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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.

),( 21 xxfy

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• Response surface: (see Figure 11.1 & 11.2)

• The function f is unknown• Approximate the true relationship between y and

the independent variables by the lower-order polynomial model.

• Response surface design

),()( 21 xxfyE

ji

jiij

k

iiii

k

iii

kk

xxxxy

xxy

1

2

10

110

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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,

• 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

k

iii xy

10

ˆˆˆ

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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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Factor 1 Factor 2 Response 1Std Run Block A:Time B:Temp yield

minutes degC percent1 7 { 1 } -1 -1 39.32 6 { 1 } 1 -1 40.93 5 { 1 } -1 1 404 2 { 1 } 1 1 41.55 9 { 1 } 0 0 40.36 4 { 1 } 0 0 40.57 1 { 1 } 0 0 40.78 3 { 1 } 0 0 40.29 8 { 1 } 0 0 40.6

1 2ˆ 40.44 0.775 0.325y x x

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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

1. Choose a step size in one process variable, xj.

2. The step size in the other variable,

3. Convert the xj from coded variables to the

natural variable

k

iii xy

10

ˆˆˆ

jj

ii

xx

/ˆ

ˆ

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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.

• 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.

2 20 1 1 2 2 12 1 2 11 1 22 2y x x x x x x

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• The second-order model:

bxy

bBx

Bbx

Bxxbx

s

1s

'0

222

11211

2

1

2

1

0

2

1ˆˆ

2

1

ˆ

2/ˆˆ2/ˆ2/ˆˆ

and

ˆ

ˆ

ˆ

,

,''ˆˆ

s

kk

k

k

kkx

x

x

y

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• Characterizing the response surface:– Contour plot or Canonical analysis– Canonical form (see Figure 11.9)

– Minimum response: i are all positive

– Maximum response: i are all negative

– Saddle point: i have different signs

2211ˆˆ kks wwyy

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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 ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 28.25 5 5.65 79.85 < 0.0001A 7.92 1 7.92 111.93 < 0.0001B 2.12 1 2.12 30.01 0.0009A2 13.18 1 13.18 186.22 < 0.0001B2 6.97 1 6.97 98.56 < 0.0001AB 0.25 1 0.25 3.53 0.1022Residual 0.50 7 0.071Lack of Fit 0.28 3 0.094 1.78 0.2897Pure Error 0.21 4 0.053Cor Total 28.74 12

1 2 1 2

2 21 2

ˆ 79.94 0.99 0.52 0.25

1.38 1.00

y x x x x

x x

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• The contour plot is given in the natural variables (see Figure 11.11)

• The optimum is at about 87 minutes and 176.5 degrees

DESIGN-EXPERT Plot

yieldX = A: timeY = B: temp

Design Points

yield

A: time

B: te

mp

80.00 82.50 85.00 87.50 90.00

170.00

172.50

175.00

177.50

180.00

76.954

77.6056

78.2573

78.2573

78.9089

79.5606

5

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• The relationship between x and w:

– 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

)(' sxxMw

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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:

– 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.

xX)(X'x'x 1 2))(ˆ( yVar

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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