Design and Analysis of Experiments (7) Response Surface Methods and Designs (2)

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

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

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

4. Obtain the maximum yield at Chemical Plant • current operating condition • time : 75 min • temperature : 130℃ 22 Factorial Design 132.5 75, 130 (3times) Temperature 127.5 80 70 Time

5. Factorial Design Factor: 2, level:2, Center Pt: 3

6. Factorial Design

7. 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 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

8. Factorial Design 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

9. Factorial Design

10. Factorial Design 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

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

12. Factorial Design

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

14. Factorial Design • 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. 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 1.9149 temperature (4.5/2.35) *2.5 (unit of temp)

17. Method of Steepest Ascent time positon temp position 75 130.000 80 134.787 85 139.574 90 144.361 95 149.148 100 153.935 105 158.722 110 163.509 115 168.296 120 173.083 125 177.870

18. Method of Steepest Ascent time positon temp position yield(S) 75 130.0 62.3 80 134.5 73.3 90 144.4 86.8 100 153.9 58.2

19. 22 Factorial Design around the maximum yield • current operating condition • time : 90 min • temperature : 145℃ 22 Factorial Design 150 90, 145 (3times) Temperature 140 100 80 Time

20. 22 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. Central Composite Design (CCD)

24. 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 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) 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 Remove “Blocks” from model 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)

28. Central Composite Design (CCD)

29. Central Composite Design (CCD)