2023年美國大學(xué)生數(shù)學(xué)建模比賽獲獎

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幫助準(zhǔn)備美賽同學(xué)作參考

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“DreamTeam〞ofCollegeCoaches

#Team24270

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mo.crotha

.mwww模數(shù)

幫助準(zhǔn)備美賽同學(xué)作參考

Team#24270Page3of26

1.Introduction

1.1.RestatementoftheProblem

Sports,bydefinition,isallformsofusuallycompetitivephysicalactivitywhichaimtouse

physicalabilitywhileprovidingentertainmenttoparticipantsandspectators[1].Nowonderthewordsportsgivesusafirstimpressionoffiercecompetition,agitatedspectators,sweatingontherunningtrack,combinedwithajoyofvictory.Itistheuncertaintythatmakesthesportsgamesointriguing.However,wherethereiscompetition,therewillalwaysbevictory,defeat,andranking.Loyalsportfanscoulddebatedayandnightoverthequestionwhoisthebestplayerorcoach.Thesedebateshavecalledforthaneedforcertaincriterionofsportscoachesandplayers.Thecriterionhastobe:(1)all-encompassingtotakeintoconsiderationavarietyoffactors;(2)applicabletovarioussports;(3)robustenoughtoremainunaffectedbyfluctuation.

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1.2.ModelOverview

Model

TheevaluationmethodinModelⅠisbasedonacomprehensivemethodsophisticallycombiningAnalyticHierarchyProcess(AHP)andGrayRelationalAnalysisGradeMethod(GRAP).Intheevaluationprocess,wetakeintoconsiderationtheinfluenceoftimehorizon,andincorporateFuzzyAnalysisMethod,whichmakeitfeasibletocomparediversefactorsonthesamelevel.Therankingresultsinthreedifferentsportsaccordwithpreviousmediareport,whichattestthevalidityofthismethod.Model

InmodelII,weassumethattheexcellenceofacertaincoachcanbereflectedfromthemediaimpactoverthespanofhistoryandthuscanbegaugedbytheimpactonanothercoachwithinorwithoutthesameperiodoftime.WeuseGooglesearchresultstoquantifytheimpactofonecoachonanother.Therelationshipbetweencoachescanbeestablishedasacross-referencematrix.Byfurthertakingintoaccounttheinfluenceoftime,influenceofreputation,andamodificationtoruleouttheredundantinformation,weobtainafinalevaluationvector.ThefinalrankingresultisroughlyapproximatetotheresultinmodelI.Tosumup,weonlyneedthesearchresultsreturnedbyGooglesearchenginetoestimatetheexcellenceofcertaincoachwithhighaccuracy.

2.Assumptions

Weassumethatthecompetitionrulesofeachsportdonotchange.

Althoughsportsaredeveloping,wedonottakeintoaccountoftimeinthecompetitionrulesinordertocomparethecoachesofdifferentyearsmorefairly.Weneglecttiedcompetitionssincetheyhavethesameeffectonthetwocompared

teams.WeonlytaketheDivisionIintoconsideration.

mo.crotha.mwww模數(shù)

幫助準(zhǔn)備美賽同學(xué)作參考

1

stmiqi

'

andx2ix2iqii1,2,,n.

3.3.3.FuzzyAnalysis

AsforSMindexes,weassumethattheycanbedividedintofivelevels:Excellent,VeryGood,Good,NotGood,Bad.Andwecorrespondthefivelevelsinto5,4,3,2,1successivelyForcontinuousquantification,weassume:AsforExcellent,wesupposef51.AsforVeryGood,f30.7.AsforBad,f10.1.

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WeemploypartiallargeCauchydistributionandthelogarithmicfunctionasthesubordinatefunction[2]:

1axb21,1x3

f(x)

clnxd,3x5

wherea,b,c,dstandsforundeterminedconstants.Weusetheinitialconditionsabovetodefinetheirvalues.Andsolutionofthesubordinatefunction(Figure2)is:

12.8049x0.441721,1x3

f(x)(1)

0.5873lnx0.0548,3x5

Figure2:Trendoff(x)

MediapopularityismeasuredbythenumberofsearchresultsviaGoogle.Theimpactofduplicationofnamescanbeneglectedbymeansofaddingsearchkeywordsinordertoruleouttheredundantinformation.

Wemapxj(j=5,6)intointerval[1,5],throughfunction(1),wecanobtain:

mo.crotha.mwww模數(shù)

幫助準(zhǔn)備美賽同學(xué)作參考

4xjimj

i1,2,,n,j5,6(2)xf1

Mjmj

'

ji

whereMjmaxxij,mjminxijj5,6.

1in

1in

Asforx3andx4,wedefinethatx3’=x3,x4’=x4.

weusex'jj1,2,,6toproceedthefollowingcalculation.

3.3.4.Nondimensionalizationprocess

Weemployextremedifferencemethodtonondimensionalizethedifferentindexessothatwecancomparethem[2]onthesamelevel.Themethodisasfollows:

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x''ji

''''''''

andx''jxj1,xj2,xj3,,xjn

T

x'jimjMjmj

(3)

whereMjmaxxij,mjminxijj1,2,,6.

1in

1in

andthenweobtainthefinalevaluationindexmatrix:

''''''''''

x1,x''x*2,x3,x4,x5,x6(4)

3.3.5.Finalresult

ByusingAHPasthesubjectiveevaluationmethodandGRAPastheobjectivemethod,thefinalrepresentsacomprehensiveevaluationcombinedthemeritsofthesetwomethods.

AnalyticHierarchyProcess[3](AHP)

Bycomparingtheeffectoftwoindexesx'j,theweightsofthetwomethodwx'j(j=1,2,,m)aregiven.ThenweconstructthepairwisecomparisonmatrixA.

WecanobtainthelargesteigenvalueofA:λ=6.0496anditsweightvector:

mo.crotha.mwww模數(shù)

j1,2,,6

T

113A

7152

11

1

3131

3

5151361513

171516117151253

53

75

11

3315

w0.1248,0.1469,0.4593,0.8125,0.0775,0.2928

T

幫助準(zhǔn)備美賽同學(xué)作參考

Team#24270Page11of26Forx5andx4,from(1)(2),wecanobtainx5’andx6’.

Secondly,from(3),wecanobtainx''jj1,2,,6.Finally,from(4),wecanobtainx*.Welistthequantitativevalueofx*inTableA4.

Obtainingtheresultviaultimateevaluationvector

Atfirst,from(5),weuseAHPandgetB.Secondly,weuseGRAPanddefinethatρ=0.3andwi=0.05(i=1,2,,n).From(6),wecangettherelationdegreevectorr.Andthen,from(7),wecanobtainC.Finally,from(8),bydefining0.6,0.4,wecanobtaintheultimateevaluationvector:

U0.293,0.288,0.468,0.241,0.422,0.160,0.521,0.868,0.713,0.311,0.481,0.836,0.168,0.998,

0.130,0.318,0.243,0.482,1.761,0.138

T

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S19

BysortingthevalueofUi(i=1,2,,n),wecanobtaintherankingresultofSi.Andtherankingvectoris:

Rank119,14,8,12,9,7,18,11,3,5,16,10,1,2,17,4,13,6,20,15

T

Therefore,welisttopfivecoachesofbasketballinthepreviouscenturyinTable3:

Table3:Top5CoachesofBasketball

No.3S8

JohnWooden

Thisresultislargelyagreementwiththewidelyacceptedresult[8][9].

3.4.2.Football

Searchingandselectingdata

Likewhatwedoinbasketball,wesearchandselectdatathroughtheInternet[5][10][11].However,wecalculatethatthenumberoffinalfoursisthesumnumberoftimesthatteamscanenterintoSuperBowl.

Determiningthefinalevaluationindexmatrix

Atfirst,wedeterminevectorx’.Forx1,weuseW(ti)=Num2,whereNumrepresentsthetotalnumberofteamsinti.

WecanobtainW(2)(t)bysimulatingandcurvefittingofdata.SowecangetpiforeachSi,andthenweobtainthevectorx1’.

Forx2,wealsoobtains(1)(t)bysimulatingandcurvefittingofdata.Sowecangetqiandthevectorx2’.

Finally,from(4),wecanobtainx*.Welistthequantitativevalueofx*.Obtainingtheresultviaultimateevaluationvector

LikewhatwedoinBasketball,wecanobtaintheultimateevaluationvector:

U0.234,0.879,0.738,0.106,0.359,0.296,0.383,0.193,0.291,0.453,0.494,0.248,0.180,0.615,

1.000,0.316,0.151,0.047,0.392,0.052

T

BysortingthevalueofUi(i=1,2,,n),wecanobtaintherankingresultofSi.Andtherankingvectoris:

mo.crotha.mwww數(shù)

No.1

No.2S14

No.4S12

No.5S9

DeanSmith

MikeKrzyzewski

AdolphRupp

BobKnight

幫助準(zhǔn)備美賽同學(xué)作參考

Team#24270Page12of26

Rank215,2,3,14,11,10,19,7,5,16,6,9,12,1,8,13,17,4,20,18

T

Therefore,welisttopfivecoachesofbasketballinthepreviouscenturyinTable4:

Table4:Top5CoachesofFootball

No.1S15JoePaterno

No.2S2BobbyBowden

No.3S3BearBryant

No.4S14TomOsborne

No.5S11DonJames

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Thisresultislargelyagreementwiththewidelyacceptedresult[12].

3.4.3.Baseball

Searchingandselectingdata

Likewhatwedoinbasketball,wesearchandselectdatathroughtheInternet[13][14].Butinthissport,weassumethatthenumberoffinalfoursisthenumberofchampionsofNCAAcompetitionsthatteamscanachieve.AndweassumethatthenumberofchampionsisthenumberofchampionsofNationalcompetitionsthatteamscanget.Determiningthefinalevaluationindexmatrix

Atfirst,wedeterminex’.Forx1,duetoscarcityofthedata,wecanonlysearchalittleinformationofseveralyears[9].WeuseW(ti)=Num2,whereNumrepresentsthetotalnumberofcompetitionsofchampioninti.

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WecanobtainW(3)(t)bysimulatingandcurvefittingofdata.SowecangetpiforeachSi,andthenweobtainthevectorx1’.

Forx2,duetolackingthestandarddifferenceofwinning-percentageineverytenyear,wechooseanotherapproachtogetx2’.Consideringtheinfluenceoftime,first,weemployextremedifferencemethodtonondimensionalizetmintotm’,wheretmiisthemiddleyearoftenureofSi.Then,wedefine

''tm

'''''

andthenwedefinex2ix2itmi.Sofrom(3),weobtainx2.

Finally,from(4),wecanobtainx*.Welistthequantitativevalueofx*.Obtainingtheresultviaultimateevaluationvector

LikewhatwedoinBasketball,wecanobtaintheultimateevaluationvector:

0.022,0.044,0.232,0.225,0.138,0.118

T

U0.353,0.278,0.587,0.223,0.205,0.302,0.208,0.189,0.314,0.494,0.203,1.000,0.214,0,

BysortingthevalueofUi(i=1,2,,n),wecanobtaintherankingresultofSi.Andtherankingvectoris:

Rank312,3,10,1,9,6,2,17,18,4,13,7,5,11,8,19,20,16,15,14

T

Therefore,welisttopfivecoachesofbasketballinthepreviouscenturyinTable5:

Table5:Top5CoachesofBaseball

mo.crotha.mwww模數(shù)

1'tm5

幫助準(zhǔn)備美賽同學(xué)作參考

Team#24270Page13of26

No.1S15JohnBarry

No.2S2MikeMartin

No.3S3RodDedeaux

No.4S14AugieGarrido

No.5S11JimMorris

Thisresultislargelyagreementwiththewidelyacceptedresult[15].

3.4.4.Sensitivityanalysis

BychangingtheweightofAHPandGRAPinequation(8),weanalyzethechangingresultofbasketball.Forexample,wedefine0.5,0.5,andtheresultislistedinTable6.Thecoacheswhoranktop5donotchange:

Table6:Top5CoachesofBasketball

No.2S12AdolphRupp

No.3S14DeanSmith

No.4S8

MikeKrzyzewski

No.5S9BobKnight

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S19S19S19

No.1

JohnWooden

Whendefined0.4,0.6,theresultchanges,whichislistedinTable7.Thecoacheswhoranktopfivechange:

JohnWooden

Whendefined0.7,0.3,theresultislistedinTable8.Thecoacheswhoranktopfivedonotchange:

JohnWooden

Ascanbeseenfromabove,whenthereisaslightchangeofweights,theresultdonotchange.Butwitharelativelygreaterchange,weightshaveaneffectontheresult.

4.Model

Howcouldonesreputationaffectanothers?Onewayistofollowtheimplicationinthesaying:YouwouldntmentionAandBinthesamebreath.Itmeansifthedifferencebetweentwopeopleistoowide,itwouldbeunlikelyformostofindividualstomentiontheminasametalk.Thesameholdstrueforthesportscoaches.Thatmeans,iftwocoachesareabsolutelynotonthesamelevel,morelikelythannot,therewillbefewreportsonthesetwocoaches.Ontheotherhand,iftwoofthemaretopcoaches,therewillbeaplethoraofreports:suchasTheGreatestCoachesEverBasketballHallofFame,onthetwocoaches.Informedbythisnaturallaw,wemayfindaninnovativeapproachtoestimateacoachslevelofexcellenceandpopularity.Theworkingflowisshownasfollows:

mo.crotha.mw數(shù)

Table7:Top5CoachesofBasketball

No.3S14

No.1

No.2S12

No.4S7

No.5S8

AdolphRupp

DeanSmith

HankIba

MikeKrzyzewski

Table8:Top5CoachesofBasketball

No.3S12

No.1No.2S14

No.4S8

No.5S9

DeanSmithAdolphRuppMikeKrzyzewskiBobKnight

幫助準(zhǔn)備美賽同學(xué)作參考

4.3.1.Originaldata

Herewedefinetiasthecharacteristicyear,theaverageoftheyearthatthecoachistartcoachingandtheyearofhisorherretirement.(Ifthecoachiisstillactive,thentistheaverageoftheyearthatthecoachistartcoachingandthisyear,thatis,2023)

Thesearchresultsvectoraistheoriginaldataweusetoestimatetheindividualinfluence,whereaiisthenumberofsearchresultsofcoachi.Particularly,thecoachesherearesortedbycharacteristicyearinadescendorder.Thiscanbeagreatconveniencetoourlaterdiscussionabouttimefactor.

4.3.2.Theinfluencecoefficientoftime

Accordingtothegrowthlawofwebinformation[16],theinformationaimingatacertainfieldissimilartoanexponentincrease.Totestthishypothesisandbetterapplyittosports,weenteredtheGooglewebsite.Using1910basketball,1920basketball，and1930basketballastheexactkeywords[17]respectively.ThenumbersofsearchresultsareshowninTable10:

Table10:TheNumbersofSearchResults

1940

1950

1960

1970

1980

16200

26400

40200

25800

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Year

Results27505160

Assumingthatthisisanexponentialfunction:y1cedt.Weusetheleastsquaredmethodtoobtaintheunknownnumbersinthefunction.SeeFigure6.

Theresultgivesasatisfyingsimulationtothenumbersofsearchresults.However,thedistinctionbetween2000sand1900sistoolarge.Inourobservation,thesearchresultsofcoachesatdifferentperiodoftimeisalmostofthesamemagnitudeofaseachother.So,hereweusethenaturallogarithmofthesearchresults.Againweobtainalinearfunctiony2asshowedinFigure7.

Thedifferencebetweenmaximumandminimumisabouthalfoftheminimumvalue.ThisisamodestvaluethatwecansafelyputintousetoestimateICT.Commonsensetoldusthatthegreaternumberoftotalreportsis,themorevaluablethesearchresultis,thegreaterweightthesearchresultwillget.So,wedefineICTas

mo.crotha

.mwww數(shù)

19101920

1930

1990

2000

2023

744011700

2700067700326000

Figure6:Trendofexponentialfunctiony1Figure7:Trendoflinearfunctiony2

ICTi(influencecoefficientoftime)=

1

i1,2,,nktib

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Team#24270Page16of26wherekandbaretheunknownvariablesrelatedtosearchingdata.

4.3.3.Theinfluencecoefficientofreputation

Asmentionedintheoverview,bysearchingdatawithdifferentmethodsandusingdiversekeywords,weobservethatonthetrackoffame,themediawillturnatfirsttotheachievementsofsportsonehasthentotheotheraspectsinhisorherlife.Therefore,themediaattentioncanbeinterpretedandquantifiedbyusingtheoverallsearchresultsofonecoachandhisorhertriumphandachievement.

Toextracttheinformationaboutnewsreportsonthetriumphofacertaincoachfromthesearchengine,first,weuseWordnet[18]asourtooltoobtainahostofsynonymsofthewordwinning,theresultis:

booming;flourishing;palmy;prospering;prosperous;roaring;thriving;in;made;no-hit;productive;self-made;sure-fire;triple-crown;victorious;successful

Usingthesewordsasouralternativekeywords,wecanobtainthenumbersofwinningsearchresultsuiforcoachi.

Theresultturnsouttobeanindicationthattheratiobetweenwinningsearchresultsandoverallsearchresultsisnegativelycorrelatedwiththedegreeofreputation.Thatis,thehigherreputationonecoachgets,themorelikelythemassmediawillconcentrateontheotheraspectsoflifeofthiscoach.Accordingtothisrule,weestablishafunctionofICR:

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4.3.3.Theindividualinfluencevector

TheindividualinfluencevectorlcanbeinterpretedastheoverallsearchresultsmodifiedbyICTandICR.

li

Theindividualinfluencevectorgivesanaccurateestimationofthemediaattentioncertaincoachgot.Itisanormalizedvector,sothatwecanconvenientlyputitintouseinthelatersection.

4.4.TheCross-ReferenceMatrix

WiththehelpoftheGooglesearchengine,thedegreeofcorrelationoftwocoachescanbemeasuredbythenumberofsearchresultsusingtwonamesasCitationKeywords[17]simultaneously.Andwedefinetheoriginalcross-referencematrixZ.Theentriesofthematrixis:

Sinceexchangeofthetwonamesdoesnotaffecttheresult,thematrixZissymmetrical.Andwesetallofthediagonalelementsofthismatrixtobe0.

mo.crotha.mwww模數(shù)

a

ICRi(influencecoefficientofreputation)=1ii1,2,,n

ui

2

aiICTiICRi

j

j

aICTICR

j1

i1,2,,n(9)

j

Zij=numberofsearchresultsnumberofcoachiandcoachj

幫助準(zhǔn)備美賽同學(xué)作參考

4.4.1.Theweightfunction

Theelementsofcross-referencematrixareinfluencedbythedistinctperiodoftimeandreputation.However,thingsgetabitmorecomplicatedhere:iftwocoachesexistinthesameperiodoftime,thentherecouldbemorereportsoncompetitionstheyengaged.Thecompetitionreports