美国大学生数学建模C题参考论文.docx

美国大学生数学建模C题参考论文.docx

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美国大学生数学建模C题参考论文

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2012MathematicalContestinModeling（MCM）SummarySheet

（Attachacopyofthispagetoeachcopyofyoursolutionpaper.）

Typeasummaryofyourresultsonthispage.Donotinclude

thenameofyourschool,advisor,orteammembersonthispage.

ModelingforCrimeBusting

Yourorganization,theIntergalacticCrimeModelers（ICM）,isinvestigatingaconspiracytocommitacriminalact.Theinvestigatorsarehighlyconfidenttheyknowseveralmembersoftheconspiracy,buthopetoidentifytheothermembersandtheleadersbeforetheymakearrests.Theconspiratorsandthepossiblesuspectedconspiratorsallworkforthesamecompanyinalargeofficecomplex.The

companyhasbeengrowingfastandmakinganameforitselfindevelopingandmarketingcomputersoftwareforbanksandcreditcardcompanies.ICMhasrecentlyfoundasmallsetofmessagesfromagroupof82workersinthecompanythattheybelievewillhelpthemfindthemostlikelycandidatesfortheunidentifiedco‐conspiratorsandunknownleaders.Sincethemessagetrafficisforalltheofficeworkersinthecompany,itisverylikelythatsome（maybemany）oftheidentifiedcommunicatorsinthemessagetrafficarenotinvolvedintheconspiracy.Theywanttodeterminehowtheymightscheduleanoptimalmixoftrips,ofvaryingduration（measuredinnightsontheriver）andpropulsion（motororoar）thatwillutilizethecampsitesinthebestwaypossible.Inotherwords,howmanymoreboattripscouldbeaddedtotheBigLongRiver’sraftingseason?

Therivermanagershavehiredyoutoadvisethemonwaysinwhichtodevelopthebestscheduleandonwaysinwhichtodeterminethecarryingcapacityoftheriver,rememberingthatnotwosetsofcamperscanoccupythesamesiteatthesametime.Inadditiontoyouronepagesummarysheet,prepareaonepagememotothemanagersoftheriverdescribingyourkeyfindings.Throughthecarefulstudying,weabstractthecollisionsystemintoasimpleharmonicmotionsystemwhichisconstitutedby4partsofobjectsandelasticforcebetweenthem.ByanalyzingtheSimpleharmonicsystem,webuilddifferentialequationsmodelforthecollision,andgaintheballleavespeedatalllocationsofthehittingpoint.Then,wecangetthelocationofsweetpointbyfindingthemaximumreturnspeedoftheball.Forproblem2and3,wechangesomecoefficientofthemodelusedinproblem1accordingtothesituation,andconcludedthatcorkingabatwillslowthespeedofball,whileusingabatmadeofaluminumwillincreasetheballspeed.

Keywords:

simpleharmonicmotionsystem,differentialequationsmodel,collisionsystem

TheProbabilityModelforPredictingtheAreaofASerialCrime

Keywords:

theBayesTheoremequation;"centerofmass";crime;theDistanceDecayFunction;reliability;model;thegeographicalprofile.

Contents

1Introduction3

1.1ComprehensiontotheProblem..................................................................3

1.2TheGoaloftheModel...............................................................................4

1.3TheAnalysisoftheProblem......................................................................4

2NotationandDefinitions4

3SimplifyingAssumptions4

4TheModel

4.1MarktheLocationsofCrime......................................................................5

4.2PredicttheLocations...................................................................................7

4.3TwoMethodstofindthe"centerofmass"...................................................8

4.3.1TheFirstMethod........................................................................8

4.3.2TheResultoftheFirstMethod...................................................8

4.3.3TheSecondMethod..................................................................10

4.3.4TheResultoftheSecondMethod............................................12

4.3.5TheCombinationoftheTwoResults.......................................13

5CaseStudy:

TheTestoftheModel14

6ModelEvaluation16

7ExecutiveSummary17

8LiteratureReferences19

9AdditionalLists20

1Introduction

Aftergettingtheproblem,wesearchthefollowingsitesforrelevantinformation.

http:

//en.wikipedia.org/wiki/Peter_Sutcliffe

PeterWilliamSutcliffe（born2June1946inBingley,WestRidingofYorkshire）isanEnglishserialkillerwhowasdubbedTheYorkshireRipper.Sutcliffecommittedhisfirstdocumentedassaultonthenightof5July1975inKeighley.On2January1981,Sutcliffewasstoppedbythepolicewith24yearoldprostituteOliviaReiversinthedrivewayofLightTradesHouse,MelbourneAvenue,Broomhill,Sheffield,SouthYorkshire.ApolicecheckrevealedthecarwasfittedwithfalsenumberplatesandSutcliffewasarrestedforthisoffenceandtransferredtoDewsburyPoliceStation,Sutcliffewasconvictedin1981ofmurdering13womenandattackingseveralothers.Detailsareasfollows:

MurderType/Practices-SerialKiller/Sadism.Method/WeaponsUsed-Stabbing,Strangulation,Bashing/Ball-PeinHammer,Knives,ClawHammer,Hacksaw,Screwdrivers,Rope.

Organization-Disorganized

Mobility-Stable..

VictimVicinity-Yorkshire,NorthernEngland.（Leeds,Bradford,Huddersfield,Manchester）

MurderTimeSpan-July1975-January1981.

VictimType-Prostitutes.

Victims-WilmaMcCann（Died30Oct1975）,EmilyJackson（Died20Jan1976）,IreneRichardson（Died5Feb1977）,PatriciaAtkinson（Died23Apr1977）,JayneMacDonald（Died26Jun1977）,JeanJordan（Died1Oct1977）,YvonnePearson（Died21Jan1978）,HelenRytka（Died31Jan1978）,VeraMillward（Died16May1978）,JosephineWhitaker（Died4Apr1979）,BarbaraLeach（Died2Sep1979）,MargueriteWalls（Died18Aug1980）,JacquelineHill（Died17Nov1980）.

Therearealotofmurderslikethis,whichleadtoplentyofvictimseveryday.Wehavedevelopedanumberofwaystopredictthemurderbasedontheseproblemsinordertomaketheinnocentpeopleawayfromthesuffering.

1.1Comprehensiontotheproblem

Thegivenproblemrequiresustosolveserialcrimes.Ononehand,wehavetopredictthepossiblelocationsinthenextstagesbasedonthetimeandlocationsofthepastcrimescenes.Ontheotherhand,estimatingthereliabilityofourpredictioniswhatwehavetodo.Inaddition,thegivenproblemalsoasksustoaccomplishadditionaltwo-pageexecutivesummarywhichwillbereadbythechiefofpolice.Aftertheaboveanalysisoftheproblem'srequirements,weunderstandthathowtodeterminethelocationsandenvironmentofthepastcrimescenesisthekeyfactortosolvetheproblem.Themodelwehaveestablishedshouldfindawaytosolvethisproblem.Onlyinthiswaycanwemakethebestprediction.

1.2Thegoalofthemodel

Thegoalofourmodelistopredictthepossiblelocationsofthecrime.Inordertoachievethisgoal,wetakethefollowingsteps.

Step1:

Analyzingthelocationsandenvironmentsofthepastcrimescenes.

Step2:

Makinguseoftwodifferentschemestogenerateageographicalprofile.

Step3:

Combiningtheresultsoftheschemesandgenerateausefulprediction.

Step4:

Estimatethereliabilityofourpredictionandaccomplishthesummary.

1.3Theanalysisoftheproblem

First,wemarkthelocationsofmurdersandattacksonthemapandfindthemurdersandtheirattacktime.Then,usinglongitudeandlatitudetorepresentthesepoints,andregardthemasthecoordinatesofthesepoints.What'smore,findingoutthe“centerofmass”iswhatwehavetodo.Wealsohavetousetwodifferentmethodstoformageographicalprofile.Finally,theprobabilityofnextmurderingineachregioncanbepredicted.Inadditiontothat,wehavetotestandverifyourmodel.Thus,theproblemanalysisiscompleted.

2NotationandDefinitions

Notation

meaning

Xi

Thepastcrimepoints

Yi

Thepredictedpoints

d（x,y）

Thedistancebetweenxandy

f（d）

Thedistancedecayfunction

S（y）

Thehitscorefunction

Lon

Longitude

Lat

Latitude

Won

Thelatitude'sactualdistance

Wat

Thelongitude'sactualdistance

Δon

Thelongitudedifference

Δat

Thelatitudedifference

Theexpectedvalue

Zi

Thedividedarea

m

The"centerofmass"point

3SimplifyingAssumptions

1.Duringashortperiod,thecriminal'smentalfeaturedoesn'talter.

2.Thecriminaldoesn'tsufferfromtheunexpectedthings,suchastheaccident,theillnessandsoon.

3.Naturedisastersdon'tappearinthisregion.

4.Thecriminalhasastableresidence.

4Themodel

4.1MarktheLocationsofcrime

Inordertoeasilymakeaforecast,wefindoutthedetailedlocationsandtimeofthemurdersandtheattacks.Thenwemarkthemoutonthemap.

Sutcliffeisconvictedofmurderingthefollowing13victims:

Table.1locationsandtimeofthecrime

Date

Nameofvictim

Ageatdeath

Bodyfound

Locationonmap

30October1975

WilmaMcCann

28

PrincePhillipPlayingFields,Leeds

1

20January1976

EmilyJackson

42

ManorStreet,Sheepscar,Leedsat

2

5February1977

IreneRichardson

28

RoundhayPark,Leedsat

3

23April1977

PatriciaAtkinson

32

Flat3,9OakAvenue,Bradford

4

26June1977

JayneMacDonald

16

Adventureplayground,ReginaldStreet,Leeds

5

1October1977

JeanJordan

20

AllotmentsnexttoSouthernCemetery,Manchester

6

21January1978

YvonnePearson

21

UnderadiscardedsofaonwastegroundoffArthing