備戰(zhàn)美國數(shù)學(xué)建模競賽與剖析

數(shù)學(xué)建模競賽講座1.美國數(shù)學(xué)建模競賽概況

2.競賽論文剖析3.一些建議主講:龔劬美國數(shù)學(xué)建模競賽概況1美國大學(xué)生數(shù)學(xué)建模競賽（MCM）始于1985年學(xué)生三人一組在四天之內(nèi)合作完成一個實際問題參賽者來自美國、中國、加拿大、德國、英國以及印尼等國家或地區(qū)我國于1989年起組織大學(xué)生參加MCM我校從1994年開始組織學(xué)生參加MCM2006年我校共有8個隊參賽，3個隊獲一等獎、2個隊獲二等獎和3個隊獲三等獎。2部分MCM/ICM題目指紋唯一嗎？（2004A)颶風(fēng)來臨時的疏散問題（2001B）信息技術(shù)安全問題（2004C)核磁共振切片成像(1998A)伽瑪?shù)吨委熡媱?2003B)機場安全檢查(2003C)特技表演(2003A)公園快速通道系統(tǒng)（2004B)風(fēng)與噴泉(2002A)過路收費站(2005B)灌木蜥蜴問題(2002C)3評閱JudgingwillbecompletedbyMarch31,2008.

ThesolutionswillberecognizedasSuccessfulParticipant,HonorableMention,Meritorious,orOutstandingWinner.Youhaveabout43%chanceofbeingrankedhigherthanSuccessfulParticipantThefirst(triage)roundEachpaperisskimmedbytwoorthreejudges,whospendbetween5and10minutseachreadingthepaper.Ifyousurvivethetriageround,youhaveaboutan80%chanceofbeingrankedhigherthanSuccessfulParticipant.4OutstandingWinnerINFORMSwinner

TheInstituteforOperationsResearchandtheManagementSciences(INFORMS)willdesignateanOutstandingteamfromeachofthethreeproblemsasanINFORMSwinnerSIAMwinner

TheSocietyforIndustrialandAppliedMathematics(SIAM)willdesignateoneOutstandingteamfromeachproblemasaSIAMwinner.MAAwinner

TheMathematicalAssociationofAmerica(MAA)willdesignateoneOutstandingteamfromeachproblemfortheMCMasaMAAwinner.

競賽論文剖析1Summary

(Themostimportantpartofthepaper)provideabriefoverviewoftheproblemdescribeclearlytheapproachtakentomodelingtheproblemstatethemostimportantresultsandconclusions.mentionanyrecommendationsdirectlyrelevanttotheproblem.Ifyoudidanythingcreative,itmustbehere.

競賽論文剖析1Summary特色和創(chuàng)新之處必須在這里強調(diào)（稍夸張地）。長度：理想長度很難說，必須包括上述要點，但簡潔也非常重要。一般掌握在半頁至2/3頁左右。是文章最重要的部分。要保證準(zhǔn)確、簡明、條理清晰，突出特色和創(chuàng)新點。注：評閱時將首先根據(jù)摘要和論文整體結(jié)構(gòu)及概貌對論文優(yōu)劣進(jìn)行初步篩選。Agoodexampleofaclear,conciseandthoroughsummary(2004A)

Fingerprintidentificationdependsontheassumptionthataperson’sfingerprintsareunique.Weassessthetruthofthisassumptionbycalculatingthetotalnumberofdistinctfingerprints.Weassumeaccuratefingerprints(ignoringproceduralerror)aredefinedby12pointsofdetailorminutiae.Thenumberofdistinctfingerprintsdependsalsoonthenumberofpotentialpositionsoftheseminutiae.Twohistoricalmethodsandageometricanalysisestimatetheretobe1,400positions,afigureconfirmedbyouralgorithmforcountingridgesinafingerprint.Wecreatetwomodelstoestimatethenumberofuniquefingerprints:Onemodelcomputesfingerprintsasarrangementsinminutiae;theotherextrapolatesthenumberoffingerprintsfromtheShannonentropyoftheinformationthatdefinesafingerprint.

Agoodexampleofaclear,conciseandthoroughsummary(2004A)Thesetwomodelsagreetowithinanorderofmagnitudethatthereare51033uniquefingerprints,acompellingvalidationofourgeneralapproach.Tohandlethelargenumberoffingerprints,weimplementanapproximationforthecalculationofprobabilities.Givenacumulativeworldpopulationof120billion,theprobabilityoftwopeopleeverhavingthesamefingerprintis1.410-6.

Theprobabilityoftwohumanslivingtodaysharingafingerprintis3.510-15,whichsuggeststhatfingerprintsareatheoreticallymorereliablemethodofidentificationthanDNAanalysis,whichhasafalsepositiveprobabilityof10-9.Noneofthesecalculationstakeintoaccountproceduralerrors.

競賽論文剖析2Introduction

重述問題，用自己的語言陳述你將要做的工作根據(jù)自己的理解，用自己的語言清楚簡明地闡述問題的背景、條件和要求。Example1：

Thepossibilityofduplicatefingerprints,orfingerprintslikelytobemistakenforeachother,hasledtorecentcriticismoffingerprintsasameansofidentification.Thekeyproblemis:Howmanydistinctfingerprintsarethere?Weapproachthisproblemwithtwogeneralmethods:“building”afingerprintfromthegroundup,usingdifferentmodels;andusingtheinformationcontentofafingerprint到2004A.docExample2：

IntroductionWedevelopsomemodels,apatternmodel,ashapemodelandapracticalmodel,toanalyzetheuniquenessoffingerprintsandoddsofmisidentificationbyfingerprintevidence.Throughcomparison,wealsodevelopasequencemodeltoanalyzeoddsofmisidentificationbyDNAevidence.Noticingthatthereareseveralpatternsoffingerprints,andtheridgesandfurrowsinfingerprintsaredifferentfromonetoanother,weusethesecharacteristicstodevelopourmodel.Consideringtheincompletenessofthefingerprints,weanalyzetheprobabilityofmisidentificationoffingerprints.BasedonourlimitedknowledgeaboutDNAandgenetics,wecreateasequencemodeltocalculatetheodds.Ignoringthosecomplicatedbiochemicalrestrictionsandbiomolecularrequirements,weestablishapuremathematicalmodelrelatedtoourproblem.競賽論文剖析Example2：

ConsideringsomefundamentalassumptionsandrequirementsofbothfingerprintandDNAidentificationsfortheiraccuracy,weproposeouropinionsonthisidentificationproblem.

Background

Fingerprintsofhumankindareformedintheperiodofafetus.Weknowthattheyaredecidedgeneticallybygeneswhichareinheritedfromtheirparents.Theridgesandfurrowsformedthepatternsorfeaturesoffingerprints,anditiscommonplacebeliefthateveryone’sfingerprintsareunique.Eventwinbornbabies,theirfingerprintsaredifferent.Tillnow,thereisn’tanycasethattwopersons,eitherChineseorAmericanpeople,havethesamefingerprintsexactlyallovertheworld.Actually,nobodyhasfoundanyrelatedrecordappearedinanygovernmentsintheprocessesoffingerprintmanagement,identification,checkandrecognition.

競賽論文剖析Example2：

In1953,WatsonandCrick[5]putforwardtheDNAdoublehelixmodelbasedonbasespairs.In1967,LinandChargaff[6]foundthatinsinglenucleotidechain,thereisanbalancebasedistributions:and.Ineverychromosomeofhuman,theratiovalueofbasepairs,bothA/TandG/Cvaryin0.999~1.001.NowaccordingtoachievementsofHumanGenomeProjection(HGP)[8]whichhasbeencompletedbyApril14th2003,thehumanDNAsequencecontainsabout3billionorinmoredetails,3164.7millionchemicalnucleotidebases(A,C,T,andG).Almostall(99.9%)nucleotidebasesareexactlythesameinallpeople.Intotalhumangenome,anyDNAconsequenceconsistsofcodingandnoncodingregions,whilethecodingregionsarecalledgenes.Genesappeartobeconcentratedinrandomareasalongthegenome,withvastexpansesofnoncodingDNAbetween.BecauseofthepolymorphismofDNAandtherandomarrangementofthebasesinDNAsequence,itisnothardtoimaginethekindsofDNAshouldbeanastronomicalfigureorananoscopicmap.AndthisgivesthefundamentofDNAidentification.

競賽論文剖析3模型假設(shè)假設(shè)要合理，在建模時會用到，不要羅列大量無關(guān)緊要的假設(shè)。有用的假設(shè)主要包括兩種：與問題相關(guān)的特別信息缺乏且在規(guī)定的時間內(nèi)得不到，因此必須做出假設(shè)，才能繼續(xù)。另外就是為了應(yīng)用你熟悉的數(shù)學(xué)或為了避免在給定的時間不能完成你的模型而簡化問題的細(xì)節(jié)所作出的簡化假設(shè)，

合理假設(shè)的作用：簡化問題，明確問題，限定模型的適用范圍

競賽論文剖析3模型假設(shè)Example1Fingerprintsareofthehighestpossiblequality,withoutdamagefromabrationandinjury.Theridgestructureofafingerprintisinoneoffivecategories:Arch,LeftLoop,RightLoop,TentedArch,orWhorl,employedintheautomaticclassificationsystemofCappellietal.[1999](derivedfromthoseoftheFBIandWatsonandWilson[1992]).Eachcategoryhasacharacteristicridgeflowtopology,whichwebreakintohomogeneousdomainsofapproximatelyunidirectionalflow.WhileCappellietal.[1999]raisetheissueof“unclassifiable”prints,andtheyandMarcialisetal.[2001]confuseclassesofridgestructures,weassumethatsuchambiguitiesstemfrompoorprintquality.競賽論文快剖析Ea廢ch幸m隊in申ut懂ia剃c寒a(chǎn)n扇b訪e差cl戲as掌si草fi廳ed悉a鮮sa既bi飾fu和rc擺at姜io怖n,賤a仗t身er賀mi悠na皮ti間on燥,責(zé)or疊a伍d渴ot.[Pa愉nk蔑an掘tiet嬌a晨l.辜2鴉00唯2;St司on效eyan板d梨Th曾or卷nt住on19藝86瓶]池.伯Th棗ou萌ghGa慎lt潔on[1漁89連2]音i知de直nt標(biāo)if系ie宋s相10快m歲in標(biāo)ut選ia患s粘tr蹈uc因tu明re誼s訂an躺d嚼ot木he螺rs濤f瞞in處d雹13輔[Os礙te竭rb析ur吊get杠a戰(zhàn)l.小1司97突7]庫,茶we謹(jǐn)c際an詢i奴gn性or倉e陽th世es巷e兔fu溉rt歇he斷r濫st源ru倍ct粉ur夜es戀(斗wh稻ic輩h郵ar輝e佛co穗mp許os跡it淘io逆ns賊o濕f目th醉e堆ba艷si亭c聲th毫re猛e)振b兔ec姻au浙se附o統(tǒng)f舟th捐ei蘋r粥l(xiāng)o鳴w梨fr嫁eq把ue牲nc皮y認(rèn)[Os倉te具rb尤ur較get杜a慌l.愿1糾97竊7]械.Lo躲ca戚ti邀on早a視nd龜o湊ri聰en扯ta延ti贊on即o返f差mi斗nu傳ti腔ae葵r飄el落at威iv傍e煩to飾e陣ac搖h脖ot密he篇r永ar狗e娃in不de翁pe雨nd畢en嫩t.Fr醬eq記ue扣nc喇ie離s給of狂r代id忠ge支s耗tr錯uc槍tu傅re墻c葬la醫(yī)ss就es清a旬nd萬c搖on嶼fi稱gu蒼ra酸t(yī)i沸on余s趟an魂d緣瑞mi叫nu爺ti早ae瓜t瓣yp兼es航d貍o占no璃t魯ch填an然ge乎a兼pp臣re愉ci觸ab拌ly順w谷it麻h觀ti闊me毛.競賽論文姻剖析3模型犬假設(shè)Ex未am骨pl講e尺2A地th辰um斥bp掩ri卡nt裕i做s貢de遞fi罰ne港dgl挺oa獅al唐lyby繁r剖id眨ge窩p賞at敵te使rn遍s哪an疑d侵lo循ca釀ll圖y浪by愈a言d召is歉tr雷ib襖ut斬io愁n言of睜m勺in隊ut謀ia超e,京w旬hi分ch伶w騾e債re葡fe懶r暴to扮a據(jù)ls騙o評as圍f不ea漏tu據(jù)re售s.Th祖e需ar玩ea走o嫂f州in袋te恢re塑st蒙t托yp倆ic嬌al燦t夸hu若mb桶pr胞in很t霸is煮2搏0m執(zhí)m2籃0m健m液sq覺ua菌re跑g剩ri惜d.Th象er收e賭ar總e派tw湖o縮慧si掃gn元if睬ic岔an最t蟲ty句pe均s席of包m凱in盼ut載ia隙e,傍t境he鴉b懶if患ur榮ca潛ti譜on售a頑nd糠t與he巡壽r箭id恩ge巾e殼nd媽in盞g:剪a輕ll間o邪th稼er伍m鴿in卷ut煤ia足e情ar乏e任co旨mp鈔os縫it判io述ns襖o禽f金th贊es決e忌[Os循te欄rb好ur棋get渡a影l(fā).恰19塘77鵲].Th筋e泰pr及ob低ab疾il誤it扣y單of桿a豈m睬in哄ut圓ia刻o峽cc地ur圍ri串ng蓬i買n災(zāi)a福gr塑id根b六ox餓i殊s喊0.陜23練4恰[Os霸te派rb還ur即get陵a幅l.補19腿77與].Th叛e扇or黎ie私nt余at求io照n侵of朽t債he脾m降in扇ut好ia儀e英wa豈s區(qū)no兄t暑ta嗽ke貴n團(tuán)in司to荷a悄cc煎ou幻玉nt故b設(shè)yOs固te給rb播ur信g;介we鳳a渠ss檢ig扇n躲a擦mi悟nu芒ti升a暢on濤e敲of蓬e督ig居ht拘a青ng軌le君s,歉f啦ro毅m膛0oto顫1殿57騾.5o,斬in不s星te掙ps憑o魂f憶22池.5o.Wh娘en楚c疾om法pa光ri肆ng蛋t逃wo南p底ri龍nt睜s,繁w睬e株kn瞇ow胃o牛ne蠅p綠ri聯(lián)nt受a荒rb喝it饒ra嫂ri單ly咱w周el最l.Th籃e喘nu遇mb派er染o納f圈pe訪op腔le極w昂ho濕h射av呢e鉛ev亂er丸l昂iv費ed型i壞s異1.撲06惠41災(zāi)011[Ha獄ub19磚95針].競賽論文奇剖析.建立碗模型模型疾的主侮要類樣型：初等膝模型嘴、微橫分方粉程模墾型、衛(wèi)差分慎方程傍模型備、概旅率模具型、涂統(tǒng)計屬預(yù)測骨模型發(fā)、優(yōu)守化模糊型、伐決策糕模型僑、排撞隊模張型、舉圖論餡模型辮等幾類吳常見別建模罷目的:（1吊）描夏述或勒解釋次現(xiàn)實邁世界踏的各焦類現(xiàn)品象，穩(wěn)常采刷用機績理分侍析的戰(zhàn)方法迅,探最索研籠究對血象的奔內(nèi)在減規(guī)律瞎性（2君）預(yù)挺測感媽興趣瞧的事個件是真否會羽發(fā)生心,或陪者事物物的王發(fā)展鑰趨勢癢，常拼采用堂數(shù)理盒統(tǒng)計箏或模溪擬的香方法（3濾）優(yōu)妥化管中理、咬決策午或者覺控制步事物充，需當(dāng)合理匙地定董義可團(tuán)量化肆的評坐價指芳標(biāo)及棄評價蝴方法復(fù).4武建管立模耗型建模云過程伯中的略幾個緒要點魂：模型薪的整霸體設(shè)把計、詠合理遮的假缺設(shè)、時建立漢數(shù)學(xué)擋結(jié)構(gòu)增、建袋立數(shù)斗學(xué)表柿達(dá)式數(shù)學(xué)騎模型頌最好份明確圓、合臺理、房誠簡潔汪，具宅有一擠般性盲；有些嬌論文米不給吸出明懼確的知模型兄，只益是就鬧賽題企所給失的特拜殊情啟況，瓶用“崗湊”能的方崇法給吵出結(jié)高果，揪雖然擱結(jié)果隙大致疑對，廢但缺檢乏一癢般性曉，不要是數(shù)蟻學(xué)建際模的潮正確另思路鼓勵播創(chuàng)新以，特別議欣賞注獨樹丈一幟傳、標(biāo)荷新立勒異，花但要暫合理。避免鞠出現(xiàn)挎羅列界一系箱列?；谛?，攻又不似作評察價的俱現(xiàn)象驢?？商б詮尿_簡單弓模型叢開始岡，然遣后改讀進(jìn)。競賽論文邀剖析5售模型倆求解旨——想最重咽要的踩部分凳之一算法帖設(shè)計啊或選割擇，疏算爪法思佩想依線據(jù)，趕步驟客；引用漆或建偵立必伏要的壁數(shù)學(xué)母命題掘和定堡理；在不叼能求強出精裹確解灶的情周況下異，需育要給皆出不狂只一臥種解伶法（牲算法陽），肢并進(jìn)趙行測跟試比們較，舊給出偉評價虎。為策了說餡明你摧的算威法好表，你逼需要桂有一愛個參遲照與暫之比逢較，迎你可訓(xùn)以從水最簡平單、死最易州得到淺的算線法開疼始，戲逐步待改進(jìn)管直到辜得到唱你的次最好寄解。通常普對于侵離散凡問題頃，最眾簡單茂的解拒可能退只是喉做隨您機選道擇，匪然后蘆用你這的算書法得貧到的節(jié)解與廢它比屠較。競賽論文氣剖析6結(jié)果倒表示欄、結(jié)層果分散析、邀模型威檢驗結(jié)果敲表示號要集抱中，琴一目粒了然探，直巧觀，凝便于消比較即.結(jié)果梢分析攀不能宇少，臟從計首算結(jié)掠果可欣得出贈什么廣結(jié)論模型說檢驗麥特別遇重要ru封n贈MU謝LT樣IP室LE詠t躁es澇ts偵a齡nd旅f厚in癢d媽ou葬t(yī)善if提y貢ou咽r掃so慣lu詠ti瓜on肺i賊s嗽st另ab孫le按!Wi帆ll踩i叫t削ho歇ld浮u巾p盒un當(dāng)de舌r漸sl臣ig慌ht敏ly鉆d恩if呈fe廣re剩nt風(fēng)c園ir效cu疾ms吉ta魂nc劇es術(shù)?艇At釘t弊hi秧s尤po坊in咱t督y(tǒng)o僅ur退c壇od黃e刷is暗d痕eb滋ug耐ge潔d劇an知d榜ge浪ne帽ra尾ti當(dāng)ng信u噴se錦fu勉l拐in艘fo倒rm紅at桃io州n是--護(hù)t覺ak笛e嫁ju昂st屢a泰no憶th比er抵2周0名mi勉nu喂te爭s翅an產(chǎn)d努ru朱n舟a巷fe健w削mo總re承c緊as羅es盤,株va破ry鋼in憤g蘿th身e筍pa蠟ra筑me綿te個rs左j仔us惜t鄰a懲bi奶t.If屠y蔑ou云'r垮e械do恭in涂g傳th秀e仙Es田ca叨pi刻ng篇a診H突ur范ri康ca憤ne法p激ro裳bl項em蘋,顛ru森n污yo伐ur郵c珍od陰e缺wi習(xí)th風(fēng)a能f精ew埋m凍or擊e沿ca羅rs朋,貼or撥a串f毛ew橡l濟(jì)es丟s礦--志s鞏ee棄h段ow望f渴le蟻xi菊bl鴉e璃yo京ur段r縫es儲ul腥ts顯a扔re叔.溝If呼y慰ou戶'r剝e練do斤in斃g毒th遼e由MR腐I糞Pr煙ob萬le灘m,址t北es抬t閣yo紛ur割r欄ou蛾ti銳ne丟s尸on碑s虹ev扇er珍al揉d劈燕if恩fe員re賞nt例s會im茶ul編at蕩ed當(dāng)o放rg垂an造s.濁I甲f添yo矩u'萬re賄d筋oi蟲ng冠t長he挖A濾ir仔T郵ra跡ff定ic冰C動on插tr通ol勁p僵ro嚴(yán)bl卷em擾,澆th乏ro艦w虹th可e殺pl挺an斥es含a畝t龍yo免ur質(zhì)s眨im隊ul倡at膠ed姐a耗ir喂po解rt棄a磁b豪it幫f狀as木te爹r,蘿a阻b捕it緣瑞s賣lo療we啞r繞an農(nóng)d祥se啞e齡ho大w銹yo番ur潮p指ro蟲gr踐am粉c崇op認(rèn)es能.渾Pr朗ov眠e擔(dān)to吼t亞he蘿j幣ud賞ge光s篩th鄭at躺y躍ou殼r魄so冬lu魔ti俗o(hù)n始i篇s醉fl萄ex景ib另le柳a橫nd拜s宋ta蝦bl往e,蔽o衰r頌ho切ne興st獸ly黎a扭dm經(jīng)it信t般ha球t夠yo宅ur泊s鵲ol傭ut扭io赤ndo炎es脈n't初w踐or盆k旱as忽w介el晚l頌ag例ai脊ns都t借ce伏rt紫ai寫n鐮ch勉al挺le匙ng川es宅.譯Yo氏ur腫p景ap疲er駐w物il嚴(yán)l加lo遷ok租a顧w柴ho寫le擇l怪ot言b脅et護(hù)te違r!7St杠re掀ng匹th是s就an勺d霸We斃ak深ne密ss野es昂o安f數(shù)th昂e韻mo殼de恭l突出甜優(yōu)點益，但晝?nèi)觞c昂也不裂能回巧避。我在寫妨模型繭弱點肚時，竊要對特照你桂建立得模型筋時所聲作的仰假設(shè)列，模惕型有桂哪些烤局限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