六西格瑪046sigma工具

DOE試驗(yàn)設(shè)計(jì)因素試驗(yàn)

FACTORIALEXPERIMENTS因素試驗(yàn)-

導(dǎo)言INTRODUCTION因素試驗(yàn)-由重復(fù)嘗試和錯(cuò)誤走向系統(tǒng)試驗(yàn)togofromtrialanderrortosystematicexperimentation因素試驗(yàn)擁有許多因素(x1,x2,x3,...xn)的試驗(yàn)，這些因素對(duì)因變量y產(chǎn)生如下影響：experimentswithmanyfactors(x1,x2,x3,...xn),influencingononedependentvariable,y,

y=f(x1,x2,x3,...xn)作用：確定關(guān)鍵因素及各因素之間的(潛在)交互作用。在試驗(yàn)中，所有的因素都發(fā)生系統(tǒng)變化。madetoestablishsignificantfactorsandthe(potential)interactionbetweenfactors.Intheexperiment,allfactorsarechanged,systematically.如果待觀察因素不是同時(shí)、在同一試驗(yàn)中發(fā)生變化，我們永遠(yuǎn)無法找出各因素之間的交互作用。Wecanneverfindinteractionbetweenfactorsinaprocess,unlessthefactorsarechangedsimultaneously,inthesameexperiment.我們永遠(yuǎn)無法從常量中找出重要性所在，因此所有的因素都有必要發(fā)生改變。Wecanneverfindtheimportanceofaconstant,thereforeallfactorshavetobechanged.完全因素試驗(yàn):對(duì)所有因素的變化進(jìn)行一切可能聯(lián)結(jié)的實(shí)驗(yàn)。不完全因素實(shí)驗(yàn)：僅對(duì)完全因素實(shí)驗(yàn)中的某一個(gè)特殊部分進(jìn)行的實(shí)驗(yàn)。Fullfactorialexperiment:Anexperimentwithrunsforallpossiblecombinationsoffactorsandlevels.Fractionalfactorialexperiments:Anexperimentwithonlya(particular)fractionoftherunsrequiredforthefullfactorialexperiment.試驗(yàn)設(shè)計(jì)=DOE(Designofexperiments)DOE=試驗(yàn)過程的計(jì)劃階段+因素試驗(yàn)。factorialexperiments+theproceedingplanningphase

試驗(yàn)設(shè)計(jì)(DOE)因素試驗(yàn)2Factorial1因素試驗(yàn)1Factorial1因素試驗(yàn)的起源(Genesis)：1928，R.A.Fisher先生在英國(guó)Rothamsted實(shí)驗(yàn)室首創(chuàng)，用于觀察、檢測(cè)都柏林釀酒廠大麥增產(chǎn)的各因素。inventedbySirR.A.Fisher,RothamstedLaboratoryUKin1928,examiningfactorsleadingtoimprovebarleygrowthfortheDublinBrewery.因素試驗(yàn)從那時(shí)起開始在農(nóng)業(yè)和生物科學(xué)領(lǐng)域得到推廣。

Factorialexperimentswerethenintroducedtotheagriculturalandbiologicalsciences.

棉花和毛織品工業(yè)/CottonandWoolIndustries

化學(xué)工業(yè)/ChemicalIndustries

日本/Japan

美國(guó)/US因素試驗(yàn)-介紹(Introduction)從以下的研究中找出日光、雨水以及它們之間的相互作用對(duì)四株相同植物生長(zhǎng)情況(高度：cm)的影響。Findoutthecontributionsofsun,rain,andtheinteractionfromthefollowingstudyregardingthegrowth(incm)offouridenticalplants.植物1，日光少，雨水少，生長(zhǎng)=2cmplant1:littlesun,littlerain,growth=2cm植物2，日光多，雨水少，生長(zhǎng)=4cmplant2:muchsun,littlerain,growth=4cm植物3，日光少，雨水多，生長(zhǎng)=6cmplant3:littlesun,muchrain,growth=6cm植物4，日光多，雨水多，生長(zhǎng)=12cmplant4:muchsun,muchrain,growth=12cm.

日光

低

高雨水

低

24高612植物1，日光少，雨水少，生長(zhǎng)=2cm植物2，日光多，雨水少，生長(zhǎng)=4

cm植物3，日光少，雨水多，生長(zhǎng)=6cm植物4，日光多，雨水多，生長(zhǎng)=12cm61224-

日光

+-

雨水

+日光 雨水

產(chǎn)量-1 -1 21 -1 4-1 1 61 112日光雨水日光*雨水產(chǎn)量-1 -1 121 -1 -14-1 1 -161 1112交互作用：一個(gè)因素的作用取決于其它因素的設(shè)置Interaction:

Theeffectofonefactordependsonthesettingoftheotherfactor(s).試驗(yàn)的最簡(jiǎn)單形式就是在其它因素保持不變的情況下改變某一因素。這種方式使用的較為頻繁而且看起來也較為合理；但是，這種方式忽略了一個(gè)非常重要的影響因素-交互作用。Thesimplestformofexperimentationischangingonefactoratatimewhileallotherfactorsarekeptconstant.Thisapproachisfrequentlyusedandseemsreasonable;however,averyimportanteffectisnotinvestigated–theinteraction.因素試驗(yàn)意味著同時(shí)改變多個(gè)因素，通過這種方法發(fā)現(xiàn)交互作用的影響。Factorialexperimentsmeansimultaneouschangingofseveralfactorssothatinteractioneffectsarefound.

系數(shù)=對(duì)比的結(jié)果/n用對(duì)比的方法為過程建立數(shù)學(xué)模型Thecontrastwayforestablishingamathematicalmodeloftheprocess產(chǎn)出

=6+2*日光+3*雨水+1*日光*雨水日光

雨水日光*雨水產(chǎn)出-2-2224-4-44-66-6612121212對(duì)比812424系數(shù)2316產(chǎn)出=6+2*日光+3*雨水+1*日光*雨水?dāng)?shù)學(xué)模型，對(duì)比的方法Amathematicalmodel,thecontrastway日光

雨水日光*雨水產(chǎn)出產(chǎn)出2-2-22244-4-4416-66-663612121212144對(duì)比812424200對(duì)比26414416576SS(對(duì)比2/n)16364144因素作用46212系數(shù)2316主要作用

Maineffects因素的主要作用在于當(dāng)它由低(-1)到高(+1)變化時(shí)對(duì)產(chǎn)出發(fā)生的影響。Themaineffectofafactoristheinfluencethatfactorhasonyield,whengoingfromlowlevel(-1)tohighlevel(+1).日光的主要作用是：雨水的主要作用是：Themaineffectofsunis:Themaineffectofrainis:8-4=49-3=6這意味著當(dāng)日光從低到高變化時(shí)，產(chǎn)出增加了4

cm。whichmeansthattheyieldincreasesby4cmwhenthesungoesfromlowtohigh61224-日光

SUN+-雨水

RAIN+8461224-日光

SUN+-雨水

RAIN+93交互作用效果

Interactioneffect交互作用效果是指當(dāng)因素間的交互作用產(chǎn)生由低到高的變化時(shí)對(duì)產(chǎn)出發(fā)生的影響；所有其它因素和交互作用都處于中心位置(在低級(jí)和高級(jí)之間)。Theinteractioneffectistheinfluenceaninteractionhasonyield,whengoingfromlowtohigh;allotherfactorsandinteractionsareincenter(betweenlowandhighlevel).當(dāng)雨水少時(shí)：日光的作用為：Whenrainislow:theeffectofsunis:4-2=2當(dāng)雨水多時(shí)：日光的作用為：Whenrainishigh:theeffectofsunis:12-6=6因?yàn)槿展獾淖饔萌Q于雨水的多少，所以我們說日光和雨水之間有交互作用。Sincetheeffectofsundependsonthelevelofrain,wesaythatthereisinteractionbetweensunandrain.交互作用效果為：(6-2)/2=2Theeffectofinteractionis

-halfthedifferencebetween6and2.即，日光少時(shí)雨水作用為：6-2=4Alternatively:effectofrainwhensunislow：6-2=4日光多時(shí)雨水作用為：

12-4=8.effectofrainwhensunishigh:12-4=8.這樣，交互作用效果為：

(8-4)/2=2Thus,theeffectofinteractionis(8-4)/2=261224-

SUN+-

RAIN+結(jié)果圖-交互作用效果圖Responseplots-interactioneffectplot61224-

SUN+-

RAIN+日光

sun+日光

sun-雨水rain+雨水

rain

--

日光

sun

+-

雨水

rain

+結(jié)果圖-主要作用圖Responseplot-maineffectplots-sun+-rain+61224-SUN+-RAIN+8461224-SUN+-RAIN+93日光雨水日光*雨水產(chǎn)出產(chǎn)出2-2-22244-4-4416-66-663612121212144對(duì)比812424200對(duì)比26414416576SS(對(duì)比2/n)16364144因素作用46212系數(shù)2316Anova-方差分析

analysisofvarianceS

產(chǎn)出2改正因素CORRECTIONFACTORSSTOTAL=SYIELD2-CORRECTIONFACTORSSRESIDUAL=SSTOTAL-(SSSUN+SSRAIN+SSSUN*RAIN)

偏差分析ANOVA資源

SourcedfSSMeanSSFcalcP雨水

Rain13636****日光

Sun11616****日光*雨水

Sun*Rain144殘差

Residual000總計(jì)

Total356“相關(guān)作用”因素餅圖

Pie-chartforfactor“relativeeffect”?aq1stopangleQ0startanglerFrBdR

FDmasscenterMandmarethemassesofthearmandball,respectively.rGisthedistancefrom0tomasscenterofarm.I0isthemomentofinertiaofthemovementarm/ballrelativetopivotpoint0.M和m分別表示發(fā)射臂和發(fā)射球的質(zhì)量。rG

表示從0點(diǎn)到質(zhì)心的距離。I0

表示發(fā)射臂/球與支點(diǎn)0點(diǎn)之間運(yùn)動(dòng)的慣性力矩。該因素試驗(yàn)包含三個(gè)因素：掛鉤設(shè)置(掛鉤)杯口位置(杯口)發(fā)射角度(角度)工藝

PROCESS發(fā)射球

Shootingball輸出Output

掛鉤

Hook杯口

Cup角度

Angle

工藝

PROCESS發(fā)射球

Shootingball輸入

Input

輸出

Output

AFactorialexperimentwiththreefactors:HOOKSETTING(HOOK)CUPPOSITION(CUP)STARTINGANGLEOFARM(ANGLE)掛鉤

HOOK杯口

CUP角度

ANGLE產(chǎn)出

YIELD-1-1-16881-1-11'883-11-153311-12'199-1-111'1531-113'246-1111'3431114'966用對(duì)比的方法為過程建立數(shù)學(xué)模型Thecontrastwayforestablishingamathematicalmodeloftheprocess工藝

PROCESS發(fā)射球

Shootingball輸出Output

掛鉤

Hook杯口

Cup角度

Angle

用對(duì)比的方法為過程建立數(shù)學(xué)模型ThecontrastwayforestablishingamathematicalmodeloftheprocessYIELD=2001+1072*HOOK+259*CUP+676*ANGLE++250*H*C+357*H*A+219*C*A+132*H*C*A用對(duì)比的方法為過程建立數(shù)學(xué)模型Thecontrastwayforestablishingamathematicalmodeloftheprocess方差分析數(shù)據(jù)的完整對(duì)比表

ThecompletecontrasttableforinputtoAnovaYIELD=2001+1072*HOOK+259*CUP+676*ANGLE++250*H*C+357*H*A+219*C*A+132*H*C*A

-

HOOK++CUP-

315541063583938124813662041256512869212200611-ANGLE+533219968818831343496611533246

-

HOOK

++CUP--

ANGLE

+立方圖-結(jié)果圖的基礎(chǔ)

Acubeplotlaysgroundforresponseplots角度的主要作用(所有其它因素的平均值)：Maineffectofangle(allotherfactorsataverage):角度低

anglelow:角度高

anglehigh:因此，角度產(chǎn)生的主要作用為1351mm。Hence,themaineffectofangleis1351mm533+2199+1883+6884=13264966+1342+1153+32464=2677533219968818831343496611533246-HOOK++CUP--ANGLE+05001000150020002500300035000500100015002000250030003500-angle+-cup+0500100015002000250030003500-hook+結(jié)果圖：主要作用圖Responseplots:maineffectplots267713261351

mm010002000300040005000-cup+hook+hook-

315541063583938124813662041256512869212200611-cup+-angle+-hook+結(jié)果圖：交互作用

Responseplots:interactionplots010002000300040005000010002000300040005000-angle+-angle+hook+hook-cup+cup-010002000300040005000-cup+hook+hook-010002000300040005000010002000300040005000-angle+-angle+hook+hook-cup+cup-交互作用圖結(jié)論：掛鉤總是對(duì)產(chǎn)出產(chǎn)生較大的影響。掛鉤位置低可使角度對(duì)產(chǎn)出的影響減小。同樣，杯口位置低也可使降低角度對(duì)產(chǎn)出的影響。Conclusionsofinteractionplots:Hookalwayshasalargeimpactonyield.Hookinlowpositionmakesyieldlesssensitivetoangle.AlsoCupatlowpositionmakesyieldlesssensitivetoangle.(當(dāng)干擾因素的作用可以忽略不計(jì)時(shí)，該工藝可認(rèn)為是優(yōu)化工藝)(Aprocessiscalledrobustwhentheeffectofnoiseisnegligible)SSTOTAL=SYIELD2-correctionfactor(改正因素)=47‘469’453-32‘044’015= 15'425'437SSRESIDUAL=SSTOT-(SSH+SSC+SSA+SSHC+SSHA+SSCA)Anova

餅圖，以方差分析的SS為基礎(chǔ)Effectpie-chart,basedonSSofAnova對(duì)標(biāo)準(zhǔn)差運(yùn)行因素試驗(yàn)，因?yàn)槠钍俏覀兊念^號(hào)敵人Runthefactorialonstdev.Becausevariationisourenemy#1YSTDEV=22.4+8.1H+2C+7.8A+0.7HC+1HA-1.7CA-0.1HCAY標(biāo)準(zhǔn)差的方差分析

AnalysisofVarianceforYstdevSourceDFSSMSFPH1520.93520.932.2E+040.004C132.6232.621378.820.017A1484.53484.532.0E+040.004H*C13.823.82161.540.050H*A17.907.90333.790.035C*A121.9221.92926.390.021Error10.020.02Total71071.7317.24338.21426.08340.2164.58321.36210.44020.817-11HOOKCUPANGEL-11-11Y標(biāo)準(zhǔn)差的立方圖(平均數(shù)據(jù))CubePlot(datameans)forYstdevHOOKCUPANGEL-11-11-111519232731YstdevY標(biāo)準(zhǔn)差的主作用圖(平均數(shù)據(jù))MainEffectsPlot(datameans)forYstdev對(duì)標(biāo)準(zhǔn)差運(yùn)行因素試驗(yàn)，因?yàn)槠钍俏覀兊念^號(hào)敵人Runthefactorialonstdev.Becausevariationisourenemy#1對(duì)標(biāo)準(zhǔn)差運(yùn)行因素試驗(yàn)，優(yōu)化工藝(產(chǎn)品)Runthefactorialonstdev.Tomakeprocesses(products)robust“產(chǎn)品和工藝只有經(jīng)過設(shè)計(jì)才是優(yōu)化的，才不會(huì)輕易受外部變化影響。試驗(yàn)設(shè)計(jì)(DOE)正是這方面的工具?！? “Productsandprocessesshallbedesignedsothattheyarerobust(notsensitive)toexternalsourcesofvariability.DOEisthetoolforthis.” --Taguchi優(yōu)化：當(dāng)(統(tǒng)計(jì))過程/工藝的產(chǎn)出不輕易受輸入的偏差和其它干擾因素的變化而變化，該過程/工藝可認(rèn)為是優(yōu)化的。Taguchi博士認(rèn)為如果過程或工藝能夠始終與目標(biāo)值一致、不易受難控因素的影響，那么它就是優(yōu)化的過程/工藝。ROBUST:A(statistical)procedure(process)iscalledrobustwhenitisnotsensitivetovariationintheinputvariablesandtheeffectofnoiseisnegligible.Dr.Taguchisaysthatastatisticalprocedureorprocessisrobustwhenitperformsconsistentlyontargetvalueandisrelativelyinsensitivetofactorsthataredifficulttocontrol.不完全因素試驗(yàn)

FRACTIONALFACTORIALS不完全因素實(shí)驗(yàn)：僅對(duì)完全因素實(shí)驗(yàn)中的某一(特殊)部分進(jìn)行的實(shí)驗(yàn)。Afractionalfactorialisanexperimentwithonlya(particular)fractionofallrunsofthefullfactorialexperiment

不完全因素試驗(yàn)

Fractionalfactorialexperiments對(duì)于完全因素試驗(yàn)：運(yùn)行的數(shù)量=2n，其中n=因素?cái)?shù)量Forfullfactorialexperiments:thenumberofruns=2n,wheren=numberoffactors“我們手頭上有32個(gè)因素”“Wehad32factors”232=4294967296units如果我們一秒可做一個(gè)部件，那么試驗(yàn)要求71582788分鐘。Ifwecandooneunitin1second,testingrequires71582788minutes=1193046hours=49710days=7101weeks=136years不完全因素試驗(yàn)

Fractionalfactorialexperiments“我們對(duì)64個(gè)部件進(jìn)行不完全因素試驗(yàn)”

Weusedafractionalfactorialwith232-26

=64units我們對(duì)所有試驗(yàn)重復(fù)4次，試驗(yàn)共用5分鐘。Wehad4replicates,testingtook5minutes.我們想進(jìn)行4個(gè)因素的因素試驗(yàn)，但卻只能進(jìn)行8次。Wewanttorunafactorialexperimentwithfourfactors,butwecanonlyafford8runs.解決方案：進(jìn)行不完全因素試驗(yàn)。將第四個(gè)因素peg

置于前三個(gè)因素交互作用(H*C*A)的一欄中，因?yàn)槲覀兇_信該交互作用最不具重要性。Solution:runafractionalfactorialexperiment.Weputthefourthfactor,peg,onthecolumnofthethreefactorinteraction(H*C*A),becausewebelievethatthatinteractionhasthesmallestsignificance. PEG8次試驗(yàn)中的4個(gè)因素，亦被稱為24-1設(shè)計(jì)：

Fourfactorsin8runs,asocalled24-1design:我們進(jìn)行這樣的試驗(yàn)：Weruntheexperiment:H*P=C*AH*A=C*P這些是“混合”因素

theyare“confounded”H*C=A*P這當(dāng)然是因?yàn)閷?duì)于8次試驗(yàn)只有7個(gè)獨(dú)立的欄，所以我們只能掌握7個(gè)因素，即：掛鉤、杯口、角度、固定栓、(H*C+A*P)，(H*P+C*P)，以及(H*P+C*A)。(因此，我們無法僅僅依靠3個(gè)因素和4個(gè)因素交互作用欄使上面的矩陣完整化。)thereason--ofcourse--isthatthereareonly7uniquecolumnsfor8runs,sowecanonlyhave7uniquefactors;theyare:HOOK,CUP,ANGLE,PEG,(H*C+A*P),(H*P+C*P),and(H*P+C*A).(Therefore,wedidn’tmakethematrixabovecompletewiththethreefactorandthefourfactorinteractioncolumns.)YIELD=1599+883*HOOK+292*CUP+634*ANGLE+535*PEG+292*(H*C+A*P)+324*(H*A+C*P)+408*(H*P+C*A)Minitab這樣表達(dá):AliasStructure I+HOOK*CUP*ANGLE*PEG HOOK+CUP*ANGLE*PEG CUP+HOOK*ANGLE*PEG ANGLE+HOOK*CUP*PEG PEG+HOOK*CUP*ANGLE HOOK*CUP+ANGLE*PEG HOOK*ANGLE+CUP*PEG HOOK*PEG+CUP*ANGLE主要作用，交互作用

Maineffect,interactionSourceDFSSMSFPhook1623045062304506.540.083cup16797786797780.710.460angle1321057832105783.370.164peg1228552222855222.400.219Error32857730952577Total715264058作用方差分析餅圖Anovaochpie-chartforeffects殘差掛鉤固定栓角度杯口包含5個(gè)因素的不完全因素試驗(yàn)-共進(jìn)行8次Afractionalfactorialwith5factorsin8runs將終止角度，即STOP，置于某一雙因素交互作用欄處。該試驗(yàn)將會(huì)出現(xiàn)多個(gè)混合因素，其根本原因就是在8次試驗(yàn)中僅可能存在7個(gè)獨(dú)立的欄。Stopangle,calledSTOPisputonthecolumnofoneofthetwofactorinteraction.Therewillbemanyconfoundedfactors,thebasicreasonbeingthatthereareonly7uniquecolumnsif8runsareallowed包含5個(gè)因素的不完全因素試驗(yàn)-共進(jìn)行8次Afractionalfactorialwith5factorsin8runs將終止角度，即STOP，置于某一雙因素交互作用欄處。該試驗(yàn)將會(huì)出現(xiàn)多個(gè)混合因素，其根本原因就是在8次試驗(yàn)中僅可能存在7個(gè)獨(dú)立的欄。Stopangle,calledSTOPisputonthecolumnofoneofthetwofactorinteraction.Therewillbemanyconfoundedfactors,thebasicreasonbeingthatthereareonly7uniquecolumnsif8runsareallowed

STOP包含5個(gè)因素的不完全因素試驗(yàn)-共進(jìn)行8次(續(xù))Afractionalfactorialwith5factorsin8runs,cont’dMaineffectsarenotaliasedwithothermaineffects.However,atleastonemaineffectisaliasedwithasecondorderinteraction.Called“resolutionIII”.AliasStructureI+HOOK*CUP*STOP+ANGLE*PEG*STOP+HOOK*CUP*ANGLE*PEGHOOK+CUP*STOP+CUP*ANGLE*PEG+HOOK*ANGLE*PEG*STOPCUP+HOOK*STOP+HOOK*ANGLE*PEG+CUP*ANGLE*PEG*STOPANGLE+PEG*STOP+HOOK*CUP*PEG+HOOK*CUP*ANGLE*STOPPEG+ANGLE*STOP+HOOK*CUP*ANGLE+HOOK*CUP*PEG*STOPSTOP+HOOK*CUP+ANGLE*PEG+HOOK*CUP*ANGLE*PEG*STOPHOOK*ANGLE+CUP*PEG+HOOK*PEG*STOP+CUP*ANGLE*STOPHOOK*PEG+CUP*ANGLE+HOOK*ANGLE*STOP+CUP*PEG*STOP包含5個(gè)因素的不完全因素試驗(yàn)-共進(jìn)行8次(續(xù))Afractionalfactorialwith5factorsin8runs,cont’dTermEffectCoefConstant204.050HOOK83.95041.975CUP46.90023.450ANGLE138.15069.075PEG21.40010.700STOP7.0503.525HOOK*ANGLE27.30013.650HOOK*PEG10.5