美国大学生数学建模一等奖31552.docx

美国大学生数学建模一等奖31552.docx

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美国大学生数学建模一等奖31552

Bestalltimecollegecoach

Abstract

Inordertoselectthe“bestalltimecollegecoach”inthelastcenturyfairly,Wetakeselectingthebestmalebasketballcoachasanexample,andestablishtheTOPSISsort-ComprehensiveEvaluationimprovedmodelbasedonentropyandAnalyticalHierarchyProcess.

Themodelmainlyanalyzedsuchindicatorsaswinningrate,coachingtime,thetimeofwinningthechampionship,thenumberofracesandtheabilitytoperceive.Firstly，AnalyticalHierarchyProcessandEntropyareintegrativelyutilizedtodeterminetheindexweightsoftheselectingindicatorsSecondly，Standardizedmatrixandparametermatrixarecombinedtoconstructtheweightedstandardizeddecisionmatrix.Finally,wecangetthecollegemen'sbasketballcompositescore,namelytheorderofmalebasketballcoaches,whichisshowninTable7.AdolphRuppandMarkFewarethelastcenturyandthiscentury's"bestalltimecollegecoach"respectively.Itisrealistic.Therankofcollegecoachescanbeclearlydeterminedthroughthismethods.

Next,ANOVAshowsthatthescoresoflastcentury’scoachesandthiscentury’scoacheshavesignificantdifference,whichdemonstratesthattimelinehorizonexertsinfluenceupontheevaluationandgenderfactorhasnosignificantinfluenceoncoaches’score.Theassessmentmodel,therefore,canbeappliedtobothmaleandfemalecoaches.Nevertheless,basedonthis,wehavedrawncoaches’coachingabilitydistributingdiagramunderidealsituationandnon-idealsituationaccordingtothedatawehavefound,throughwhichwegetthatiftimelinehorizonischosenreasonably,itwillnotaffecttheselectingresults.Inthisproblem,thetimelinehorizonoftheyear2000willnotinfluencetheselectingresults.

Furthermore,weputthedataofthethreetypesofsports,whichhavebeenfoundbyus,intotheaboveModel,andgetthetop5coachesofthethreesports,whichareillustratedinTable10,Table11,Table12andTable13respectively.TheseresultsarecomparedwiththeresultsontheInternet[7],soastoexaminethereasonablenessofourresults.Wechoosethesportsrandomlywhichundoubtedlyshowsthatourmodelcanbeappliedingeneralacrossbothgendersandallpossiblesports.Atthesametime,italsoshowsthepracticalityandeffectivenessofourmodel.

Finally,wehaveprepareda1-2pagearticleforSportsIllustratedthatexplainsourresultsandincludesanon-technicalexplanationofourmathematicalmodelthatsportsfanswillunderstand.

Keywords:

TOPSISImprovedModel;Entropy;AnalyticalHierarchyProcess;

ComprehensiveEvaluationModel;ANOVA

Contents

I.Introduction

Thepaperistohelp"SportsIllustrated"tofindthe“bestalltimecollegecoach”maleorfemale.

Wetacklefivemainproblems:

●Buildamathematicalmodeltochoosethebestcollegecoachorcoaches（pastorpresent）fromamongeithermaleorfemalecoachesinsuchsportsascollegehockeyorfieldhockey,football,baseballorsoftball,basketball,orsoccer,andclearlyarticulateourmetricsforassessment.

●Doesitmakeadifferencewhichtimelinehorizonthatyouuseinyouranalysis,i.e.,doescoachingin1913differfromcoachingin2013?

●Presentourmodel’stop5coachesineachof3differentsports.

●Discusshowourmodelcanbeappliedingeneralacrossbothgendersandallpossiblesports.

●InadditiontotheMCMformatandrequirements,preparea1-2pagearticleforSportsIllustratedthatexplainsourresultsandincludesanon-technicalexplanationofourmathematicalmodelthatsportsfanswillunderstand.

Totacklethefirstproblem,wesearchedtheindicatorsofTop600men’sbasketballcoachesoftheAmericancolleges.Takeselectingthebestmalebasketballcoachasanexample:

fortheexplicitfactorsthataffectassessmentstandards,wecalculateeachindicator’sweightbyusingEntropymethod;forthoseimplicitfactors,wecalculatetheweightthroughexperts’evaluation.Thedeterminationofeachindicator’sscoreshouldbegivenbyexpertsevaluationofeachindicator.Theseindicatorsarethennumericalized,andtheimportanceofeachindicatorisdeterminedthroughweightcoefficients.Thenthroughthemultiplicationofthescoresofcoaches’differentabilityindicatorwithcorrespondingweightcoefficients,wegetthecorrespondingscores,andthehighestscoreindicatesthebestchoice.

Forthesecondquestion,wefirstuseANOVAtodeterminewhethersignificantdifferenceexistsbetweenthescoresofcoachesinthelastcenturyandthiscenturyandthegenderfactorSignificancedifferenceshowsthatthetimelinehorizon,thegenderfactorhasinfluenceontheassessment,whereasinsignificantdifferenceshowsnoinfluence.Andbasedonthis,wehavedrawncoaches’coachingabilitydistributingdiagramunderidealsituationandnon-idealsituationaccordingtothedatawehavefound,whichhelpusfurtherresearchtheinfluenceoftimelinehorizonontheassessment.

Forquestion3and4,weputthedataofthethreetypesofsports,whichhavebeenfoundbyus,intotheModel,andgetthetop5coachesofthethreesports,whichareillustratedinTable10,Table11,Table12andTable13respectively.TheseresultsarecomparedwiththeresultsontheInternet,soastoexaminethereasonablenessofourresults.Wechoosethesportsrandomly,whichundoubtedlyshowsthatourmodelcanbeappliedingeneralacrossbothgendersandallpossiblesports.Atthesametime,italsoshowsthepracticalityandeffectivenessofourmodel.

Figure1.Thesourceofthebestcollegecoaches

П.TheBasicAssumption

●Expertsrecessivefactorsevaluationcriteriaevaluationisfairandequitable.

●Coaches’coachinglevelwillincreasewithincreasingage,butitwilldeclineduetomentaldeclinationandthelackofthephysicalstrength.

●Assessmentexpertsarefullyknownoncollegecoaches.

●Theevaluationcriteriaonlyconsiderthefactorsenumeratedinthispaper,withoutconsideringotherfactors.

●Theevaluationcriteriaapplyequallytomenandwomencoaches.

●Weusedthegeneraldatafromareliablewebsite,Website（seeAppendix）.

Ⅲ.Nomenclature

Variable

Meaning

Indexdatanormalizationmatrix

Indexweights

Transformednormalizedmatrix

"Positiveidealsolution"

"Negativeidealsolution"

icomprehensiveevaluationindexvaluesofbeingevaluated

Indexentropy

IndexInformationutility

Fstatistic

Ⅳ.Model

4.1DataProcessing

Inordertobetterassesstheextentofoutstandingcoaches,weselectedanumberofindicatorstodeterminethecoachforthe"bestalltimecollegesportscoach".Wefoundinformationonthevariousindicatorsofdataonthesiteandgetsomereliableindicatorsdataofthesecollegecoaches.Duetothedimensionsofeachindexinconsistenciesexist,sowetransformedthedatatoeliminatetheeffectsofdimensionless.Andthroughpoorconversiongetanormalizedmatrix

isadimensionlessquantityand

.

4.2Modelanalysis

Inordertoaddresstheproblemsmentionedaboveandprovideavalid,feasibleassessmentstrategyforSportsIllustrated,wedecidetoselectsoftball,basketballandfootballbyreviewingtherelevantliterature.Coachingtime,Competitionwinningrate,Culturalqualities,Athleticability,Socialskills,Abilitytowithstand,Innovationcapacity,Abilitytoperceive,andsoon,whichareevaluationindexes.Theseevaluationindexesaredividedintodominantfactorsandrecessivefactors.SpecificfactorsofaffectingtheevaluationcriteriaareshowninFigureX.Theseindicatorswillbequantifiedanddeterminethedegreeofimportanceofeachindexbyweightcoefficient.Whenselectingcoaches,thescoresoftheindicatorsmultiplycorrespondingweightcoefficient,gettingcorrespondingscores,andthepersonwiththehighestscoreisthebestcandidate.

Multi-levelanalysismethodtodeterminetheweightismoresubjective.Itissuitabletodeterminetheweightsforhiddenfactors,whicharenotusedwidelyinbothsexesandallpossiblerequirementsforsport.Weneedtobuildamorereasonablemodeltodeterminetheweightforthedominantfactorandrecessivefactors.Finally,wedeterminethe“bestalltimecollegecoach”.

4.3Modelbuilding

Welookforthe“bestalltimecollegecoach”byestablishingamathematicalmodelinTechniqueforOrderPreferencebySimilaritytoIdealSolution.Takechoosingthebestcollegecoachorcoachesfromamongmalecoachesinsuchsportsasbasketballasanexample.Forthedominantfactor,wecalculatetheweightofeachindicatorinEntropyMethod;Forthehiddenfactors,wecalculatetheweightofeachindicatorinexpertassessmentmethod.Accordingtothesituationofthecoaches,thescoresofalllevelsshouldbedeterminedbyexperts,andtheseindicatorsshouldbequantified.Weightingcoefficientsrepresenttheimportanceofeachindicator.Thescoresoftheindicatorsmultiplycorrespondingweightcoefficienttoobtainthetotalscore,andthepersonofhighestscoreisthebestcandidate.Thismethodismoreobjective,comprehensive,accurateandwide-applicablethanthepreviousevaluationmodel.

Flowchartoflookingforthe“bestalltimecollegecoach”isshowninFigure2.

Figure2.FlowchartofModel

TOPSISModel（TechniqueforOrderPreferencebySimilaritytoanIdealSolution）wasfirstlyintroducedbyC.L.HwangandK.Yoonin1981.TOPSISModelisbasedontheproximityofalimitednumberofevaluationobjectsandidealisticgoalsandevaluatetherelativemeritsofexistingobjects.Meanwhile,TOPSISModelisanapproximationoftheidealsolutioninordermodel,themodelrequiresonlyamonotonicallyincreasing（ordecreasing）ofeachUtilityfunction.Furthermore,TOPSI