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14-1 Introduction
An experiment is a test or series of tests.
The design of an experiment plays a major role in
the eventual solution of the problem.
In a factorial experimental design, experimental
trials (or runs) are performed at all combinations of
the factor levels.
The analysis of variance (ANOVA) will be used as
one of the primary tools for statistical data analysis.
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14-2 Factorial Experiments
Definition
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14-2 Factorial Experiments
Figure 14-3 Factorial Experiment, no interaction.
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14-2 Factorial Experiments
Figure 14-4 Factorial Experiment, with interaction.
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14-2 Factorial Experiments
Figure 14-5 Three-dimensional surface plot of the data from
Table 14-1, showing main effects of the two factorsA and B.
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14-2 Factorial Experiments
Figure 14-6 Three-dimensional surface plot of the data from
Table 14-2, showing main effects of theA and B interaction.
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14-2 Factorial Experiments
Figure 14-7 Yield versus reaction time with temperature
constant at 155 F.
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14-2 Factorial Experiments
Figure 14-8 Yield versus temperature with reaction time
constant at 1.7 hours.
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14-2 Factorial Experiments
Figure 14-9
Optimizationexperiment using the
one-factor-at-a-time
method.
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14-3 Two-Factor Factorial Experiments
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14-3 Two-Factor Factorial Experiments
The observations may be described by the linear
statistical model:
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
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14-3 Two-Factor Factorial Experiments
To test H0: Xi = 0 use the ratio
14-3.1 Statistical Analysis of the Fixed-Effects Model
To test H0: Fj = 0 use the ratio
To test H0: (XF)ij = 0 use the ratio
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Definition
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
Figure 14-10 Graph
of average adhesionforce versus primer
types for both
application methods.
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14-3 Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Minitab Output for Example 14-1
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14-3 Two-Factor Factorial Experiments
14-3.2 Model Adequacy Checking
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14-3 Two-Factor Factorial Experiments
14-3.2 Model Adequacy Checking
Figure 14-11
Normal probabilityplot of the residuals
from Example 14-1
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14-3 Two-Factor Factorial Experiments
14-3.2 Model Adequacy Checking
Figure 14-12 Plot of residuals versus primer type.
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14-3 Two-Factor Factorial Experiments
14-3.2 Model Adequacy Checking
Figure 14-13 Plot of residuals versus application method.
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14-3 Two-Factor Factorial Experiments
14-3.2 Model Adequacy Checking
Figure 14-14 Plot of residuals versus predicted values.
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14-4 General Factorial Experiments
Model for a three-factor factorial experiment
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14-4 General Factorial Experiments
Example 14-2
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Example 14-2
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14-4 General Factorial Experiments
Example 14-2
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14-5 2kFactorial Designs
14-5.1 22 Design
Figure 14-15 The 22 factorial design.
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14-5 2kFactorial Designs
14-5.1 22 Design
The main effect of a factor A is estimated by
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14-5 2kFactorial Designs
14-5.1 22 Design
The main effect of a factor B is estimated by
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14-5 2kFactorial Designs
14-5.1 22 Design
The AB interaction effect is estimated by
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14-5 2kFactorial Designs
14-5.1 22 Design
The quantities in brackets in Equations 14-11, 14-12, and 14-
13 are called contrasts. For example, theA contrast is
ContrastA = a + abb (1)
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14-5 2kFactorial Designs
14-5.1 22 Design
Contrasts are used in calculating both the effect estimates and
the sums of squares forA,B, and theAB interaction. The
sums of squares formulas are
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14-5 2kFactorial Designs
Example 14-3
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14-5 2kFactorial Designs
Example 14-3
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14-5 2kFactorial Designs
Example 14-3
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14-5 2kFactorial Designs
Residual Analysis
Figure 14-16
Normal
probability plot of
residuals for the
epitaxial process
experiment.
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14-5 2kFactorial Designs
Residual Analysis
Figure 14-17 Plot
of residualsversus deposition
time.
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14-5 2kFactorial Designs
Residual Analysis
Figure 14-18 Plot
of residualsversus arsenic
flow rate.
k
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14-5 2kFactorial Designs
Residual Analysis
Figure 14-19 The standard deviation of epitaxial layer
thickness at the four runs in the 2
2
design.
k
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14-5 2kFactorial Designs
14-5.2 2kDesign for ku 3 Factors
Figure 14-20T
he 2
3
design.
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Figure 14-21 Geometric
presentation of contrasts
corresponding to the
main effects andinteraction in the 23
design. (a) Main effects.
(b) Two-factor
interactions. (c) Three-
factor interaction.
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14 5 2k F i l D i
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14-5 2kFactorial Designs
14-5.2 2kDesign for ku 3 Factors
The main effect ofC is estimated by
The interaction effect ofAB is estimated by
14 5 2k F t i l D i
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14-5 2kFactorial Designs
14-5.2 2kDesign for ku 3 Factors
Other two-factor interactions effects estimated by
The three-factor interaction effect,ABC, is estimated by
14 5 2k F t i l D i
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14-5 2kFactorial Designs
14-5.2 2kDesign for ku 3 Factors
14 5 2k F t i l D i
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14-5 2kFactorial Designs
14-5.2 2kDesign for ku 3 Factors
14 5 2k F t i l D i
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14-5 2kFactorial Designs
14-5.2 2kDesign for ku 3 Factors
Contrasts can be used to calculate several quantities:
14 5 2k F t i l D i
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14-5 2kFactorial Designs
Example 14-4
14 5 2k F t i l D i
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14-5 2kFactorial Designs
Example 14-4
14 5 2k F t i l D i
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14-5 2kFactorial Designs
Example 14-4
14 5 2k F t i l D i
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14-5 2kFactorial Designs
Example 14-4
14 5 2k Factorial Designs
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14-5 2kFactorial Designs
Example 14-4
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Example 14-4
14 5 2k Factorial Designs
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14-5 2kFactorial Designs
Residual Analysis
Figure 14-22 Normal
probability plot of
residuals from thesurface roughness
experiment.
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14-5 2kFactorial Designs
14-5.3 Single Replicate of the 2kDesign
Example 14-5
14 5 2k Factorial Designs
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14-5 2kFactorial Designs
14-5.3 Single Replicate of the 2kDesign
Example 14-5
14 5 2k Factorial Designs
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14-5 2kFactorial Designs
14-5.3 Single Replicate of the 2kDesign
Example 14-5
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14 5 2k Factorial Designs
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14-5 2kFactorial Designs
14-5.3 Single Replicate of the 2kDesign
Example 14-5
14 5 2k Factorial Designs
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14-5 2kFactorial Designs
14-5.3 Single Replicate of the 2kDesign
Example 14-5
Figure 14-23
Normal probability
plot of effects from
the plasma etchexperiment.
14-5 2kFactorial Designs
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14-5 2 Factorial Designs
14-5.3 Single Replicate of the 2kDesign
Example 14-5
Figure 14-24 AD (Gap-Power) interaction from the
plasma etch experiment.
14-5 2kFactorial Designs
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14-5 2 Factorial Designs
14-5.3 Single Replicate of the 2kDesign
Example 14-5
14-5 2kFactorial Designs
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14-5 2 Factorial Designs
14-5.3 Single Replicate of the 2kDesign
Example 14-5
Figure 14-25
Normal probability
plot of residuals
from the plasmaetch experiment.
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14-5 2kFactorial Designs
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14-5 2 Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Figure 14-26 A 22
Design with center
points.
14-5 2kFactorial Designs
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14 5 2 Factorial Designs
14-5.4 Additional Center Points to a 2k Design
A single-degree-of-freedom sum of squares
for curvature is given by:
14-5 2kFactorial Designs
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14 5 2 Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Example 14-6
Figure 14-27 The
22 Design with five
center points for
Example 14-6.
14-5 2kFactorial Designs
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14 5 2 Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Example 14-6
14-5 2kFactorial Designs
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14 5 2 Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Example 14-6
14-5 2kFactorial Designs
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14 5 2 Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Example 14-6
14-6 Blocking and Confounding in the 2k
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14 6 Blocking and Confounding in the 2
Design
Figure 14-28 A 22 design in two blocks. (a) Geometric view. (b)
Assignment of the four runs to two blocks.
14-6 Blocking and Confounding in the 2k
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14 6 Blocking and Confounding in the 2
Design
Figure 14-29 A 23 design in two blocks withABCconfounded. (a)
Geometric view. (b) Assignment of the eight runs to two blocks.
14-6 Blocking and Confounding in the 2k
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14 6 Blocking and Confounding in the 2
Design
General method of constructing blocks employs a
defining contrast
14-6 Blocking and Confounding in the 2k
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14 6 Blocking and Confounding in the 2
Design
Example 14-7
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Example 14-7
Figure 14-30 A 24 design in two blocks for Example 14-7. (a)
Geometric view. (b) Assignment of the 16 runs to two blocks.
14-6 Blocking and Confounding in the 2k
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14 6 Blocking and Confounding in the 2
Design
Example 14-7
Figure 14-31 Normal
probability plot of the effects
from Minitab, Example 14-7.
14-6 Blocking and Confounding in the 2k
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14 6 Blocking and Confounding in the 2
Design
Example 14-7
14 7 F ti l R li ti f th 2k D i
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14-7 Fractional Replication of the 2kDesign
14-7.1 One-Half Fraction of the 2kDesign
14 7 F ti l R li ti f th 2k D i
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14-7 Fractional Replication of the 2kDesign
14-7.1 One-Half Fraction of the 2kDesign
Figure 14-32 The one-half fractions of the 23 design. (a) The
principal fraction, I = +ABC. (B) The alternate fraction, I = -ABC
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14 7 F ti l R li ti f th 2k D i
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14-7 Fractional Replication of the 2kDesign
Example 14-8
Figure 14-33 The 24-1 design for the experiment of Example 14-8.
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14-7 Fractional Replication of the 2kDesign
Example 14-8
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14 7 F ti l R li ti f th 2k D i
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14-7 Fractional Replication of the 2kDesign
Example 14-8
14 7 Fractional Replication of the 2k Design
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14-7 Fractional Replication of the 2kDesign
Example 14-8
Figure 14-34 Normal probability plot of the effects from
Minitab, Example 14-8.
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14-7 Fractional Replication of the 2kDesign
Projection of the 2k-1 Design
Figure 14-35 Projection of a 23-1 design into three 22 designs.
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14-7 Fractional Replication of the 2kDesign
Projection of the 2k-1 Design
Figure 14-36 The 22 design obtained by dropping factors B
and Cfrom the plasma etch experiment in Example 14-8.
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14-7 Fractional Replication of the 2kDesign
Design Resolution
14 7 Fractional Replication of the 2k Design
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14-7 Fractional Replication of the 2kDesign
14-7.2 Smaller Fractions: The 2k-p
FractionalFactorial
14 7 Fractional Replication of the 2k Design
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14-7 Fractional Replication of the 2kDesign
Example 14-9
Example 14 8
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Example 14-8
14 7 Fractional Replication of the 2k Design
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14-7 Fractional Replication of the 2kDesign
Example 14-9
14 7 Fractional Replication of the 2kDesign
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14-7 Fractional Replication of the 2 Design
Example 14-9
Figure 14-37 Normalprobability plot of effects
for Example 14-9.
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14-7 Fractional Replication of the 2 Design
Example 14-9
Figure 14-38 Plot ofAB(mold temperature-screw
speed) interaction for
Example 14-9.
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14-7 Fractional Replication of the 2kDesign
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14-7 Fractional Replication of the 2 Design
Example 14-9
Figure 14-39 Normalprobability plot of
residuals for Example
14-9.
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14-7 Fractional Replication of the 2 Design
Example 14-9
Figure 14-40 Residualsversus holding time (C)
for Example 14-9.
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14-7 Fractional Replication of the 2 Design
Example 14-9
Figure 14-41 Average shrinkage and range of shrinkage in
factorsA, B, and Cfor Example 14-9.
14-8 Response Surface Methods and Designs
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14 8 Response Surface Methods and Designs
Response surface methodology, orRSM , is a
collection of mathematical and statistical techniques
that are useful for modeling and analysis in
applications where a response of interest is
influenced by several variables and the objective isto optimize this response.
14-8 Response Surface Methods and Designs
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14 8 Response Surface Methods and Designs
Figure 14-42 A three-dimensional response surface showing
the expected yield as a function of temperature and feed
concentration.
14-8 Response Surface Methods and Designs
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14 8 Response Surface Methods and Designs
Figure 14-43 A contour plot of yield response surface in Figure
14-42.
14-8 Response Surface Methods and Designs
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14 8 Response Surface Methods and Designs
The first-ordermodel
The second-ordermodel
14-8 Response Surface Methods and Designs
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14 8 Response Surface Methods and Designs
Methodof Steepest Ascent
14-8 Response Surface Methods and Designs
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14 8 Response Surface Methods and Designs
Methodof Steepest Ascent
Figure 14-41 First-order
response surface andpath of steepest ascent.
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14-8 Response Surface Methods and Designs
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14 8 Response Surface Methods and Designs
Example 14-11
Figure 14-45 Response surface plots for the first-order
model in the Example 14-11.
14-8 Response Surface Methods and Designs
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14 8 Response Surface Methods and Designs
Example 14-11
Figure 14-46 Steepest ascent experiment for Example
14-11.
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