2017美赛数学建模M奖论文

ForofficeuseonlyT1________________T2________________T3________________T4________________TeamControlNumber70028ProblemChosenBForofficeuseonlyF1________________F2________________F3________________F4________________2017MCM/ICMSummarySheet(Yourteam'ssummaryshouldbeincludedasthefirstpageofyourelectronicsubmission.)Typeasummaryofyourresultsonthispage.Donotincludethenameofyourschool,advisor,orteammembersonthispage.SummaryTheperformanceofhighwaytollplazadirectlyaffectsthecapacityofthehighway,sothedesignofroadtollplazaisimperative.Inthispaper,weconductperformanceanalysisforaspecifictollplazainNewJersey,USA,includingaccidentprevention,throughputandcost.Firstofall,weusegreymodeltopredictthefutureoutputofthetollplaza,andcomparedwiththerealdata,theaveragevalueoftheresidualvalueis0.429.Thenwecandrawaconclusionthatthethroughputperformanceofthetollplazaissecondary.Next,weusequeuingtheorytogettheserviceindexofthetollplazainthelightandheavytraffic,andthecellularautomatonmodelisusedtoconsiderthechangingcircumstancesofservicelevel,usesregressionmodeltoestablishafunctionrelationbetweentrafficaccidentandfourfactors.Then,wefindthattherateofchangehasthegreatestinfluenceonitandthepavementperformancehastheleastinfluence.Intermsofcost,weconsiderthetollplazalandandroadconstruction.Andthecostofroadconstructionisdividedintothelaborcostandmaterialcost.Next,accordingtotheinfluenceofroadgeometryonthetrafficperformanceofTollPlaza,weselectthetransitioncurvetrajectorymodeltoimprovethetollplazatransition,whichcanalsohaveanimprovementonthesizeandshapeofthetollplazaandmergemode.Finally,wedoaseriesofperformancestudiesforourimprovedtollplaza.Firstofall,theimprovementinthesquareflowandcarflowundertheconditionofservicelevelaredeterminedrespectivelythroughsimulation.Next,wedrawaconclusionthattheserviceperformanceofthetollplazaisnotobviousinsmallcarflow,butthereisamarkedincreaseinlargeflow.Then,duetothefactthattheunmannedvehiclecoulddealwithavarietyofroadconditions,itundoubtedlyexpandsourimprovedoptionalscheme.Eventually,weobtainthethroughputoftollbeforeandaftertheimprovementunderthedifferentproportionofmixedchargemodeandfindthattheimprovedthroughputinthetollplazahasbeenincreasedontheperformance.contents1Introduction:............................................................................................................11.1Problembackground:...................................................................................11.2Steps:............................................................................................................11.3Ourwork:.....................................................................................................12Assumptions............................................................................................................23Nomenclature..........................................................................................................24Throughputanalysisofgreyforecastingmodel......................................................35erroranalysis...........................................................................................................46Serviceleveloftollstation.....................................................................................57VehiclelanechangingrulesbasedonCellularAutomata.......................................68Securityanalysisbasedonmultivariatestatisticalregressionmode......................88.1StudyontherateofchangeofTollPlaza.....................................................88.2StudyonthelongitudinalslopeofentrancesectionofTollPlaza...............98.3Researchonserviceleveloftollstation.....................................................108.4Studyonpavementperformanceoftollstation..........................................109Safetyperformanceevaluationmodeloftollstation............................................1110Costanalysismodeloftollstation....................................................................1111Analysisoftheinfluenceoflanegeometryparametersonitscapacity............1211.1Determinationoflanechangingrate..........................................................1211.2Influenceofgeometricparametersontheflowofthecarlane..................1411.3Energyconsumptionanalysisbasedoncellularautomatamodel..............15Definitionofenergyconsumption:................................................................16Numericalsimulationandanalysisoftheresults:.........................................17Influenceofcurvatureradiusonenergyconsumption...................................17Influenceofarclengthonenergyconsumption.............................................1812Theeffectoftrafficflowonserviceperformancebasedonimprovedqueuingtheory1913TheinfluenceofunmannedvehiclesontheimprovedmodelofTollPlaza..2114TheinfluenceofchargingmethodonimprovingmodelofTollPlaza..........2115StrengthsandWeaknesses.............................................................................2215.1Strengths:....................................................................................................2215.2Weaknesses:................................................................................................2215.3FutureModelDevelopment:......................................................................22#Team70028page1of28Comprehensiveimprovementstrategyoftollplaza1Introduction:1.1Problembackground:Highwaytollandtollplazaistoensuretrafficsafetyandunimpeded,howeverbecauseoflackofunifieddesignspecification,tollstationanditssquareconstructionexistsmanyproblems.Suchas:lowvaluebecauseofthetechnicalindicatorstomakesquareconstructionscaletoosmallandcausethetollplazaopenedonlyfewyearsasthetrafficbottleneck,andusethehighvalueontheonehand,becauseofthetechnicalindicatorsandmakethetollstationconstructionscaleistoolarge,wastealotofmoneyandresources.Duetoincorrectlinearindicators,ortooshort,thegradualsquaresquarelengthisinsufficient,squareroadcenterlineoffset,etc.,itistoodifficulttouseafterthecompletionofthesquare.soestablishingthetollgatesandthetollplazadesignnorms,assoonaspossible,hastheveryvitalsignificanceinstandardizingtheconstructionofthetollstation,ensuringthesmoothgeneralcharacteristicoftollplazaandtrafficsafety,improvingthechargingefficiencyandmanagementlevel,reducingthelandacquisitionandcontrollinginvestmentandsoon.1.2Steps:·Aperformanceanalysisofanyparticulartollplazadesignthatmayalreadybeimplementedthroughthefollowingthreefactors:accidentprevention,throughputandcost.·Determineiftherearebettersolutions(shape,size,andmergingpattern)thananyincommonuse.·Considertheperformanceofyoursolutioninlightandheavytraffic.·Considerthesituationwheremoreautonomous(self-driving)vehiclesareaddedandhowthesolutionisaffectedbytheproportionsofconventional(human-staffed)tollbooths,exact-change(automated)tollbooths,andelectronictollcollectionbooths(suchaselectronictollcollectionviaatransponderinthevehicle)1.3Ourwork:·Basedontheavailabledata,wemakeaperformanceanalysisofanyparticular#Team70028page2of28tollplazadesignthatmayalreadybeimplemented.·Accordingtotheproblemfromtheperformanceanalysis,wemakeoutabettersolutions(shape,size,andmergingpattern)thananyincommonuse.·Determinetheperformanceofthesolutioninlightandheavytraffic,howthesolutionchangeasmoreautonomous(self-driving)vehiclesareaddedtothetrafficmixandhowthesolutionisaffectedbytheproportionsofconventional(human-staffed)tollbooths,exact-change(automated)tollbooths,andelectronictollcollectionbooths.2AssumptionsTosimplifytheproblemandmakeitconvenientforustosimulatereal-lifeconditions,wemakethefollowingbasicassumptions.1.Eachsectionofroadsisone-waytraffic2.Vehiclesintheretentionperiodoftollstationcanbeneglected3.Inanyhourofthevehiclearrivalrateisproportionaltothelengthoftime4.Theprobabilityofanyvehiclearrivalinonehouroftimeisnotaffectedbytheprevioushistory.5.ThevehiclesarriveinlinewiththePoissondistribution,namelytheheadwayisnegativeexponentialdistribution3Nomenclature𝜺(𝟎)(𝒕)theresidualerror𝐪(𝐭)therelativeerror𝐜thevarianceratio𝐏thesmallerrorprobability𝐫thecurvatureofthebendradius𝐮thestaticfrictioncoefficient𝐥thegradualchangeratio𝐤thenumberofservingdriveway𝛒/𝐤trafficintensity𝐰meantimetostayatatollstation𝐝automotivebrakingdistance𝐟thetireandroadsurfacefrictioncoefficient𝐘thenumberoftrafficaccidentsintollstationsperyear∆𝐖Widthofthegradual#Team70028page3of28𝛂𝟏curveangle𝐑𝟏theradiusofconvexcurvepoints𝐩delayprobability𝐞(𝐧,𝐭)energyconsumptionofthefirstnvehiclesfromtimettot+14ThroughputanalysisofgreyforecastingmodelFigure4-0-1SchematicdiagramofNewJerseytollplazaFirstofall,wechoseatollplazaontheNewJerseyintheUnitedStatesforaspecificperformanceanalysisoftollplaza,anditincludestheaccidentprevention,throughput,andcost.Inviewofthethroughputofthetollplaza,wechoosethegreyforecastingmodelGM(1,1),topredictthethroughputofthetollplaza.Duetotheproblemofuncertainty,sowetakethegreypredictionmodeltodealwithit.Supposex(0)(1),x(0)(2)…,x(0)(M)Inordertoovercometheirregular,weuseaccumulationprocessx(1)(t)=∑x(0)(i)Mi<1Sucharelativelysmoothnewseriesapproximationcanbedescribedbythefollowingdifferentialequation:dx(1)dt+ax(1)=μItsanalbinoformdiscretesolutionofdifferentialequationis:x̂(1)(i+1)=.x(1)−ua/e;ai+uaThetypeoftheparametera、ubedeterminedbytheleastsquaresfittingmethodisasfollows:（1）（2）（3）#Team70028page4of28Â=0au1=(BTB);1BTYNAmongthemthematrixis:B=[−12,x(1)(1)+x(1)(2)-1−12,x(1)(2)+x(1)(3)-1⋯⋯−12,x(1)(m−1)+x(1)(m)-1]YN=(x(0)(2),x(0)(3),⋯,x(0)(m))TSotheoriginaldatafittingsequenceis:x̂(0)(1)=x(0)(1)x̂(0)(i+1)=x(1)(i+1)−x(1)(i)Table4-0-1Trafficflowpredictiontable5erroranalysisInequation(11),andregulations,theoriginaldataofreducingvalueanditsresidualerrorandrelativeerrorbetweenobservedvalueisasfollows{ε(0)(t)=x(0)(t)−x′(0)(t)q(t)=ε(0)(t)x(0)(t)×100%Thefollowinginspectionoftheaccuracy:x(0)=1M∑x(0)(t)Mt<0ε(0)=1M;1∑(ε(0)(t)−ε0Mt<2)2Second,calculatethevarianceratioc=s2s1andsmallerrorprobabilityP=2|ε(0)(t)−ε(0)|<0.6745s135.52831.84283.68560.92142.76425.06765.98906.44977.831820.731323.956225.798932.248740.080536.394933.170134.552127.181019.809919.809912.438810.13535.06767.8318（4）（5）（6）（7）（8）（9）（10）（11）#Team70028page5of28Figure5-0-2comparisonchartofgreypredictionmodelWeusem、p、vmaxtorepresentqualityofthevehicle,randomdelayprobabilityandmaximumspeedrespectively,grepresentsthelocalaccelerationofgravity,rrepresentscurvatureofthebendradiusandurepresentsthestaticfrictioncoefficient.Withtheroadstatisticalanalysiscarriedoutontherealvalueandtheerrorofpredictedvalue,weobtainthefollowingresults:Table5-0-2ResidualanalysistablemeanVarianceSkewnessKurtosis0.4291.1660.307-0.245ItshowsthattheGM(1,1)modelpredictionresultshaveabetterresponseto.reflecttheactualsituation.6ServiceleveloftollstationThedirectfeelingofthedrivertothetrafficenvironmentofthetollstationisfromthequeuelengthofthetolllane,andthelengthofthequeuedependsontheservicelevelofthetollstationV/C.Inthisregard,weusethequeuingtheorymodelofmultichannelQueuingservice,inwhichthevehiclearrivaltimeisinaPoissondistribution,whichisthenegativeexponentialdistribution;Supposemisrandomarrivalrate,c𝑖isoutputrate,kisthenumberofservingdriveway,ρ=𝑚𝑐.Thereistheprobabilityofhavingnovehicleinthequeuingtheoryρ(0)=1,∑1𝑛!𝑘−1𝑛=0𝑝𝑛:1𝑘!𝜌𝑘𝑘𝑘−𝜌-Averagenumberofvehiclesinqueueingtheory:n=ρ+𝑝𝑛𝜌(0)𝑘!𝑘𝑛−𝑘(1;𝜌𝑘)2（12）（13）#Team70028page6of28queuelength:q=n−ρ=𝑝𝑛𝜌(0)𝑘!𝑘𝑛−𝑘(1;𝜌𝑘)2Averagenumberofwaitingvehiclesperlanea=𝑞𝑘Averagewaitingtimeinqueuesystems:d=𝑛𝑚=𝑞𝑚+1𝑐Averagewaitingtimeinqueue:W=𝑞𝑚MeantardinessDecelerationtimeofvehicleenteringtollstation𝑡1=𝑣03.6𝑎1Meantimetostayatatollstationw=E,S-+𝑊𝑞Vehicleaccelerationtimeofleavingtollstation𝑡2=𝑣03.6𝑎2Inthisequation,v0isthenormaltrafficflow(km/h);a1、a2aredecelerationofthevehicle(m/s2);Wqisaveragequeuetime(s)；E,S-isexpectedservicetime(s);7VehiclelanechangingrulesbasedonCellularAutomataWeapplythepreviouscellularautomatamodel,whichisnowextendedtomultiLanecase.Themaindifferencebetweenmultilaneandsinglelaneistoconsiderthemodeloflanechanging.Inthispaper,wetake4lanesasanexample.Inreality,itmaybepossibletochangelaneswhenthedriverisfoundtobeclosetotheexitandthefrontoftheadjacentlaneisempty.Ifyouwanttochangelanes,youshouldconsiderthevehiclebehindtheadjacentlane.Whenthedistance（14）（15）（16）（17）（18）（19）（20）#Team70028page7of28totherearoftheadjacentlanereachestoacertainlength,youcanchangetheroad.Lanechangescenarioscanbeshowninfigure(),whentheccaronthe1Laneisblockedbythe𝑐1car,whilethe𝑐2and𝑐3carsonthe2lanesarerelativelylarge.inordertomaintainthespeed,ccarwillchangetotheroadlane2.Figure7-0-3SchematicdiagramoflanechangingWhetherornotthedriverchoosesthelanechangeismainlydecidedbythe𝑑0,𝑑𝑛,𝑜𝑡ℎ𝑒𝑟、𝑑𝑛,𝑏𝑎𝑐𝑘threeindicators,throughthepreviousresearch,thispaperthinksthatthelanechangingruleis:When𝑑𝑛,𝑏𝑎𝑐𝑘>𝑣𝑚𝑎𝑥𝐶𝑛={1−𝐶𝑛𝑑𝑛<min{𝑣𝑛+1，𝑣𝑚𝑎𝑥}𝑑𝑛,𝑜𝑡ℎ𝑒𝑟>𝑑𝑛,𝑑𝑛,𝑏𝑎𝑐𝑘>𝑣𝑚𝑎𝑥𝑐𝑛𝑂𝑡ℎ𝑒𝑟𝑐𝑖𝑟𝑐𝑢𝑚𝑠𝑡𝑎𝑛𝑐𝑒𝑠When𝑑𝑛,𝑏𝑎𝑐𝑘≤𝑣𝑚𝑎𝑥,𝐶𝑛={1−𝐶𝑛𝑑𝑛<𝑚𝑖𝑛{𝑣𝑛+1，𝑣𝑚𝑎𝑥}𝑑𝑛,𝑜𝑡ℎ𝑒𝑟>𝑑𝑛𝑣𝑚𝑎𝑥−𝜃(−∆𝑥)𝛼>1+min{𝑑𝑛,𝑜𝑡ℎ𝑒𝑟+1,𝑣𝑚𝑎𝑥}−min*𝑉𝑛+1，𝑣𝑚𝑎𝑥+𝑐𝑛𝑂𝑡ℎ𝑒𝑟𝑐𝑖𝑟𝑐𝑢𝑚𝑠𝑡𝑎𝑛𝑐𝑒𝑠Amongthem,𝐶𝑛isthencarinthelane,𝐶𝑛=0𝑜𝑟1,𝑑𝑛、𝑑𝑛,𝑜𝑡ℎ𝑒𝑟𝑎𝑛𝑑𝑑𝑛,𝑏𝑎𝑐𝑘arethedistancebetweenthefirstnvehicleandthefrontvehicle,thedistancefromtheadjacentlaneandthedistancefromthevehicleintheadjacentlane,respectively.𝑑𝑠𝑎𝑓𝑒issafetylanechangemodel.𝑑𝑛,𝑏𝑎𝑐𝑘−𝑣𝑚𝑎𝑥,∆𝑥<0,𝑣𝑚𝑎𝑥−𝜃(−∆𝑥)𝛼isthedistancebetweenthevehicleandthevehicleintheadjacentlaneaftercorrectionbythevaluefunction,1+min{𝑑𝑛,𝑜𝑡ℎ𝑒𝑟+1,𝑣𝑚𝑎𝑥}−min*𝑉𝑛+1，𝑣𝑚𝑎𝑥+isLimitLanedistance.Theparametersαandθchangeaccordingtothepsychologicalstatusofdriver.Ifα>1,thegreaterαis,themorecarefulthedriveris.Ifθ>1,thegreaterθis,themorecarefulthedriveris.Whenα=1，θ=1,that’sLanechangingmodel.（21）（22）#Team70028page8of28Inordertodiscusstheαandθ,weuseCellularautomatasimulation.Inatwolaneroadwithalengthof7.5km,adoptingtheopenboundarycondition,eachlaneiscomposedof1000cellswithalengthof7.5km,themaximumspeedofvehicle𝑣𝑚𝑎𝑥=5.Therandomslowingdownratewas0.2.8SecurityanalysisbasedonmultivariatestatisticalregressionmodeAimedatthepreventionoftheaccident,weusemultiplelinearregressiontoestablishafunctionbetweenthenumberoftrafficaccidentsandthefollowingfourfactors:tollsquaregradient,servicelevel,Tollplazaentrancesectionofthelongitudinalslope,thePavementperformanceofTollstation.Figure8-1Causeanalysisofaccident8.1StudyontherateofchangeofTollPlazaFaninandfanoutareaoftollplazaaredesignedtomakethegradualvehiclesmorenaturalsmoothlyinandoutofthetollplaza.Inordertodrivevehicleeasily,therehasarequirementonitsgentlegradientchange.Otherwisethedrivercouldproducedrivingdeviation,whichmaycauseimproperoperationandendangerssafety.Therelationshipisasfollow:l=𝑏𝐿,Accordingtotheexperience,thevehicleswithstraightintoanotherlanedeviationthanataround0.9𝑚𝑠⁄,driversusuallyhavenomovefeelinganduncomfortablefeeling.（23）#Team70028page9of28Figure8-1TherelationshipbetweenAccidentnumberandTollplazaramprateFigure8-1showstherelationcurvebetweenhighwaytollplazaramprateandtrafficaccident,thefiguredemonstratesthatasthetollplazaramprateincreases,thetrafficaccidentswillincrease,whereasthesecurityofthetollplazawilldecrease.Throughthedataregressionanalysis,wegettherelatedmodelsbetweentollplazaramprateandthenumberoftrafficaccidentsY=1.423𝑒.0064𝑥Inthisequation,Yistheforecastednumbersoftrafficaccidentcorrespondingtothetollplazaramprate,xisthetollplazaramprateoftollplaza.Thecorrelationcoefficientinthemodel𝑅2=0.8621,itshowsthatdescriptionmodelofcorrelationishigher,Fromthemodel,wecanlearnthattheoccurrenceoftrafficaccidentfrequencyisproportionaltothetollplazaramprate.Gradientlengthisinsufficient,soitcan'tmeettoslowdownandchangelanesenteringthetollplazavehiclesafetyrequirements,resultingintheoccurrenceoftrafficaccidents.8.2StudyonthelongitudinalslopeofentrancesectionofTollPlazaHighwaytollentrancesectionofthelongitudinalslopedesignwithoutfullyconsideringthecharacteristicsofvehiclesenteringthetollplaza,alongdownhillorturndownhillandsoonbadroadalignment,thosewillaffectthenormaloperationofthepilotandmakethevehiclesenteringthetollplazaslowdownnotsufficient,longitudinalsafedrivingdistancenotenoughanddrivingdirectioncan'tadjusttothechargelane,whichwillcausetrafficaccidents.Thiswillleadtoseriouslosses.（24）#Team70028page10of28Figure8-0-4entrancesectionofthelongitudinalslopeandaccidentnumberThroughregressionanalysis,wegettherelevantmodelbetweenthetollplazaentrancesectionoflongitudinalwaveandtrafficaccidentsY=2.6254𝑒0.638𝑥Inthisequation,Yistheforecastednumbersoftrafficaccidentcorrespondingtothetollplazaramprate,xisthelongitudinalwaveoftollplaza’sentrysection.Thecorrelationcoefficientinthemodel𝑅2=0.9219,itshowsthecorrelationofthismodelisrelativelyhigh.Butwecanlearnthattollstation‘strafficaccidentanditsentrancesectionoflongitudinalwavehaveapositivecorrelationfromfiguremodelrepresentation,.Thegreatertheslope,thelowerchargewarsecurity.8.3ResearchonserviceleveloftollstationBasedonthepreviousresearchofserviceperformanceoftollstation,wetake𝑽𝑪asthemeasureofservicelevelandCitepreviousresults.8.4StudyonpavementperformanceoftollstationAccordingtothevehicledynamics,thevehicle'sbrakingdistancecanbeexpressedasfollows:d=𝑢257.9(𝑓:𝐼)Inthisequation,disautomotivebrakingdistance,uisthespeedatthebeginningoftheautomobilebrake,fisthetireandroadsurfacefrictioncoefficient,Iisroadlongitudinalslope（25）#Team70028page11of289SafetyperformanceevaluationmodeloftollstationBasedontheaboveanalysis,theevaluationmodelofdescriptivecanbewrittenastheequationform,usingmultiplelinearregressionmodel.Yisthenumberoftrafficaccidentsintollstationseveryyear,𝑥1=1𝑙,𝑥2=𝑉𝐶,𝑥3=𝑖,则Y=𝛽0+𝛽1𝑥1+𝛽2𝑥2+𝛽3𝑥3Nissamplesize,𝑌𝑖(𝑖=1,2,…,𝑁)representtheYvalueofsamplei,𝑥𝑖1,𝑥𝑖2,…𝑥𝑖𝑛(𝑖=1,2…，𝑁)representthevalueofeachvariableinsampleI,respectively.令Y=[𝑌1𝑌2⋮𝑌𝑛],X=[11⋮1𝑥11𝑥21⋮𝑥𝑛1⋯⋯⋮⋯𝑥1𝑛𝑥2𝑛⋮𝑥𝑛𝑛]β=[𝛽0𝛽1⋮𝛽𝑛]Y=Xβ,makingmaximumlikelihoodestimateofeachvariablecoefficient𝛽1，𝛽2，…𝛽𝑛,itcangetanormalequations:𝑋𝑇𝑋𝛽=𝑋𝑇𝑌SoyoucangetthefollowingregressionequationY=−4.4012−9.947511𝑙+10.098𝑉𝐶+11.25𝑖10CostanalysismodeloftollstationWeselectedtheAmericanNewJerseyatollplazatomakecostanalysisFirstly,accordingtorelevantdata,welearnthatNewJersey’saveragepriceis（26）（27）（28）（29）（30）#Team70028page12of28$3500permu,Andthetollplazawhichweanalyzedoccupiesabout5mu,therefore,thelandpriceofthetollplazaisabout$17500;Second,theroadconstructioncostsincludelaborandmaterialcost,andthelocalconstructionindustry’saveragemonthlysalaryis$3000,weuseittocalculatelabor,thisoccupiesthelargestintheroadconstructioncosts;Asformaterialcost,wecalculatebythecurrentpricesintheUnitedStates,isabout$40percubicmeter,thenaccordingtothesizeofthetollplaza,itwillcostabout$45000.Inconclusion,thecostoftollplazaspendmainlyonthelaborcostofhighwayconstruction,thematerialcostalsoaccordinglyaccountforpartofit.11Analysisoftheinfluenceoflanegeometryparametersonitscapacity11.1DeterminationoflanechangingrateAccordingtotheanalysisofvehicletrajectoryandrunningstateofvehicle,vehicletrajectoryinthemiddleofthegradualpathissimilartovehiclelanechangingtrajectory,andconsideringthefactorswhenthedriverturns,weselecttheeasementcurvetrajectorymodeltodesignthegradualchangesectionoftollplaza.Andinthemiddleofthetwoconvextypecurve,wejoinalongforLstraightsection,itisshowninthefigurebelowFigure11-0-5TollplazaimprovementsAccordingtocharacteristicsofconvexcurvegeometricelements,wecanusethefollowingformulatocalculatethefirstperiodofconvexcurveofeasementcurvetangentlengthT1:𝑇1=(𝑅1+𝑝1)tanα12+𝑞1（31）#Team70028page13of28Inthisequation,R1istheradiusofth