Taguchi Robust Design for Product Improvement: Part II
During the past several decades many methods on experimental design and data
analysis for achieving product/process robustness have been proposed. In this
article, we will use
ReliaSoft’s DOE++ to illustrate
the location and dispersion modeling method for doing so. Two models (transfer
functions) will be built: one for the location (the mean of the observations)
and the other for the dispersion (the standard deviation of the observations).
The optimal setting of each control factor will be identified using these two
transfer functions. This article provides a complement
to Issue
122 of Reliability HotWire where we presented a case study on using
the signaltonoise ratio as the response for the regression model. The optimal
setting of each control factor was identified using the transfer function
between the control factors and the signaltonoise ratio.
Philosophy
In Design of Experiments (DOE), the key to having a successful experiment is
to carefully plan the design. In order to have a well designed
experiment, engineering knowledge should be integrated into the design process
from the very beginning, such as determining the number and values of the control
factors and the noise factors. As pointed out by many researchers and engineers,
the secret of robust design is the interactions between the control factors
and the noise factors. If there are no interactions between these two types of
factors, then it is very difficult to achieve a robust design. Let us use a
simple example to illustrate this statement. Assume that there is only one
control factor and one noise factor, and both are 2 level factors. The two
possible scenarios of the interaction effect of these two factors are
given below.
The plot of (a) shows that there is no interaction between the two factors.
The changes of the response y caused by the change of the noise factor
are the
same at the high and low levels of the control factor. In other words, it is
impossible to improve the robustness to the noise of the product by adjusting
the value of the control factor.
The plot of (b) shows that there is a strong interaction between the control
factor and the noise factor. The changes of the response y caused by
the change
of the noise factor are different at the high and low levels of the control
factor. In other words, it is possible to improve the robustness to the noise
factor of the product by adjusting the setting of the control factor. For
plot (b) we can see that the variation of y caused by the noise factor
is smaller
when the control factor is set to its low level. Therefore, setting the control
factor to its low level will improve the robustness of the product to this
noise factor.
From the above discussion we can see that reducing the response variation is
one of the major goals of robust designs. Using the signaltonoise ratio as the
response can help in reaching this goal. Another widely used method in robust
design is the so called location and dispersion modeling method, or
the mean and standard deviation modeling method. This method assumes the
mean and variation are independent. We will use the same
example from
the previous article to illustrate this method. Note that this example is
only used for illustration purposes and the data presented here have been
modified from a real application.
An automobile manufacturer wants to improve the quality of the painted surface
of its cars. The quality is measured by the gloss reading of the surface. The
manufacturer wants the painted surface to have a higher gloss reading
(i.e., maximize the response) and to be robust to the environment. Environmental
factors, particularly temperature and humidity, are known to affect the painted
surface. Thus, a robust design will be used here. For a detailed background on
robust design, please see:
http://reliawiki.org/index.php/Robust_Parameter_Design.
The control factors have been identified to be the flow rate of the paint,
the pressure in the paint gun, the viscosity of the paint and the cure
temperature (i.e., the surrounding air temperature).
The factors and levels are as follows:
Controllable Factors 
Name 
Low Level 
High Level 
A 
Flow Rate 
30 
50 
B 
Pressure 
3 
5 
C 
Viscosity 
10 
15 
D 
Cure Temperature 
120 
160 
Two noise factors will be taken into account: the air temperature
(ambient = room temperature) and humidity (ambient = room humidity).
Noise Factors 
Name 
Low Level 
High Level 
A' 
Air Temperature 
15 
30 
B' 
Humidity 
30 
90 
The design matrix for the control factor in robust design is called the
inner array, while the design matrix for the noise factor is called the
outer array.
For the inner (control) array, a 2 level fractional factorial design with D=ABC
is used. For the outer (noise) array, a 2 level full factorial design is
applied.
Assume that the experiment has been conducted properly and the data set was
entered into DOE++, as shown below.
In the last
issue, we used the signaltonoise ratio method and found out that factors
A and D
are significant. In this article, we will build two models: one is for the Y mean
and the other is for the standard deviation (Y Std) as shown in the above
figure. The strategy is:
 Build a model for the mean response (location) of each treatment
(control factor combination).
 Build a model for the standard deviation (dispersion) of
each treatment.
 Adjust the factors in these two models to optimize the response and
reduce the variance.
The modeling process in DOE++ is given below.
Step 1: Click the Design tab.
Step 2: Include all twoway interactions in the model.
To do this, click the Select Effects icon. In the Effects window, use
the settings shown next, then click OK.
Click the Options page of the control panel and notice
that Y Mean is showing in the Response dropdown list and that
the Individual Terms check box is selected.
Step 3: In the Response dropdown list
choose Y Std and select the Individual Terms check box.
Step 4: Return to the Main page of the control panel and click
the Calculate icon.
Notice that the ANOVA table and the Regression Information table for the
model are
displayed on the Analysis tab, which is added to the folio upon calculation.
Step 5: Click the Plot icon to add the Plot sheet to the folio.
From the normal probability plot for the Y Mean response (i.e., for the
location model), we can see that effects A and D are significant, as
shown next.
Step 6: To view the normal probability plot for the Y Std response
(i.e., for the dispersion model), select Y Std from the Response
field in the control panel. Effect A is identified as significant, as
shown next.
From Step 5 and Step 6, the significant effects are identified for
the Y Mean and the Y Std. Only these statistically significant
effects are used to build the final models. The optimal values of the significant
factors can be determined using the final models given in DOE++.
Optimization
Step 1: Click the Select Effects icon and select only the
significant factors for the Y Mean response, the Y Std response
and the Signal Noise Ratio response, as shown next, and then
click OK.
Step 2: Click Calculate.
The Regression Information table for the Y Mean response is
shown next.
Using the values found in the Coefficient column, the location model is found
to be:
y_{mean} = 79.3438 + 2.5312A + 2.6562D
Because factors A and D both have positive coefficients, both should be set
to their high levels in order to maximize the mean.
The Regression Information table for the Y Std response is
shown next.
Using the values found in the Coefficient column, the dispersion model is
found to be:
ln(y_{std} ) = 0.7903  0.5425A
Because factor A has a negative coefficient, in order to minimize the
variance, it should be set to its high level.
So the best settings for this simple example are: A = 50 and D = 160, with
factors B and C at their most economical settings.
In cases where the models are complicated (e.g., a case where changing the
value of a factor to maximize the mean will also maximize the standard
deviation), the optimization tool in DOE++ can be used to find a
compromise solution.
Step 3: Click the Optimization icon on the control panel.
In the Optimization Settings window, use the settings shown next then
click OK.
The optimization results are:
The conclusion from the above modeling method is the same as the results we
obtained through the signaltonoise ratio method used in the last issue.
Sometimes these two analysis methods will provide different results. If this
occurs, use your engineering judgment on which results to use.
Conclusion
In this article, we introduced the location and dispersion modeling method
for robust design and focused on presenting the statistical
modeling procedure. In reality, when applying a DOE method, engineering knowledge
must be integrated into the design and analysis process.
Besides statistical modeling, you should also use other important factors
such as design decomposition, measurement
improvements, verification testing and the gains made by running a wellplanned
engineering experiment. The design layout used
in this article and the article in the last issue is called the cross
array design method. This is because the inner array for the control
factors is crossed with the outer array for the noise factors. In other
words, each combination of the control factors is tested under all the noise
factor combinations, as seen in the data sheet in
the Example section. Sometimes
this will lead to a very large sample size. To overcome this drawback,
you can also conduct robust design in a single array manner where noise
factors are treated the same as control factors so the fractional factorial
designs can be applied. You can include control factors only or you can
include both noise factors and control factors in the transfer functions. If
only the control factors
are included in the transfer functions, the location and dispersion modeling
method discussed in this article can still be used. Complicated statistical
models also have been developed for the case when both types of factors are
in the transfer function. For details, please see
reference [1].
Acknowledgment: We would like to express our appreciation for
the useful comments and
suggestions from Louis Lavallee. He has many years of experience with Design
of Experiments and is a senior reliability engineering consultant at Ops
A La Carte.
References
[1] R. H. Myers and D. C. Montgomery, Response Surface
Methodology: Process and Product Optimization Using Design Experiments, 2nd
Edition, New York, NY: Wiley & Sons, 2002.
