design and analysis of experiments (7) response surface methods and designs (2)
DESCRIPTION
Design and Analysis of Experiments (7) Response Surface Methods and Designs (2). Kyung-Ho Park. Steps to optimize a process. ③. Region of the optimum. ②. Temperature. Path of Improvement. current operating condition. 90%. ①. 80%. 60%. 60%. Time. Steps to optimize a process - PowerPoint PPT PresentationTRANSCRIPT
Design and Analysis of Experiments (7)
Response Surface Methods and Designs
(2)Kyung-Ho Park
Steps to optimize a processTe
mpe
ratu
re
Time
currentoperatingcondition
Regionof theoptimum
60% 60% 80%
90%Path ofImprovement
①
③
②
Steps to optimize a process
1. Sequential ExperimentsFactorial Design
2. Method of Steepest Ascent
3. Augmenting DesignResponse Surface Methods and Designs
current operating condition time : 75 min temperature : 130℃
Obtain the maximum yield at Chemical Plant
22 Factorial Design
Time
Tem
pera
ture
8070127.5
75, 130 (3times)
132.5
Factorial DesignFactor: 2, level:2, Center Pt: 3
StdOrder RunOrder CenterPt Blocks time temperature Yield
1 5 1 1 70 127.5 54.3
2 4 1 1 80 127.5 60.3
3 1 1 1 70 132.5 64.6
4 6 1 1 80 132.5 68
5 2 0 1 75 130 60.3
6 3 0 1 75 130 64.3
7 7 0 1 75 130 62.3
Factorial Design
Term
Standardized Effect
AB
A
B
543210
4.303Factor NameA timeB temperature
Pareto Chart of the Standardized Effects(response is Yield, Alpha = .05)
Factors: 2 Base Design: 2, 4Runs: 7 Replicates: 1Blocks: 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 PConstant 61.8000 1.000 61.80 0.000time 4.7000 2.3500 1.000 2.35 0.143temperature 9.0000 4.5000 1.000 4.50 0.046time*temperature -1.3000 -0.6500 1.000 -0.65 0.582Ct Pt 0.5000 1.528 0.33 0.775
Factorial Design
Analysis of Variance for Yield (coded units)
Source DF Seq SS Adj SS Adj MS F PMain Effects 2 103.090 103.090 51.5450 12.89 0.0722-Way Interactions 1 1.690 1.690 1.6900 0.42 0.582 Curvature 1 0.429 0.429 0.4286 0.11 0.775Residual Error 2 8.000 8.000 4.0000 Pure Error 2 8.000 8.000 4.0000Total 6 113.209
Factorial Design
Factorial Design
Estimated Effects and Coefficients for Yield (coded units)
Term Effect Coef SE Coef T PConstant 62.014 0.6011 103.16 0.000time 4.700 2.350 0.7952 2.96 0.042temperature 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 PMain Effects 2 103.090 103.090 51.5450 20.38 0.008Residual 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 4.0000Total 6 113.209
Estimated Coefficients for Yield using data in uncoded units
Term CoefConstant -207.236time 0.470000temperature 1.80000
Factorial Design
Yield = 62.014 + 2.350*time +4.500*Temperature (code)time : 70 min – 80 mintemperature : 127.5℃ - 132.5℃
Deleted Residual
Perc
ent
3.01.50.0-1.5-3.0
99
90
50
10
1
Fitted Value
Del
eted
Res
idua
l
70656055
2
1
0
-1
Deleted Residual
Freq
uenc
y
2.01.51.00.50.0-0.5-1.0
3
2
1
0
Observation Order
Del
eted
Res
idua
l
7654321
2
1
0
-1
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Histogram of the Residuals Residuals Versus the Order of the Data
Residual Plots for Yield
Factorial Design
Mea
n of
Yie
ld
807570
66
64
62
60
58
56132.5130.0127.5
time temperature Point TypeCornerCenter
Main Effects Plot (data means) for Yield
temperature
Mea
n
132.5130.0127.5
67.5
65.0
62.5
60.0
57.5
55.0
time
Center80 Corner
Point Type70 Corner75
Interaction Plot (data means) for Yield
Factorial Design
132.5
127.58070
temperature
time
62.3
68.0
60.354.3
64.6CenterpointFactorial Point
Cube Plot (data means) for Yield
optimum condition time 80 min, temperature 132.5℃, yield = 68%
Factorial Design
Factorial Design
conclusion• optimum : 80 min. 132.5℃• no evidence for curvature – not arrive at no optimum value• path of steepest ascent is required
Method of Steepest Ascent
time
tem
pera
ture
807876747270
132
131
130
129
128
Yield
58 - 6060 - 6262 - 6464 - 6666 - 68
<
> 68
5656 - 58
Contour Plot of Yield vs temperature, time
time
tem
pera
ture
1.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
yield
58 - 6060 - 6262 - 6464 - 6666 - 68
<
> 68
5656 - 58
Contour Plot of yield vs temperature, time
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)
Method of Steepest Ascent
time positon temp position75 130.00080 134.78785 139.57490 144.36195 149.148100 153.935105 158.722110 163.509115 168.296120 173.083125 177.870
temp position
tim
e po
sito
n
180170160150140130
130
120
110
100
90
80
70
Scatterplot of time positon vs temp position
Method of Steepest Ascenttime positon temp position yield(S)75 130.0 62.380 134.5 73.390 144.4 86.8100 153.9 58.2
10090
yield
60
70
80
time
90
130 80140 150temp
3D Scatterplot of yield vs time vs temp
current operating condition time : 90 min temperature : 145℃
22 Factorial Design around the maximum yield
22 Factorial Design
Time
Tem
pera
ture
10080140
90, 145 (3times)
150
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 PMain Effects 2 23.425 23.425 11.713 5.40 0.1562-Way Interactions 1 95.062 95.062 95.062 43.81 0.022 Curvature 1 45.027 45.027 45.027 20.75 0.045Residual Error 2 4.340 4.340 2.170 Pure Error 2 4.340 4.340 2.170Total 6 167.854
Central Composite Design (CCD)
Stat > DOE > Modify DesignClick Add Axial Points
Central Composite Design (CCD)
Stat > DOE > Modify DesignClick Randomize
StdOrder RunOrder CenterPt Blocks time temp yield1 2 1 1 80 140 78.82 5 1 1 100 140 84.53 4 1 1 80 150 91.24 6 1 1 100 150 77.45 3 0 1 90 145 86.86 1 0 1 90 145 87.87 7 0 1 90 145 89.7
8 12 -1 2 75.85786 145 83.3
9 9 -1 2 104.1421 145 81.2
10 10 -1 2 90 137.9289 81.2
11 14 -1 2 90 152.0711 79.5
12 8 0 2 90 145 8713 11 0 2 90 145 8614 13 0 2 90 145 89.3
Central Composite Design (CCD)
Central Composite Design (CCD)
Stat > DOE > Response surface > Analysis Response surfaceClick Randomize
Analysis of Variance for yield
Source DF Seq SS Adj SS Adj MS F PBlocks 1 5.406 5.406 5.406 1.75 0.228Regression 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 0.001Residual 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 2.517Total 13 250.735
Central Composite Design (CCD)
Remove “Blocks” from model
Analysis of Variance for yield
Source DF Seq SS Adj SS Adj MS F PBlocks 1 5.406 5.406 5.406 1.75 0.228Regression 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 0.001Residual 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 2.517Total 13 250.735
Analysis of Variance for yield
Source DF Seq SS Adj SS Adj MS F PRegression 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 0.001Residual 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 2.147Total 13 250.74
Central Composite Design (CCD)Estimated Regression Coefficients for yield
Term Coef SE Coef T PConstant 87.7667 0.7507 116.906 0.000time -1.3837 0.6502 -2.128 0.066temp 0.3620 0.6502 0.557 0.593time*time -2.3396 0.6767 -3.457 0.009temp*temp -3.2896 0.6767 -4.861 0.001time*temp -4.8750 0.9195 -5.302 0.001S = 1.839 R-Sq = 89.2% R-Sq(adj) = 82.5%
Estimated Regression Coefficients for yield using data in uncoded units
Term CoefConstant -4138.6980time 18.2104temp 47.0066time*time -0.0234temp*temp -0.1316time*temp -0.0975
Yield = -4139 + 18.21*time + 47.01*temp -0.0234*time+time -0.1316*temp*temp – 0.0975*time*temp
Central Composite Design (CCD)
Deleted Residual
Perc
ent
420-2-4
99
90
50
10
1
Fitted Value
Del
eted
Res
idua
l
90858075
4
2
0
-2
Deleted Residual
Freq
uenc
y
3210-1-2-3
4.8
3.6
2.4
1.2
0.0
Observation Order
Del
eted
Res
idua
l
1413121110987654321
4
2
0
-2
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Histogram of the Residuals Residuals Versus the Order of the Data
Residual Plots for yield
Central Composite Design (CCD)
time
tem
p
10095908580
150.0
147.5
145.0
142.5
140.0
yield
77 - 8181 - 8585 - 89
> 89
< 7373 - 77
Contour Plot of yield vs temp, time
Central Composite Design (CCD)
152148
yield
70
80
144 temp
90
80 14090 100time
Surface Plot of yield vs temp, time