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[(i)g, (4)andtheviolationofthesumoftheoverallroutedurationsas:L(R)=Xr2Rmaxf0,PfAvfrlrg)]TJ /F5 11.955 Tf 20.59 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[(r)g. (4)TocomparethequalityofneighboringsolutionsofR2Kwithinthetabusearch,foreach2N(R)wecalculateFP()=F()+zB()+zL() (4)forpositivepenaltycoefcientszandz.Toencouragediversicationofthesolution,thevaluesofzandzareadjusteddynamicallythroughoutthetabusearch,asin[ 23 , 110 ].Denotethesolutionacceptedattheendofatabusearchiterationas2N(R).Thevaluesofzandzareinitiallysetto1.IfB()>0,thevaluezisdividedby=1.5,otherwiseitismultipliedby=1.5.IfL()>0,thevaluezisupdatedsimilarly.Thus,repeatedfailuretoobtainasolutionfeasiblewithrespecttotheseconstraints,impliesfuturesolutionsaremorelenientlypenalizeduntilafeasiblesolutionsisdiscovered.Whiletheperturbationsoftheiterativetabusearchhelptodiversifythesearch,thelocalsearchperformedineachtabuiterationstilltendstofocusinalocalsearchregion.Weemployacontinuousdiversicationmethodwithineachiterationofthetabusearch[ 38 , 91 ].Solutionneighbor2N(R)isobtainedbyremovingacustomerfromonerouteandplacingitintoanewroute.Callthefrequencythatthiscustomerhasbeenplacedintothisrouteintheprioriterationsofthiscalltothetabusearch.WecalculateFD()=FP()+0.015F()(jCjK)1 2. (4) 141

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Thisdiversicationtermwasalsousedby[ 110 ].Suchamethodbiasesanon-improvingsolutionbyaddingasmalltermtothepenaltyobjectivefunction. 4.6.2.4TabucriteriaMuchoftheeffectivenessofthetabusearchcomesfromitsabilitytoavoidcyclingbyexcluding(declaring“tabu”)previouslyusedsolutionsforagiventenure.Thistabulistofexclusionsiskeptinshorttermmemory.Ifinaniterationofthetabusearch,neighbor2N(R)iscreatedbyremovingcustomercfromitscurrentroute,andplacingitinrouter,thepair(c,r)isaddedtothetabulist.Asitisexpensivetostoreanentiresolution,onlythesepairsthatdictatethetransformationsofpriorsolutionsarestoredinthetabulist.Improvingsolutionsgeneratedbythesametransformareunabletobeselectedforatenurethatisafunctionoftheproblemsize,namelyanumberofiterationsb(jCjK)1 2c,wherebxcrepresentsthenearestintegernotmorethanx,andjCjisthenumberofcustomers.Suchtabulistsofxedlengtharecommoninpractice[ 38 ]. 4.6.2.5AspirationcriteriaToavoidexcludingimprovingsolutions,anaspirationcriterionisusedthatmayoverridethetabucriteria.Onecommonaspirationcriterionissuchthatiftheobjectivevalueofasolutiongeneratedbyacustomer-to-routeinsertionpresentonthetabulistisanimprovementoverthecurrentbestknownsolution,thenewsolution(feasible,orotherwise)isaccepted.Thisisanoftenusedaspirationcriterionas,iftheobjectivevalueisanimprovement,theinsertiondoesnotresultincycling[ 38 ].However,asourobjectivevalueisdeterminedviaasimulationprocedure,itispossiblethatapreviouslyvisitedsolutioncouldprovideanimprovedobjectiveinasubsequentiteration.Motivatedbyasimilarmethod[ 82 ],ifnonon-tabusolutionisfoundduringaniterationofthetabusearch,weselectanimprovingtabusolutionbasedontheimprovementrelativetothecurrentbest-knownsolution,andthetimethattheexclusionaryinsertionhasbeentabu.Foreachtabusolution2Rwecalculatetheratio nwhere=F(R))]TJ /F6 11.955 Tf 12.24 0 Td[(F(),.HereRisthecurrentbest-knownsolution,andnisthenumberoftabuelementsthatmust 142

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beremovedfromthetabulistbeforewouldnolongerbetabu.Thetabusolutionwiththemaximumratioisselected.BecausethecalculationofFrequiresacomputationallyexpensivesimulation,westorethevalueofF(R)foreachsolutionRinthetabulist. 4.7ComputationalResultsWeimplementedtheiterativetabusearchheuristicfortheVRTWSCprobleminMicrosoft'sC#programminglanguage.AstheVRTWSCwasdesignedtoassistinrouteplanningatCSXTransportation,Inc.theresultingcodemusteasilyintegratewiththecompany'sother.NETapplicationsandwebservices.C#wasselectedforthisreason,althoughwerecognizethatitmaynotbeanidealchoiceofprogramminglanguageforsuchacomputationalexpensivesolutionapproachwhentheprobleminstancesarelarge.AllinstanceswererunonaLenovoThinkPadwithIntelCorei5-2520Mwithtwo2.5GHzprocessors,with6GBRAM,runninga64bitWindows7OperationSystem. 4.7.1TestProblemsIntotal15testinstanceswererandomlygeneratedwith10,15,and20customers.Foreachtestinstance,thecustomerservicedistributions,traveltimedistributions,beginningofeachcustomer'stimewindow,andprobabilitiesofservingeachcustomerwererandomlygenerated.Forthesecomputationalstudiesweassumethatthecustomerservicedistributionsareeithernormalorlognormal.Weconsideralltraveltimesbetweencustomers,traveltimesfromtheinitialstationtoallcustomers,andthetraveltimesfromallcustomerstothenalstationtobelognormallydistributed.AllnormaldistributionswererandomlygeneratedbyobtainingmeanvaluesfromauniformdistributionU(25,120),andstandarddeviationsfromU(15,45).TheshapeandscaleofallthelognormaldistributionsweregeneratedusingU(3.3,4.4),andU(0.34,0.75),respectively.ThesevaluesareconsistentwiththedistributionsoftheactualcustomerserviceandtraveltimedistributionsobtainedfromtheCSXTransportation,Inc.railroadservicedatainChapter3.WewereunabletotesttheheuristicusingtheactualCSXdata,aswehavenotyetobtainedcustomerservicedataforasufcientlylargenumber 143

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ofcustomers.Notethatitispossiblethatvaluesdrawnfromanormaldistributionwithalargestandarddeviation,andsmallermean,maybenegative.Wedidnotincludeadditionallogictoaddressthesenon-intuitiveservicetimevalues.Thenumerousiterationsofthesimulationshouldresultinmeaningfulvaluesintheaveragebehavior,andsuchadistributionwithahighprobabilityofgeneratingnegativevaluesisunlikelyinpractice.Weassumethatallcustomersarereachablefromallothercustomers.Namely,wehaveavalidtraveltimedistributionTijforalli,j2Cwherei6=j.Thisisareasonableassumption,inthatinmosttraditionaltransportationnetworks,avehiclemayreachanylocationfromanyother,althoughthepathmaybelongandcircuitous.Forallcustomers,theprobabilitythatacustomerisservicedwasrandomlygeneratedbyauniformdistributionU(0.0,1.0).Thecustomerservicelevelsofallcustomersweresetto80%,meaningthatafeasiblesolutionensuresthatvehiclesarenotlatetoservicecustomerswithaprobabilityof0.80.Thestartofeachcustomer'stimewindowwasalsorandomlygeneratedbyU(6AM,7PM).Weseteachtimewindowtobeexactly4hourslong.Whentimewindowsareshortrelativetotravelandservicetimes,ndingfeasiblesolutionswhereinvehiclesarrivewithinthetimewindowbecomesdifcult.Likewise,ifalltimewindowsareverylarge,allsolutionsaretriviallyfeasiblewithrespecttocustomerservicelevelconstraints.Thedesiredmaximumroutelengthlrwassetto10hoursforallroutes.Becauseofthelargercostsincurredifvehicleroutesrunlongerthanlr,forr2RwithR2K,wesetthedesiredprobabilitythatrouterrunslongerthanlrtobelessthanorequalto0.10.ForinstanceswithanumberofcustomersjCj=10,weappliedthemodelusinganumberofroutesK=f3,4,5g.ForjCj=15,K=f4,5,6g,andforjCj=20,K=f5,6,7g.WhensolvingtheVRTWSC,wecomparedtwodifferentsetsofpenaltyparameters.Inonecaseweconsideredearliness,lateness,andexcessiveroutedurationequivalent, 144

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andset(i,r), (i,r),and(r)equalto10foralli2C,andr2RforR2K.Secondly,wecreatedasetofpenaltiesbasedonanintuitiveunderstandingofshippingandtransportationoperations.Overtimeisparticularlycostlynotonlyduetoovertimewages,butalsothedangersofcrewfatigue.Also,logically,customersmayappreciatelatenessmuchlessthantheydoearliness.Thus,inoursecondpenaltyschemeweset(i,r)=1, (i,r)=10,and(r)=100.Tables 4-1 – 4-6 showtheresultsfortheiterativetabusearchoneachofthegeneratedinstances.Table 4-1 andTable 4-2 showtheresultsfortheinstanceswithjCj=10for(i,r)=1, (i,r)=10,and(r)=100,andfor(i,r)=10, (i,r)=10,and(r)=10,respectively.Similarly,Table 4-3 andTable 4-4 showtheresultsforjCj=15,andTable 4-5 andTable 4-6 forjCj=20.Eachrowofatablecorrespondstotheresultingsolutionforaprobleminstance,wheretheinstanceIDisgivenintheInst.columnofthetable.TheKcolumnindicatesthetotalnumberofroutesallowedinthesolution.TheTot.Obj.columncontainsthesumofthepenaltytermsthatformtheobjectivefunctionascalculatedbyequation 4 .TheEarlyPen.columnindicatesthesumofthepenaltiesincurredforearlyarrivalsoverallroutesandcustomers.SimilarlytheLatePen.andDur.Pen.columnsindicatethesumofpenaltiesincurredforlatearrivals,androuteduration,respectively.TheAvg.ServicecolumncontainstheaveragePfAirbigoverallcustomersi2C,androutesrinthesolution.Notethat,inafeasiblesolutioneachindividualprobabilitywillbelessthantheserviceleveliforeachi2C.TheProb.OvercolumncontainstheaveragePfAvfrlrgoverallroutesrinthesolution.Inafeasiblesolutioneachoftheseprobabilitieswillbelessthanrforeachroute.TheTot.Travelcolumncontainssumofthetotalexpectedtraveltimeforallroutesgiventhatallcustomersassignedtotherouteareworked.Thisvalueiscalculatedafterthesolutionroutesareobtained,astheexpectedtraveltimebetweenallpairsofcustomerscanbeknownapriori.Whilethesevaluesdonotfactorintothedeterminationofthesolutions,thesevaluesmaybeofinterestformanagementwhen 145

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comparingsolutionsofdifferentnumberofroutes.TheRunTimecolumnindicatestheamountoftimeinsecondsittookfortheiterativetabusearchtoterminate,andnallytheIter.columnindicatestheiterationnumberduringwhichthesolutionwasfound.ConsiderTable 4-1 thatcontainstheresultswhenC=10,withparametersr,i=1, r,i=10and,r=100.Feasiblesolutionswerefoundquicklyinallcases.ThevaluesofAvg.Serviceshowallcustomertimewindowsweremetwithhighpercentages.Whenthenumberofroutesallowed(K)islarger,theaveragepercentofinstancesinwhichallcustomers'timewindowsweremetincreases.Thisisintuitivelycorrect,asistheobservationthatadditionalroutesincreasethelikelihoodofallroutesreturningtotheendingterminalwithinthetotaldurationtimelimit.Table 4-1 showsthatinallfeasiblesolutions,alllatepenaltyandoverroutedurationpenaltieswerezero.Thisisintuitiveas r,iand,rhorarelargerelativetor,i,asweexpectthemodeloutputtoavoidsolutionsthatprovidelateservicetocustomers,aswellaslongerroutedurationsolutions.Notethatthelatepenaltyandexcessivedurationpenaltytermsarealsozerointhecasewhenr,i=10, r,i=10and,r=10(Table 4-2 ).Here,thechanceconstraintsforservicelevel 4 anddurationlevel 4 ensurethatthesevaluesarelow,andthislimitsthelateanddurationpenaltiesoffeasiblesolutions.Whilefeasiblesolutionswerefoundforall10customersinstances,thiswasnotthecasewhenC=15orC=20.InTables 4-4 – 4-5 ,weindicatethatnofeasiblesolutionwasfoundbythetimethetabusearchendedwithtwoasterisks()abovethetotalpenaltyincolumnTot.Obj.Whennofeasiblesolutionwasfound,thevaluesreportedintherowfortheinstancecorrespondtothebestinfeasiblesolutionfoundduringtheiterativetabusearch.Whilethisreportedsolutionprovidesthelowestobjectivevalue,itmaybeveryfarfromfeasiblewithrespecttothechanceconstraints( 4 )and( 4 )thatimposeserviceandovertimelevels.Therouteconnectivityconstraintsofthemodelarealwayspreserved.Inthecasethatnofeasiblesolutionisfound,theusermayopt 146

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toreviewtheintermediatesolutionstondasolutionthatcomesclosesttosatisfyingthechanceconstraintsattheexpenseoftheobjectivefunctionvalue.Wedonotethattheservicelevelandexcessdurationparameters(and)maybeexcessivelystrictcomparedtowhatisacceptableinpractice.Feasibilitywouldbemoreeasilymaintainedifthesevalueswererelaxed,oralargeKconsidered.Althoughitispossiblethatfeasiblesolutionsmayexistfortheseinstances,whenthenumberofroutesissmallrelativetothenumberofcustomers,itismoreunlikely.Thisisespeciallytrueifseveralcustomershaveoverlappingservicetimewindows.Inthatcaseitisespeciallydifculttoserveallcustomerson-timewithasmallnumberofroutes. 4.7.2PerformanceofthePermutationMethodInSection 4.6.1 wedescribedthemethodusedtointelligentlypermutethecurrentsolutionaftereachtabusearchiterationtoencouragediversication.Todeterminetheeffectivenessofthisperturbationscheme,wealsoimplementedasecondpermutationmethodthatrandomlyreassignedcustomerstoroutes.Werantheiterativetabusearchusingbothpermutationmethodsonthe10customertestinstances.Table 4-7 showsthebestfeasibleobjectivefunctionvalueFfortheiterativetabusearchusingbothmethods.Italsocomparesthenumberofiterationsrequiredtoreachthebestfeasiblesolution.Whilethespecializedpermutationmethodshowedgreatimprovementovertherandompermutationschemeinsomecases,itwasnotconclusivelybetterinallinstances.Thespecializedpermutationschemeprovidesagreaterchancethatthenextiterationofthetabusearchbeginswithasolutionthatis.Yet,theentirelyrandomizedpermutationmethodmayallowtheiterativetabusearchtoescapelocalminimumsmoreeffectively.However,neithermethodprovedsuperiortotheother. 4.8ProblemExtensionsInordertofurthergeneralizetheproblem,wemayalsowishtoincludeseveralotherconceptscommoninvehicleroutingproblems.Belowwesummarizeseveralenhancementsthatmaybeincludedinfuturemodels. 147

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Table4-1. Resultsfor10customerinstanceswith(i,r)=1, (i,r)=10,and(r)=100foralli2C,r2RforsolutionR. Inst.KTot.Obj.EarlyPen.LatePen.Dur.Pen.Avg.ServiceProb.OverTot.TravelRunTimeIter. C10 1333.5733.570.000.000.940.96768.392445C10 23190.11190.110.000.000.950.99806.9325787C10 331362.681362.680.000.000.950.97865.2029158C10 4346.9846.980.000.000.940.93807.7127119C10 5361.7861.780.000.000.990.98707.2526495C10 1429.0229.020.000.000.970.95782.4423927C10 242.162.160.000.000.980.93894.8123839C10 3421.4621.460.000.000.950.95949.4026415C10 4439.4839.480.000.000.970.94925.8025016C10 5482.5882.580.000.000.980.98852.4623518C10 150.000.000.000.000.980.95969.9719472C10 2540.7840.780.000.000.980.97964.7121781C10 350.000.000.000.000.980.96930.9322634C10 450.000.000.000.000.980.97944.8121382C10 550.000.000.000.000.990.981042.5519667 148

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Table4-2. Resultsfor10customerinstanceswith(i,r)=10, (i,r)=10,and(r)=10foralli2C,r2RforsolutionR. Inst.KTot.Obj.EarlyPen.LatePen.Dur.Pen.Avg.ServiceProb.OverTot.TravelRunTimeIter. C10 13256.77256.770.000.000.970.96772.1024958C10 23931.07931.070.000.000.960.99776.5224827C10 331489.961489.960.000.000.970.96715.4628168C10 43752.66752.660.000.000.970.97762.5528071C10 53259.38259.380.000.000.970.96779.9025725C10 14150.87150.870.000.001.000.99876.4722657C10 24115.38115.380.000.000.980.98927.172318C10 34856.14856.140.000.000.970.94905.582669C10 44113.69113.690.000.000.970.96908.1325377C10 5419.0819.080.000.000.980.961017.292330C10 150.000.000.000.000.980.97915.121941C10 257.227.220.000.001.000.97941.1220292C10 35130.01130.010.000.000.970.98897.4321812C10 45174.19174.190.000.000.970.98923.0220721C10 550.000.000.000.001.000.98905.711931 149

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Table4-3. Resultsfor15customerinstanceswith(i,r)=1, (i,r)=10,and(r)=100foralli2C,r2RforsolutionR. Inst.KTot.Obj.EarlyPen.LatePen.Dur.Pen.Avg.ServiceProb.OverTot.TravelRunTimeIter. C15 140.00**0.000.000.000.820.881238.217629C15 24979.15979.150.000.000.960.951244.0565560C15 3478.31**13.3964.910.000.800.891216.2576044C15 440.00**0.000.000.000.890.891285.357187C15 54440.33440.330.000.000.970.981226.8968629C15 151587.671587.670.000.000.950.971249.797108C15 251062.701062.700.000.000.950.991315.0463074C15 3584.85**84.850.000.000.910.941467.9970816C15 45166.47166.470.000.000.980.981250.9666161C15 55177.95177.950.000.000.970.961261.0863678C15 16132.14132.140.000.000.960.961281.756281C15 26438.86438.860.000.000.970.971354.9058248C15 36192.24**192.240.000.000.890.951444.9263529C15 46168.19168.190.000.000.980.981319.8658829C15 56199.58199.580.000.000.990.981386.0558415 150

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Table4-4. Resultsfor10customerinstanceswith(i,r)=10, (i,r)=10,and(r)=10foralli2C,r2RforsolutionR. Inst.KTot.Obj.EarlyPen.LatePen.Dur.Pen.Avg.ServiceProb.OverTot.TravelRunTimeIter. C15 140.00**0.000.000.000.910.941020.4175932C15 24363.83**337.3726.460.000.880.901186.4664415C15 34320.18**0.00320.180.000.810.821200.437420C15 440.00**0.000.000.000.920.811210.177005C15 54598.27**573.440.0024.830.860.941233.4367538C15 152162.432162.430.000.000.960.971165.787048C15 252898.742898.740.000.000.970.971194.1761472C15 35553.97**0.00553.970.000.940.901307.1169428C15 450.00**0.000.000.000.890.831249.6265213C15 553578.963578.960.000.000.980.991229.5762742C15 163605.793605.790.000.000.960.961197.9163432C15 261246.931246.930.000.000.970.971321.3655557C15 363063.923063.920.000.000.950.971478.1163716C15 461574.731574.730.000.000.960.961355.5657942C15 561204.411204.410.000.000.970.971426.2656673 151

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Table4-5. Resultsfor20customerinstanceswith(i,r)=1, (i,r)=10,and(r)=100foralli2C,r2RforsolutionR. Inst.KTot.Obj.EarlyPen.LatePen.Dur.Pen.Avg.ServiceProb.OverTot.TravelRunTimeIter. C20 150.00**0.000.000.000.850.901417.4315424C20 251194.65**591.56603.090.000.860.911544.19133458C20 351159.67**882.23277.440.000.860.881698.8614907C20 45183.72**125.4458.290.000.850.891592.0414134C20 5572.24**72.240.000.000.870.941637.67138653C20 16132.59**132.590.000.000.930.891658.6223:4129C20 26171.57**71.48100.100.000.860.901732.25124563C20 36228.87**228.870.000.000.900.951574.27139224C20 46168.09**168.090.000.000.920.941694.91129156C20 5667.37**67.370.000.000.930.941552.48127228C20 1713.41**13.410.000.000.940.921612.19131565C20 27473.43**453.3720.050.000.880.861785.71114464C20 37121.77**121.770.000.000.880.941676.33128532C20 4759.55**55.593.970.000.910.921595.5911751C20 57467.62467.620.000.000.960.961884.99114711 152

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Table4-6. Resultsfor20customerinstanceswith(i,r)=10, (i,r)=10,and(r)=10foralli2C,r2RforsolutionR. Inst.KTot.Obj.EarlyPen.LatePen.Dur.Pen.Avg.ServiceProb.OverTot.TravelRunTimeIter. C20 15385.25**67.19318.070.000.800.911693.19150849C20 251346.20**36.911309.290.000.840.851572.9813432C20 351791.55**244.271547.280.000.860.861475.47148633C20 45890.89**890.890.000.000.850.931669.90139260C20 55503.52**162.08341.440.000.930.871403.80138623C20 16756.90**474.52115.33167.050.840.891705.69141058C20 262096.69**1816.24280.440.000.920.871906.4512531C20 361812.01**1039.98772.030.000.900.901499.81138924C20 46373.50**373.500.000.000.950.941820.46128140C20 561000.57**982.7017.870.000.950.931724.54125245C20 170.00**0.000.000.000.920.911648.15130911C20 272578.21**779.630.001798.580.840.931892.9611464C20 371365.28**0.00601.56763.720.850.891722.91127037C20 47663.76**663.760.000.000.890.931801.56115726C20 572016.202016.200.000.000.980.971762.04113932 153

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Table4-7. ComparisonofrandomizedpermutationsandenhancedpermutationsforjCj=10withpenalties(i,r)=10, (i,r)=10,and(r)=10foralli2C,r2RforsolutionR. Inst.RRandomizedPermutationsEnhancedPermutationsTot.Obj.Iter.Tot.Obj.Iter. C10 1305256.7758C10 231168.8256931.0727C10 332936.05551489.9668C10 431819.4249752.6671C10 53159.763259.3825C10 14020150.8757C10 24114.829115.388C10 34842.7917856.149C10 44676.141113.6977C10 54230.371919.080C10 150201C10 250237.2292C10 3502130.0112C10 45059174.1921C10 55152.363901 4.8.1CapacityConstraintsWhenthevehiclesassignedtorouteshavenitecapacityvalues,additionalfeasibilityconstraintsmustbeincludedtoensurethatthevehiclehassufcientremainingcapacitywhenitarrivestoserviceacustomer.Anyviolationofcapacityonaroutemayrequirethatthevehiclereturntotheterminal,oranintermediarypointoftheroute,tounloadafullvehicle.Alternatively,anyviolationofcapacitymaybeallowedbutwouldrequireanadditional(costly)modicationtothevehicleassignedtotheroute.Forexamplealargertruckmaybeneeded,orinthecaseofrailtransit,anadditionallocomotivemaybeaddedtoincreasethetowingcapacity.Wheneachrouter2RhasamaximumcapacityofQr,andeachcustomerhasaxedvolumeofworkqi,thiscanbeenforcedbyincludingthefollowingconstraintintheexistingFormulation 4.4 Xr2RXi,j2CqixijrQr. (4) 154

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Theseconstraintsmaybeenforcedintwowayswhenincludedinthetabusearchprocedure.Firstly,theymaybeenforcedbyrestrictingtheneighborhoodofasolutiontoonlyincludeneighborsthatrespecttheconstraints.Secondly,thecapacityconstraintsmayberelaxedandpenalizedinthemodiedobjectiveFP.Unfortunately,thenotionthatallcustomershaveaxedvolumeofworkforeveryserviceeventmaynotbepractical.Thequantityofworkatanycustomerisastochasticvaluesubjecttosomedistributionthatmayvarywidely,orevenseasonally.Theinclusionofstochasticcustomervolumeintoamodelinvolvingstochasticcustomers,aswellasstochastictravelandservicetimesiscomplex.However,usingaxedvaluetoestimateqimayresultinsolutionsthatfailinpractice.Usingtheexpectedvolumeatacustomertoapproximatethevolumeresultsinroughlya50%chancethatarouteexceedsitscapacityonagivenday.Returntripstoterminals,orincreasingroutecapacity,iscostly.Toavoidthis,itislogicaltomodelcapacityusinganoverestimateofcustomervolume.Arstapproachmaybetoestimatethevolumeateachcustomerusingthemaximumvolumeforacustomerobservedinhistoricaldata.However,thislogicmayresultinasuboptimalsolution.Ifacustomerwithveryhighvolumehasaverylowlikelihoodofrequiringservice,thisapproximationwillresultinaconsistentlylargeexcessofavailablevolumeonthisrouteforthemanydaysonwhichthiscustomerisnotserviced.Theinclusionofcapacityconstraintsinthepresenceofstochasticcustomerswillmostlikelyrequireproblemspecicknowledgeinordertoobtainasolutionthatisusefulinpractice. 4.8.2MinimizingtheTotalNumberofRoutesOnecommongoalofvehicleroutingproblemsistominimizethenumberoftotalvehiclesrequired.Notethatthisisnotcurrentlyincludedinourmodel;infact,thenumberofroutesremainsaxedconstantthroughouttheiterativetabusearch.TominimizethenumberofroutesinFormulation 4.4 ,thevalueofKisconsideredavariable.AtermZKwouldbeaddedtotheobjective,whereZisthexedcosttousea 155

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vehicle,andKisthetotalnumberofvehiclesused.AppropriateconstraintswouldbealsoneedtobeaddedtodeneK.Tominimizethenumberofroutesintheiterativetabuprocedurea“routereducing”procedurecanbeincludedafter(and/orbefore)eachcalltotheTABUprocedure.Suchaprocedureattemptstoremoveallcustomersfromonerouteandinsertthemintootherrouteswithminimalchangeintheobjectivevalue.Wedidnotincludethisminimizationofroutesinthecurrentversionofourproblemsolutionfortworeasons.Firstly,thevalueofZisoftenlargecomparedwiththepenaltyvaluesof, ,and.WhenthevalueofZislarge,thistermdrivesthesearchuntilthevalueofKminimizedtoafeasiblelevel;theotherconsiderationsareessentiallysecondaryobjectives.Whiletheobjectivecoefcientsvalues, ,andmaybeincreased,ndingappropriate,anduniversallyapplicable,valuesforthemmaybetricky.Ourprimaryobjectiveconsiderationwastominimizecustomerarrivalearliness/lateness,aswellastheoverallrouteduration.Thesecondreasonisrelatedtotheproblemapplication.Consideringthattheuserislikelyoperatingunderanexistingroutingplan,adrasticchangetothenumberofvehicleroutesmaybedifculttoimplementimmediatelyandthusunacceptable.Fixingthetotalnumberofroutesallowstheusertodeterminepossiblechangesincustomerservicelevelsusingonlyoneortwofewer(ormore)routes.Ascrewingandobtainingequipmentforarouteisexpensive,ifnewcustomersareaddedto,orremovedfromroutes,theoptimalassignmentandsequenceofcustomersmaybecomputedwithoutchangingtheexistingnumberofroutes. 4.8.3MinimizingTotalTravelDistanceAnotheroftenusedobjectivecriteriaistheminimizationofthetotaldistancetraveledbyeachroute.Ifcustomersarelocatedfarfromoneanother,minimizingthistermmaybeimportanttoreducefuelcosts.Stochasticvehicleroutingproblemsmodelthisbyincludingtheterm(r)Pr2RPi,j2CE(Tij)xijrintheobjectivefunction,where(r)forallr2R,andR2K,weightstheimportanceofminimizingthetotaldistanceof 156

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routerrelativetotheotherobjectivegoals.Inthecaseofstochasticcustomers,thisexpectationmaybethesumoftheaveragetraveltimesforthesequenceofcustomersontheroute.Or,itmaybethesumoftheexpectedtotaltraveltimeconditionedontheprobabilityofservingeachcustomerontheroute.Inthissecondcase,thesumoftheexpectedvaluesisidenticaltothatofE((r))inobjectivefunction( 4 ).Intherstcase,whilethesumoftheaveragetraveltimesissimilartoE((r)),itisnotequivalent.Minimizingthesumofaveragetraveltimesminimizesthemaximumtotaldistancethatthevehiclemaytravelinthecasethatallcustomersinthesequenceareserviced.Iftheroutecontainsoneormorerarelyvisitedcustomers,Pr2RPi,j2CE(Tij)xijrmaygreatlyexceedtheactualtraveltimeformostinstancesoftheproblem.Asolutionthatminimizesthetotaldistancetraveledoneachroutemaybeatoddswiththeotherobjectivegoalsofmeetingthecustomerarrivaltimeconstraints.Itmaybepreferentialtohavethevehicletakeaslightlylongerroutetoavoidarrivingearlyatacustomerifearlypenaltycostsarehigh.Thepracticalgoaloftheroutingplanderivedshouldweighthistrade-offwhendeterminingwhethertoincludethistermintheobjectivefunction.Notethattheaveragetraveltimebetweenallpairsofcustomersisknownapriorifromthegiventraveldistributions.Asitdoesnotdependupontherealizationofthestochasticcustomers,itistrivialtoincludeinthecurrentiterativetabusearchapproach. 4.9ConclusionInconclusion,weintroducedaStochasticVehicleRoutingProblemwithsofttimewindows.Stochastictravelandservicetimesareconsidered,aswellasstochasticcustomers.Asinmanysuchstochasticvehicleroutingproblems,deterministicsolutionmethodsaredifculttoemploy.Theinclusionofthestochasticcustomersmakesthedeterminationofthearrivaltimesparticularlydifculttoestimate.Toobtainthearrivaltimesforasinglegivenroute,weprovideasimulationprocedure.Wesolvedthelargerproblemofdetermininganappropriateassignment,andsequence,ofcustomersfor 157

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axednumberofroutesusinganiterativetabusearchmethodinwhichasimulationprovidedarrivaltimeestimates.Wepresentedpreliminarycomputationalresultsusinganiterativetabuheuristic.Asthesimulationprocedurethatestimatesthearrivaltimesiscurrentlycalledforeachneighboringsolutionineachiterationofthetabusearch,asignicantportionofthecomputationaltimeisspenthere.Whileitisnotyetintuitivehowtoproceedinthecaseofstochasticcustomers,anapproximationforthearrivaltimesisneeded,inlieuofthesimulationprocedure.SuchanapproximationwouldallowforlargerVRTWSCproblemstobesolved.Wealsopresentedseveralpossibleextensionsoftheproblemthatcouldbeconsideredinlaterversionsoftheproblem.Ascertainoftheseextensionsrequiremodifyingthefeasibleregionviaadditionalconstraints,thecomplexityofdeterminingsolutionsmayincrease.Inthiscaseobtainingalowerboundontheproblemwillhelptoguaranteesolutionquality. 158

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BIOGRAPHICALSKETCH ShantihSpanton'sDoctorofPhilosophywasobtainedfromtheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridain2014.ShereceivedBachelorofSciencedegreesinphysicsandmathematicsfromtheUniversityofWisconsin–EauClaire,in2006. 168