chapter 11 response surface methods and other appr

1Chapter11ResponseSurfaceMethodsandOtherApproachestoProcessOptimization211.1IntroductiontoResponseSurfaceMethodology•ResponseSurfaceMethodology(RSM)isusefulforthemodelingandanalysisofprogramsinwhicharesponseofinterestisinfluencedbyseveralvariablesandtheobjectiveistooptimizethisresponse.•Forexample:Findthelevelsoftemperature(x1)andpressure(x2)tomaximizetheyield(y)ofaprocess.),(21xxfy3•Responsesurface:(seeFigure11.1&11.2)•Thefunctionfisunknown•Approximatethetruerelationshipbetweenyandtheindependentvariablesbythelower-orderpolynomialmodel.•Responsesurfacedesign),()(21xxfyEjijiijkiiiikiiikkxxxxyxxy12101104•Asequentialprocedure•Theobjectiveistoleadtheexperimenterrapidlyandefficientlyalongapathofimprovementtowardthegeneralvicinityoftheoptimum.•First-ordermodel=Second-ordermodel•Climbahill511.2TheMethodofSteepestAscent•Assumethatthefirst-ordermodelisanadequateapproximationtothetruesurfaceinasmallragionofthex’s.•Themethodofsteepestascent:Aprocedureformovingsequentiallyalongthepathofsteepestascent.6•Basedonthefirst-ordermodel,•Thepathofsteepestascent//theregressioncoefficients•Theactualstepsizeisdeterminedbytheexperimenterbasedonprocessknowledgeorotherpracticalconsiderationskiiixy10ˆˆˆ7•Example11.1–Twofactors,reactiontime&reactiontemperature–Useafullfactorialdesignandcenterpoints(seeTable11.1):1.Obtainanestimateoferror2.Checkforinteractionsinthemodel3.Checkforquadraticeffect•ANOVAtable(seeTable11.2)•Table11.3&Figure11.5•Table11.4&11.58Factor1Factor2Response1StdRunBlockA:TimeB:TempyieldminutesdegCpercent17{1}-1-139.326{1}1-140.935{1}-114042{1}1141.559{1}0040.364{1}0040.571{1}0040.783{1}0040.298{1}0040.612ˆ40.440.7750.325yxx9Thestepsizeis5minutesofreactiontimeand2degreesFWhathappensattheconclusionofsteepestascent?10•Assumethefirst-ordermodel1.Chooseastepsizeinoneprocessvariable,xj.2.Thestepsizeintheothervariable,3.Convertthexjfromcodedvariablestothenaturalvariablekiiixy10ˆˆˆjjiixx/ˆˆ1111.3AnalysisofaSecond-orderResponseSurface•Whentheexperimenterisrelativeclosedtotheoptimum,thesecond-ordermodelisusedtoapproximatetheresponse.•Findthestationarypoint.Maximumresponse,Minimumresponseorsaddlepoint.•Determinewhetherthestationarypointisapointofmaximumorminimumresponseorasaddlepoint.22011221212111222yxxxxxx12•Thesecond-ordermodel:bxybBxBbxBxxbxs1s'0222112112121021ˆˆ21ˆ2/ˆˆ2/ˆ2/ˆˆandˆˆˆ,,''ˆˆskkkkkkxxxy13•Characterizingtheresponsesurface:–ContourplotorCanonicalanalysis–Canonicalform(seeFigure11.9)–Minimumresponse:iareallpositive–Maximumresponse:iareallnegative–Saddlepoint:ihavedifferentsigns2211ˆˆkkswwyy14•Example11.2–ContinueExample11.1–Centralcompositedesign(CCD)(Table11.6&Figure11.10)–Table11.7ANOVAforResponseSurfaceQuadraticModelAnalysisofvariancetable[Partialsumofsquares]SumofMeanFSourceSquaresDFSquareValueProbFModel28.2555.6579.850.0001A7.9217.92111.930.0001B2.1212.1230.010.0009A213.18113.18186.220.0001B26.9716.9798.560.0001AB0.2510.253.530.1022Residual0.5070.071LackofFit0.2830.0941.780.2897PureError0.2140.053CorTotal28.741212122212ˆ79.940.990.520.251.381.00yxxxxxx15•Thecontourplotisgiveninthenaturalvariables(seeFigure11.11)•Theoptimumisatabout87minutesand176.5degreesDESIGN-EXPERTPlotyieldX=A:timeY=B:tempDesignPointsyieldA:timeB:temp80.0082.5085.0087.5090.00170.00172.50175.00177.50180.0076.95477.605678.257378.257378.908979.5606516•Therelationshipbetweenxandw:–MisanorthogonalmatrixandthecolumnsofMarethenormalizedeigenvectorsofB.•Multipleresponse:–Typically,wewanttosimultaneouslyoptimizeallresponses,orfindasetofconditionswherecertainproductpropertiesareachieved–Overlaythecontourplots(Figure11.16)–Constrainedoptimizationproblem)('sxxMw1711.4ExperimentalDesignsforFittingResponseSurfaces•Designsforfittingthefirst-ordermodel–Theorthogonalfirst-orderdesigns–X’Xisadiagonalmatrix–2kfactorialandfractionsofthe2kseriesinwhichmaineffectsarenotaliasedwitheachothers–Besidesfactorialdesigns,includeseveralobservationsatthecenter.–Simplexdesign18•Designsforfittingthesecond-ordermodel–Centralcompositedesign(CCD)–nFrunson2kaxialorstarpoints,andnCcenterruns–Sequentialexperimentation–Twoparameters:nCand–Thevarianceofthepredictedresponseatx:–Rotatabledesign:Thevarianceofpredictedresponseisconstantonspheres–ThepurposeofRSMisoptimizationandthelocationoftheoptimumisunknownpriortorunningtheexperiment.xX)(X'x'x12))(ˆ(yVar19–=(nF)1/4yieldsarotatablecentralcompositedesign–ThesphericalCCD:Set=(k)1/2–CenterrunsintheCCD,nC:3to5centerruns–TheBox-Behnkendesign:three-leveldesigns(seeTable11.8)–Cuboidalregion:•face-centeredcentralcompositedesign(orface-centeredcube)•=1•nC=2or3