美赛A题论文Word文件下载.docx
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2014MathematicalContestinModeling(MCM)SummarySheet
(Attachacopyofthispagetoyoursolutionpaper.)
Typeasummaryofyourresultsonthispage.Donotinclude
thenameofyourschool,advisor,orteammembersonthispage.
Summary
TheaimofthispaperistoevaluatetheperformanceofthechanginglanerulenamedKeep-Right-Except-To-Passandcompareotherruleswe’veconstructedwiththegivenone.
Theperformanceofchanginglanerulesmainlymanifestinsafetyandtrafficflow.Meanwhile,safetyisinfluencedbypostedspeedlimitsandtrafficdensityandtrafficflowisinfluencedbyaveragespeed.
FirstweconstructModel1todescribetherelationshipbetweenpostedspeedlimits,trafficdensityandsafety,notingthatsafetyhasnegativecorrelationwithcollisiontimesandcanbefullyexpressedbyit.ThereforeweuseMatlabtoimitatethechanginglaneandcollisionprocess.ThenweconstructModel2todescribetherelationshipbetweentrafficflowandaveragespeed.CombiningtheresultofModel1and2,wecanconcludethatthehigherthepostedspeedlimitsandthelowerthetrafficdensity,thehigherlevelsafetyantrafficflowcouldreach,andtheKeep-To-Right-Except-To-Passrulehasthebestperformance.
ThenweconstructModel3tocomparesomenormalchanginglaneruleswiththegivenoneandwitheachother.Thusweimitatealltherulesusingalgorithmgivenbymodel1.WeintroduceanewconceptnamedStandardTimetoexpresstrafficflowandstillcollisiontimestoexpresssafety.Theresultisthatwhenafreewayroadcontains2lanes,thegivenruleshowsthebestperformance.Andwhenthenumberoflanesbecomes3,thentheChoose-Speedrule,averycommontrafficruleactsthebest.
Asfortheleft-drivingcountries,wefirstchoosethebestrulesandmirrorsymmetricallymodifythem.ThenusethemethodofModel2toimitateperformancesoftheserules.Wehaveobservedthatthesetwobestrulescanbesimplycarriedover.
Finally,wehaveashortdiscussionabouttheintelligentsystem.Sincespeedcanbefastestandchanginglanescanbewellplanned,weregardthissystemasabsolutelysafetrafficwhichhasthelargesttrafficflow.
1.Introduction
1.1Analysistotheproblem
Thisproblemcanbedividedinto4requirementsthatwemustmeet,listedasfollows:
(1)BuildamathematicalmodeltodemonstratetheessenceofKeep-Right-Except-To-Passruleinlightandheavytraffic;
(2)Showtheeffectsofotherreasonablechanging-lanerulesorconditionsbyusingthemodifiedmodelbuiltintheformerrequirement;
(3)Determinewhethertheleft-side-drivingcountriescanusethesamerule(s)withsimplemirrorsymmetricmodification,ormustaddsomerequirementstoguaranteethesafetyortrafficflow;
(4)Constructanintelligenttrafficsystemwhichdoesnotdependonhumans’compliance,andthencomparetheeffectswithearlieranalysis.
Therefore,thisproblemrequiresustoevaluatetheperformancesofseveralchanging-lanerules,especiallytheKeep-Right-Except-To-Passrulewhileperformancesmainlymanifestintrafficflowandsafety.Itisobviousthatspeedlimitandthenumberofcarsinfluentcars’speedandthentheperformancesofthoserules.
Tosolverequirement
(2),wewillusethesamemethodasabove.Butallrulesweconstructedshouldbediscussed,imitatedandcomparewitheachother.
Todiscusstheleft-drivingcountries,wewillconcludethebestrulesfromrequirement2andapplytothosecountries.Thencompareresultswiththatofright-drivingcountries.Iftheresultsaretotallythesame,thenthoserulescanbesimplycarriedoverwithsimplechange.Ifnot,additionalrequirementareneeded.
Asfortheintelligentsystem,weassumenocollisionwillhappen.Soasimplediscussionisfeasible.
1.2Crucialmethodfortheproblem
Toimitatetheprocessofchanginglanesandcollision,wewillconstructanalgorithmnamedChanging-lineandCollision.Thiswillshowthestateofanumberofcaronthesamestretchoffreewayroad,includinggoingstraight,changinglanetopassandtrafficcollision.TheimitationprocesswillbedeliveredonMatlab;
2.Symbolsanddefinition
Symbol
Definition
Unit
FLO
TrafficFlow
\
MU,
AverageSpeed
miles/h
StandardDeviationofSpeed
85%PercentileofNormalDistribution
70%PercentileofNormalDistribution
TrafficDensity
(Note:
expressedbythenumberofcars)
StandardTime
CollisionTimes
3.Generalhypothesesforallmodels
(1)Allfreewayroadscontaineithertwoorthreelanes.
(2)