数学建模美赛o奖论文.pdf

数学建模美赛o奖论文.pdf

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ForofficeuseonlyT1_T2_T3_T4_TeamControlNumber55069ProblemChosenAForofficeuseonlyF1_F2_F3_F4_2017MCM/ICMSummarySheetTheRehabilitationoftheKaribaDamRecently,theInstituteofRiskManagementofSouthAfricahasjustwarnedthattheKaribadamisindesperateneedofrehabilitation,otherwisethewholedamwouldcollapse,putting3.5millionpeopleatrisk.Aimedtolookforthebeststrategywiththethreeoptionslistedtomaintainthedam,weemployAHPmodeltofilterfactorsanddeterminetwomostinfluentialcriteria,includingpotentialcostsandbenefits.Withtheweightofeachcriterionworkedout,ourmodeldemonstratesthatoption3istheoptimalchoice.Accordingtoourchoice,wearerequiredtooffertherecommendationastothenumberandplacementofthenewdams.Regardingitasasetcoveringproblem,wedevelopamulti-objectiveoptimizationmodeltominimizethenumberofsmallerdamswhileimprovingthewaterresourcesmanagementcapacity.ApplyingTOPSISevaluationmethodtogetthedemandoftheelectricityandwater,wesolvethisproblemwithgeneticalgorithmandgetanapproximateoptimalsolutionwith12smallerdamsanddeterminethelocationofthem.Takingthestrategyformodulatingthewaterflowintoaccount,weconstructajointoperationofdamsystemtosimulatetherelationshipamongthesmallerdamswithgeneticalgorithmapproach.WedefinefourkindsofyearbasedontheKaribasclimatedataofclimate,namely,normalflowyear,lowflowyear,highflowyearanddifferentialyear.Finally,thesestatisticscouldhelpussimulatethewaterflowofeachmonthinoneyear,thenweobtainthewaterresourcesplanningandmodulatingstrategy.Thesensitivityanalysisofourmodelhaspointedoutthatsmallalterationinourconstraints（includingremovinganimportantcityofthecountriesandchangingthemeasurementoftheeconomicdevelopmentindexetc.）affectsthelocationofsomeofourdamsslightlywhilethenumberofdamsremainsthesame.Alsowefindthattheoutputcoefficientisnotanimportantfactorforjointoperationofthedamsystem,forthereasonthatthedischargeindexandthecapacityindexwouldnotchangealotwiththeoutputcoefficientchanging.Team#55069Page2of2Contents1Overview.11.1Background.11.2RestatementoftheProblem.11.3LiteratureReview.22AssumptionsandJustifications.23Notation.34ModelOverview.45ModelTheory.55.1DeterminationoftheNumberandLocationoftheDams.55.2Jointoperationofdamsystemmodel.96ModelImplementationandResults.116.1TheNumberandLocation.116.2TheStrategyofModulatingWaterFlow.147SensitivityAnalysis.157.1TheModelofDeterminationoftheNumberandLocation.157.2TheModelofModulatingWaterFlow.178Furtherdiscussion.179StrengthsandWeaknesses.189.1Strengths.189.2Weaknesses.1910Conclusion.1911TheEvaluationofThreeOptions.2011.1EstablishaHierarchicalModel.2011.2AnalysisandResults.21Team#55069Page1of281Overview1.1BackgroundAZambeziRiverAuthorityconferencewasheldinMarch2014,engineerswarnedthatthefoundationsofthedamhadweakenedandtherewasapossibilityofdamfailureunlessrepairsweremade.On3October2014theBBCreportedthat“TheKaribaDamisinadangerousstate.Openedin1959,itwasbuiltonaseeminglysolidbedofbasalt.However,inthepast50years,thetorrentsfromthespillwayhaveerodedthatbedrock,carvingavastcraterthathasundercutthedamsfoundations,engineersarenowwarningthatwithouturgentrepairs,thewholedamwillcollapse.Ifthathappened,atsunami-likewallofwaterwouldripthroughtheZambezivalley,reachingtheMozambiqueborderwithineighthours.ThetorrentwouldoverwhelmMozambiquesCahoraBassaDamandknockout40%ofsouthernAfricashydroelectriccapacity.Alongwiththedevastationofwildlifeinthevalley,theZambeziRiverAuthorityestimatesthattheliveof3.5millionpeopleareatrisk.”OnFebruary2015,EngineershavestartedonaR3.3bnrescuemarathontopreventthe“catastrophicfailure”oftheKaribaDam.AccordingtoaWorldBankspecialreportonthebeleagueredstructure-oneofthebiggestman-madedamsintheworld-apotentialwallcollapsethreatensthelivesofabout3-millionpeoplelivingontheZambeziRiverfloodplainbetweenthehydroschemeontheZambia-ZimbabweborderandtheMozambiquecoast.11.2RestatementoftheProblemWearerequiredtoprovideanoverviewofpotentialcostsandbenefitswiththethreeoptionsalreadylisted.ThenweneedtoestablishamodeltodeterminethenumberandplacementofthenewdamswhenremovingtheKaribadamalongtheZambeziRiver.Thesameoverallwatermanagementcapabilitiesarealsoneeded.Inaddition,weshouldconsideremergencywaterflowsituationsandrestrictionsregardingthelocationsandtime,sothatwecouldgiveoutthestrategyformodulatingthewaterflowthroughournewmultipledamsystem.Inordertosolvethoseproblems,wewillproceedasfollows:

Buildamodeltodeterminethenumberandlocationofthemultipledams.Givethecorrespondingstrategyofmodulatingwaterflowindifferentconditions.Inourmodel,wefirstestablishamulti-objectivemodelandusegeneticalgorithmdeterminethenumberandlocationofthemultipledams.Therearetwogoalsimprovingthewaterresourcesmanagementcapacityandreducingthecost.Besides,weaddsomeconstraintssuchaswaterbalance,waterlevel,safetyandwaterprotection.Wechoosetwentysuitabledamsitesandemploythegeneticalgorithmtosolvetheoptimalproblemtodeterminethenumberandthelocation.Afterdeterminingthenumberandlocationofthedams,weconstructourjointoperationofdamTeam#55069Page2of28systemmodelandemploythegeneticalgorithmtosolvetheproblembasedonthethoughtofdynamicprogramming.AccordingtotheKaribasclimatedataforabout30years,weabstractnormalflowyear,lowflowyear,highflowyearanddifferentialyear.Weusethemtoworkoutthewaterresourcesplanningandschedulingstrategy.Theconstructionofthedischargeindexandthecapacityindexbenefitsananalysisandevaluationforjointoperationofthedamsystemsperformanceindifferentmonthandyear.1.3LiteratureReviewDatingbackto2004,theUnitedStatesremoved72damsintotal,whichcreatedahistoricalrecord.Therefore,itishightimeforustofocusontheconstructionofdamsconcerningtheirnumberandplacement.PlentyofresearchershavealreadymadeanumberofnotablepaperstoaddresstheseproblemsAlferWeber（1909）firstproposedaframeworkforlocationproblem,whichisanallocationquestionwithrespecttospaceresource.Amongthethreeclassicallocationmodel,setcoveringproblemisasignificantbranchofsitingissues.Theyexploredamulti-objectivelocationmodeltotackleproblemswithsitingoptimalpoints.Intheirmodel,maximizingcoveragerateinordertosatisfyeveryplacesneedisthetargetfunction,theconcentricpointandthecapacityrestrictionsareconstraintconditions.Thus,theycouldconverttheoptimizationproblemtothemixedintegerlinearprogrammingquestion.Afterthesetcoveringmodelwasestablished,wecanoptimizeourchoiceofsitingthedams.Thenseveralscientistsweredevotedtobuildinganoptimaloperationmodeltoprovideareasonablebalancebetweensafetyandbalance.Theybegintofigureouthowthemultipledamsystemwouldbenefitoraffecteachotherwithinitssystem.Masse（1940）firstillustratedtheconceptofit;theirmaincomputingmethodwastooptimizewatermodulatingstrategyduringdispatchingperiod.Furtherstudiesarecarriedoutaboutdifferentmethodstoinvestigatetheoptimaloperationmodel,includingdynamicprogrammingalgorithmandneuralnetworkalgorithmbasedonimprovedcomputertechnique.Also,thereismuchtheoreticalanalysisaboutlocationproblemsince1990.JohnCurrentandMortonOKelly（1992）suggestedusingamodifiedversionofthesetcoveringlocationmodel,whichstilldidnttaketherealityintoaccount.2AssumptionsandJustificationsTosimplifyourproblems,wemakethefollowingbasicassumptions,eachofwhichisproperlyjustified.ThedamsystemisbuiltdownstreaminthevalleyoftheKaribaDam.Becauseitsmoreconvenienttobuildandalsowithlesscost,whichcanbeeasilyimplemented.Thecostofthedamismostlythesame.Owingtothefactthatthelengthofthecanyonisnotlarge（24km）,geologicalconditionsandclimateconditionsaremostlythesame.Team#55069Page3of28Eachdamswatersupplyismostlythesame.Takingintoaccountofsafeoperationoftheentiremulti-damsystem,weshouldmaketheburdenofeachdamtobethesameasmuchaspossible.Thewaterqualityofthedamsystemistheaverageofthewaterqualitybetweenthetworeservoirs.Theriverisflowing,sothewaterqualityismostlysimilar.Waterofthedamsdownstreamonlycomesfromdamsupstreamandnaturalprecipitation.AccordingtoGoogleMaps,therearenotributariesnearthecanyon.Alsowiththeprincipleofconservationofwater,theformulashouldbemaintained.3NotationAbbreviationDescription12,mYyyyThesetofcities12=,nXxxxThesetofdams（,）indyXThedistancefromthethicitytothenearestsmallerdamiEleTheelectricitydemandofthicityiWatThewaterdemandofthicity（,）WtiThedischargeamountofthethidamattheendofperiodt,（,）ZtiThetotalamountofwaterreleasedduringperiodt（,）TtiTheamountofnaturalwaterinperiodtijVThevolumeinthethjperiodofthethidamijQTheinflowinthethjperiodofthethidamTeam#55069Page4of28ijqThedischargevolumeinthethjperiodofthethidamijQJTherunoffvolumebetweenthethidamandthe

（1）thidamijSTheactualwatersupplythidaminthethjperiodijDTheplannedwatersupplyofthethidaminthethjperiodijHThehydraulicheadofthidaminthethj