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Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,*,Response Surface Methodology,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 variables(or factors),with the objective of optimizing the response.,2,Response Surface,The yield of a process(Y)was determined to be influenced by the amount of nitrogen(X,1,)and phosphoric acid(X,2,),i.e.,Y =(X,1,X,2,)+,where is the noise or error observed in the response.,If we denote the expected response by,E(Y)=,(X,1,X,2,)=,then the surface represented by,=,(X,1,X,2,),is called a,response surface,.,3,Response Surface Plots,Response Surface Plots,show how a response variable relates to two quantifiable factors based on a model equation.,4,Response Surface Designs,Designs for fitting response 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,5,Response 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,6,Response 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 quality,model a relationship between the control factors and the response,7,Response Surface Functions,First-Order Model,Response surface will be planar.,Second-Order Model,Response surface will be curvi-planar,8,Response Surface Functions,RSM seeks to identify the relationship between the response 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 model will be appropriate.,At the region of the optimum,curvature is often present,and the second-order model will become necessary.,9,Example,An engineer has determined that two factors reaction time(X,1,)and reaction temperature(X,2,)have significant effect on the yield(Y)of a process.,The process 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.,10,Example,The experimental design and accompanying results(available in,Response Surface Methodology.MTW,)are shown below:,11,Example,S,tat,D,OE,F,actorial,A,nalyze Factorial Design,12,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 0.000,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.,13,Example,Session Window,Fractional Factorial Fit:Yield 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-Order Model is valid.,14,Example,15,Analysis of Second-Order Models,Methods to analyze Second-Order Response Surfaces include:,3,k,Factorial Designs,Box-Behnken Designs,Central Composite Designs,We will compare 3-factor variants of these designs.,16,3,k,Factorial Designs,17,3,k,Factorial Designs,Each of 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 81,5 243,6 729,18,3,k,Factorial Designs,S,tat,D,OE,F,actorial,C,reate Factorial Design,(2),(3),(1),(4),19,3,k,Factorial Designs,C,reate Factorial Design,D,esign,F,actors,20,Box-Behnken Designs,21,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 3,k,Factorial Designs.,c)Does not include any corner points.,Con:a)Number of runs is large enough to estimate all quadratic and
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