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1、Response Surface MethodologyWhat is Response Surface Methodology(RSM)What is Response Surface Methodology(RSM)Response Surface Methodology(RSM)is a collection of mathematical and statistical techniques that are useful for the modeling and analysis of problems in which a response of interest is influ
2、enced by several quantifiable variables(or factors),with the objective of optimizing the response.2 2Response SurfaceThe yield of a process(Y)was determined to be influenced by The yield of a process(Y)was determined to be influenced by the amount of nitrogen(Xthe amount of nitrogen(X1 1)and phospho
3、ric acid(X)and phosphoric acid(X2 2),i.e.),i.e.Y =(XY =(X1 1,X,X2 2)+)+where where is the noise or error observed in the response.is the noise or error observed in the response.If we denote the expected response byIf we denote the expected response byE(Y)=E(Y)=(X(X1 1,X,X2 2)=)=then the surface repr
4、esented bythen the surface represented by =(X(X1 1,X,X2 2)is called a is called a response surfaceresponse surface.3 3Response Surface PlotsResponse Surface PlotsResponse Surface Plots show how a response variable show how a response variable relates to two quantifiable factors based on a model rela
5、tes to two quantifiable factors based on a model equation.equation.4 4Response Surface DesignsDesigns for fitting response surfaces are called Designs for fitting response surfaces are called response response surface designssurface designs.When choosing a designWhen choosing a designn nidentify the
6、 number of control factors under investigationidentify the number of control factors under investigationn ndetermine the limiting number of experimental runsdetermine the limiting number of experimental runsn nensure adequate coverage of the region of interestensure adequate coverage of the region o
7、f interestn ndetermine the impact of economics cost,time,availability,determine the impact of economics cost,time,availability,etcetc5 5Response Surface Methodology Why?Response Surface Methods are usedare usedn nto examine the relationship between one or to examine the relationship between one or m
8、ore responses and a set of quantifiable factorsmore responses and a set of quantifiable factorsn nto search for the setting of critical control factors to search for the setting of critical control factors that would optimize the response that would optimize the response n nwhen curvature in the res
9、ponse surface is when curvature in the response surface is suspectedsuspected6 6Response Surface Methodology When?Response Surface MethodsResponse Surface Methods may be employed tomay be employed ton nfind factor settings that produce the“best”responsefind factor settings that produce the“best”resp
10、onsen nfind factor settings in which operating or process find factor settings in which operating or process specifications are satisfied specifications are satisfied n nidentify new operating conditions that would produce the identify new operating conditions that would produce the required improve
11、ment in product qualityrequired improvement in product qualityn nmodel a relationship between the control factors and the model a relationship between the control factors and the responseresponse7 7Response Surface FunctionsFirst-Order ModelFirst-Order ModelResponse surface will be planar.Response s
12、urface will be planar.Second-Order ModelSecond-Order ModelResponse surface will be curvi-planarResponse surface will be curvi-planar8 8Response Surface FunctionsRSM seeks to identify the relationship between the response and RSM seeks to identify the relationship between the response and the control
13、 factors.It is a sequential procedure,starting from the control factors.It is a sequential procedure,starting from current operating conditions and moving towards the optimum current operating conditions and moving towards the optimum condition.condition.Points on the response surface that are remot
14、e from the Points on the response surface that are remote from the optimum condition,such as current operating conditions,often optimum condition,such as current operating conditions,often exhibit little curvature.A first-order model will be appropriate.exhibit little curvature.A first-order model w
15、ill be appropriate.At the region of the optimum,curvature is often present,and At the region of the optimum,curvature is often present,and the second-order model will become necessary.the second-order model will become necessary.9 9ExampleAn engineer has determined that two factors An engineer has d
16、etermined that two factors reaction time(Xreaction time(X1 1)and reaction temperature(X)and reaction temperature(X2 2)have significant effect on the yield(Y)of a have significant effect on the yield(Y)of a process.process.The process is currently operating with a reaction The process is currently op
17、erating with a reaction time of 35 minutes and reaction temperature of time of 35 minutes and reaction temperature of 155C,resulting in yields of about 40%.155C,resulting in yields of about 40%.The engineer decides to explore the process region The engineer decides to explore the process region of 3
18、0,40 minutes and 150,160C.of 30,40 minutes and 150,160C.1010ExampleThe experimental design and accompanying The experimental design and accompanying results(available in results(available in Response Surface Response Surface Methodology.MTWMethodology.MTW)are shown below:)are shown below:1111Example
19、S Stat tat D DOE OE F Factorial actorial A Analyze Factorial Designnalyze Factorial Design1212ExampleSession WindowSession WindowFractional Factorial Fit:Yield versus Time,TemperatureFractional Factorial Fit:Yield versus Time,TemperatureEstimated Effects and Coefficients for Yield(coded units)Estima
20、ted Effects and Coefficients for Yield(coded units)Term Effect Coef SE Coef T PTerm Effect Coef SE Coef T PConstant 40.4250 0.1037 389.89 0.000Constant 40.4250 0.1037 389.89 0.000Time 1.5500 0.7750 0.1037 7.47 0.002Time 1.5500 0.7750 0.1037 7.47 0.002Temperature 0.6500 0.3250 0.1037 3.13 0.035Temper
21、ature 0.6500 0.3250 0.1037 3.13 0.035Time*Temperature -0.0500 -0.0250 0.1037 -0.24 0.821Time*Temperature -0.0500 -0.0250 0.1037 -0.24 0.821Ct Pt 0.0350 0.1391 0.25 0.814Ct Pt 0.0350 0.1391 0.25 0.814Ignore“time-temperature”interaction,i.e.analyze as a First-Order Model.1313ExampleSession WindowSessi
22、on WindowFractional Factorial Fit:Yield versus Time,Temperature(Interaction Excluded)Fractional Factorial Fit:Yield versus Time,Temperature(Interaction Excluded)Estimated Effects and Coefficients for Yield(coded units)Estimated Effects and Coefficients for Yield(coded units)Term Effect Coef SE Coef
23、T PTerm Effect Coef SE Coef T PConstant 40.4250 0.09341 432.78 0.000Constant 40.4250 0.09341 432.78 0.000Time 1.5500 0.7750 0.09341 8.30 0.000Time 1.5500 0.7750 0.09341 8.30 0.000Temperature 0.6500 0.3250 0.09341 3.48 0.018Temperature 0.6500 0.3250 0.09341 3.48 0.018Ct Pt 0.0350 0.12532 0.28 0.791Ct
24、 Pt 0.0350 0.12532 0.28 0.791The First-Order Model is valid.1414Example1515Analysis of Second-Order ModelsMethods to analyze Second-Order Response Methods to analyze Second-Order Response Surfaces include:Surfaces include:n n 3 3k k Factorial Designs Factorial Designsn n Box-Behnken Designs Box-Behn
25、ken Designsn n Central Composite Designs Central Composite DesignsWe will compare 3-factor variants of these designs.We will compare 3-factor variants of these designs.16163k Factorial Designs17173k Factorial Designsn nEach of the k factors are run at 3 levels.Each of the k factors are run at 3 leve
26、ls.n nPro:Pro:a)Able to estimate all linear and quadratic effects,and a)Able to estimate all linear and quadratic effects,and all possible simple and higher order interactions.all possible simple and higher order interactions.n nCon:Con:a)Number of runs can be excessive.a)Number of runs can be exces
27、sive.k kRunsRuns2 2 9 93 3 27 274 4 81 815 5 243 2436 6 729 72918183k Factorial DesignsS Stat tat D DOE OE F Factorial actorial C Create Factorial Designreate Factorial Design(2)(3)(1)(4)19193k Factorial DesignsCreate Factorial Design Design Factors2020Box-Behnken Designs2121Box-Behnken Designsn nEa
28、ch of the k factors are run at 3 levels.Each of the k factors are run at 3 levels.n nPro:Pro:a)Able to estimate all linear and quadratic effects,and a)Able to estimate all linear and quadratic effects,and 2-factor interactions.2-factor interactions.b)Less runs required,compared vs 3 b)Less runs requ
29、ired,compared vs 3k k Factorial Designs.Factorial Designs.c)Does not include any corner points.c)Does not include any corner points.n n Con:a)Number of runs is large enough to estimate all quadratic Con:a)Number of runs is large enough to estimate all quadratic and 2-factor interactions,regardless o
30、f need.and 2-factor interactions,regardless of need.b)Cannot be built-up from a 2 b)Cannot be built-up from a 2k-pk-p Factorial Design.Factorial Design.2222Box-Behnken DesignsS Stat tat DDOE OE R Response Surface esponse Surface C Cr reate Response Surface Designeate Response Surface Design(2)(3)(1)
31、2323Central Composite(Box-Wilson Design)=nfwhere nf is number of runsin factorial portion of CCD2424Central Composite(Box-Wilson Design)Factorial PointsFactorial Points(8 runs)+Center Points Center Points&Axial Points&Axial Points(6+6 runs)=Central Composite Central Composite(Box-Wilson)(Box-Wilson)
32、DesignDesign(20 runs)2525Central Composite(Box-Wilson Design)n nEach of the k factors can be run at 5 levels.Each of the k factors can be run at 5 levels.n nPro:Pro:a)Able to estimate all linear effects,and selected quadratic a)Able to estimate all linear effects,and selected quadratic effects and 2
33、-factor interactions.effects and 2-factor interactions.b)Can be built-up from a 2 b)Can be built-up from a 2k-qk-q screening design,by adding screening design,by adding axial points.axial points.n nCon:Con:a)Best suited for quantitative factors.a)Best suited for quantitative factors.b)Some axial poi
34、nts may be in non-desirable conditions.b)Some axial points may be in non-desirable conditions.2626Central Composite(Box-Wilson Design)S Stat tat DDOE OE R Response Surface esponse Surface C Cr reate Response Surface Designeate Response Surface Design(2)(3)(1)2727Comparison of 3-Level Design Numbers
35、in parenthesis =the number of replicated center points.For CCD,Source:Understanding Industrial Designed Experiments Stephen R Schmidt&Robert G Launsby2828Contour/Surface PlotsS Stat tat D DOE OE R Response Surfaceesponse Surface Co Con ntour/Surface(Wireframe)Plotstour/Surface(Wireframe)Plots2929End of Presentation Rev 1:17 July 02 Rev 1:17 July 02谢谢观看/欢迎下载BY FAITH I MEAN A VISION OF GOOD ONE CHERISHES AND THE ENTHUSIASM THAT PUSHES ONE TO SEEK ITS FULFILLMENT REGARDLESS OF OBSTACLES.BY FAITH I BY FAITH