chapter 11 response surface methods and other approaches to

ˆ y = 79.94 + 0.99 x1 + 0.52 x2 + 0.25 x1 x2 −1.38 x + −1.00 x
2 1 2 2
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Pபைடு நூலகம்E R T P lo t
1 8 0 .0 0
y i e ld
1 7 7 .5 0
• The contour plot is given in the natural variables (see Figure 11.11)
y = f ( x1 , x 2 ) + ε
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• Response surface: (see Figure 11.1 & 11.2)
η = E ( y ) = f ( x1 , x 2 )
• The function f is unknown • Approximate the true relationship between y and the independent variables by the lower-order polynomial model.
2 y = β 0 + β1 x1 + β 2 x2 + β12 x1 x2 + β11 x12 + β 22 x2 + ε
• 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.
Response 1 yield percent 39.3 40.9 40 41.5 40.3 40.5 40.7 40.2 40.6
ˆ y = 40.44 + 0.775 x1 + 0.325 x2
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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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•
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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78.2573 79.5606 78.9089 77.6056
1 7 5 .0 0
• The optimum is at about 87 minutes and 176.5 degrees
78.2573
B: te m p
1 7 2 .5 0
76.954
1 7 0 .0 0 8 0 .0 0 8 2 .5 0 8 5 .0 0 8 7 .5 0
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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 = β 0 + β1 x1 + L + β k x k + ε y = β 0 + ∑ β i xi + ∑ β ii x + ∑ β ij xi x j + ε
i =1 i =1 2 i i< j k k
• 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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• Based on the first-order model,
ˆ ˆ ˆ y = β 0 + ∑ β i xi
i =1 k
• 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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•
Assume the first-order model k ˆ ˆ ˆ y=β + β x
0
∑
i =1
i
i
1. Choose a step size in one process variable, xj. ˆ βi 2. The step size in the other variable, ∆xi = ˆ β j / ∆x j 3. Convert the xj from coded variables to the natural variable
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• Characterizing the response surface: – Contour plot or Canonical analysis – Canonical form (see Figure 11.9)
2 ˆ ˆ y = y s + λ1 w12 + L + λk wk
– Minimum response: i are all positive – Maximum response: i are all negative – Saddle point: i have different signs
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• The second-order model:
ˆ ˆ y = β 0 + x' b + x' Bx, ˆ ˆ β1 β11 x1 ˆ x 2 , b = β 2 and B = x= M M ˆ β k xk 1 −1 xs = − B b 2 ˆ 1 ˆ s = β 0 + x 's b y 2 ˆ ˆ β12 / 2 L β1k / 2 ˆ ˆ / 2 β 22 L β 2 k O ˆ β kk
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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
9 0 .0 0
A : ti m 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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• 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 1.78 Prob > F < 0.0001 < 0.0001 0.0009 < 0.0001 < 0.0001 0.1022 0.2897