2008国际生数学建模一等奖历年优秀

20082008LessResources, Key Choosingoutputmetricsformeasuringhealthcare GoalsofHealthCare Characteristicsofagoodhealthcare Outputmetricsformeasuringhealthcare Determiningtheweightofthemetricsanddata Weightsfromstatistical Data InputandOutputofHealthCare Aspectsof Aspectsof EvaluationSystemI:AbsoluteEffectivenessof Twoapproachesfor ApproachA:WeightedAverageEvaluationBased ApproachB:FuzzyComprehensiveEvaluationBasedModel ApplyingtheEvaluationofAbsoluteEffectiveness EvaluationsystemII:RelativeEffectivenessof Onlyoutputdoesn’t Constructingthe ApplyingtheEvaluationofRelativeEffectiveness EAEVSERE:whichis USAVS USAVS LessResources, MultipleLogisticRegression Outputasfunctionof Constructingthe Estimationof 10.1.5Howthesixmetricsinfluence TakingUSAinto AllocationCoefficient Scenario1:Lessexpendituretoachievethesame Objective 10.3.2Constraints 2/Optimizationmodel Solutionsofthe eswiththesame Optimizationmodel Solutionstothe Strengthsand 3/3/20082008LessResources, Inthispaper,weregardthehealthcaresystem(HCS)asasystemwithinputandoutput,representingtotalexpenditureonhealthanditsgoalattainmentrespectively.Ourgoalistominimizethetotalexpenditureonhealthtoarchivethesameorizetheattainmentundergivenexpenditure.First,fiveoutputmetricsandsixinputmetricsarespecified.Outputmetricsareoveralllevelofhealth,distributionofhealthinthepopulation,etc.Inputmetricsarephysiciandensityper1000population,privateprepaidplansas%privateexpenditureonhealth,etc.Second,toevaluatetheeffectivenessofHCS,twoevaluationsystemsareemployedinthisEvaluationofAbsoluteThisevaluationsystemonlydealswiththeoutputofHCS，andwedefineAbsoluteTotalScore(ATS)tofytheeffectiveness.Duringtheevaluationprocess,weightedaveragesumofthefiveoutputmetricsisdefinedasATS,andthefuzzytheoryisalsoemployedtohelpassessHCS.EvaluationofRelativeThisevaluationsystemdealswiththeoutputaswellasitsinput,andalsowedefineRelativeTotalScore(RTS)tofytheeffectiveness.ThemeasurementtoATSisunitsofoutputproducedbyunitofinput.ApplyingthetwokindsofevaluationsystemtoevaluateHCSof34countries(USAincluded),wecanfindsomecountrieswhichrankinahigherpositioninEAEgetarelativelylowerrankinERE,suchasNorwayandUSA,indicatingthattheirHCSshouldhavebeenabletoarchivemoreundertheircurrentresources.Therefore,takingUSAintoconsideration,wetrytoexplorehowtheinputinfluencestheoutputandarchivethegoal:lessinput,moreoutput.Thenthreemodelsareconstructedtoourgoal:MultipleLogisticWemodeltheoutputasfunctionofinputbythelogisticequation.Inmoredetains,wemodelATS(output)asthefunctionoftotalexpenditureonhealthsystem.Bycurvefitting,weestimatetheparametersinlogisticequation,andstatisticaltestpresentsusasatisfactoryresult.LinearOptimizationModelonminimizingthetotalexpenditureonWetrytominimizethetotalexpenditureandatthesametimearchivethesame,thatistogetaATSof0.8116.Weemploysoftwaretosolvethemodel,andbytheysisoftheresults.Wecutitto2023.2billiondollars,comparedtotheoriginaldata2109.8billiondollars.LinearOptimizationModel izingthe Wetryto izetheattainment(absolutetotalscore)underthesametotalexpenditurein2007.AndweoptimizetheATSto0.8823,comparedtotheoriginaldata0.8116.Finally,wediscussstrengthsandweaknessesofourmodelsandmakenecessary mendationstothe4/4/20082008Todayandeveryday,thelivesofvastnumbersofpeoplelieinthehandsofhealthsystems.Fromthesafedeliveryofahealthybabytothecarewithdignityofthefrailelderly,healthsystemshaveavitalandcontinuingresponsibilitytopeoplethroughoutthelifespan.Theyarecrucialtothehealthydevelopmentofindividuals,familiesandsocietieseverywhere.Duetotheirreplaceablerolethatthehealthcaresystemsplayinresidents’life,betterhealthcaresystemisneeded.“Improvingperformance”istherefore However,nowadayshealthcaresystemsinmanycountriesdonotexhibitenougheffectivenessinguaranteeingresidents’goodhealthandalonglifeexpectancy.Insomecountries, ernmentinvestslargeamountofmoneyonthehealthcaresystems,however,theydidn’tarchivewhattheyshouldhavebeentoarchive.WetrytoexploreanoptimizedsysteminthisKeyHealthCareSystemHealthCareSystemissuchasystemthathasitsinputandoutput,representingtotalexpenditureonhealthanditsgoalattainmentrespectively.EvaluationofAbsoluteEffectivenessofHealthCareSystemItisakindofevaluationsystemthatonlyconsidersthe esofthehealthcaresystem,sayingnothingtodowiththeinput(resources),anptsthe esasmeasurementtoEvaluationofRelativeEffectivenessofHealthCareSystemItisakindofevaluationsystemthatconsidersthe esofthehealthcaresystemaswellitsinputs,anptsunitsofoutputproducedbyunitofinputasmeasurementtoAbsoluteTotalScoreOverallscorefortheevaluationofabsoluteeffectivenessofhealthcareRelativeTotalScoreOverallscorefortheevaluationofrelativeeffectivenessofhealthcareInputMetricsMetricsthatarespecifiedtoassessinputofOutputMetricsMetricsthatarespecifiedtoassessoutputofChoosingoutputmetricsformeasuringhealthcare5/5/20082008Table1.NotationforgoalsandGoalsofHealthFairnessinFinance

MetricsforOveralllevelofhealthDistributionofhealthinthepopulationOveralllevelofresponsivenessDistributionofresponsivenessDistributionoffinancialcontribution

GoalsofHealthCareAccordingtotheWorldHealthReportin2000,theWHOpointedoutthethreegoalofhealthcaresystem,eachgoalwithdifferentpriority[WHO2000].BetterBetterhealthisunquestionablytheprimarygoalofahealthsystem,withthehighestFairnessinfinancialFairnessinfinancialcontributionisthesecondgoal,witharelativelylowerprioritytoResponsivenesstopeople’sexpectationsinregardtonon-healthmattersreflectsimportanceofrespectingpeople’sdignity,autonomyandtheityofandisthethirdgoal,withthelowestCharacteristicsofagoodhealthcareGoodness&&Fairness[WHOAstheWHOdefinedwhatagoodhealthcaresystemwasinitsWorldHealthReportin2000,agoodhealthcaresystemisacombinationofGoodnessandFairness.Agoodhealthsystem,aboveall,contributestogoodhealth.Butitisnotalwayssatisfactorytoprotectorimprovetheaveragehealthofthepopulation,ifatthesametimeinequalityworsensorremainshighbecausethegainaccruesdisproportionaytothosealreadyenjoyingbetterhealth.Thehealthsystemalsohastheresponsibilitytotrytoreduceinequalitiesbypreferentiallyimprovingthehealthoftheworse-off,wherevertheseinequalitiesarecausedbyconditionsamenabletointervention.Theobjectiveofgoodhealthisreallytwofold:thebestattainableaveragelevel–goodness–andthesmallestfeasibledifferencesamongindividualsandgroups–fairness.Againineitheroneofthese,withnochangeintheother,constitutesanimprovement,butthetwomaybein.OutputmetricsformeasuringhealthcareToassessahealthcaresystem,wemustmeasurethefollowingfiveoutput20082008Overalllevelof 6/Weusethemeasureofdisability-adjustedlifeexpectancy–DALEtoassesstheoveralllevelofpopulationhealth.Thismeasureconvertsthetotallifeexpectancyforapopulationtotheequivalentnumberofyearsof‘goodhealth’.DistributionofhealthintheWeusetheindexofequalityofchildsurvivaltoassessdistributionofhealthinthepopulation.Itisbasedonthedistributionofchildsurvivalacrosscountries,andtakesadvantageofthewidelyavailableandextensiveinformationoncompletebirthhistoriesinthedemographicandhealthsurveysandsmallareavitalregistrationdataonchildmortality.WHOdefineditasfollows[WHO2000]：Equalityofchildsurvival

n 3 02n

WherexisthesurvivaltimeofagivenchildOveralllevelof

isthemeansurvivaltimeacrossResponsivenessincludestwomajorRespectforpeople(includingdignity,ityandautonomyofindividualsandfamiliestodecideabouttheirownhealth);Clientorientation(includingpromptattention,accesstosocialsupportnetworksduringcare,qualityofbasicamenitiesandchoiceofprovider).Thelevelofresponsivenesswasbasedonasurveyofkeyinformantsinselectedcountries.AndWHOdefinedtheindexofOveralllevelofresponsivenessasweightedaverageofitssevencomponents:[WHO2000]

Qualityof

Choiceof

disadvantagedgroups(inmostinstancesminorities).Thekeyintensityscoresforthesefourgroupsweremulti-pliedbytheactualpercentageofthepopulationwithinthesevulnerablegroupsinacountrytocalculateasimplemeasureofresponsiveneresponsivenessinequalityrangingfrom0to1.ThetotalscorewascalculatedtakingaccountthefactthatsomeindividualsbelongtomorethanonedisadvantagedWeusea

Distributionoffinancial

7/onofThatisrespondentsinthekeysurveywereaskedtoidentifygroupswhowerewithregardto

1

i1j1xi2s.Thenumberoftimesagroupwasidentifiedaswasusedtocalculateakeyintensityscore.Fourgroupshadhighkeyscores:poorwomen,oldpeople,andgroupsor

LevelofResponsivenessDignitityPromptattentionAccesstosocialSupportnetwork200820088/8/Thefairfinancingmeasureestimatesthedegreetowhichhealthfundingisraisedaccordingtotheabilitytopayforallmembersofthepopulation.Itcapturesconcernssuchasprogressivity,andprotectionfromcatastrophichealthcosts.Fairfinancingisonlyconcernedwithdistribution.Inorderthatcompleteequalityofhouseholdcontributionsis1and0isbelowthelargestdegreeofinequalityobservedacrosscountries,WHOdefinedtheinfairnessindex.Andtheindexisoftheform:[WHO2000] Fairnessoffinancecontribution1

3

WhereHFCisthefinancialcontributionofagivenhouseholdandHFCistheaveragefinancialcontributionacrosshouseholds.DeterminingtheweightofthemetricsanddataWeightsfromstatisticalThekeyinformantsurvey,consistingof1791interviewsin35countries,yieldedscores(from0to10)oneachelementofresponsiveness,aswellasoverallscores.Asecond,Internet-basedsurveyof1006participants(halffromwithinWHO)generatedopinionsabouttherelativeimportanceoftheelements,whichwereusedtocombinetheelementscoresintoanoverallscoreinsteadofjusttakingthemeanorusingthekeyinformants’overallresponses[WorldHealthReport2000].Seefigure1and2:

FairnessinFinanceFigure1Weightsforthethreegoalsofhealthi1HFCiOveralllevelof

DistributionofhealthinthepopulationOveralllevelofDistributionofDistributionoffinancialcontributionFigure2WeightsofthefiveFigure1andfigure2illustratetheweightsofthreegoalsofhealthsystemandfivemetricsDataDataWegetourdatafromWHOStatisticalInformationSystemontheofficialwebsiteof(Anddatain‘THEWORLDHEALTHSTATISTICSREPORT’from2005to2007‘WorldHealthReport‘in2000isnowToensurecomparabilityofeffectivenessofhealthcaresystem,metricsmustbenormalizedbythefollowinggivenformulation:NormalizedData

RawDatamax(RawData) WheremaxgreatestnumberofRawDataandministheleastInputandOutputofHealthCareInthispaper,weconsiderHealthCareSystemasystemwithbothinputandoutput(seeFiveoutputmetricsandsixinputmetricsarespecifiedinthis9/9/20082008 Figure3:HowahealthcaresystemAspectsofTable2 forInputandPhysiciandensityper1000populationNursedensityper1000populationSocialSecurityexpenditureonhealth%ofernmentexpenditureonPrivateprepaidplansas%of expenditureonExternalresourcesforhealth%oftotalexpenditureonhealthOut-of-Pocketexpenditureas%ofprivateexpenditureon

OveralllevelofhealthDistributionofhealthinthepopulationOveralllevelofresponsivenessDistributionofresponsivenessDistributionoffinancial

WedefineInputVectorasasetofthefourelementsofinput,thatisInputVector{m1,m2,m3,m4,m5,Physiciandensityper1000Nursedensityper1000SocialSecurityexpenditureonhealthas%ofernmentexpenditureonPrivateprepaidplansas%ofprivateexpenditureonhealthPhysiciandensityper1000Externalresourcesforhealthas%oftotalexpenditureonOut-of-Pocketexpenditureas%ofprivateexpenditureon20082008110/29AspectsofAlso,wedefineOutputVectorasasetofthefiveelementsofOutput,thatisOutputVector{u1u2u3,u4,OveralllevelofDistributionofhealthintheOveralllevelofDistributionofDistributionoffinancialEvaluationSystemI:AbsoluteEffectivenessofInthispart,wedealwiththeevaluationofhealthcaresystembythewayof“absolute”,athatonlyconsiderstheoutputofthesystem.Thenfivetypicalmetricsthatcanwellrepresenttheesofthesystemarechosenforevaluation.Basedonthefivemetrics,twoempiricalapproachesareemployedforevaluation.Theformeroneisweightedaveragesumasacomprehensiveindicatoroftheeffectiveness,andthelatteroneisbasedonthetheoryoffuzzyWeconsiderusingoutputofthehealthsystemtoevaluatetheeffectivenessacceptableThefivemetricscanrepresentenoughinformationforevaluationofthehealthcaresystem,thusweconsideritreasonableandenoughforustousethemetrics.Wedon’tconsidertheinteractioneffectofmetricsontheThereissimplylinearrelationshipbetweenthemetricsandtheresultofevaluation,thusweightedaveragesumapproachcanreasonablyreflecthowthemetricsinfluencetheresults.AsthereisnospecificdefinitiononhowwellahealthsystemisortheextentofthusfuzzycomprehensivebasedapproachemployedhereisMostthe ollectedisreliable,neglectingitsTwoapproachesforApproachA:WeightedAverageEvaluationBasedWedefineAbsoluteTotalScore(ATS)asanindicatorthatcanbeusedtodescribehowheathsystemworks.Basedontheassumptionsabove,wecanformulatetheAbsoluteTotalScoreasfollows:11/200820085ui

Whereuirepresentstheithoutputmetric iistheweightcorrespondingtotheBycomparingtheAbsoluteTotalScoreofasystem,wecancomparesystemsamongcountries.Meanwhile,bycalculatingthevalueoffivemetrics,wecsogettherankofsystemswithrespecttoeaetric.ApproachB:FuzzyComprehensiveEvaluationBasedModelAsthereisnospecificdefinitiononhowwellahealthsystemisortheextent“effectiveness”,weemploythetheoryoffuzzymathematicstoassessCombinationofToassesstheabsoluteeffectivenessofhealthcaresystem,wefocusonthreeaspectsofhealthcaresystemthatishealth,responsivenessandfairfinancialcontribution.Healthcanbedividedintotwomajorparts,theoveralllevelofhealth;thedistributionofhealthinthepopulation.Responsivenesscanbedividedintotwomajorpart,theoveralllevelofresponsiveness;thedistributionofresponsiveness.Thefollowingfigureillustratestherelationshipsandlevelsofthosehealthcare

theoveralllevelofthedistributionofhealthintheoveralllevelofthedistributionofthedistributionofFigure5:HierarchystructureofWeusefuzzy U{u1u2 Whereu1u2 u5istheindicationforthefivebasicmetricsToincludeallthefivebasicmetrics,anddivideditintothreegroups,weU{U1U2U3}12/AbsoluteTotalScorei20082008WherefuzzysubsetU1U2U3representshealth,responsivenessandfairfinancialcontributionThenwehave U1{u1u2} U2{u3u4} ,andU3{u5}.TheweightsetforUis W(12), 123istheweightofU1,U2andU3AndtheweightsetforU1isindicatedbyW1(1,11，),wherew11w12isweightthatmetricsandu2accountforrespectively.TheweightsetforU2isindicatedbyW2(2,12，),2，12，2isweightthatmetricsu3andu4accountforDeterminemembershipdegreefor Assumethattherearencountriesoftobecomparedintermsofabsoluteeffectivenessoftheirhealthcaresystem.Wetakenormalizedformmembershipfunctionsforeaetricsothatvaluesofallthemetricsofdifferentlevelscanbeconstrainedbetween0and1.Bythemembershipdegreefunction（ui,k）

ui,jmin(ui,k1kmax(ui,k)min(ui,k1k 1k

ai,j Whereui,kindicatestheithmetricofthekthDeducingofForthefuzzysetU1,thesinglefactorjudgmenta1，a1，...a1,naByweightedaveragemethod,wecaneasilyhavematrixB1[b1,1b1,2...b1,n ,

(j1,ForthelevelU1,thesinglefactorjudgmenta3，a3，...a3,naSimilarly,wehavematrix 13/322 R1 2，2...a2,nb1,j1,ii, R2 4，2...a4,n20082008B2[b2,1b2,2...b2,n

4i

(j1, B3[b3, 3.b2.n]u[ 3,un3 ]3Finally,weperformcomprehensiveevaluationonthetoplevel.ThentheRB1b1,1b1,...bn1 b2,1b2...b33, 3,2...bn3Byweightedaveragemethod,wehaveoverallsyntheticjudge

B[b1b2...bn

(j1,ThevalueofeachelementinBcanbelookedonastheabsoluteeffectivenessofhealthcaresystemforeachcountry.SothelargerthevalueofelementinmatrixBis,moreeffectivethehealthcaresystemofthecountrytowhichthisvalueiscorrespondingis.ApplyingtheEvaluationofAbsoluteEffectivenessApplyingApproachApplyapproachAto34countries(USAincluded),andtherankisgiveninthefollowingtable.Wefocusonthethreegoalsofhealthsystem,thefiveoutputmetricsaswellastheoverallrank.Table3AbsoluteEffectivenessof34countries,rankby5outputmetrics,estimatesfor14/

b2,j2,i2.ai, 3 1..2R 2 ,B ,wherebjiib,20082008Fromtable3,wecanWithrespecttooverallhealth,JapanranksthefirstandRwandathelowest,whiletheUSAranksinthelowerlevel.WithrespecttoResponsiveness,theUSAisleadinginthe23developedcountries,whileUgandarankslast.WithrespecttoAbsoluteEffectiveness,Japanleadsfirst,whiletheUSAranks3,arelativelowerlevel.ComparisonbetweenApproachAandApproachBytheEvaluationofAbsoluteEffectiveness(EAE)method,thepolicymakersandotherrelateddepartmentcanjudgewhetherthecurrentsystemapproachesitsgoal,inotherwords,wecanidentifywhetherthesystemcansatisfyresidents’requirementofhealth.AndtheEvaluationofRelativeEffectiveness(ERE)methodcanevaluatetheefficiencyofusageofresources,whichcangiveguidanceforadjustingandimprovinghealthcaresystem.Table4HorizontalandverticalcomparisonofHCSbyEAE,estimatesfor2006andApproach Approach Approach Approach66789

Republic

Fromtable4,wecanThroughcomparingtheranksofcountriesusingthetwoapproachesrespectivelyinthesameyear,wefindthattheresultsoftwodifferentapproachestodetermineEvaluationof15/20082008AbsoluteEffectiveness(EAE)donotchangesignificantly,withranksofmostcountriesinterestedinhavingnotbigchange.Thecomparingbetweenthetwoapproachesprovescorrectnessandrationalityofeachother.Throughcomparingtheranksofcountriesusingthetwoapproachesrespectivelyinthedifferentyear,wefindtheranksofcountriesarenearlystable.ComparingtoJapanwhichhasaquitegoodhealthsystem,theUSA’seffectivenessofhealthcaresystemisnotashighasEvaluationsystemII:RelativeEffectivenessofOnlyoutputdoesn’tTheoverallindicatorofattainment,likethefivespecificmetricswhichcomposeit,isanabsolutemeasure.Itsayshowwellacountryhasdoneinreachingthedifferentgoals,butitsaysnothingabouthowthat ecomparestowhatmighthavebeenachievedwiththeresourcesavailableinthecountry.Itisachievementrelativetoresourcethatisthecriticalmeasureofahealthsystem’sForexample,ifSwedenenjoysbetterhealththanUganda–lifeexpectancyisalmostexactlytwiceaslong–thatisinlargepartbecauseitspendsexactly35timesasmuchpercapitaonitshealthsystem.ButPakistanspendsalmostpreciselythesameamountper asUganda,outofan eper thatisclosetoUganda’s,andyetithasalifeexpectancyalmost25yearshigher.Thisisthecrucialcomparison:whyarehealth esinPakistansomuchbetter,forthesameexpenditure?Anditishealthexpenditurethatmatters,notthecountry’stotal becauseonesocietymaychoosetospendlessofagiven eonhealththananother.Therefore,eachhealthsystemshouldbejudgedaccordingtotheresourcesactuallyatitsdisposal,notaccordingtootherresourceswhichinprinciplecouldhavebeendevotedtohealthbutwereusedforsomethingelse.Therefore,correspondingtotheEvaluationofAbsoluteEffectiveness,weintroduceanotherevaluationsystem,theEvaluationofRelativeEffectiveness(ERE).WecanassesstheinputofhealthcaresystembythetotalmoneyitneedstoTotalexpenditureonhealthas%ofGDPalonecanbeusedtofytheinputofhealthcaresystem.ConstructingtheTheconceptofValueEngineeringwasintroducedtodescribetherelationshipbetweencosts,functionandvalue[L·D 1943].Itdefinesvalueasfunctionofcostsandfunctioninthe 16/Similarly,wedefineRelativeTotalScore

RelativeTotalScore

AbsoluteTotal WhereRelativeTotalScoreisdefinedtoassessrelativeeffectivenessofhealthcareBycomparingtheRelativeTotalScore,wecanassesshowahealthcaresystemworksaccordingtowhatitshouldhavebeenarchived.Here,tobesimplified,weuseTotalexpenditureonhealthas%ofGDPtofytheinput.ApplyingtheEvaluationofRelativeEffectivenessTable5RelativeEffectivenessofHCS,rankedbytheRelativeEvaluationsystem,estimatesforTotalonhealthas%ofR-Totalonhealthas%ofR-12345678695Fromthetable(5),wecanfindPakistanranksthefirst,andRwandaislast.Especiallysomedevelopedones,suchasAmerica,ranksinthelowerlevel.AmericahasthelargestpercentageofGDPspentonhealthcare,whilePakistanhasonlyEAEVSERE:whichis Applythetwoevaluationsystemto34countries,wefocusonthedifferentranksfromthetwo 17/

Table6EAEVSERE,rankFromtable6,wecansee:ComparingtoranksintermsofAbsoluyEvaluationofEffectiveness,thenewranksofthesecountrieschangesignificantly.RanksofcountrieshavinglargepercentofGDPspentonhealthcaresuchasUSA,Norway,Australia,Canada,Austria,Francedecreasebymorethen15,especiallyforUSAofwhichrankdeclinesfrom7to30.Thismeansthatthesecountriesdonotmakethemostoftheirinputs.RanksofcountrieshavingsmallpercentofGDPspentonhealthcaresuchasPakistanfrom28to1.Thismeansthatthiscountrymakesthemostofitsinputs.ThismaybeagoodexamplethatthosedevelopedcountriesliketheUSAshouldlearnfrom.Butfordevelocountries,especiallythosehavingpoorhealthcaresystem,nomatterhowefficienttheirhealthcaresystemis,theystillcannotsupplygoodenoughhealthservice,simplybecausetheyhavenotenoughresourcestoinputintohealthcaresystem.USAVSFromtheaspectofEvaluationofAbsoluteEffectiveness,wecanseethatUSAranks7th,whileNorwayranks2,whilefromtheaspectofEvaluationofRelativeEffectiveness,theUSAranks30th,andNorway20th.USAVSHealthcaresystemoftheUSAisbetterthanPakistanfromtheaspectofEvaluationofAbsoluteEffectivenessobviously.However,PakistanranksfirstfromaspectofEvaluationofRelativeEffectiveness,whileAmericaranksonly30th,aquiowrank.2008200818/LessResources, MultipleLogisticRegressionOutputasfunctionofWeneedtodeterminewhethervariouschangescanimprovetheoverallqualityofacountry’shealthcaresystem.Thus,wefocusonhowtheoutputofasystemchangesduetovariationofinput.Weemploythelogisticequationtomodeltherelationshipbetweenoutputandinput[Goli1998].Bytheequation,wecanclearlyseehowinputinfluencestheoutput.Inputcanbequalifiedbyweightedaveragesumofthesixinputmetrics,andtheweightreflectshowthemetriccontributestotheinput.Outputcanbequalifiedbyweightedaveragesumofthefiveoutputmetrics(ATS),andtheweightreflectshowthemetriccontributestotheinput,Relationshipbetweeninputandoutputofhealthsystemcanbezedaslogisticequation,thatistheoutputgrowsastheinputsgrowth,andthegrowthrateisrisingatfirst,butastheoutputapproachesacertainvalue,itsgrowthratewillgraduallydecreasetozero.ConstructingtheHerewesettheAbsoluteTotalScoreastheficationofoutput,andthelogisticalequationisgivenas: ATS WhereRisthegrowthrate,KistheupperboundofoutputandMistheficationofForsimplicity,weleta=Randb=R/K,so

aATSbATS WiththeinitialconditionATS(M0)ATS0,theequationhasclosed-formATS(M aeabATSbeAccordingtotheassumptionthatinputcanbefiedbythelinearweighedaveragesumofinputmetrics,wecanfyinputas:6

(1i

iistheweightandmiistheithinput

19/RATS(1Mii20082008Thenfrom(11)and(12),wecanATS(M)

a

a(m

ThefigurebelowillustrateshowoutputchangesasinputFigure6Solutiontothelogisticequation,withoutputplottedasafunctionaEstimationofWeestimatetheparametersfor(13)bycurvefit,statisticald ollectedfromthe34countriesmentionedaboveisemployedtohelpthecurvefit,andweget(M)

1.0958e

M2998220498m1593923m25778m384232m418556m59 Also,wedostatisticaltestsforourmodel,anditpresentsusasatisfactoryresult:Residual=0.051,andConfidenceDegree=1-Residual=0.949,indicatingthatitpassesthestatistical10.1.5Howthesixmetricsinfluence Sincewehaveequation(14)and(15),wecanATSf(M)f(m1,m2,...,m6 Letus Mm nd(11)and(12)can

m abATS

220/ATSATSMM 20082008

Thenfrom(17),(18)and(19),wecan M

(aATSbATS2 Andthevalueofpartial

showhowmetricmiinfluencestheAlso,bycontrollingvariablem2,m3,m4,m5,m6,andvaryvariablem1,wecanseehowm1influencestheoutput;similarlywecangethowm2,m3,m4,m5,andm6influencestheoutputAsfigure7

Figure7HowinputmetricinfluencestheWithrespecttoprivateprepaidItisnegativelycorrelatedtoAST.Thatisastheincreaseofprivateprepaidplans,ASTdecreases.Thereasonforthisismainlyduetopeopleoftheirowncountrydonottrustthehealthcaresystem,theystorealargeamountofmoneytospendbythesickandhospitalized,whichreflectsthehealthcaresystemisfarfromperfect,solowerscores.WithrespecttotheotherfiveWecanseethattheASTincreasesastheotherfiveinputmetricsincrease,onlythattheincreasingrateisdifferent.21/29ATSATSTakingUSAintoAswehaveyzedabove,USAranks3rdbytheevaluationofabsoluteeffectivenesswhileranks7thbytheevaluationofrelativeeffectiveness.ThedifferencebetweentheranksindicatesthathealthsystemofUSAshouldhavearchivedmoreunderthecurrenttotalexpenditureonhealth.Inthispart,wetrytoexploreanoptimizedcombinationofinputmetricstominimizetheinputorizetheoutput.ThuswefocustheUSAin2007,tryingtominimizethe