RESPONSE SURFACE METHODOLOGY

RESPONSESURFACEMETHODOLOGY•RSMCANBEDEFINESASASTATISTICALMETHODTHATUSESQUANTITATIVEDATAFROMAPPROPRIATEEXPERIMENTSTODETERMINE&SIMULTANEOUSLYSOLVEMULTIVARIENTEQUATIONSFACTORSTOCONSIDER•CRITICALFACTORSAREKNOWN•REGIONOFINTEREST,WHEREFACTORLEVELSINFLUENCINGPRODUCTISKNOWN•FACTORSVARYCONTINUOUSLYTHROUGH-OUTTHEEXPERIMENTALRANGETESTED•AMATHEMATICALFUNCTIONRELATESTHEFACTORSTOTHEMEASUREDRESPONSE•THERESPONSEDEFINEDBYTHEFUNCTIONISASMOOTHCURVELIMITATIONSTORSM•LARGEVARIATIONSINTHEFACTORSCANBEMISLEADING(ERROR,BIAS,NOREPLICATION)•CRITICALFACTORSMAYNOTBECORRECTLYDEFINEDORSPECIFIED•RANGEOFLEVELSOFFACTORSTONARROWORTOWIDE--OPTIMUMCANNOTBEDEFINED•LACKOFUSEOFGOODSTATISTICALPRINCIPLES•OVER-RELIANCEONCOMPUTER--MAKESURETHERESULTSMAKEGOODSENSEUSESOFRSM•TODETERMINETHEFACTORLEVELSTHATWILLSIMULTANEOUSLYSATISFYASETOFDESIREDSPECIFICATIONS•TODETERMINETHEOPTIMUMCOMBINATIONOFFACTORSTHATYIELDADESIREDRESPONSEANDDESCRIBESTHERESPONSENEARTHEOPTIMUM•TODETERMINEHOWASPECIFICRESPONSEISAFFECTEDBYCHANGESINTHELEVELOFTHEFACTORSOVERTHESPECIFIEDLEVELSOFINTERESTUSES-CONTINUED•TOACHIEVEAQUANTITATIVEUNDERSTANDINGOFTHESYSTEMBEHAVIOROVERTHEREGIONTESTED•TOPRODUCTPRODUCTPROPERTIESTHROUGHOUTTHEREGION-EVENATFACTORCOMBINATIONSNOTACTUALLYRUN•TOFINDCONDITIONSFORPROCESSSTABILITY=INSENSITIVESPOTPROCESSMODELS•Ym=fm(x1,x2,….,xp)•Polynomialswithasmallnumberoftermsaremostdesirable•Mostprocessoutputsaresomesortofsmoothfunctionoftheinputs•Second-degreepolynomialsaregenerallyadequatePOLYNOMIALMODELS•SECONDDEGREE-ONEINDEPENDENTVARIABLEY=bo+b1x1+b11x12constantterm,+linearterm+quadraticterm•FORpFACTORS,THEREWILLBEONECONSTANTTERM,pLINEARTERMSpQUADRATICTERMSANDp(P-1)CROSSPRODUCTTERMSPOLYNOMIALMODELS•POLYNOMIALMODELDOESAPOORJOBOFPREDICTINGRESPONSEOUTSIDETHEREGIONOFEXPERIMENTATIONSIZEOFRESPONSESURFACEMODELSNUMBEROFFACTORSRUNSIN3LEVELFACT.COEF.INFULLQUANDTRIALSINFACE-CENTEREDTRIALSINBOX-BEHNKEN296111132710171548116272752432145466729288154721873662DESIGNS•PREDICTIONSALWAYSHAVESOMEDEGREEOFUNCERTAINTY•SHOULDHAVEREASONABLEPREDICTIONTHROUGHOUTTHEEXPERIMENTALRANGE•UNIFORMPREDICTIONSERRORISOBTAINEDBYUSINGADESIGNTHEFILLSOUTTHEREGIONOFINTEREST•THECHOICEOFEXPERIMENTALDESIGNISAFFECTEDBYTHESHAPEOFTHEEXPERIMENTALREGIONDESIGNS-CONTINUED•INMOSTCASES,THEREGIONISDETERMINEDBYTHERANGESOFTHEINDEPENDENTVARIABLE.INTHISCASETHEREGIONISCUBICAL(INCODEDVALUESOFX)ANDTHEBESTDESIGNINFACECENTERED•IF“STANDINGTHETHECENTER”ANDONEITISDESIREDTHATTHEPRECISIONOFPREDICATIONSBEINDEPENDENTOFDIRECTIONFROMCENTER-THENTHEREGIONISSPHERICALANDDESIGNOFCHOICEISBOX-BEHNKENDESIGNS-CONTINUED�BOX-BEHNKENDESIGNSEXCLUDETHECORNERS,WHEREALLVARIABLEARESIMULTANEOUSLYATTHEMAXIMUMLEVELS-THEREFOREBOX-BEHNKENDESIGNPERMITSAWIDERRANGEOFINDIVIDUALRANGES.�IFTHESHAPEOFTHEEXPERIMENTISNEITHERSPHERICALORCUBICALANDHASSTRONGCONSTRAINTS-THENTHEREGIONMAYBEANIRREGULARTETRAHEDRONANDWILLREQUIREASPECIALDESIGNFACECENTEREDCUBEFOR3FACTORS-•TWO-LEVELFACTORIAL•TWOFACECENTEREDPOINTSFOREACHFACTOR•THREEORMORECENTERPOINTS•WHENRUNINBLOCKS,CENTERPOINTSARERUNWITHEACHBLOCK•FACEPOINTSARERUNSFORWHICHALLFACTORSEXCEPTONEAREATTHEMIDDLESETTING-ANDPROVIDETHEINFORMATIONNEEDEDTODETERMINECURVATUREBLOCKING•INLARGESIZES,BOTHFACE-CENTEREDCUBEANDBOX-BEHNKENPERMITBLOCKING.•DIFFERENCE(ORBIASES)INTHELEVELOFTHERESPONSESBETWEENBLOCKSWITHNOTAFFECTESTIMATESOFCOEFFICIENTSNORESTIMATESOFTHEFACTORANDINTERACTIONEFFECTSFACECENTEREDCUBE•THEMAINPARTOFTHEFACE-CENTEREDCUBEDESIGNISATWO-LEVELFACTORIAL,WHICHFILLSOUTACUBICREGION•THEFACEPOINTSCONSTITUTEASEPARATEBLOCK-SOTHATTHEFIRSTTWOBLOCKS,WHICHCOMPRISEATWOLEVELFACTORIAL,CANBERUNFIRST.•THEFACEPOINTSAREADDEDIFSERIOUSCURVITUREISFOUND•“PIGGYBACK”APPROACHGIVESFLEXIBILITYFACECENTEREDCUBE•CENTERPOINTSARENEEDTOPROVIDEGOODPREDICTORSOFCENTEROFREGION•FOR3ORMOREFACTORS,ITISBESTTOUSEBLOCKS-FIRSTHALF-FRACTION-SECONDHALF-FRACTION-FACEPOINTSTHREEFACTORFACECENTEREDCUBETRIALX1X2X3BLOCK1---12+--23-+-24++-15--+26+-+17-++18+++29-00310+003110-03120+031300-31400+315000*16000*17000*BOX-BEHNKENDESIGN•THEBOX-BEHNKENDESIGNFILLSOUTAPOLYHEDRON,APPROXIMATINGASPHERE•FOR3FACTORS(15RUNS)THEDESIGNCONSISTOFTHREEFOUR-RUN,TWO-LEVELFACTORIALSINTWOFACTORS,WITHTHETHIRDFACTORATITSMID-LEVELANDTHREECENTERPOINT-RUNINTHREEBLOCKSOF10RUNSBOX-BEHNKENDESIGN•FORA3FACTOREXPERIMENT,THE15RUNSCONSISTOFTHREEFOUR-RUN,TWO-LEVELFACTORIALSINTWOFACTORS-WITHTHETHIRDFACTORATITSMID-LEVEL,ANDTHREECENTERPOINTS.•BOX-BEHNKENANDFACE-CENTEREDCUBICDESIGNSARESUBSETSOFTHEFULLTHREELEVELFACTORIALDESIGNS.EXCEPTFORCENTERPOINTS,THEYARECOMPLETMENTARYFRACTIONSINTHATNOPOINTINONEDESIGNISINTHEOTHERDESIGNDESIGNCHOICE•FACECENTEREDCUBEANDBOX-BEHNKENTAKEABOUTTHESAMENUMBEROFEXPERIMENTS•IFTIMEORMONEYDICTATESFEWERTHATTHEREQUIREDNUMBEROFINDEPENDENTVARIABLES,THENCONSIDER--REDUCENUMBEROFFACTORS-TRYASIMPLEXDESIGN-CONSIDERRUNNINGATWO-LEVELFACTORIALDESIGNTHATISTHEFIRSTTWOBLOCKSOFTHEFACE-CENTEREDCUBEANDCOMPLETETHELASTBLOCKSWHENADDITIONALEXPERIMENTATIONISPOSSIBLEDESIGNCHOICES•UNREPLICATEDRESPONSESURFACEDESIGNSCANDETECTEFFECTSABOUT1-2TIMESEXPERIMENTALERROR.•AFEWRUNSMAYBEINCLUDEDINTHEPROGRAMTOTESTHUNCHES,SPECIALCASES,“POLITICALPREFERENCES”ORSTANDARDORREFER