Transcript Chapter 11 Response Surface Methods and Other Approaches

Chapter 11 Response Surface
Methods and Other Approaches to
Process Optimization
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11.1 Introduction to Response
Surface Methodology
• Response Surface Methodology (RSM) is useful
for the modeling and analysis of programs in
which a response of interest is influenced by
several variables and the objective is to optimize
this response.
• For example: Find the levels of temperature (x1)
and pressure (x2) to maximize the yield (y) of a
process.
y  f ( x1 , x2 )  
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• Response surface: (see Figure 11.1 & 11.2)
  E( y)  f ( x1 , x2 )
• The function f is unknown
• Approximate the true relationship between y and
the independent variables by the lower-order
polynomial model.
y   0  1 x1     k xk  
k
k
i 1
i 1
y   0    i xi    ii xi2    ij xi x j  
i j
• Response surface design
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• A sequential procedure
• The objective is to lead
the experimenter rapidly
and efficiently along a
path of improvement
toward the general
vicinity of the optimum.
• First-order model =>
Second-order model
• Climb a hill
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11.2 The Method of Steepest Ascent
• Assume that the first-order model is an adequate
approximation to the true surface in a small ragion
of the x’s.
• The method of steepest ascent: A procedure for
moving sequentially along the path of steepest
ascent.
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• Based on the first-order
model,
k
yˆ  ˆ0   ˆi xi
i 1
• The path of steepest ascent //
the regression coefficients
• The actual step size is
determined by the
experimenter based on
process knowledge or other
practical considerations
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•
Example 11.1
– Two factors, reaction time & reaction
temperature
– Use a full factorial design and center points
(see Table 11.1):
1. Obtain an estimate of error
2. Check for interactions in the model
3. Check for quadratic effect
• ANOVA table (see Table 11.2)
• Table 11.3 & Figure 11.5
• Table 11.4 & 11.5
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Std
Run
1
2
3
4
5
6
7
8
9
Block
7
6
5
2
9
4
1
3
8
{1}
{1}
{1}
{1}
{1}
{1}
{1}
{1}
{1}
Factor 1
A:Time
minutes
-1
1
-1
1
0
0
0
0
0
Factor 2
B:Temp
degC
-1
-1
1
1
0
0
0
0
0
Response 1
yield
percent
39.3
40.9
40
41.5
40.3
40.5
40.7
40.2
40.6
yˆ  40.44  0.775x1  0.325x2
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The step size is 5 minutes of reaction time and 2 degrees F
What happens at the conclusion of steepest ascent?
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•
Assume the first-order model
k
yˆ  ˆ  ˆ x
0

i 1
i
i
1. Choose a step size in one process variable,
xj.
ˆi
2. The step size in the other variable, xi 
ˆ j / x j
3. Convert the xj from coded variables to the
natural variable
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11.3 Analysis of a Second-order
Response Surface
• When the experimenter is relative closed to the
optimum, the second-order model is used to
approximate the response.
y  0  1x1  2 x2  12 x1x2  11x12  22 x22  
• Find the stationary point. Maximum response,
Minimum response or saddle point.
• Determine whether the stationary point is a point
of maximum or minimum response or a saddle
point.
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• The second-order model:
yˆ  ˆ0  x' b  x' Bx,
 ˆ1 
 ˆ11
 x1 
ˆ 

x 
2 
2


x
,b 
and B  
 


 

 
 ˆ k 

 xk 
1 1
xs   B b
2
1 '
ˆ
yˆ s   0  x s b
2
ˆ12 / 2  ˆ1k / 2 

ˆ
ˆ
 22   2 k / 2




ˆ kk
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• Characterizing the response surface:
– Contour plot or Canonical analysis
– Canonical form (see Figure 11.9)
ˆy  yˆ s  1w12    k wk2
– Minimum response: i are all positive
– Maximum response: i are all negative
– Saddle point: i have different signs
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• Example 11.2
– Continue Example 11.1
– Central composite design (CCD) (Table 11.6 &
Figure 11.10)
– Table 11.7
ANOVA for Response Surface Quadratic Model
Analysis of variance table [Partial sum of squares]
Source
Model
A
B
A2
B2
AB
Residual
Lack of Fit
Pure Error
Cor Total
Sum of
Squares
28.25
7.92
2.12
13.18
6.97
0.25
0.50
0.28
0.21
28.74
DF
5
1
1
1
1
1
7
3
4
12
Mean
Square
5.65
7.92
2.12
13.18
6.97
0.25
0.071
0.094
0.053
F
Value
79.85
111.93
30.01
186.22
98.56
3.53
Prob > F
< 0.0001
< 0.0001
0.0009
< 0.0001
< 0.0001
0.1022
1.78
0.2897
yˆ  79.94  0.99 x1  0.52 x2  0.25 x1 x2
1.38x  1.00 x
2
1
2
2
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ot
yield
180.00
• The contour plot is given
in the natural variables
(see Figure 11.11)
B: temp
177.50
• The optimum is at about
87 minutes and 176.5
degrees
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175.00
78.2573
79.5606
78.9089
172.50
77.6056
76.954
78.2573
170.00
80.00
82.50
85.00
87.50
90.00
A: tim e
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• The relationship between x and w:
w  M' (x  x s )
– M is an orthogonal matrix and the columns of
M are the normalized eigenvectors of B.
• Multiple response:
– Typically, we want to simultaneously optimize
all responses, or find a set of conditions where
certain product properties are achieved
– Overlay the contour plots (Figure 11.16)
– Constrained optimization problem
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11.4 Experimental Designs for
Fitting Response Surfaces
• Designs for fitting the first-order model
– The orthogonal first-order designs
– X’X is a diagonal matrix
– 2k factorial and fractions of the 2k series in
which main effects are not aliased with each
others
– Besides factorial designs, include several
observations at the center.
– Simplex design
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• Designs for fitting the second-order model
– Central composite design (CCD)
– nF runs on 2k axial or star points, and nC center
runs
– Sequential experimentation
– Two parameters: nC and 
– The variance of the predicted response at x:
Var( yˆ (x))   2 x'(X' X) 1 x
– Rotatable design: The variance of predicted
response is constant on spheres
– The purpose of RSM is optimization and the
location of the optimum is unknown prior to
running the experiment.
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–  = (nF)1/4 yields a rotatable central composite
design
– The spherical CCD: Set  = (k)1/2
– Center runs in the CCD, nC: 3 to 5 center runs
– The Box-Behnken design: three-level designs
(see Table 11.8)
– Cuboidal region:
• face-centered central composite design (or
face-centered cube)
•=1
• nC=2 or 3
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