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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 problems 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.

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.

),( 21 xxfy

),()( 21 xxfyE

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Response surface design First-order model:

Second-order model:

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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

From Figure 11.3,analysis can be thought as 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 region of the x’s.

The method of steepest ascent: A procedure for moving sequentially along the path of steepest ascent (in the direction of the maximum increase in the response).

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Based on the first-order model,

From Figure 11.4, the contours of is a series of parallel lines. The direction is parallel to the normal to fitted response surface.

The path of steepest ascent // the regression coefficient

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 error2. Check for interactions in the model3. 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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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

/ˆˆ

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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

21ˆˆ

21

ˆ

2/ˆˆ2/ˆ2/ˆˆ

and

ˆ

ˆˆ

,

,''ˆˆ

s

kk

k

k

kkx

xx

y

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Characterizing the response surface: Characterize to determine whether the stationary point is a point

of max or min response or saddle point. Contour plot or Canonical analysis Canonical form (see Figure 11.9)

Minimum response: are all positive Maximum response: are all negative Saddle point: 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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DESIGN-EXPERT Plot

yieldX = A: timeY = B: temp

Design Points

yield

A: time

B: t

emp

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

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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

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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)- mainly useful when we have two or maybe three controllable factors but in higher dimensions it loses its efficiency. This method simply consists of overlaying contour plot for each of the responses one over another in the controllable factors space and finding the area which makes the best possible value for each of the responses.

Constrained optimization problem-treat one of the responses as the objective of a constrained optimization problem and other responses as the constraints where the constraint’s boundary is to be determined by the decision maker (DM).

)(' 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 The class includes 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 because 2k design doesn’t afford an estimate of experimental error unless some runs are replicated.

Simplex design

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Designs for fitting the second-order model Central composite design (CCD) Consist of nF runs on 2k axial or star points, and nC center

runs Sequential experimentation Two parameters: nC and must be specified. Rotatability: 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