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# Chapter 10 Optimization Designs - PowerPoint PPT Presentation

Chapter 10 Optimization Designs. C. S. R. O. R. Optimization Designs. Focus : A Few Continuous Factors Output : Best Settings Reference : Box, Hunter & Hunter Chapter 15 . Optimization Designs. C. S. R. O. 2 k with Center Points Central Composite Des. Box-Behnken Des. R.

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Chapter 10Optimization Designs

S

R

O

R

Optimization Designs

Focus: A Few Continuous Factors

Output: Best Settings

Reference: Box, Hunter & Hunter

Chapter 15

C

S

R

O

2k with Center Points

Central Composite Des.

Box-Behnken Des.

R

A Strategy of Experimental Design for finding optimum setting for factors. (Box and Wilson, 1951)

• When you are a long way from the top of the mountain, a slope may be a good approximation

• You can probably use first–order designs that fit a linear approximation

• When you are close to an optimum you need quadratic models and second–order designs to model curvature

Maximum

Minimum

Stationary Ridge

• Full Factorial

• w/Center points

• Main effects

• Interactions

• Curvature check

• Central Composite Design

• Fractional Factorial

• Linear model

• Fit a first order model

• Do a curvature check to determine next step

Use a 22 Factorial Design With Center Points

x1x2

1 – –

2 + –

3 – +

4 + +

5 0 0

6 0 0

7 0 0

135

64.6

68.0

60.3

64.3

62.3

130

Temperature (C)

54.3

60.3

125

70

80

Time (min)

• Are there any large two factor interactions? If not, then there is probably not significant curvature.

• If there are many large interaction effects then we should not follow a path of steepest ascent because there is curvature.

Here, time=4.7, temp =9.0, and time*temp=-1.3, so the main effects are larger and thus there is little curvature.

• Compare the average of the factorial points, , with the average of the center points, . If they are close then there is probably no curvature?

Statistical Test:

If zero is in the confidence interval then there is no evidence of curvature.

No Evidence of Curvature!!

Path of Steepest Ascent: Move 4.50 Units in x2 for Every 2.35 Units in x1

Equivalently, for every

one unit in x1 we

move 4.50/2.35=1.91

units in x2

x

2

4.50

1.91

x

1.0

2.35

1

145

140

135

Temperature (C)

64.6

68.0

60.3

64.3

62.3

130

54.3

60.3

125

60

70

80

90

Time (min)

155

58.2

150

145

86.8

Temperature (C)

140

135

73.3

64.6

68.0

60.3

64.3

62.3

130

54.3

60.3

125

110

60

70

80

90

100

Time (min)

155

150

91.2

77.4

86.8

89.7

145

Temperature (C)

140

78.8

84.5

135

64.6

68.0

60.3

64.3

62.3

130

54.3

60.3

125

110

60

70

80

90

100

Time (min)

Here, time=-4.05, temp=2.65, and time*temp=-9.75, so the interaction term is the largest, so there appears to be curvature.

155

79.5

150

91.2

77.4

87.0

86.0

145

83.3

81.2

Temperature (C)

140

78.8

84.5

81.2

135

64.6

68.0

60.3

64.3

62.3

130

54.3

60.3

125

110

60

70

80

90

100

Time (min)

The Quadratic Model in Coded units

The Quadratic Model in Original units

155

79.5

150

91.2

77.4

87.0

86.0

145

83.3

81.2

Temperature (C)

140

78.8

84.5

81.2

135

110

70

80

90

100

Time (min)

Maximum

Minimum

Stationary Ridge

### Canonical Analysis

Enables us to analyze systems of maxima and minima in many dimensions and, in particular to identify complicated ridge systems, where direct geometric representation is not possible.





Two Dimensional Example

x

2

x

1

a is a constant

• Any two quadratic surfaces with the same eigenvalues (’s) are just shifted and rotated versions of each other.

• The version which is located at the origin and oriented along the axes is easy to interpret without plots.

• The shape of the surface is determined by the signs and magnitudes of the eigenvalues.

Eigenvalues

Type of Surface

Minimum

All eigenvalues positive

Some eigenvalues positive and some negative

Maximum

All eigenvalues negative

Ridge

At least one eigenvalue zero

• The existence of stationary ridges can often be exploited to maintain high quality while reducing cost or complexity.

Tools

Goals

Select Factors and Levels

and Responses

Brainstorming

Eliminate Inactive Factors

Fractional Factorials (Res III)

Fractional Factorials (Higher Resolution from projection or additional runs)

Find Path of Steepest Ascent

Single Experiments until Improvement stops

Center Points (Curvature Check)

Fractional Factorials (Res > III) (Interactions > Main Effects)

Find Curvature

Central Composite Designs

Model Curvature

Continued on Next Page

Goals

Canonical Analysis

• A-Form if Stationary Point is outside Experimental Region

• B-Form if Stationary Point is inside Experimental Region

Understand the shape of the surface

Find out what the optimum looks like

Point, Line, Plane, etc.

Reduce the Canonical Form with the DLR Method

If there is a rising ridge, then follow it.

Translate the reduced model back and optimum formula back to original coordinates

If there is a stationary ridge, then find the cheapest place that is optimal.

Translate the reduced model back and optimum formula back to original coordinates