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

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

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

4. current operating condition time : 75 min temperature : 130 ℃ Obtain the maximum yield at Chemical Plant 2 2 Factorial Design Time Temperature 80 70 127.5 75, 130 (3times) 132.5

5. Factorial Design Factor: 2, level:2, Center Pt: 3 StdOrder RunOrde r CenterPtBlockstime temperat ure Yield 151170127.554.3 241180127.560.3 311170132.564.6 461180132.568 52017513060.3 63017513064.3 77017513062.3

6. Factorial Design

7. 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 61.8000 1.000 61.80 0.000 time 4.7000 2.3500 1.000 2.35 0.143 temperature 9.0000 4.5000 1.000 4.50 0.046 time*temperature -1.3000 -0.6500 1.000 -0.65 0.582 Ct Pt 0.5000 1.528 0.33 0.775 Factorial Design

8. Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 2 103.090 103.090 51.5450 12.89 0.072 2-Way Interactions 1 1.690 1.690 1.6900 0.42 0.582 Curvature 1 0.429 0.429 0.4286 0.11 0.775 Residual Error 2 8.000 8.000 4.0000 Pure Error 2 8.000 8.000 4.0000 Total 6 113.209 Factorial Design

10. Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 62.014 0.6011 103.16 0.000 time 4.700 2.350 0.7952 2.96 0.042 temperature 9.000 4.500 0.7952 5.66 0.005 S = 1.59049 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 2 103.090 103.090 51.5450 20.38 0.008 Residual Error 4 10.119 10.119 2.5296 Curvature 1 0.429 0.429 0.4286 0.13 0.740 Lack of Fit 1 1.690 1.690 1.6900 0.42 0.582 Pure Error 2 8.000 8.000 4.0000 Total 6 113.209 Estimated Coefficients for Yield using data in uncoded units Term Coef Constant -207.236 time 0.470000 temperature 1.80000 Factorial Design

11. Yield = 62.014 + 2.350*time +4.500*Temperature (code) time : 70 min – 80 min temperature : 127.5 ℃ - 132.5 ℃ Factorial Design

13. optimum condition time 80 min, temperature 132.5 ℃, yield = 68% Factorial Design

14. conclusion optimum : 80 min. 132.5 ℃ no evidence for curvature – not arrive at no optimum value path of steepest ascent is required

15. Method of Steepest Ascent

16. 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 1.9149 temperature (4.5/2.35) *2.5 (unit of temp)

17. Method of Steepest Ascent time positontemp position 75130.000 80134.787 85139.574 90144.361 95149.148 100153.935 105158.722 110163.509 115168.296 120173.083 125177.870

18. Method of Steepest Ascent time positontemp positionyield(S) 75130.062.3 80134.573.3 90144.486.8 100153.958.2

19. current operating condition time : 90 min temperature : 145 ℃ 2 2 Factorial Design around the maximum yield 2 2 Factorial Design Time Temperature 100 80 140 90, 145 (3times) 150

20. 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 2 23.425 23.425 11.713 5.40 0.156 2-Way Interactions 1 95.062 95.062 95.062 43.81 0.022 Curvature 1 45.027 45.027 45.027 20.75 0.045 Residual Error 2 4.340 4.340 2.170 Pure Error 2 4.340 4.340 2.170 Total 6 167.854

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

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

23. StdOrder RunOrde r CenterPtBlockstimetempyield 12118014078.8 251110014084.5 34118015091.2 461110015077.4 53019014586.8 61019014587.8 77019014589.7 812275.8578614583.3 992104.142114581.2 10 290137.928981.2 1114290152.071179.5 128029014587 1311029014586 1413029014589.3 Central Composite Design (CCD)

24. 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 1 5.406 5.406 5.406 1.75 0.228 Regression 5 223.681 223.681 44.736 14.47 0.001 Linear 2 16.366 202.338 101.169 32.71 0.000 Square 2 112.253 112.253 56.126 18.15 0.002 Interaction 1 95.062 95.062 95.062 30.74 0.001 Residual Error 7 21.647 21.647 3.092 Lack-of-Fit 3 11.581 11.581 3.860 1.53 0.336 Pure Error 4 10.067 10.067 2.517 Total 13 250.735

25. Central Composite Design (CCD) Remove “Blocks” from model Analysis of Variance for yield Source DF Seq SS Adj SS Adj MS F P Blocks 1 5.406 5.406 5.406 1.75 0.228 Regression 5 223.681 223.681 44.736 14.47 0.001 Linear 2 16.366 202.338 101.169 32.71 0.000 Square 2 112.253 112.253 56.126 18.15 0.002 Interaction 1 95.062 95.062 95.062 30.74 0.001 Residual Error 7 21.647 21.647 3.092 Lack-of-Fit 3 11.581 11.581 3.860 1.53 0.336 Pure Error 4 10.067 10.067 2.517 Total 13 250.735 Analysis of Variance for yield Source DF Seq SS Adj SS Adj MS F P Regression 5 223.68 223.68 44.736 13.23 0.001 Linear 2 16.37 202.34 101.169 29.92 0.000 Square 2 112.25 112.25 56.126 16.60 0.001 Interaction 1 95.06 95.06 95.062 28.11 0.001 Residual Error 8 27.05 27.05 3.382 Lack-of-Fit 3 16.32 16.32 5.440 2.53 0.171 Pure Error 5 10.73 10.73 2.147 Total 13 250.74

26. Central Composite Design (CCD) Estimated Regression Coefficients for yield Term Coef SE Coef T P Constant 87.7667 0.7507 116.906 0.000 time -1.3837 0.6502 -2.128 0.066 temp 0.3620 0.6502 0.557 0.593 time*time -2.3396 0.6767 -3.457 0.009 temp*temp -3.2896 0.6767 -4.861 0.001 time*temp -4.8750 0.9195 -5.302 0.001 S = 1.839 R-Sq = 89.2% R-Sq(adj) = 82.5% Estimated Regression Coefficients for yield using data in uncoded units Term Coef Constant -4138.6980 time 18.2104 temp 47.0066 time*time -0.0234 temp*temp -0.1316 time*temp -0.0975 Yield = -4139 + 18.21*time + 47.01*temp - 0.0234*time+time -0.1316*temp*temp – 0.0975*time*temp

27. Central Composite Design (CCD)