A Response Surface Approach for Robust Parameter
Design
[Editor's Note: This article has been updated
since its original publication to reflect a more recent
version of the software interface.]
Robust parameter designs have been used widely in
industry. In
Issue 122 of the Reliability Hotwire, we
discussed using Taguchi’s method to design robust
products and processes. An example with a crossarray
design using ReliaSoft’s
DOE++
was given. In this article, we will use a singlearray
design with a response surface method approach for robust
parameter design.
The Secret of Robust Parameter Design
The following diagram illustrates the relation
between input variables and output response for a
system. The system could be either a product or a process.
Figure 1  System diagram
There are two types of factors in the above figure:
control factors and noise factors. Control factors are
variables that can be controlled, such as design
parameters or process variables. Noise factors are
variables that are very difficult to control, such as
ambient temperature and humidity. Both types of factors will
affect the response. For example, consider a system
where, when the input
variable is 5 volts, the system’s response is 10 mA at
a room temperature of 20°C. However, when the room
temperature is 30°C, the system response is 13 mA with
the same input of 5 volts. This system is not robust to
room temperature, although ideally it should be.
Therefore, we need to adjust the control factors to make
the system robust.
The secret of robust design is understanding the control and noise
factor interactions. The following plot illustrates this
point.
Figure 2  Control and noise factor interaction
There is one noise factor and one control factor in
Figure 2. The value of the noise factor can change from
low (1) to high (+1). The change of this noise factor
causes variation of response y.
In Figure 2(a), no matter what level the control
factor is, the amount of change for y is the same.
Therefore, it is impossible to adjust the level of
the control factor to reduce the effect of the noise factor.
In Figure 2(b), when the control factor is at its low
level, the change of y is much smaller than that when
the control factor is at its high level. Therefore,
setting the value of the control factor at its low level
will reduce the effect of the noise factor, or in other
words, will make the system robust to the noise factor.
In summary, Figure 2(a) shows a situation where there is no interaction
between the control and the noise factor, while Figure
2(b) shows a situation where they do have interaction. Because of this
interaction, it is possible to have robust design for
the system.
The Single Array Method for Robust Design
Let x be the control factor and z be the noise
factor. The relationship between them and the response
y is: [1]

(1) 
The variation of the noise factor z will be
transformed to y. As discussed above, if x and z
have interactions, then the variation in y can also
be reduced by changing the settings of x. For
example, consider a simple model such as:

(2) 
Eqn. (2) has two control factors, x_{1} and
x_{2}, and
one noise factor, z. ε is the error term,
following a normal distribution with mean of 0 and
standard deviation of Var(ε). Assume Eqn. (2) is in
terms of the coded values of each variable and z follows a
normal distribution with mean of 0 and standard
deviation of Var(z). Then, the expected value and variance
of y are:

(3) 
We can adjust x_{1} to bring y to its target
value and adjust x_{2} to reduce the variance of y. Eqn. (3) clearly shows the importance of the
controlbynoise interaction in reducing the
variation in y.
Example: The Pilot Plant Experiment
A singlereplicate factorial
experiment is conducted to study the factors
affecting the filtration rate of a chemical product.
The four factors are: temperature (A), pressure (B),
concentration of formaldehyde (C) and stirring rate
(D). Factor A is treated as noise, since it is hard
to control. Each factor has two levels. The
engineer wants to maintain the filtration rate at 75
gallons per hour and reduce its variation. The design
and results are given below. [2]
Figure 3

Design matrix and response
The initial model is shown next.
Figure 4
 Results of the initial model
The results show that factors A, C and D are
significant, as are the interactions of AC and AD. The
engineer pools the nonsignificant effects into error by
excluding those effects from the model, as shown next.
The recalculated final model is:
Figure 5  Results of the final model
Therefore,
the final equation for response y is:

(4) 
where z is the noise factor A (temperature), x_{2}
is factor C (concentration) and x_{3} is factor D
(stirring
rate). Using Eqn. (3), the expected value of y and its
variance are:

(5) 
In Figure 5, S (located below the ANOVA table) is
the estimated standard deviation. Its square is the
estimated variance (i.e., Var(ε) =
4.4173^{2} = 19.5125). Therefore, the problem in this experiment becomes an
optimization problem:

(6) 
Using the solver in Micrososft Excel®, the optimal values
for x_{2} and x_{3} are 0.8861 and 0,
respectively. At these settings,
the estimated variance of y is 22.29498.
In this
example, the engineer wanted to maintain the
filtration rate at 75 gallons per hour. If the engineer
wanted to maximize the filtration rate and at the
same time to reduce the variance, the multiple response
optimization method explained in the Experiment
Design and Analysis
reference book could be used (http://ReliaWiki.org/index.php/Response_Surface_Methods_for_Optimization#Multiple_Responses).
Conclusion
In this article, a single array
and a response surface modeling method are used for
robust parameter design. This is an alternative to the
traditional crossarray robust design method. The
optimization method in this article can be used when
the objective is to maintain the response at a given
value and reduce its variation. Otherwise, the
location and dispersion modeling strategy should be
used with either a crossarray or a singlearray
design. For
more detail, please refer to the Reliability
Hotwire articles on
variability
analysis in DOE++ and on
robust parameter design.
References
[1] C. F. Wu and M.
S. Hamada, Experiments: Planning, Analysis,
and Optimization, 2nd ed. New York, NY: John
Wiley & Sons, Inc., 2009, pp. 518520. [2] R. H. Myers
and D. C. Montgomery, Response Surface
Methodology, 2nd ed. New York, NY: John
Wiley & Sons, Inc., pp. 557560.
