AutomatedOptimalBalancingofTrafficVolumeDataforLargeAccess‐ControlledHighwayNetworksandFreeway‐to‐FreewayInterchanges

JohnW.Shaw,PE(CorrespondingAuthor)ResearcherTrafficOperations&SafetyLaboratoryUniversityofWisconsin1415EngineeringDriveMadison,WI53706Phone:(414)227‐2150jwshaw@wisc.eduDavidA.NoyceProfessorDepartmentofCivil&EnvironmentalEngineeringUniversityofWisconsin1204EngineeringHall1415EngineeringDriveMadison,WI53706Phone:(608)265‐1882noyce@engr.wisc.eduWordCount(4342+2Figures×250+3Tables×250)=5592words

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ABSTRACTTrafficanalystsandmodelersfrequentlyneedtocombinestreetandhighwayvolumedatafrommultipledaysandsourcesinordertoassembleacompletedatasetforastudyarea.Pro‐ratamethodshavetraditionallybeenusedto“balance”inconsistentdatasetsforlimited‐accessfacilities,butthepro‐rataprocedurehasseverallimitationsandcannotreadilybeappliedtocomplicatedroadwaygeometriessuchasfreeway‐to‐freewayinterchanges.MathematicaloptimizationofthedatasetusingalgorithmssuchastheGRG2solvercanaddressmanyofthesesituationsandallowsrapidbalancingoflargeroadwaynetworksformultipletimeperiods.Optimizationalsoallowsasecond(perhapslessreliable)setofmainlineobservationstobeconsideredinthebalancingprocess.WeselectedminimizationofthesumoftheGEHforeachobservationastheobjectivefunctionandimplementedtheprocedureinaspreadsheet(itcouldalsobeimplementedasadatabase‐drivenapplication).OptimizationproducedmorefavorableGEHvaluesthanpro‐ratamethods,indicatingbetterfittotheoriginaldataset.OptimizationresultscombiningGRG2andGEHappeartobereasonablystableandtolerantoffluctuationsintheinputdatavaluesattributabletotypicaltrafficcountingdevices.Additionalresearchisnecessarytoquantifytheeffectsofsevereoutliersontheoptimizationresultsanddeterminewhichcombinationofobjectivefunctionandsearchalgorithmis“best”forvarioustypesofnetworks.INTRODUCTIONTrafficanalystsandmodelersfrequentlyneedtocombinestreetandhighwayvolumedatafrommultipledaysandsourcesinordertoassembleacompletedatasetforastudyarea.Theunderlyingdataoftenvariesconsiderablyintermsofageandquality.Asaresult,field‐collectedtrafficvolumesforafreewaysystem(orotheraccess‐controlledcorridor)frequentlycontainmathematicalinconsistencieswhichcanhampertheanalyticalprocessandslowtheconvergenceofdata‐intensiveoperationssuchasOrigin‐DestinationMatrixEstimation(ODME).Manuallybalancingalargefreewaynetworkisextremelylabor‐intensiveandhasnouniquemathematicalsolution.Figure1showssomeobserveddailyvolumesalongonedirectionofafreewaysectioninsoutheasternWisconsin.IfwestartwiththevolumeattheAutomaticTrafficRecorder(ATR)stationontheleftsideofthefigureandproceedinthedirectionoftravel,addingtheon‐rampvolumesandsubtractingtheoff‐rampvolumes,therunningtotal(showninpurpleonthebottomline)doesnotmatchtheobservedvolumeatthedownstreamATRontherightsideofthefigure(inblueontheupperline).

Figure1:ExampleofanimbalancebetweenupstreamATRvolume(representedbytriangle)andrunningtotalofentranceandexitvolumesatnextdownstreamATR.

840008300 7100 6200 7500 4800 4400 80600

84000 – 8300 + 7100 – 6200 + 7500 – 4800 + 4400 = 83700

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TheinconsistencieshighlightedinFigure1ariseprimarilyfromthefactthatthenetworkwasnotcountedsimultaneously.Instead,thefigurerepresentstypicalWisconsindatacollectionpractice:acombinationofannualaveragesfrommainlineATRs(continuouscounts)and48‐hourrampcountscollectedonvariousdaysusingportablepneumatictubecounters.Alocationsuchasthismightalsohaveshort‐durationmainlinecounts,volumesfromFreewayTrafficManagementSystem(FTMS)sensors,toll‐tagreaderdata,weigh‐in‐motionstations,intersectionturningmovementcounts,etc.Sincedataiscollectedatdifferenttimeswithvarioustechnologies(anddifferingquality‐acceptancecriteria),itmustbeadjustedor"balanced"toobtainamathematicallyconsistentdataset.Thispaperdiscussesaprocedureforbalancingtrafficvolumedataforanaccess‐controlledfacilitysuchasafreeway.Ifdirectionaldataisavailable,balancingcouldalsobeappliedtoarterialsorothernon‐access‐controlledfacilities,providedthattrafficgeneratorssuchassidestreetsandmajordrivewaysaretreatedinamannersimilarto"ramps"inthediscussionthatfollows.GOALSOFTHEBALANCINGPROCESSOurprimaryapplicationsforthebalanceddataareHCMcapacityanalysis,workzonetrafficplanning,andODmatrixestimationformicrosimulationmodeling(wherepriorexperienceindicatedthatbalancingtheinputdataresultedinfasterconvergenceandreducedtheriskofoscillationbetweenconflictingnumericaltargets).Thereforeourgoalsincludethefollowing:

Computebalancedvolumesforallaccess‐controlledroadwaygeometries,includingfreeway‐to‐freewayinterchanges(systeminterchanges)andfrontageroadsystems,aswellasordinaryfreewaysegments.

Facilitatesimultaneousbalancingofallaccess‐controlledfacilitiesinalargenetwork(i.e.multi‐countyregionsorstatewide).

Supportbalancingovermultipletimeperiods,e.g.12months×24hours×4daytypes(weekday,Friday,Saturday,Sunday).

Utilizeaspreadsheetoradatabaseapplication,withoutrequiringexternaltoolssuchasatraveldemandforecastingmodel.

ReconciledatafromallavailablesourcesincludingATRs,FTMSsensors,andshort‐durationcounts.

Imputemissingdataforshortsegmentssuchasbetweenrampsatadiamondinterchange. Tempertheeffectsofoutliers(suchascountscollectedonnon‐representativedays)and

errorssuchasequipmentproblems. Utilizeprior‐yeardataasasurrogateincaseofdetectorfailureorsimilarcircumstances. Provideanindicatorofpossiblesensorproblems.

PREVIOUSAUTOMATEDBALANCINGMETHODSSeveralauthorshaveaddressedthequestionofbalancingturningmovementcountdataalonganarterialcorridor.Hauer,etal(1)developedatechniqueforsolvingandbalancingturningmovementvolumesbasedonaninitialestimateofturningproportionssuppliedbytheuser.ThemethodwasfurtherelaboratedbyLin&Rasp(2)anddevelopedintoaspreadsheetapplicationcalledTURNS5fortheFloridaDepartmentofTransportation(3).Ren&Rahman(4)appliedafuzzylogicapproachtotheproblemofforecastingfutureturningmovementcounts,givenamainline

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forecastandanexistingturningpattern.Xin(5)developedadesktopapplicationcalledArtBalTforbalancingexistingintersectionturningmovementcountsalongashortarterialcorridor(upto6typicalintersections)by“proportioningtheextralinkvolume(i.e.thevolumedifferencebetweenthetotalinflowandtotaloutflowofalink)amonganymovementsthatcanbechanged.”Liao(6)expandedXin’slogictoallowpro‐ratabalancingalonga“lineararterialnetworkwithvirtuallynointersectionsizelimit.”Acaveattothesemethodsisthepotentialexistenceofunmonitoredaccesspointsbetweenintersections,potentiallyviolatingtheassumedequalitybetweenupstreamoutputvolumesanddownstreaminputvolumes.Inthecontextofaccess‐controlledfacilities,the2001FHWATrafficMonitoringGuide(TMG)(7)suggeststheuseofapro‐rataadjustmentprocessforfreewaymainlinebalancing.IntheTMGprocedure,pairsofATRsareselectedandusedas“anchorpoints”;countsattheanchorsareassumedtobeaccurateandpro‐ratarampvolumeadjustmentsaremadeinordertobalancevolumesalongthesegment.

(1)

| |

(2)

Where:, =Mainlinevolumesatanchorpoints

=Observedvolumeonrampi(>0forentrancerampsand<0forexitramps)

=Observedvolumeonrampj(>0forentrancerampsand<0forexitramps)

=Adjustedvolumeonrampi(>0forentrancerampsand<0forexitramps)

=Adjustedvolumeonrampj(>0forentrancerampsand<0forexitramps)

InTable1,ColumnCshowsthewaytheTMG’spro‐ratamethoddistributestheimbalanceof+3100vehiclesfromtheFigure1data(inthiscasethe3.1%mainlineimbalanceresultsinan8.1%adjustmenttoeachentranceandexitramp).Xin(5)utilizedthismethodinadesktopdata‐preparationapplicationcalledTradaX.Themethodwasalsoutilizedonaregionalscaleinanunpublishedspreadsheet‐basedtoolcalledBALTdevelopedbyNam&OrnekfortheSoutheastWisconsinFreewaySystemOperationalAssessment.Inthemid1990sZhaoetal(8)developedatrafficdatareconciliationsystemfortheConnecticutDOT.TheiralgorithmfocusesprimarilyontheuseofbalancingtechniquestoimputetrafficvolumesforunmonitoredmainlinelinkswhenthereareasmallnumberofATRsservingasanchorpointsandafullsetofcountsforfreewayramps.Inaddition,theyprovidedlogicusingleastsquaresregressiontoresolvediscrepanciesatmainlinecountstationswhereredundantdataisavailable.Asignificantlimitationoftheirmethodisthattheobjectivefunctionrequiresanaprioriestimateofthevariance(orstandarddeviation)ofeachsite’svolumes,andassumesthattheerrorsarenormallydistributed,whichisnotalwaysthecase.

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ArelatedapproachwastakenbyKwonetal(9),whousedmatrix‐basedweightedleastsquaresregressiontosolvethebalancingproblemandimputemissingdatavaluesforaccess‐controlledlinearcorridors.ThemethodisprimarilyintendedforusewithFreewayTrafficMonitoringSystems(FTMS)thathaverelativelyclosedetectorspacingandcontinuousdatafeeds.Alimitationofthemethodisthatitrequiresestablishingconfidencelevels(excellent/good/fair/poor)foreachtrafficdetectorstationinadvance.Settingconfidencelevelsmaybedifficultinthecaseof“planning”typedatathatcombinesobservationsfromdifferentdays,sincetheanalystmaybeunawareofsensorproblemsorephemeralundercountsorovercountscausedbyincidentsorevents.Themethodalsorequiresacorridor‐widestandardizedflowvalueand,liketheZhaomethod,assumesthaterrorsarenormallydistributed.

TABLE1.ExamplesofbalancingcalculationmethodsandresultingGEHscores.

A  B  C  D  E 

Segment Description 

Raw Observed Data  Pro‐Rata Adjustment Two Constrained Anchors Branch & Bound Solver 

No Anchor ConstraintsGRG2 Solver 

Volume Running 

Total Volume

Running Total

GD VolumeRunning 

TotalGD  Volume 

Running Total

GD

ATR on Segment A  84000  84000  84000    0.0 84000 84000 0.0  81376  81376 2.9

Exit Ramp 1  ‐8300     ‐8972    2.3 ‐8764    1.6  ‐8300     0.0

Between Ramps at Exit 1     75700     84672       75236       73076   

Entrance Ramp 1  7100     6525    2.2 6501    2.3  6896     0.8

Basic Segment B     82800     85246       81737       79971   

Exit Ramp 2  ‐6200     ‐6702    2.0 ‐6659    1.8  ‐6187     0.1

Between Ramps at Exit 2     76600     85748       75078       73784   

Entrance Ramp 2  7500     6893    2.3 6896    2.3  7284     0.8

Basic Segment C  82000  84100     86355 4.7    81974 0.0     81069 1.0

Exit Ramp 3  ‐4800     ‐5189    1.7 ‐5261    2.1  ‐4799     0.0

Between Ramps at Exit 3     79300     86744       76713       76270   

Entrance Ramp 3  4400     4044    1.7 3887    2.5  4251     0.7

ATR on Seg D (Calculated)     83700  80600    0.0    80600 0.0     80521 0.1

ATR on Seg D (Observed)  80600                               

Imbalance     +3100   0       0       0   

Total              16.9       12.6        6.3

 

Currentlythepro‐ratatechniquediscussedintheTMGappearstobethemostwidelyappliedbalancingmethodology.Whilepro‐rataadjustmentsareexpedientandrelativelysimpletoimplement,themethodhassomeimportantlimitations:

1. Inpracticeitisunlikelythatcountingimbalancesareactuallyproportionalforallsitesalongacorridor.Oftenalargepartofanimbalanceisattributabletoafewsiteswheredatawasaffectedbyaproblemsuchaspoorsensorplacement,equipmentmalfunction,incidents,orconstruction.Stressingtheneedforcheckingtheresultofbalancing,theTMGcautions,“anequipmenterrorinanyoftheinitialcountsmayhavecausedtheproblemwiththeendingdifference,andtheerroristhenfurtheraggravatedbytheadjustment.”

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2. Thepro‐rataprocessplaces100%confidenceintheanchorpointdata.Inpractice,evenwhenATRsareusedasanchorsthedataisnotcompletelyaccurate,especiallyassensorsdegradewithage.

3. TheTMGrecommendsthattherebenomorethan5interchangesbetweenanchors,butthisisnotalwaysfeasibleduetolimitedATRinfrastructure(especiallyinruralareas).

4. Thepro‐rataprocessisdifficulttoimplementforcomplicatedroadwaygeometries,suchasfreeway‐to‐freewayinterchanges(whereeachrampaffectsthreeormorecomputations).Frontageroadsthatre‐jointhemainlinealsopresentcomputationalchallenges.

5. Thepro‐rataprocessrequiresanATRstationorothertrustedobservationateachendofsectionthatisbeinganalyzed.Suchdatamaynotexist,forexample,atstub‐endsofthenetworkorjurisdictionalboundaries.

6. Thepro‐rataprocesscannottakeadvantageofnon‐anchorobservationsonbasicroadwaysegments.Consequentlyitcannotuseadditionaldatathathelpsclarifywhichofthemanypossiblesolutionsisthemostappropriate.

SomeoftheconsiderationsenumeratedabovealsoapplytotheZhaoandKwontechniques.THEGEHFORMULADevelopmentofanautomatedbalancingprocessrequiresarobustmeasureofthedifferencebetweenadjustedandunadjustedtrafficcounts.Whilepercentdifferenceisafrequently‐usedmetric,itcanbeproblematicbecauseofthewiderangeofvolumevaluestypicallyencounteredinhighwaysystems.Forexample,adjustingarampvolumefrom40veh/hrto50veh/hrgeneratesthesamepercentdifferenceasadjustingamainlinevolumefrom4000veh/hrto5000veh/hr,buttheeffectonthesystemasawholeisquitedifferent.Thus,apotentialpitfallwithusingpercentdifferenceastheobjectivefunctionforoptimizationisthatarampwithasevereundercountmightnotbeadjustedsufficiently.Toaddressthisissue,inthe1970sGeoffE.HaversoftheGreaterLondonCouncil(10)proposedaheuristicmeasureofvolumedifferencenowknownastheGEHformula.TheBritishHighwaysAgencyDesignManualforRoads&Bridgesdescribesitas“aformofChi‐squaredstatisticthatincorporatesbothrelativeandabsoluteerrors[and]canbecalculatedforindividuallinksorgroupsoflinks.”(11).IntheUnitedKingdomitisacceptedas“astandardmeasureofthe‘goodnessoffit’between[twosetsoftraffic]flows.Unlikecomparingflowsusingpercentagedifference,theGEHstatisticplacesmoreemphasisonlargerflowsthanonsmallerflows.”(12)LowGEHvaluesindicatesimilaritybetweentheoriginalandadjustedvalues,whilehighGEHindicatesgreaterchange.Table2showsrule‐of‐thumbinterpretationsoftheGEHvaluethatweredevelopedforcomparingobservedandmodeledvaluesintraveldemandforestingmodels;weapplysimilarinterpretationstothedifferencesbetweentheoriginalandadjustedvaluesfromthebalancingprocess.(DetectorsthatgeneratehighGEHvaluesinthebalancingprocessmayrequireevaluationorrepair).

TABLE2.InterpretationsofGEHvaluesfromtheSaturntraveldemandforecastingsoftwareuser’smanual(8).

ValueGEH=1.0GEH=2.0GEH=5.0GEH=10.0

Comment“Excellent”“Good”“Acceptable”“Rubbish!”

Example1±65in4,000±130in4,000±325in4,000±650in4,000

Example2±25in500±45in500±120in500±250in500

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TheGEHformulaforhourlytrafficflowsis:

G2 v vv v

Where:G =GEHforhourlyvolumev=Unadjustedhourlyvolumev=Adjustedhourlyvolume

(3)

TheGEHformulawasoriginallyintendedforusewithhourlytrafficflows.TopreservethetypicalGEHacceptancethresholds(“5isOK”,“10isnot”)whenbalancingdailytrafficvolumes(e.g.AADT),researcherscanusetheapproximationthatpeakhourlyvolumeisontheorderof10%ofthedailyvolume.InthiscasetheGEHformulabecomes:

G0.2V 0.4VV 0.2V

V V

Where:G =GEHfordailyvolumeV=UnadjusteddailyvolumeV=Adjusteddailyvolume

(4)

OPTIMIZATIONFORBALANCINGTomeetthepreviously‐statedgoalsandovercomethelimitationsofpro‐rataadjustment,wedevelopedanautomatedbalancingmethodthatisbasedonmathematicaloptimization.Inthisoptimizationproblem,ourgoalistominimizethedifferencebetweentheoriginalandadjustedtrafficvolumes,subjecttotheconstraintthatthevolumesinthenetworkmustbalance(trafficinputsequaltrafficoutputs).TocomputetheobjectivefunctionweusedthesumoftheGEHforalllocationswhereanobservedcountwasavailable;thisallowsustoconsiderdatafromasecond(perhapslessreliable)setofmainlineobservations(suchasshort‐durationmainlinecountsorFTMSsensoroutput),withouttreatingthosevolumesasanchors.WeappliedtheGHformulatohourlydata(suchastheAMpeakhour)andtheGDformulatodailydata(equation5illustratestheuseofGH).Optionally,wemayconstraincertainvolumestomatchknownvaluesathighly‐trustedanchorsites,butincontrasttotheTMGprocedurewedonotneedtouseeveryATRasananchor.Theprimaryadvantageofsettinganchorsistoallowthebalancingprocesstobecompletedpiecewise,forexamplebalancingfreeway‐to‐freewayinterchangesbeforemovingontogeneralfreewaysegments.WithGEHminimizationastheobjective,thisprocesscanbeexpressedmathematicallyas:

min ∑

∑

subjectto: ∑

andoptionallysubjectto: fortrustedsites

5

Where:=Unadjustedmainlinevolume(typicallyfromATRs)

=Adjustedmainlinevolume

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=Unadjustedrampvolume =Adjustedrampvolume

t=StationIDatterminusoftheanalysissectionIMPLEMETINGTHEBALANCINGPROCESSProceduralStepsThemainstepsinthisprocessareasfollows:

1. Importingtherawdatafromvarioustrafficdatabases.2. Computingtheimbalanceintheimporteddatasetbymeansofrunningtotals.3. Adjustingthecountsusingmathematicaloptimization,subjecttotheconstraintthatthe

imbalancemustbezero.Weimplementedvolumebalancinginaspreadsheetforthefreewaysystemsinsomerelativelylargemetropolitanareas(i.e.metroMilwaukeeandmetroGreenBay).Amainframeapplicationthatcandirectlyaccesstherawtrafficdataalsowarrantsconsideration,especiallyifbalancingofmultipletimeperiodsorverylargeareasisdesired.Ineithercase,anoptimizationalgorithmneedstobeavailable.Forspreadsheet‐basedbalancingweusedtheSolveradd‐inforMicrosoftExcel,whichusestheGeneralizedReducedGradient(GRG2)algorithmforoptimizingnonlinearproblemsandtheBranch&Bound(B&B)methodforoptimizingintegerproblems(13).InourdatasetstheGRG2algorithmtendedtoproduceresultsmorequicklythantheB&Bmethod,butGRG2hadagreaterchanceofstoppingonalocalminimum(insteadoftheglobalminimum).OutputExamplesTheresultsofoptimizingtheFigure1datasetareshowninColumnsDandEofTable1.Forillustrativepurposes,itisassumedthata48‐hourmainlinecountwasavailableinSegmentC,withanobservedaveragedailyvolumeof82,000(sinceitisincludedintheGEHcalculationsthissupplementalobservationinfluencestheoptimization).ColumnDshowstheresultswiththeB&BalgorithmandtheconstraintthatbothATRsaretreatedasunchangeableanchorsthatmustexactlymatchtheoriginalobservations(asinthepro‐ratamethod).Comparedtothepro‐ratamethod,totalGDforthisfreewaysectiondropsfrom16.9to12.6(averageGDdropsfrom1.9to1.4).ColumnEshowstheresultofrelaxingtheanchorconstraintandusingtheGRG2algorithm;inthiscasetotalGDdropsto4.4(averageGDdropsto0.5).InbothcasestheoriginalunadjustedvolumeswereusedasthetrialsolutionandGEHminimizationwasusedastheobjectivefunction.Table3showstheresultsofaGRG2basedoptimizationforathree‐legfreeway‐to‐freewayinterchangelocatednearBellevue,Wisconsin.ATRsitesidentifiedbygreenshadingweretreatedasunchangeableanchors.Greytextidentifiesvaluesthatappearinthecalculationsasecond(orthird)timebecausetheyinvolvetwo(ormore)legsoftheinterchange.Thisproblemcannotbesolvedusingthepro‐ratamethod,butanoptimization‐basedsolutionisfeasible.Thetablereflectssomerealitiesofthis2006dataset:the“raw”I‐43North‐to‐SouthandSouth‐to‐Norththroughmovementsareestimatesbecausenorelevantfielddatawasavailable,buttheoptimizationrefinestheseestimatesaspartoftheprocessofestablishingamathematicallyconsistentvolumeset.AdditionalConsiderationsIfacorridorconsistsonlyofserviceinterchangesandordinaryfreewaysegments,eachtraveldirectioncanbebalancedseparately.Ifthestudyareaincludesfreeway‐to‐freewayinterchanges,theentireareamustbebalancedsimultaneously.(Specialcaremustbetakenwhencodingfreeway‐

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to‐freewaymovementstoassurethattheadjustedvolumesareconsistentforallroutesinvolvingthatmovement.)Theprocedureistolerantofalimitedamountofmissingorsurrogatecountdata.Forexample,mostWisconsinfreewayshavecountsonrampsandbasicfreewaysegments,butnotontheshortsegmentsbetweenrampsataninterchange,andthissituationiseasilyaccommodatedasintheTable1example.Theprocessisalsoreasonablytolerantoftheuseofolddataasasurrogatewhensomeofthenewerdataisunavailable(forexample,combiningcurrentmainlinecountswithacombinationofoldandnewrampcounts).Nevertheless,therearesomeinstanceswherebalancingisthwartedbyincompletedata.Forexample,ifanexitsplitsintotwodownstreamrampsandonlyoneofthemhasbeencounted,themissingramp'svolumemustbecollectedinordertodeterminethetotalvolume(withoutthisdatathebalancingprocesscanproducemisleadingresultsorfailentirely).

Figure2.MapofI‐43&WIS172Interchange,Bellevue,Wisconsin

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TABLE3.Balancingexampleforthree‐legfreeway‐to‐freewayinterchange(seeFigure2formap).BELLEVUE INTERCHANGE Southbound Thru Raw Balanced Change % Chng GD

I-43 SB Btwn Manitowoc Rd & Bellevue System Interchange [Anchor] 21945 21945 0 0.0% 0.0System Exit Ramp I-43 SB to STH 172 WB 18000 18245 245 1.4% 0.6I-43 North-South Thru Movement 3700 3700 0 0.0% 0.0System Ramp STH 172 EB to I-43 SB 9526 9247 -279 -2.9% 0.9I-43 SB Btwn Bellevue IC and Elm View Rd (CTH MM) [Anchor] 12948 12948 0 0.0% 0.0Calculated Volume 13471 12948 1.5Imbalance 524 0

Southbound Turning Westbound Raw Balanced Change % Chng GD

I-43 SB Btwn Manitowoc Rd & Bellevue System Interchange [Anchor] 21945 21945 0 0.0% 0.0I-43 North-South Thru Movement 3700 3700 0 0.0% 0.0System Exit Ramp I-43 SB to STH 172 WB 18000 18245 245 1.4% 0.6System Ramp from I-43 NB to STH 172 WB (Bellevue IC) 10077 9462 -615 -6.1% 2.0STH 172 WB Btwn Bellevue IC and Monroe Rd 22500 27707 5207 23.1% 10.4Monroe Rd (CTH GV) Exit Ramp 5300 5300 0 0.0% 0.0Between Ramps at Monroe Rd Interchange 22407 Monroe Rd (CTH GV) Entrance Ramp 4550 4550 0 0.0% 0.0STH 172 WB Bridge over East River [Anchor] 26957 26957 0 0.0% 0.0Calculated Volume 27572 26957 12.9Imbalance 615 0

Eastbound Turning Northbound Raw Balanced Change % Chng GD

STH 172 EB Bridge over East River [Anchor] 26957 26957 0 0.0% 0.0Monroe Rd (CTH GV) Exit Ramp 4550 4995 445 9.8% 2.0Between Ramps at Monroe Rd Interchange 21962 Monroe Rd (CTH GV) Entrance Ramp 5500 4988 -512 -9.3% 2.2STH 172 EB Btwn Monroe Rd and Bellevue Interchange 22500 26950 4450 19.8% 8.9System Ramp STH 172 EB to I-43 SB 9526 9247 -279 -2.9% 0.9System Ramp from STH 172 EB to I-43 NB (Bellevue IC) 18600 17702 -898 -4.8% 2.1I-43 South-North Thru Movement 3700 3486 -214 -5.8% 1.1I-43 NB Btwn Bellevue Interchange & Manitowoc Rd [Anchor] 21188 21188 0 0.0% 0.0Calculated Volume 22081 21188 17.4Imbalance 893 0

Northbound Thru Raw Balanced Change % Chng GD

I-43 NB Btwn Elm View Rd (CTH MM) and Bellevue IC [Anchor] 12948 12948 0 0.0% 0.0System Ramp from I-43 NB to STH 172 WB (Bellevue IC) 10077 9462 -615 -6.1% 2.0I-43 South-North Thru Movement 3700 3486 -214 -5.8% 1.1System Ramp from STH 172 EB to I-43 NB (Bellevue IC) 18600 17702 -898 -4.8% 2.1I-43 NB Btwn Bellevue Interchange & Manitowoc Rd [Anchor] 21188 21188 0 0.0% 0.0Calculated Volume 21471 21188 5.2Imbalance 283 0

Northbound Turning Westbound Raw Balanced Change % Chng GD

I-43 NB Btwn Elm View Rd (CTH MM) and Bellevue IC [Anchor] 12948 12948 0 0 0I-43 South-North Thru Movement 3700 3486 -214 -5.8% 1.1System Ramp from I-43 NB to STH 172 WB (Bellevue IC) 10077 9462 -615 -6.1% 2.0System Exit Ramp I-43 SB to STH 172 WB 18000 18245 245 1.4% 0.6STH 172 WB Btwn Bellevue IC and Monroe Rd 22500 27707 5207 23.1% 10.4Monroe Rd (CTH GV) Exit Ramp 5300 5300 0 0.0% 0.0Between Ramps at Monroe Rd Interchange 22407 Monroe Rd (CTH GV) Entrance Ramp 4550 4550 0 0.0% 0.0STH 172 WB Bridge over East River [Anchor] 26957 26957 0 0 0Calculated Volume 26498 26957 14.1Imbalance -460 0

Eastbound Turning Southbound Raw Balanced Change % Chng GD

STH 172 EB Bridge over East River [Anchor] 26957 26957 0 0.0% 0.0Monroe Rd (CTH GV) Exit Ramp 4550 4995 445 9.8% 2.0Between Ramps at Monroe Rd Interchange 21962 Monroe Rd (CTH GV) Entrance Ramp 5500 4988 -512 -9.3% 2.2STH 172 EB Btwn Monroe Rd and Bellevue Interchange 22500 26950 4450 19.8% 8.9System Ramp from STH 172 EB to I-43 NB (Bellevue IC) 18600 17702 -898 -4.8% 2.1System Ramp STH 172 EB to I-43 SB 9526 9247 -279 -2.9% 0.9I-43 North-South Thru Movement 3700 3700 0 0.0% 0.0I-43 SB Btwn Bellevue IC and Elm View Rd (CTH MM) [Anchor] 12948 12948 0 0.0% 0.0Calculated Volume 13007 12948 16.2Imbalance 60 0

Grand Total 80.2

 

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VALIDATIONWhilevalidationofthismethodologyisdifficultbecausethe“true”trafficvolumesforanunbalanceddatasetareinherentlyunknown,weconductedsomesensitivitytestingtoexplorethestabilityoftheoptimization.ExpectedLevelsofInputDataVariationIn2011McGowen&Sanderson(14)comparedvolumeresultsfrompneumatictubetrafficcounterswithmanualcountsandoutputfromportablemagenetometers;theyreportedthatfordailytrafficvolumes,“thetotalerroroftheroadtubecountswaslessthan4percent.”OlderBritishresearchsummarizedinDMRB(15)providesasimilarassessment,“thecurrentbest‘working’estimateoftheaccuracyofmeasurementinthenumberofvehiclesthatpassedanautomatictrafficcounteristhatthe95%confidenceintervalofacountoflongerthan12hoursdurationisontheorderof±5%ofthetotalcount.Thisassumesthatthecounterwasinstalledandmaintainedtothestandardslaiddowninthe‘ManualofAutomaticTrafficCountingPractice.’”SensitivityTestResultsUsingtheGRG2solverandtreatingthebalancedBellevuedatafromTable3asthe“groundtruth”,weincreasedanddecreasedtheinputobservedvalueforoneofthefreeway‐to‐freewayramps(STH172EBtoI‐43SB)by‐5%and+5%;inbothcasestheoptimizationrestoredthevolumestotheirpreviousvalues.Wealsoappliedsimulatedrandomfluctuationsintherangeof‐5%to+5%tonineinputvolumesintheBellevuedataset.Amongst10simulationruns,theGDbetweentheoriginalvaluesandthere‐optimized“withfluctuation”valuesrangedfrom0.0to2.2(average0.6);theRMSEbetweentheoriginalandre‐optimizedvalueswas3%.Therefore,theoptimizationprocessappearstobereasonablyconsistentandstable,atleastiftheinputdatasetisfreeofsevereoutliers.CONCLUSIONSAutomatedtrafficvolumecountbalancingusingmathematicaloptimizationcanbeappliedsuccessfullytoresolvedatadiscrepanciesinaccess‐controlledcorridorsandnetworks.GEHvaluesaregenerallylowerthantheresultsofthe“standard”pro‐rataprocedurerecommendedinthe2001TrafficMonitoringGuide,indicatingbetterfittotheoriginaldataset.Ingeneralthereisnouniquesolutiontotheoptimizationproblem,sotheresultswilldifferdependingontheoptimizationalgorithmsandobjectivefunctionthatareselected.WefoundthatcombiningaGEHbasedobjectivewithaGRG2searchproducedstableresults.AshortcomingofGRG2isthatitissensitivetothetrialsolutionandsometimesselectsalocaloptimumratherthantheglobaloptimum.AdditionalresearchisnecessarytodeterminewhetheraGEH‐basedobjectivefunctionandaGRG2searchisthe“best”combinationforallnetworks.Furtherworkisnecessarytoquantifytheeffectsofsevereoutliersontheoptimizationresults.ACKNOWLEDGEMENTSTheauthorswishtoacknowledgetheongoingsupportandassistanceoftheWisconsinDepartmentofTransportationandcolleaguesattheUniversityofWisconsinTrafficOperations&SafetyLaboratory.SpecialthankstoProf.AlanHorowitzoftheUniversityofWisconsin‐MilwaukeeandtothedevelopersoftheoriginalBALTtool,Mr.ErtanOrnek,andDr.DoNam.

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                                                                                                                                                                                                15  HighwaysAgency(1991),DesignManualforRoads&Bridges,Volume12,Section1,Chapter

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TRB 2014 Annual Meeting Paper revised from original submittal.