Design and Analysis of Experiments (7) Response Surface Methods and Designs (2) Kyung-Ho Park

Steps to optimize a process Temperature Time current operating condition Region of the optimum 60% 80% 90% Path of Improvement ① ③ ②

Steps to optimize a process 1.Sequential Experiments Factorial Design 2.Method of Steepest Ascent 3.Augmenting Design Response Surface Methods and Designs

current operating condition time : 75 min temperature : 130 ℃ Obtain the maximum yield at Chemical Plant 2 2 Factorial Design Time Temperature , 130 (3times) 132.5

Factorial Design Factor: 2, level:2, Center Pt: 3 StdOrder RunOrde r CenterPtBlockstime temperat ure Yield

Factorial Design

Factors: 2 Base Design: 2, 4 Runs: 7 Replicates: 1 Blocks: 1 Center pts (total): 3 Results for: example7-1.XLS Factorial Fit: Yield versus time, temperature Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant time temperature time*temperature Ct Pt Factorial Design

Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Curvature Residual Error Pure Error Total Factorial Design

Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant time temperature S = R-Sq = 91.06% R-Sq(adj) = 86.59% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Residual Error Curvature Lack of Fit Pure Error Total Estimated Coefficients for Yield using data in uncoded units Term Coef Constant time temperature Factorial Design

Yield = *time *Temperature (code) time : 70 min – 80 min temperature : ℃ ℃ Factorial Design

optimum condition time 80 min, temperature ℃, yield = 68% Factorial Design

conclusion optimum : 80 min ℃ no evidence for curvature – not arrive at no optimum value path of steepest ascent is required

Method of Steepest Ascent

select key factor: time key factor : factor which can not be controlled easily increase of one unit (5 minutes) of key factor (time) increase of temperature (4.5/2.35) *2.5 (unit of temp)

Method of Steepest Ascent time positontemp position

Method of Steepest Ascent time positontemp positionyield(S)

current operating condition time : 90 min temperature : 145 ℃ 2 2 Factorial Design around the maximum yield 2 2 Factorial Design Time Temperature , 145 (3times) 150

2 2 Factorial Design around the maximum yield Factorial Fit: yield versus time, temp Analysis of Variance for yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Curvature Residual Error Pure Error Total

Central Composite Design (CCD) Stat > DOE > Modify Design Click Add Axial Points

Central Composite Design (CCD) Stat > DOE > Modify Design Click Randomize

StdOrder RunOrde r CenterPtBlockstimetempyield Central Composite Design (CCD)

Stat > DOE > Response surface > Analysis Response surface Click Randomize Analysis of Variance for yield Source DF Seq SS Adj SS Adj MS F P Blocks Regression Linear Square Interaction Residual Error Lack-of-Fit Pure Error Total

Central Composite Design (CCD) Remove “Blocks” from model Analysis of Variance for yield Source DF Seq SS Adj SS Adj MS F P Blocks Regression Linear Square Interaction Residual Error Lack-of-Fit Pure Error Total Analysis of Variance for yield Source DF Seq SS Adj SS Adj MS F P Regression Linear Square Interaction Residual Error Lack-of-Fit Pure Error Total

Central Composite Design (CCD) Estimated Regression Coefficients for yield Term Coef SE Coef T P Constant time temp time*time temp*temp time*temp S = R-Sq = 89.2% R-Sq(adj) = 82.5% Estimated Regression Coefficients for yield using data in uncoded units Term Coef Constant time temp time*time temp*temp time*temp Yield = *time *temp *time+time *temp*temp – *time*temp

Central Composite Design (CCD)