美国数学建模.docx
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美国数学建模.docx
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美国数学建模
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Summary
Facedwiththepopularizationofcompetitivesports,teamcoacheshaveneverbeenputundersuchanenormousstagethaneverbefore.Atthesametime,theyareundergoingmorepressuregivenbycriticsandenthusiasticfans.Expertsaretirelesslyexplainingtheircriterionstothepublic.Theyhopetofindoutwhoisthebestcoach.Wearerequiredtofindoutthe“bestalltimecollegecoach”maleorfemaleforthepreviouscentury.
Inordertounveilthesecretlegendarycoach,webuildamathematicalmodeltoassesstheaccomplishmentofacollegecoachofallfields.WeareinspiredbythegameLOL(LeagueofLegend),whichevaluatespeople’sstrengthbyaspecificalgorithmandgetarankpoint.Wewanttomakeasimilaralgorithmtoarticulateascoringsystemforcoaches.
First,tofigureoutwhethertimelinehorizonthatweuseinouranalysismakesanydifferences,wedecidedtofocusononespecifickindofsports.Forexample,wechoosebasketball.Obviously,therulesofbasketballchangedalotastimegoesby.What’smore,nowadaysBasketballplayersaremuchstronger,fasterandtallerthaneverbefore.Timemakesconsideratechangestoouranalysis.
Ontheotherhand,weconsiderthatbothmaleandfemalecoachescomefromallsports,whichvariesfromhockeyorfieldhockey,football,etc.Wecan’tsimplymaketheconclusionwiththegamestheyhavewon.Inaddition,withtheaccumulationoftheyearofcoachesthemselves,thenumberofwinninggamesgrows.Sodoestheirexperience.
Wehavemadeourownmathematicalmodeltojudgecoachesbyscoringforthemwiththestatistics.Therefore,wearenowcapabletorepresenttop5coachesineachof3differentsports.Thestrongtheoreticalbasisandsuitableassumptionmakeitestimableforfurtherstudyofthisquestion.Combiningourmodelwithmoreaccurateinformationofthisquestionwillmaximizeitsutility.
CollegeCoachingLegends
Summary
Facedwiththepopularizationofcompetitivesports,teamcoacheshaveneverbeenputundersuchanenormousstagethaneverbefore.Atthesametime,theyareundergoingmorepressuregivenbycriticsandenthusiasticfans.Expertsaretirelesslyexplainingtheircriterionstothepublic.Theyhopetofindoutwhoisthebestcoach.Wearerequiredtofindoutthe“bestalltimecollegecoach”maleorfemaleforthepreviouscentury.
Inordertounveilthesecretlegendarycoach,webuildamathematicalmodeltoassesstheaccomplishmentofacollegecoachofallfields.WeareinspiredbythegameLOL(LeagueofLegend),whichevaluatespeople’sstrengthbyaspecificalgorithmandgetarankpoint.Wewanttomakeasimilaralgorithmtoarticulateascoringsystemforcoaches.
First,tofigureoutwhethertimelinehorizonthatweuseinouranalysismakesanydifferences,wedecidedtofocusononespecifickindofsports.Forexample,wechoosebasketball.Obviously,therulesofbasketballchangedalotastimegoesby.What’smore,nowadaysBasketballplayersaremuchstronger,fasterandtallerthaneverbefore.Timemakesconsideratechangestoouranalysis.
Ontheotherhand,weconsiderthatbothmaleandfemalecoachescomefromallsports,whichvariesfromhockeyorfieldhockey,football,etc.Wecan’tsimplymaketheconclusionwiththegamestheyhavewon.Inaddition,withtheaccumulationoftheyearofcoachesthemselves,thenumberofwinninggamesgrows.Sodoestheirexperience.
Wehavemadeourownmathematicalmodeltojudgecoachesbyscoringforthemwiththestatistics.Therefore,wearenowcapabletorepresenttop5coachesineachof3differentsports.Thestrongtheoreticalbasisandsuitableassumptionmakeitestimableforfurtherstudyofthisquestion.Combiningourmodelwithmoreaccurateinformationofthisquestionwillmaximizeitsutility.
Contents
Summary................................................................................................1
Introduction.............................................................................................4
Restatementoftheproblem....................................................................4
TheBasicAssumption.............................................................................5
Coachesarededicatedtoeachgame............................................5
Allthegamesare“fair”...................................................................5
Playersoncourtare“diligent”........................................................5
Wegivecertainpointswhenqualifyingcoaches............................5
Question1:
MathematicModelforscoringsystem...................................6
1.1Introductiontoourscoringsystem............................................6
1.2Derivingofvariousindexesofevaluation’sweight...................6
First-gradeindex....................................................................6
Second-gradeindex.........................................................…...7
1.3Ourmathematicmodelanditsapplication...............................8
Question2:
ImprovedModelone………………………………….…….......10
Question3:
ImprovedModeltwo………………………………………........12
StrengthsandWeaknesses..................................................................13
References...........................................................................................14
Attachment...........................................................................................15
AnarticleforSportsIllustrated...............................................................16
Introduction
Webuildamathematicalmodeltoassesstheaccomplishmentofacollegecoachofallfields.Thereforeweareabletodistinguishtopcoachalltime.
Ononehand,inordertofigureoutwhethertimelinehorizonthatweuseinouranalysismakesanydifferences,wedecidedtofocusononespecificgame.Forexample,wechoosebasketballwhilemodeling.Obviously,therulesofbasketballchangedalotastimegoesby.What’smore,nowadaysBasketballplayersaremuchstronger,fasterandtallerthaneverbefore.Timemakesconsideratechangestoouranalysis.
Ontheotherhand,weconsiderthatbothmaleandfemalecoachescomesfromallsports,whichvariesfromhockeyorfieldhockey,football,baseballorsoftball,basketball,orsoccer,etc.Theyhavedifferentrulesandfeatures.Wecan’tsimplymaketheconclusionwiththegamestheyhavewon.Inaddition,withtheaccumulationoftheyearofcoachesthemselves,thenumberofwinninggamesgrows.
Furthermore,wecanuseourownmathematicalmodeltojudgecoachesbygivingthosepointswiththestatisticsofallcoaches.Therefore,wearenowcapabletorepresenttop5coachesineachof3differentsports.OurmathematicmodelshowsthatJohnWoodenisthemalecollegecoachinglegendofbasketball,whichisinaccordancewithwhatisuniversallyacknowledgedbymostofthepeople.
Restatementoftheproblem
Theproblemneedsustomakeamathematicalmodeltodescribeandclassifytheachievementsofcoaches.Estimationofacoachhasmanyaspects,andisfullofuncertainty.SomeinstitutionsoftheU.S.carryoutdifferentcriterionsoftheevaluationandgiveouttheirownopinions.Thentheydraftalistofthetopcoaches.However,howtogetaflawlesscriterionhavepuzzledeveryexpertsforalongtime.WeprefertochooseanalgorithmwhichcombinesFCE(FuzzyComprehensiveEvaluation)andAHP(AnalyticHierarchyProcess)toscoreforacoach.
Ourscoringsystemconcludes5first-gradeindexes,andeachindexhastwosecond-gradeindexes.Inourfirst-gradeindexes,thereare
·Coachingcompetence
·Coachingexperience
·Coachingeffect
·Coachingmethods
·Publicevaluation
Basicassumptions
1.Coachesarededicatedtoeachgame
Coachesarewishedtodevoteintoeverygame.However,sometimescoachmightdecidetogiveupafewgamesforsomespecificpurposes,suchastoavoidinjury,ortogivethebencheschancesthenhidingstartingplayers.Thisisacommonactionbetweencollegegames.Asaresult,weprefertoignoretheinfluencecausedbycoachingstrategy.
2.Allthegamesare“fair”
Throughoutthehistoryofcompetitivegames,wehavefoundmanyscandalsofjudges.Someofthemviolatetheprincipleofafairgameandchangetheresultofagame.Withnodoubt,thiswilldefinitelyincapacitateusfromthecorrectassessofacoach.Fortunately,allofthemonlystandforaminoraspectofthegame.Undermostcircumstance,judgesarefairandthegameresultisreliable.
3.Playersoncourtare“diligent”
Becauseofthecomplexityanduncertaintyofindividuality,weinvoketheassumptionthattheplayersoncourtare“diligent”,whichdenotesthatplayersabidebytacticsoftheircoach.Theyworkhardanddon’trebelagainstcoachesoncourt.
4.Wegivecertainpointswhenqualifyingcoaches
Sinceu11,u12,u21,u22,u31,u32areobjectivecriterions,wecangiveoutexactpointsofthem,andwillnotbefuzzified.Thereforewedefined4classesofpoints:
onepoint,fourpoints,sevenpointsandtenpoints.Thismeansweassumethatthesecertainpointswillgivetothemregardingtotheircriterions.Table1showstheircorrelation.(DefinitionsofuwillbelistedinthenextTable2)
Table1
1point
4points
7points
10points
u11
[0,8]
[9,16]
[17,24]
[25,30]
u12
[0,3)
[3,6)
[6,9)
[9,+∞)
u21
[0,250)
[250,500)
[500,750)
[750,+∞)
u22
[0,13)
[13,26)
[26,39)
[39,+∞)
u31
(0,0.25)
[0.25,0
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