Executing Robust Design
Definition of Robust Design
Robustness is defined as a condition in which the
product or process will be minimally affected by
sources of variation.
A product can be robust:
Against variation in raw materials
Against variation in manufacturing conditions
Against variation in manufacturing personnel
Against variation in the end use environment
` Against variation in end-users
Against wear-out or deterioration
Why We Need to Reduce Variation
LSL
Nom
USL
Cost
Low Variation;
Minimum Cost
Cost
LSL
Nom
USL
High Variation;
High Cost
Purpose of this Module
 To introduce a variation improvement investigation
strategy
– Can noise factors be manipulated?
 To provide the MINITAB steps to design, execute,
and analyze a variability response experiment
 To provide the MINITAB steps to optimize a design
for both mean and variation effects
Objectives of this Module
At the end of this module, participants will be able to :
 Identify possible variation effects from residual plots
 Create a variability response from replicates
 Identify possible mean and variance adjustment
factors from noise-factor interaction plots
 Use the MINITAB Response Optimizer to achieve a
process on target with minimum variation
Strategies to Detect Variation Effects
 Passive Approach
– Noise factors are NOT included, manipulated or controlled in
the experimental design
– Possible variation effects are identified through analysis of
the variability of replicates from an experimental design
 Active Approach
– Noise factors ARE included in the experimental design in
order to force variability to occur
– Analysis is similar to the passive approach
The Passive Approach
 A factorial experiment is performed using Control
factors. Noise factors are not explicitly manipulated
nor is an attempt made to control them during the
course of the experiment.
 Pros
– Simple extension of standard experimental techniques
– Does not require explicit identification of noise factors
 Cons
– Requires larger number of replicates than would typically be
required to determine mean effects
– Requires “true” randomization and replication
– Requires that noise factors be “noisy” during the execution of
the experiment
How to ensure that noise is noisy?
 Let excluded factors vary
 Compare noise factor variations prior to and within
DOE
– Monitor noise factor levels during normal process conditions
– Monitor noise factor variation during course of experiment
– Compare before/during levels
 Run DOE over a longer period of time with :
– More replicates
– Full randomization
A Passive Example
A
B
 A and B are control factors. Within each treatment combination,
noise factors are allowed to naturally fluctuate. Within treatment
variation is largely driven by this background noise.
Example output from a Passive Design
Mean Y
Variation Y
 The graphs at right
illustrate the type of output
which might be obtained
from a Robust Parameter
Design Experiment. Both
are Main Effects plots with
the top row showing the
main effects of factors A
and B on the mean and
the bottom row showing
the main effects of factors
A and B on the variation.
 Note that in this example
the mean and variation
can be adjusted
independently of each
other!
The Models
 Our objective in performing a designed experiment is to develop
a transfer function between the factors (X’s) and the Y. Thus
far, we have only addressed the mean of Y.
y  b0  b1 x1  b 2 x 2  b12 x1 x 2  
 Now we must also consider the variability of Y
s
2
y
 c 0  c1 x1  c 2 x 2  c12 x1 x 2  
 If our experiments are successful at identifying a variation effect,
we now have an opportunity to simultaneously optimize both
equations!
Example: A Passive Noise Experiment
 A design engineer has evaluated the output performance of a
circuit design and performed an initial capability analysis of this
design to determine if there is a problem with the mean and/or
the variability.
Stat > Quality Tools > Capability Analysis > Normal
Y = Y1Initial; Lower Spec = 58; Upper Spec = 62
Design Capability Analysis
 Is there a problem?
P r o c e s s C a pa bi lity o f Y 1 Initi a l
LS L
USL
P ro ce ss D a ta
W ith in
LS L
58
T a rg e t
*
USL
62
S a m p le M e a n
5 6 .2 5 3
S a m p le N
100
S tD e v (W ith in )
1 .4 7 2 1 7
S tD e v (O v e ra ll)
1 .5 4 1 4
O v er all
P o te n tia l (W ith in ) C a p a b ility
Cp
C PL
0 .4 5
-0 .4 0
C PU
1 .3 0
C pk
-0 .4 0
O v e ra ll C a p a b ility
Pp
PPL
PPU
1 .2 4
P pk
-0 .3 8
C pm
5 2 .5
O b se rv e d P e rfo rm a n ce
P P M < LS L
PPM > USL
P P M T o ta l
8 6 0 0 0 0 .0 0
0 .0 0
8 6 0 0 0 0 .0 0
5 4 .0
5 5 .5
5 7 .0
5 8 .5
6 0 .0
E xp . W ith in P e rfo rm a n ce
E xp . O v e ra ll P e rfo rm a n ce
P P M < LS L
P P M < LS L
PPM > USL
P P M T o ta l
8 8 2 3 2 4 .0 2
4 7 .3 5
8 8 2 3 7 1 .3 7
PPM > USL
P P M T o ta l
8 7 1 4 7 4 .4 8
9 6 .3 3
8 7 1 5 7 0 .8 1
6 1 .5
0 .4 3
-0 .3 8
*
24 Full Factorial Experiment
 A 24 full factorial has been designed to determine if four factors
have an effect on the mean and/or variability of voltage drop
(Y1). There are five replicates for a total of 80 runs, with no
center points or blocks.
–
–
–
–
Resistor R33 (A)
Inductor L3 (B)
Capacitor C23 (E)
Capacitor C29 (F)
 Worksheet: “Passive Design”
Passive Analysis Roadmap - Part 1
(for Mean Only)
 Analyze the response of interest
– Factorial Plots (Main Effects, Interaction)
– Statistical Results (ANOVA table and p-values)
– Residual Plots by factor
 Reduce model using statistical results
 Use the residuals plot to evaluate potential existence
of variation effects
– If residuals plot indicates a possible variation effect, go to
Passive Analysis Roadmap - Part 2
Interaction Plot
 Based on the interaction plot, a few of the interactions may be
significant. Check the statistical output for verification.
Inte r a c tio n P lo t ( da ta me a ns ) fo r Y 1
1
5
12.5
13.5
63
69
60.0
57.5
A
10
20
A
55.0
60.0
57.5
B
1
5
B
55.0
60.0
57.5
E
12.5
13.5
E
55.0
F
Stat > DOE > Factorial > Factorial Plots > Interaction Plot
Main Effects Plot
 The main effects plot indicates that factors B and E have the
largest effects. Factor A also has a moderate positive effect.
Factor F does not seem to be important. Let’s look at the results.
M a i n E f f e c ts P l o t ( da ta me a ns ) f o r Y 1
A
58
B
57
56
Mean of Y 1
55
54
10
20
1
E
58
5
F
57
56
55
54
1 2 .5
1 3 .5
63
69
Stat > DOE > Factorial > Factorial Plots > Main Effects Plot
Factorial Analysis
 A preliminary look at the statistical output of the experiment
indicates factor F may not be significant. Did we make a
mistake by including it in the experimental design?
Estimated Effects and Coefficients for Y1 (coded units)
Term
Constant
A
B
E
F
A*B
A*E
A*F
B*E
B*F
E*F
Effect
2.363
-3.192
3.312
0.463
0.178
0.002
-0.138
-0.933
-0.662
-0.007
Coef
55.991
1.181
-1.596
1.656
0.231
0.089
0.001
-0.069
-0.466
-0.331
-0.004
SE Coef
0.2041
0.2041
0.2041
0.2041
0.2041
0.2041
0.2041
0.2041
0.2041
0.2041
0.2041
T
274.27
5.79
-7.82
8.11
1.13
0.43
0.01
-0.34
-2.28
-1.62
-0.02
P
0.000
0.000
0.000
0.000
0.262
0.665
0.995
0.737
0.026
0.110
0.985
* Note that the 3-way and 4-way interactions are still in the model but not
presented in the output above
Stat > DOE > Factorial > Analyze Factorial Design
Reduce Model to Significant Terms
 Our final model indicates that factors A, B, and E are
significant, along with interactions BE, BF, ABF, and
BEF, using a p-value cut-off of 0.2
Estimated Effects and Coefficients for Y1 (coded units)
Term
Constant
A
B
E
F
B*E
B*F
A*B*F
B*E*F
Effect
2.363
-3.192
3.312
0.463
-0.932
-0.662
-0.513
0.838
S = 1.74046
Coef
55.991
1.181
-1.596
1.656
0.231
-0.466
-0.331
-0.256
0.419
SE Coef
0.1946
0.1946
0.1946
0.1946
0.1946
0.1946
0.1946
0.1946
0.1946
R-Sq = 73.11%
T
287.74
6.07
-8.20
8.51
1.19
-2.40
-1.70
-1.32
2.15
P
0.000
0.000
0.000
0.000
0.239
0.019
0.093
0.192
0.035
R-Sq(adj) = 70.08%
Stat > DOE > Factorial > Analyze Factorial Design
The Role of Residual Plots in RD
 In Robust Parameter Design, the residual plots can show the
possibility for a variation effect
 Remember from ANOVA and Regression, we stated one of the
assumptions on the residuals was constant variance and we
checked this via plots
Stat > DOE > Factorial > Analyze Factorial Design
Choose Graphs > Residuals vs Variables > A B E F
What Next?
 After reducing the model, the “Residuals versus Factor F” plot
still indicates that F contributes to a variation effect. This finding
should encourage us to move further in the analysis of this data
to create a variability response and analyze the data. Thus we
move on to Part 2 of the roadmap.
R e s idua ls V e r s us F
(r e s po ns e is Y1)
Stat > DOE > Factorial >
Analyze Factorial Design
Choose Graphs >
Residuals vs Variables > F
S t a nd a r d iz e d R e s id ua l
3
2
1
0
-1
-2
-3
63
64
65
66
F
67
68
69
Passive Analysis Roadmap - Part 2
(if Variation effect present)
 Create a Variability Response
 Analyze Variability
– Factorial Plots (Main Effects, Interaction)
– Statistical Results (ANOVA table and p-values)
– Reduce model using statistical results
 Compare main effects plots for mean and variability
to determine which are Mean Adjustment Factors and
which are Variance Adjustment Factors (or both)
 Use the Multiple Response Optimizer to find optimal
settings of the factors
– Mean on target
– Minimum variability
 Perform a capability study / analysis on the resulting
factor settings
Create a Variability Response
 We are now going to use the replications to make a
new response in order to model the variability. Once
we have modeled the variability, we can use the
MINITAB Response Optimizer to find the settings of
the control factors that will put Y on target with
minimum variation.
 MINITAB makes this easy with a pre-processing of
the responses in preparation for a variability analysis
 You will see that MINITAB will use the standard
deviation as the measure of variability, rather than
the variance
– the results are equivalent
Uses Natural Log (Standard Dev of Y)
 All of the statistical techniques that we are using to analyze this
DOE assume that the data is symmetric (because we are testing
for mean differences)
 Unfortunately, when we use a calculated standard deviation as
a response, we do not meet this assumption because the
sampling distribution of variances is expected to be skewed,
hence the distribution of standard deviations would also be
skewed
Raw St Dev
Ln (St Dev)
Create a Variability Response
Stat > DOE > Factorial > Pre-Process Responses for Analyze Variability
Create a Variability Response
 Worksheet should now contain the following new columns
Analyze the Variability
 Analysis of the variability will be essentially identical to the
analysis for the mean
 Will select the “Terms” to estimate in the model
 Will use the Pareto of Effects “Graph” in order to facilitate the
first model reduction
Stat > DOE > Factorial >
Analyze Variability
Analyze the Variability
TERMS
GRAPHS
Pareto Chart of the Effects
 Because we don’t have any degrees of freedom for error, we
must look at the Pareto of effects to decide which term to drop
into the error and begin to reduce the model
P a r e to C ha r t o f the E ffe c ts
(R e s po ns e is na tur a l lo g o f S tDe vY1 , A lpha = 0 .2 0 )
0.130
F a cto r
N am e
A
A
B
B
AC
C
E
B
D
F
D
A
AD
Te r m
C
A BC
BC D
BD
A BC D
AB
Drop ABD
interaction first
ACD
CD
BC
A BD
0.0
0.2
0 .4
0 .6
0 .8
Effe c t
Le nth's PS E = 0.0878 31 3
1.0
1 .2
1.4
Final Model for ln StDevY1
 Once the insignificant terms have been eliminated using a pvalue cut-off of 0.2, the reduced model is shown below
Regression Estimated Effects and Coefficients for Natural Log
of StDevY1 (coded units)
Term
Constant
A
B
E
F
A*B
A*E
A*F
B*F
A*B*E
B*E*F
A*B*E*F
Effect
Ratio
Effect
0.5139
-0.2953
-0.1521
-1.3384
0.0371
0.4451
-0.2573
-0.0781
0.1323
0.0850
0.0390
1.6718
0.7443
0.8589
0.2623
1.0378
1.5606
0.7731
0.9249
1.1414
1.0887
1.0398
R-Sq = 99.93%
Coef
0.0747
0.2569
-0.1477
-0.0760
-0.6692
0.0185
0.2225
-0.1287
-0.0390
0.0661
0.0425
0.0195
R-Sq(adj) = 99.75%
SE Coef
0.01013
0.01013
0.01013
0.01013
0.01013
0.01013
0.01013
0.01013
0.01013
0.01013
0.01013
0.01013
T
7.38
25.37
-14.58
-7.51
-66.08
1.83
21.98
-12.71
-3.86
6.53
4.20
1.93
P
0.002
0.000
0.000
0.002
0.000
0.141
0.000
0.000
0.018
0.003
0.014
0.126
Interaction Plot for StDevY1
 The interaction plot indicates a moderately strong interaction between
factors A & E and A & F
Inte r a c tio n P l o t ( da ta me a ns ) fo r S tD e v Y 1
1
5
12.5
13.5
63
69
3
A
10
2
A
20
1
3
B
1
2
B
5
1
3
E
12.5
2
E
13.5
Where should
factors A, B, E
and F be set in
order to
minimize the
variability in
voltage drop,
Y1?
1
F
Stat > DOE > Factorial > Factorial Plots > Interaction Plot
Main Effects Plot for StDevY1
 Factor F has the largest effect on the variability. Increasing F should
reduce variability. But what did the interaction plot show?
 Factor A is the next strongest. Set A = 10. What did the interaction
plot show?
 Factors B and E are weak but what did the interaction plot show?
M a in E ffe c ts P l o t ( da ta me a ns ) fo r S tD e v Y 1
A
2 .5
B
2 .0
M e a n o f S t De v Y 1
1 .5
1 .0
0 .5
10
20
1
E
2 .5
5
F
2 .0
1 .5
1 .0
0 .5
1 2 .5
1 3 .5
63
69
Stat > DOE > Factorial > Factorial Plots > Main Effects Plot
Determine Mean & Variation Effects
A
58
M ain Ef f ec ts P lot (data m eans ) f or Y 1
B
F
57
Me a n o f Y1
57
Me a n o f Y1
E
58
56
55
56
55
54
54
10
20
Affects Both
1
5
12.5
A
2.5
13.5
63
69
Affects Mean Affects Variation
Affects Mean
M ain Ef f ec ts P lot (data m eans ) f or StDev Y 1
M ain Ef f ec ts P lot (data m eans ) f or StD ev Y 1
B
E
2.5
F
2.0
2.0
M e a n o f S t D e vY 1
– Factors B and E are
Mean Adjustment
Factors since they
affect the mean with
little or no effect on the
variability
– Factor F is a Variance
Adjustment Factor
since it affects the
variability with little or
no effect on the mean
– Factor A appears to
affect both mean and
variability
M ain Ef f ec ts P lot (data m eans ) f or Y 1
M e a n o f S t D e vY 1
 The graphs at right allow
us to directly compare
each factor’s singular
effect on both the mean
and variation
 Based on these graphs,
1.5
1.5
1.0
1.0
0.5
0.5
10
20
1
5
12.5
13.5
63
69
Quality Check: Status of Your Models
 Use the “Show Design” icon
to check on the status of the
analysis. You should make
sure that the correct model
has been fit for each
response that you intend to
specify in the response
optimizer.
 As shown in this window,
models have been fit for both
the Y1 and StDevY1
responses
Multiple Response Optimizer
 Set the Weight for Y1 to 10 to ensure hitting 60, tight lower & upper range
 Read “first-guess” target & upper values for StDevY1 from interaction plots
– Note that StDevY1 is in regular units here, NOT logged units!
 For StDevY1, set weight low to protect against a bad first guess
Stat > DOE > Factorial > Response Optimizer
Multiple Response Optimizer
 We use the multiple response optimizer to provide a stacked
main effects plot. This plot allows us to interactively manipulate
the values of each factor in the model and see the effect on both
the mean and the variation.
Optimal
Hi
D
Cur
0.93698 Lo
A
20.0
[13.4878]
10.0
B
5.0
[1.0]
1.0
E
13.50
[13.50]
12.50
F
69.0
[69.0]
63.0
Y1
Targ: 60.0
y = 60.0001
d = 0.99884
StDevY1
Minimum
y = 0.5725
d = 0.87896
 You can use the red sliders to tune each of the factors
Use the Equations to Confirm Y1
 Let’s use the model coefficients to predict and see
that it matches (make sure to use un-coded)!
 From the optimized solution, A = 13.4878, B = 1, E =
13.5, F = 69
Y1 = -22.0994 + 0.214670*A + 45.2619*B + 4.71125*E
+ 0.242708*F - 3.26279*B*E - 0.607676*B*F +
0.000108988*A*B*F + 0.0423718*B*E*F
Y1 = -22.0994 + 0.214670*13.4878 + 45.2619*1 +
4.71125*13.5 + 0.242708*69 - 3.26279*1*13.5 0.607676*1*69 + 0.000108988*13.4878*1*69 +
0.0423718*1*13.5*69
Y1 = 60.0001, as seen in the optimizer window
Use the Equations to Confirm StDevY1
 Again, make sure to use the un-coded coefficients!
 Again, A = 13.4878, B = 1, E = 13.5, F = 69
lnStDevY1 = 15.4015 - 0.001778*A + 0.727714*B 0.927015*E - 0.0566905*F - 0.152626*A*B +
0.0533652*A*E - 0.00978998*A*F + 0.0242198*B*F +
0.00983241*A*B*E - 0.00282966*B*E*F +
0.0000317*A*B*E*F
lnStDevY1 = 15.4015 - 0.001778*13.4878 + 0.727714*1
- 0.927015*13.5 - 0.0566905*69 0.152626*13.4878*1 + 0.0533652*13.4878*13.5 0.00978998*13.4878*69 + 0.0242198*1*69 +
0.00983241*13.4878*1*13.5 - 0.00282966*1*13.5*69
+ 0.0000317*13.4878*1*13.5*69
lnStDevY1 = -0.5577
StDevY1 = e-0.5577 = 0.5725, as seen in the optimizer
Final Design Capability Analysis
 Did we achieve our objectives?
P r o c e s s C a pa bility o f Y 1 F ina l
LS L
USL
P ro ce ss D a ta
LS L
58
T a rg e t
*
USL
62
S a m p le M e a n
6 0 .3 1 8 3
S a m p le N
100
S tD e v (W ith in )
0 .1 2 0 0 8 4
S tD e v (O v e ra ll)
0 .1 1 5 9 5 5
W ith in
O v er all
P o te n tia l (W ith in ) C a p a b ility
Cp
5 .5 5
C PL
6 .4 4
C PU
4 .6 7
C pk
4 .6 7
O v e ra ll C a p a b ility
worksheet “passive capability”
Stat > Quality Tools > Capability
Analysis > Normal
Y = Y1Final; Lower Spec = 58;
Upper Spec = 62
Pp
5 .7 5
PPL
6 .6 6
PPU
4 .8 3
P pk
4 .8 3
C pm
58.30 58.85 59.40 59.95 60.50 61.05 61.60
O b se rv e d P e rfo rm a n ce
E xp . W ith in P e rfo rm a n ce
E xp . O v e ra ll P e rfo rm a n ce
P P M < LS L
0 .0 0
P P M < LS L
0 .0 0
P P M < LS L
0 .0 0
PPM > USL
0 .0 0
PPM > USL
0 .0 0
PPM > USL
0 .0 0
P P M T o ta l
0 .0 0
P P M T o ta l
0 .0 0
P P M T o ta l
0 .0 0
*
Remember the Two Strategies?
 We just reviewed the Passive Approach
– Noise factors are NOT included, manipulated or controlled in
the experimental design
– We analyzed the variability of replicates from an
experimental design
 Now we will look at the Active Approach
– Noise factors ARE included in the experimental design in
order to force variability to occur
– We will see that the analysis is similar to the passive
approach
The Active Approach
 A factorial experiment is performed using Control AND Noise
factors in the same experiment. Analysis can be performed by
characterizing Control*Noise interactions only or by moving
forward to analyze the variability by dropping the noise factors
into the error term.
 Pros
–
–
–
–
Simple extension of standard experimental techniques
Guarantees noise in the Noise factors
Provides for flexibility in analysis methods
Can allow for reduced replication
 Cons
– Requires ability to manipulate and control Noise factors
– Optimal designs for minimization of unneeded effects (noise by
noise interactions) can be difficult to create
Example: An Active Noise Experiment
 3 control factors
 1 noise factor
 24 full factorial design
 Two Approaches to Analysis
– Use only interpretation of interaction plots to choose settings
of the control factors to minimize effect of noise
– Model the variability by dropping the noise factors into the
error and analyze like the passive approach
Active Analysis Roadmap – Plots Only
 Create and execute with noise included as a factor
 Analyze the response of interest
– Factorial Plots (Main Effects, Interaction)
– Statistical Results (ANOVA table and p-values)
– Reduce model using statistical results
 Review Interactions Plot
– Interpret the interaction plots to look for evidence of variation
effects
 Review Main Effects Plot (if applicable)
 Use the Multiple Response Optimizer to find the optimal settings
of the factors such that the mean is on target
– Will force in settings obtained from the interaction plots
 Perform a capability study / analysis on the resulting factor
settings
Example: An Active Noise Experiment
 An engineer is interested in improving the stability and
robustness of a filtration product
 Review the capability of the current performance to determine
the opportunity to apply robust design techniques
Stat > Quality Tools > Capability Analysis > Normal
Y = Y4Initial; Lower Spec = 60; Upper Spec = 80
Design Capability Analysis
 What conclusions can you draw from this graph?
P r o c e s s C a pa bi lity o f Y 4 Initi a l
LS L
USL
P ro ce ss D a ta
W ith in
LS L
60
T a rg e t
*
USL
80
S a m p le M e a n
7 0 .3 2 1
S a m p le N
100
S tD e v (W ith in )
4 .0 0 6 3 8
S tD e v (O v e ra ll)
3 .9 7 6 6 9
O v er all
P o te n tia l (W ith in ) C a p a b ility
Cp
0 .8 3
C PL
0 .8 6
C PU
0 .8 1
C pk
0 .8 1
O v e ra ll C a p a b ility
Pp
0 .8 4
PPL
0 .8 7
PPU
0 .8 1
P pk
0 .8 1
C pm
60
64
68
72
O b se rv e d P e rfo rm a n ce
E xp . W ith in P e rfo rm a n ce
P P M < LS L
P P M < LS L
4 9 9 5 .4 6
P P M < LS L
7 8 4 8 .2 0
PPM > USL
0 .0 0
PPM > USL
1 0 0 0 0 .0 0
PPM > USL
P P M T o ta l
1 0 0 0 0 .0 0
P P M T o ta l
1 2 8 4 3 .6 6
76
E xp . O v e ra ll P e rfo rm a n ce
P P M T o ta l
4 7 2 4 .4 0
7 4 6 7 .8 7
1 2 1 9 2 .2 8
80
*
Example: An Active Noise Experiment
 The device contains several control factors from
which three were identified as DOE candidates
– Pressure (A)
– Concentration (B)
– Stir Rate (C)
 Ambient Temperature was identified as being
significant, but not economically controllable
– Temperature would not change appreciably during the time
in which it would take to execute a three factor experiment
– Decided to include it as a factor in the design to force it to
change
– Call this Factor G
The Experimental Design
 A 24-1 fractional design (Res IV) was rejected because
the 2-way interactions are of great interest in this
experiment
 A 24 full factorial design was used
 Because of time constraints, only 1 replicate was
performed
 The variables are listed below:
–
–
–
–
A = Pressure
B = Concentration
C = Stir Rate
G = Temperature
 The data is in worksheet “Active Design”
 Open worksheet “Active Design” within “Robust
Design.mpj”
Factorial Analysis
 This DOE is an unreplicated, 4-factor full factorial
– We need to create the “Pareto of Effects” chart
P a r e to C ha r t o f the E ffe c ts
(r e s po ns e is Y4, A lpha = .2 0 )
0.0 8
F a cto r
N am e
A
A
B
B
CD
C
C
C
D
G
D
BD
B
Te r m
A
A BD
AD
A BC
BC
AB
A BC D
BC D
ACD
AC
0
5
10
15
20
Effe c t
Le nth's PS E = 0 .05 62 5
Stat > DOE > Factorial > Analyze Factorial Design
Fit the Reduced Model
 Based on the p-values, the A*G and B*G and C*G interactions
are important. Since we have identified some important
control*noise interactions, the next step is to examine the
interaction plots.
Estimated Effects and Coefficients for Y4 (coded units)
Term
Constant
A
B
C
G
A*B
A*G
B*C
B*G
C*G
A*B*C
A*B*G
A*B*C*G
Effect
5.987
9.787
14.612
21.387
-0.038
-0.087
-0.062
-18.038
16.638
0.062
0.137
0.037
Coef
70.081
2.994
4.894
7.306
10.694
-0.019
-0.044
-0.031
-9.019
8.319
0.031
0.069
0.019
SE Coef
0.006250
0.006250
0.006250
0.006250
0.006250
0.006250
0.006250
0.006250
0.006250
0.006250
0.006250
0.006250
0.006250
T
11213.00
479.00
783.00
1169.00
1711.00
-3.00
-7.00
-5.00
-1443.00
1331.00
5.00
11.00
3.00
P
0.000
0.000
0.000
0.000
0.000
0.058
0.006
0.015
0.000
0.000
0.015
0.002
0.058
Stat > DOE > Factorial > Analyze Factorial Design
Interaction Plot
 The interaction plot show the valuable interactions available to
the designer. Let’s take a closer look at the strongest ones.
Inte r a c tio n P lo t ( da ta me a ns ) fo r Y 4
5
1
200
150
1
-1
100
A
10
75
A
20
50
100
B
1
75
B
5
50
100
C
150
75
C
200
50
G
Stat > DOE > Factorial > Factorial Plots > Interaction Plot
Interaction Plot – A Closer Look
Int e r a c t io n P lo t ( d a t a m e a ns ) f o r Y 4
100
B
1
5
Me an
80
60
40
-1
1
G
100
C
150
200
80
Me an
 The two interaction plots at right
indicate that both B and C can be
exploited to desensitize Y4 to the
noise variable G
– If B is set at its high level, the
slope of the G effect line is
minimized
– If C is set at its low level, the
slope of the G effect line is also
minimized
 To minimize output variation due to
noise in ambient temperature (G),
the above two settings should be
controlled in the design
60
40
-1
1
G
Main Effects Plot
 Since factor A is not involved in a significant interaction and its
main effect is significant, we should take a look at its main effect
plot to see if there is some potential value in controlling A
M a i n E f f e c ts P l o t ( d a ta m e a ns ) f o r Y 4
73
Mean of Y 4
72
71
70
69
68
67
10
20
A
 This plot indicates that factor A has about +/- 3 units of control
over the nominal value of Y4. Thus, if manipulating one of the
other factors takes the mean value off target, this factor could be
used to exert some control over the mean value of Y.
Stat > DOE > Factorial > Factorial Plots > Main Effects Plot
New Settings : Capability Analysis
 B was set to 5, C was set to 150, A was left at nominal (15)
 It appears that the changes to B & C were successful in reducing
variation in Y4 but the mean is now off target
 Use factor A to adjust back to target!
P r o c e s s C a pa bility o f Y 4 V a li d
LS L
USL
P ro ce ss D a ta
LS L
60
T a rg e t
*
USL
80
S a m p le M e a n
6 7 .8 9 8
S a m p le N
100
S tD e v (W ith in )
2 .0 6 5 8 7
S tD e v (O v e ra ll)
2 .1 5 0 4 2
W ith in
O v er all
P o te n tia l (W ith in ) C a p a b ility
Cp
1 .6 1
C PL
1 .2 7
C PU
1 .9 5
C pk
1 .2 7
O v e ra ll C a p a b ility
Pp
1 .5 5
PPL
1 .2 2
PPU
1 .8 8
P pk
1 .2 2
C pm
60
O b se rv e d P e rfo rm a n ce
63
66
E xp . W ith in P e rfo rm a n ce
P P M < LS L
0 .0 0
P P M < LS L
PPM > USL
0 .0 0
PPM > USL
P P M T o ta l
0 .0 0
P P M T o ta l
6 5 .9 0
0 .0 0
6 5 .9 0
69
72
75
E xp . O v e ra ll P e rfo rm a n ce
P P M < LS L
PPM > USL
P P M T o ta l
1 1 9 .9 7
0 .0 1
1 1 9 .9 8
78
*
How much should we shift factor A?
 Set up the response
optimizer to target 70. The
lower and upper limits are
not important since we will
manually manipulate this.
 Set factor B=5 and factor
C=150. The optimizer
indicates a nominal
Y4=67.7, very close to that
observed in the validation
study.
 Finally, slowly slide the bar
for factor A to the right
while observing the
predicted value of Y4.
This indicates that a
nominal setting of A=18.8
should achieve Y4=70.
New
Hi
D
Cur
0.00000 Lo
A
20.0
[15.0]
10.0
B
5.0
[5.0]
1.0
C
200.0
[150.0]
150.0
A
20.0
[18.80]
10.0
B
5.0
[5.0]
1.0
C
200.0
[150.0]
150.0
G
1.0
[-0.0076]
-1.0
Y4
Targ: 70.0
y = 67.7506
d = 0.00000
New
Hi
D
Cur
0.99934 Lo
Y4
Targ: 70.0
y = 69.9987
d = 0.99934
Worksheet “active design”
Stat > DOE > Factorial > Response Optimizer
G
1.0
[-0.0076]
-1.0
New Setting for A : Capability Analysis
 When we shift factor A to this new nominal value, we succeed at
shifting the response to put it on target without degrading the
variation
P r o c e s s C a pa bil ity o f Y 4 R e duc e d
LS L
USL
P ro ce ss D a ta
LS L
60
T a rg e t
*
USL
80
S a m p le M e a n
7 0 .2 9 0 2
S a m p le N
100
S tD e v (W ith in )
2 .0 9 1 0 3
S tD e v (O v e ra ll)
2 .2 0 5 8 3
W ith in
O v er all
P o te n tia l (W ith in ) C a p a b ility
Cp
1 .5 9
C PL
1 .6 4
C PU
1 .5 5
C pk
1 .5 5
O v e ra ll C a p a b ility
Pp
1 .5 1
PPL
1 .5 6
PPU
1 .4 7
P pk
1 .4 7
C pm
60
O b se rv e d P e rfo rm a n ce
63
66
E xp . W ith in P e rfo rm a n ce
69
72
75
E xp . O v e ra ll P e rfo rm a n ce
P P M < LS L
0 .0 0
P P M < LS L
0 .4 3
P P M < LS L
1 .5 4
PPM > USL
0 .0 0
PPM > USL
1 .7 1
PPM > USL
5 .3 7
P P M T o ta l
0 .0 0
P P M T o ta l
2 .1 4
P P M T o ta l
6 .9 1
78
*
Summary
 Variation improvement strategies can take two forms:
– Passive Approach
– Active Approach
 The choice of which strategy to use depends on the ability to
control or manipulate noise factors (at least for the duration of
the experiment)
 Standard full and fractional designs can be used
 Variation effects must be calculated using replications
 A log transform of the variability response is automatically used
to minimize the effects of asymmetry in the variance distribution
 The response optimizer can be used to simultaneously optimize
both mean and variability responses
 A validation study must be made at the end of a Robust
Parameter Design study
Objectives Revisited
At the end of this module, participants should be able to :
 Identify possible variation effects from residual plots
 Create a variability response from replicates
 Identify possible mean and variance adjustment
factors from noise-factor interaction plots
 Use the MINITAB Response Optimizer to achieve a
process on target with minimum variation
 Complete validation capability studies
Copyright 2003 Cummins, Inc. All Rights Reserved.
Copyright 2000-2002 Sigma Breakthrough Technologies, Inc. Used with permission.
DFSS-57