ResponseSurfaceology.ppt

1、Response Surface Methodology,2,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 influenced by several quantifiable vari

2、ables (or factors), with the objective of optimizing the response.,3,Response Surface,The yield of a process (Y) was determined to be influenced by the amount of nitrogen (X1) and phosphoric acid (X2), i.e. Y = (X1, X2) + where is the noise or error observed in the response. If we denote the expecte

3、d response by E(Y) = (X1, X2) = then the surface represented by = (X1, X2) is called a response surface.,4,Response Surface Plots,Response Surface Plots show how a response variable relates to two quantifiable factors based on a model equation.,5,Response Surface Designs,Designs for fitting response

4、 surfaces are called response surface designs. When choosing a design identify the number of control factors under investigation determine the limiting number of experimental runs ensure adequate coverage of the region of interest determine the impact of economics cost, time, availability, etc,6,Res

5、ponse Surface Methodology Why?,Response Surface Methods are used to examine the relationship between one or more responses and a set of quantifiable factors to search for the setting of critical control factors that would optimize the response when curvature in the response surface is suspected,7,Re

6、sponse Surface Methodology When?,Response Surface Methods may be employed to find factor settings that produce the “best” response find factor settings in which operating or process specifications are satisfied identify new operating conditions that would produce the required improvement in product

7、quality model a relationship between the control factors and the response,8,Response Surface Functions,First-Order Model Response surface will be planar. Second-Order Model Response surface will be curvi-planar,9,Response Surface Functions,RSM seeks to identify the relationship between the response

8、and the control factors. It is a sequential procedure, starting from current operating conditions and moving towards the optimum condition. Points on the response surface that are remote from the optimum condition, such as current operating conditions, often exhibit little curvature. A first-order m

9、odel will be appropriate. At the region of the optimum, curvature is often present, and the second-order model will become necessary.,10,Example,An engineer has determined that two factors reaction time (X1) and reaction temperature (X2) have significant effect on the yield (Y) of a process. The pro

10、cess is currently operating with a reaction time of 35 minutes and reaction temperature of 155C , resulting in yields of about 40%. The engineer decides to explore the process region of 30,40 minutes and 150,160C.,11,Example,The experimental design and accompanying results (available in Response Sur

11、face Methodology.MTW) are shown below:,12,Example,Stat DOE Factorial Analyze Factorial Design,13,Example,Session Window Fractional Factorial Fit: Yield versus Time, Temperature Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 40.4250 0.1037 389.89 0.00

12、0 Time 1.5500 0.7750 0.1037 7.47 0.002 Temperature 0.6500 0.3250 0.1037 3.13 0.035 Time*Temperature -0.0500 -0.0250 0.1037 -0.24 0.821 Ct Pt 0.0350 0.1391 0.25 0.814,Ignore “time-temperature” interaction, i.e. analyze as a First-Order Model.,14,Example,Session Window Fractional Factorial Fit: Yield

13、versus Time, Temperature (Interaction Excluded) Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 40.4250 0.09341 432.78 0.000 Time 1.5500 0.7750 0.09341 8.30 0.000 Temperature 0.6500 0.3250 0.09341 3.48 0.018 Ct Pt 0.0350 0.12532 0.28 0.791,The First-O

14、rder Model is valid.,15,Example,16,Analysis of Second-Order Models,Methods to analyze Second-Order Response Surfaces include: 3k Factorial Designs Box-Behnken Designs Central Composite Designs We will compare 3-factor variants of these designs.,17,3k Factorial Designs,18,3k Factorial Designs,Each of

15、 the k factors are run at 3 levels. Pro: a) Able to estimate all linear and quadratic effects, and all possible simple and higher order interactions. Con: a) Number of runs can be excessive. kRuns 2 9 3 27 4 81 5 243 6 729,19,3k Factorial Designs,Stat DOE Factorial Create Factorial Design,(2),(3),(1

16、),(4),20,3k Factorial Designs,Create Factorial Design Design Factors,21,Box-Behnken Designs,22,Box-Behnken Designs,Each of the k factors are run at 3 levels. Pro: a) Able to estimate all linear and quadratic effects, and 2-factor interactions. b) Less runs required, compared vs 3k Factorial Designs.

17、 c) Does not include any corner points. Con: a) Number of runs is large enough to estimate all quadratic and 2-factor interactions, regardless of need. b) Cannot be built-up from a 2k-p Factorial Design.,23,Box-Behnken Designs,Stat DOE Response Surface Create Response Surface Design,(2),(3),(1),24,C

18、entral Composite (Box-Wilson Design),25,Central Composite (Box-Wilson Design),26,Central Composite (Box-Wilson Design),Each of the k factors can be run at 5 levels. Pro: a) Able to estimate all linear effects, and selected quadratic effects and 2-factor interactions. b) Can be built-up from a 2k-q screening design, by adding axial points. Con: a) Best suited for quantitative factors. b) Some axial points may be in non-desirable conditions.,27,Central Composite (Box-Wilson Design),Stat DOE Response Surface Create Response Surface Design,(2),(3),(1),28,Comparison of 3-Lev