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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 10 optimization designs
Chapter 10Optimization Designs


Optimization designs

C

S

R

O

R

Optimization Designs

Focus: A Few Continuous Factors

Output: Best Settings

Reference: Box, Hunter & Hunter

Chapter 15


Optimization designs1
Optimization Designs

C

S

R

O

2k with Center Points

Central Composite Des.

Box-Behnken Des.

R


Response surface methodology
Response Surface Methodology

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


Response surface methodology1
Response Surface Methodology

  • 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





Quadratic models can only take certain forms
Quadratic Models can only take certain forms

Maximum

Minimum

Saddle Point

Stationary Ridge


Sequential assembly of experimental designs as needed
Sequential Assembly of Experimental Designs as Needed

  • Full Factorial

  • w/Center points

  • Main effects

  • Interactions

  • Curvature check

  • Central Composite Design

  • Full quadratic model

  • Fractional Factorial

  • Linear model



Strategy
Strategy

  • Fit a first order model

  • Do a curvature check to determine next step


Use a 2 2 factorial design with center points
Use a 22 Factorial Design With Center Points

x1x2

1 – –

2 + –

3 – +

4 + +

5 0 0

6 0 0

7 0 0


Plot the data
Plot the Data

135

64.6

68.0

60.3

64.3

62.3

130

Temperature (C)

54.3

60.3

125

70

80

Time (min)




Curvature check by interactions
Curvature Check by Interactions

  • 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.


Curvature check with center points
Curvature Check with Center Points

  • 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.


Curvature check with center points1
Curvature Check with Center Points

No Evidence of Curvature!!



Path of steepest ascent move 4 50 units in x 2 for every 2 35 units in x 1
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


The path in the original factors
The Path in the Original Factors

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)




Exploring the Path of Steepest Ascent

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)


A Second Factorial

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)



Curvature check by interactions1
Curvature Check by Interactions

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.



A Central Composite Design

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)



Must use regression to find the predictive equation
Must Use Regression to find the Predictive Equation

The Quadratic Model in Coded units


Must use regression to find the predictive equation1
Must Use Regression to find the Predictive Equation

The Quadratic Model in Original units


The Fitted Surface in the Region of Interest

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)


Quadratic models can only take certain forms1
Quadratic Models can only take certain forms

Maximum

Minimum

Saddle Point

Stationary Ridge


Canonical analysis

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







Two Dimensional Example

x

2

x

1

a is a constant


Quadratic response surfaces
Quadratic Response Surfaces

  • 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 importance of the eigenvalues
The Importance of the Eigenvalues

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

Eigenvalues

Type of Surface

Minimum

All eigenvalues positive

Saddle Point

Some eigenvalues positive and some negative

Maximum

All eigenvalues negative

Ridge

At least one eigenvalue zero


Stationary ridges in a response surface
Stationary Ridges in a Response Surface

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


Response surface methodology2
Response Surface Methodology

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

Follow Path

Single Experiments until Improvement stops

Center Points (Curvature Check)

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

Find Curvature

Central Composite Designs

Fit Full Quadratic Model

Model Curvature

Continued on Next Page


Tools

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


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