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ResponseSurfaceMethodologyWhatisResponseSurfaceMethodology((RSM)ResponseSurfaceMethodology(RSM)isacollectionofmathematicalandstatisticaltechniquesthatareusefulforthemodelingandanalysisofproblemsinwhicharesponseofinterestisinfluencedbyseveralquantifiablevariables(orfactors),withtheobjectiveofoptimizingtheresponse.2ResponseSurfaceTheyieldofaprocess(Y)wasdeterminedtobeinfluencedbytheamountofnitrogen((X1)andphosphoricacid((X2),i.e..Y==ƒƒ((X1,X2)+whereisthenoiseorerrorobservedintheresponse..IfwedenotetheexpectedresponsebyE(Y))==ƒ(X1,X2)==thenthesurfacerepresentedby=ƒ(X1,X2)iscalledaresponsesurface.3ResponseSurfacePlotsResponseSurfacePlotsshowhowaresponsevariablerelatestotwoquantifiablefactorsbasedonamodelequation..4ResponseSurfaceDesignsDesignsforfittingresponsesurfacesarecalledresponsesurfacedesigns.Whenchoosingadesignidentifythenumberofcontrolfactorsunderinvestigationdeterminethelimitingnumberofexperimentalrunsensureadequatecoverageoftheregionofinterestdeterminetheimpactofeconomics––cost,time,availability,,etc5ResponseSurfaceMethodology–Why?ResponseSurfaceMethodsareusedtoexaminetherelationshipbetweenoneormoreresponsesandasetofquantifiablefactorstosearchforthesettingofcriticalcontrolfactorsthatwouldoptimizetheresponsewhencurvatureintheresponsesurfaceissuspected6ResponseSurfaceMethodology–When?ResponseSurfaceMethodsmaybeemployedtofindfactorsettingsthatproducethe“best”responsefindfactorsettingsinwhichoperatingorprocessspecificationsaresatisfiedidentifynewoperatingconditionsthatwouldproducetherequiredimprovementinproductqualitymodelarelationshipbetweenthecontrolfactorsandtheresponse7ResponseSurfaceFunctionsFirst-OrderModelResponsesurfacewillbeplanar.Second--OrderModelResponsesurfacewillbecurvi--planar8ResponseSurfaceFunctionsRSMseekstoidentifytherelationshipbetweentheresponseandthecontrolfactors.Itisasequentialprocedure,,startingfromcurrentoperatingconditionsandmovingtowardstheoptimumcondition.Pointsontheresponsesurfacethatareremotefromtheoptimumcondition,suchascurrentoperatingconditions,,oftenexhibitlittlecurvature.Afirst--ordermodelwillbeappropriate.Attheregionoftheoptimum,curvatureisoftenpresent,andthesecond--ordermodelwillbecomenecessary.9ExampleAnengineerhasdeterminedthattwofactors–reactiontime((X1)andreactiontemperature((X2)–havesignificanteffectontheyield(Y)ofaprocess..Theprocessiscurrentlyoperatingwithareactiontimeof35minutesandreactiontemperatureof155°C,resultinginyieldsofabout40%..Theengineerdecidestoexploretheprocessregionof[[30,,40]]minutesand[150,,160]°C.10ExampleTheexperimentaldesignandaccompanyingresults((availableinResponseSurfaceMethodology.MTW)areshownbelow:11ExampleStatDOEFactorialAnalyzeFactorialDesign12ExampleSessionWindowFractionalFactorialFit::YieldversusTime,TemperatureEstimatedEffectsandCoefficientsforYield((codedunits))TermEffectCoefSECoefTPConstant40..42500..1037389.890.000Time1.55000..77500..10377.470.002Temperature0.65000..32500..10373.130.035Time*Temperature--0.0500--0..02500..1037--0.240.821CtPt0..03500..13910.250.814Ignore““time--temperature””interaction,i.e..analyzeasaFirst-OrderModel.13ExampleSessionWindowFractionalFactorialFit::YieldversusTime,Temperature((InteractionExcluded))EstimatedEffectsandCoefficientsforYield((codedunits))TermEffectCoefSECoefTPConstant40..42500.09341432.780.000Time1.55000..77500.093418.300.000Temperature0.65000..32500.093413.480.018CtPt0..03500.125320.280.791TheFirst--OrderModelisvalid.14Example15AnalysisofSecond--OrderModelsMethodstoanalyzeSecond-OrderResponseSurfacesinclude::3kFactorialDesignsBox--BehnkenDesignsCentralCompositeDesignsWewillcompare3-factorvariantsofthesedesigns.163kFactorialDesigns173kFactorialDesignsEachofthekfactorsarerunat3levels.Pro::a)Abletoestimatealllinearandquadraticeffects,,andallpossiblesimpleandhigherorderinteractions.Con::a)Numberofrunscanbeexcessive..k Runs2932748152436729183kFactorialDesignsStatDOEFactorialCreateFactorialDesign(2)(3)(1)(4)193kFactorialDesignsCreateFactorialDesignDesignFactors20Box--BehnkenDesigns21Box--BehnkenDesignsEachofthekfactorsarerunat3levels.Pro::a)Abletoestimatealllinearandquadraticeffects,,and2-factorinteractions..b)Lessrunsrequired,comparedvs3kFactorialDesigns.c)Doesnotincludeanycornerpoints.Con::a)Numberofrunsislargeenoughtoestimateallquadraticand2--factorinteractions,regardlessofneed.b)Cannotbebuilt-upfroma2k-pFactorialDesign.22Box--BehnkenDesignsStatDOEResponseSurfaceCreateResponseSurfaceDesign(2)(3)(1)23CentralComposite((Box-WilsonDesign))=nf¼wherenfisnumberofrunsinfactorialportionofCCD24CentralComposite((Box-WilsonDesign))FactorialPoints(8runs)+CenterPoints&AxialPoints(6+6runs)=CentralComposite(Box-Wilson)Design(20runs)25CentralComposite((Box-WilsonDesign))Eachofthekfactorscanberunat5levels.Pro::a)Abletoestimatealllineareffects,andselectedquadraticeffectsand2-factorinteractions..b)Canbebuilt-upfroma2k-qscreeningdesign,byaddingaxialpoints..Con::a)Bestsuitedforquantitativefactors.b)Someaxialpointsmaybeinnon-desirableconditions.26CentralComposite((Box-WilsonDesign))StatDOEResponseSurfaceCreateResponseSurfaceDesign(2)(3)(1)27Comparisonof3--Le