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Robust Parameter Design of Multiple-Target
Systems
V Roshan Josepha & C F Jeff Wua
a Department of Statistics, University of Michigan, Ann Arbor, MI 48109
Published online: 01 Jan 2012.
To cite this article: V Roshan Joseph & C F Jeff Wu (2002) Robust Parameter Design of Multiple-Target Systems,
Technometrics, 44:4, 338-346, DOI: 10.1198/004017002188618536
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Robust Parameter Design
of Multiple-Target Systems
V. Roshan Joseph and C. F. Jeff Wu
Department of Statistics
University of Michigan
Ann Arbor, MI 48109
( roshanjv@umich.edu)
( jeffwu@umich.edu)
In multiple-target systems, the signal factor is varied based on the signal-response relationship to achieve
different targets for the response speciŽ ed by the customer. Robust parameter design aims to make
the signal-response relationship insensitive to the noise variation by choosing appropriate levels for
the control factors. Taguchi’s dynamic signal-to-noise ratio has several limitations for the optimization
in multiple target systems. We give a theoretical formulation of the problem and develop a practical
approach for optimization that overcomes these limitations. The methodology is illustrated using a
temperature controller example.
KEY WORDS: Experiments; Optimization; Signal-response system; Signal-to-noise ratio.
2
1. INTRODUCTION
…
‘
‚
‘
where has mean 0 and variance . The and are eval-
uated for different control factor settings, and a setting is
selected that will optimize a performance measure of varia-
tion from the linear relationship. Taguchi used the (dynamic)
signal-to-noise (SN) ratio given by
Robust parameter design aims to Ž nd the levels of control
factors that will make the effect of noise factors on the perfor-
mance of the system as small as possible. When the customer
puts forth different requirements on the system performance,
all of the control factors cannot be kept at a constant level.
‚2=‘ 2
D
SN log
(2)
signal factor
In such cases, a factor known as a
is selected
as the performance measure. Miller and Wu (1996) conŽ rmed
from the set of control factors and is changed continuously
depending on the customer’s intent, to meet his requirements
(Fig. 1). For example, different turning radii for a car can be
achieved by changing the steering wheel angle, different hole
measure-
the suitability of this measure for optimization in
ment systems
, but commented on its inadequacy for dealing
multiple-target systems
with
. They proposed a response func-
diameters in a drilling operation can be obtained by using dif- tion modeling approach for the analysis. Lunani, Nair, and
ferent drill bit sizes, and different line widths for the circuitry
in a printed circuit board can be produced by giving different
line widths in the artwork used in the image transfer process.
The system user has to specify a value for the signal factor to
achieve a speciŽ c customer intent and hence needs to know the
relationship between the signal factor and the output response.
The system will work well if this relationship is robust to
the noise factors. The approach of making the signal-response
relationship insensitive to noise was introduced by Taguchi
Wasserman (1997) advocated using a generalized SN (GSN)
ratio,
‚ƒ =‘ 21
D
GSN log
(3)
as the performance measure and gave some graphical methods
ƒ
for obtaining from the data.
Y
In an engineered system, the actual relationship between
M
and
need not be linear. Taguchi (1990) treated the nonlin-
earity as undesirable, considering it a “distortion in engineer-
dynamic parameter design
and is known as
(Taguchi 1987)
(Miller and
‘
ing Ž elds.” The
calculated from model (1) is a combined
parameter design in signal-response systems
or
measure of variation due to noises and departure from linear-
ity. There are many situations in which a system can work
well with a nonlinear relationship as long as the relation-
ship is well understood and is easily implementable in the
system. Hence, forcing a system to behave with a linear signal-
response relationship can result in a suboptimal system design.
But modeling a nonlinear relationship in an experiment can
be expensive, because the data need to be collected for many
levels of the signal factor for every control factor setting. We
show that the optimization procedure is very complex with a
Wu 1996). The recent developments in this area have been
discussed by Miller and Wu (1996), Wu and Hamada (2000),
and Wu and Wu (2000). The increasing competition in the
market and changing customer demands have forced manufac-
turers to design and produce products with varied functional
requirements. It has become necessary not only to meet cur-
rent requirements, but also to develop a process by which to
anticipate future requirements. Dynamic parameter design has
emerged as an important engineering statistical tool in this
technology development process. Many interesting case stud-
ies from industries have been given by Taguchi, Chowdhury,
and Taguchi (2000).
Taguchi (1987, 1993) used a linear relationship between
© 2002 American Statistical Association and
the American Society for Quality
TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4
DOI 10.1198/004017002188618536
M
Y
signal ( ) and response ( ) given by
D
C
‚M …1
Y
(1)
338

ROBUST PARAMETER DESIGN
339
2
2
2
D
D
E4…5
4…5 ‘ R
0 and var
‚
‘
are func-
where
3. The and
R 1R 1R 1E 1
1
E
tions of the nominal values of
and z. We use
2
4
0
a modiŽ ed version of this example in Section 6 to illustrate
the approach of this article.
2.2 Electroplating Example
Consider an electroplating process used to deposit metal
onto another metallic substrate. The plating time can be varied
to achieve different thicknesses of the plated metal. By Fara-
day’s law, the plating thickness (mean) is proportional to the
Figure 1. Multiple-Target System.
nonlinear relationship. We develop some methods to address
nonlinearity while keeping the optimization simple.
V 4’5
plating time. Let
represents the variability in thickness
’
’=n
4 1 ’5
n
during time . Divide the interval 0
into equal parts
5’=n1’5
. The amount
M
In model (1), we see that as
is reduced to 0, the mean of
ƒ
4 1 ’=n51: : : 1 44n
as 0
of length
1
Y
Y
is also reduced to 0. If is allowed to take only nonnega-
of metal deposited in each of these intervals can be assumed
Y
tive values, then the variation in will also reduce to 0. The
D
V 4’5
to be independently and identically distributed. Then
statistical model in (1) does not capture this property, leading
D
nV 4’=n5
/
n
1 1 1: : :
1 2 3
for all
. This is true if and only if
‚
‘
to inefŽ cient estimates of and . We stress the importance
V 4’5
’
. Hence the statistical model for this system can be
Y
of modeling variation in as a function of the signal factor,
written as
which is crucial in obtaining the right performance measure
for robust design optimization.
D
4…5 ‘ ’
C
Y
‚’ …1
The article is organized as follows. In Section 2 we give
some typical examples to show that variance will usually be a
function of the signal factor. In Section 3 we describe a general
approach to the robust design optimization of multiple-target
systems. In Section 4 we develop two practical approaches to
modeling, and in Section 5 we identify the underlying sta-
tistical model for the SN ratio. We illustrate the proposed
methodology with an example in Section 6, and give conclu-
sions in Section 7.
2
D
D
E4…5
V 4’5
. The variability is
where
0 and var
caused by the  uctuations in some of the noise factors during
’
time . If we also absorb the other noise factors into the vari-
2
2 2
C
‘ ’ ‘2 ’
ance term, then the variation can be modeled as
which can be approximated as
between 1 and 2, depending on the magnitudes of
in the experimental region.
,
1

2
N
‘ ’

is a constant
. The
2
1
2
2
‘
‘
and
3. GENERAL FORMULATION
2. VARIANCE FUNCTION
Two types of systems are usually encountered in dynamic
parameter design problems: measurement systems and
multiple-target systems. In a measurement system, the true
value is estimated with minimum variability based on the
measured value. This is a problem of inverse regression.
Using the concept of Fieller interval, Miller and Wu (1996)
proved that the SN ratio in (2) is an appropriate performance
measure for optimization. In a multiple-target system, the
signal factor is to be adjusted depending on the target for
the response. The objective is to design the system so that the
signal-response relationship is robust to noise factors. In this
section we describe a general approach to the robust design
of multiple-target systems.
A signal factor can be used to cause large changes in the
mean of a response. Changes in the variation are bound to
happen when the mean is varied greatly and hence the variance
of the response is a function of the signal factor. We illustrated
this situation with two examples.
2.1 Temperature Controller Example
Phadke (1989) described the robust design of a temperature
controller. The controller maintains the desired room temper-
ature by turning a heating element on and off to control the
heat input. The target for the room temperature is set by the
user, and therefore the temperature controller is a multiple-
target system. The thermistor resistance at which the heater
will be turned on is given by
D
D
EZ4Y 5 f4 1M5
Z4Y 5 V 4 1 M5
and var , where X
Let
X
X
is the set of control factors. The expectation and variance are
taken over the distribution of the noise factors Z. The expected
quality loss assuming a quadratic loss function is proportional
to
C
R3R24R4 R1E0=EZ5
D
RT ƒ
1
(4)
on
ƒ
ƒ
R14R4 R24E0=EZ
55
1
R11R21R31
R4
E0
and
E
where
and
are the resistances and
2
z
D
ƒ
C
L4 1M1 t5 4f4 1 M5 t5 V 4 1M51
X
X
X
are the voltages in the controller circuit. More details about
the functioning of this circuit have been given by Phadke where is the customer intent. Given , the signal factor can be
t
t
R
(1989) and McCaskey and Tsui (1997). The resistance
RT ƒ
can adjusted to minimize the loss. Cost considerations or system
3
be set to achieve a desired value for the
on, and there-
is taken as the signal factor. The nominal values of and H . Let
R 1 R 1 R 1 E 1
M
ML
be between
limitations may require that the value of
R3
M
fore
E
and
0
are the control factors, and the variation
ü
z
1
2
4
D
L4 1 M1t5 L4 1M 1 t50
2min
X
X
(5)
in them within their manufacturing tolerance is the noise in the
M
6ML1 MH 7
2
/
4R on5
system. From (4), var
model for this system is
R
3. An approximate statistical
ƒ
T
4a1b5
t
W 4t5
Let
be the range of values of and be the prob-
ability distribution function. Because X cannot be varied with
ü
D
C
‚R3 …1
D
RT ƒ
t
M
M
, the expected loss at
with respect to customer intent
on
TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4

340
V. ROSHAN JOSEPH AND C. F. JEFF WU
4. MODELING
can be used as a measure to evaluate the control factor settings.
Thus the performance measure for a given X is
In this section we develop two different approaches to mod-
Z
b
mean-variance modeling
, we model mean and vari-
eling. In
PM
L4 1Mü1 t5dW 4t50
D
X
(6)
ance functions separately. This modeling approach is ideal
when the noise levels are randomly chosen for every observa-
a
2
D
PM D
, where is
Our objective is to Ž nd an X
minimizing
response modeling
, we directly model the response in
tion. In
the feasible region for X. Minimization of the loss in (5) need
terms of the control, noise, and signal factors. This approach
is suitable when the noise factors have Ž xed levels in an exper-
iment.
M
not result in a value of
to get the expected value of the
t
response at target , but in engineering applications, an unbi-
ased adjustment strategy makes more sense. Therefore, we can
M
h4 1t5
simplify (6) by avoiding the minimization over . Let
X
4.1 Mean-Variance Modeling
D
M
f4 1M5
t
. A unique solution
exists because the signal-response relationship is assumed to
be a continuous monotonic function. Now the
be the solution of
from
X
We explain the ideas behind mean-variance modeling by
choosing some special functional forms for both mean and
variance. Consider an additive noise model
PM
can be
D
C
Y
f4 1M5
X
average variability in Y after adjustment
viewed as the
. This
simpliŽ cation is possible only if the evaluation of the mean
h4 1t5 h4 1t5
D
D
…
E4…5
4…5 V4 1 M5
0 and var . Assume that the
, where
X
response and signal factor are nonnegative variables and that
and variance functions at
6M 1M 7
X
is meaningful. The
X
D
f 4 1 5
X 0
0. Also assume that the mean and variance are
need not lie in
for all X. Thus we need to search
for an X satisfying this requirement, and consequently, the
L
H
M
increasing functions of , perhaps after a suitable transforma-
tion of the signal factor. Based on the discussions in Section 2,
optimization problem becomes

2
Z
b
D
V 4 1M5 ‘ 4 5M
X
X
(10)
is a reasonably good approximation for the variance function.
f4 1M5
D
PM
V 4 1h4 1t55dW 4t5
X X
min
(7)
(8)
(9)
2
D
X
a
We Ž rst consider a special form of
X
and develop
subject to
and
the optimization procedure. As we discuss later, this form is
helpful in dealing with a more general case using lack of Ž t.
Let
h4 1 t5 µ M
X
max
H
2
t
6a1b7
D
f4 1 M5 f 4‚4 5M51
X
X
(11)
ˆ
h4 1 t5 ¶ M 0
min
X
L
2
t
6a1b7
f
ˆ
is a known function up to a set of parameters .
where
ˆ
Because the mean and variance functions are meaningful
Because the objective function and constraints are not given
explicitly as functions of X, the optimization is difŽ cult to per-
form. Therefore, the formulation must be re-cast into a more
tractable form. Moreover, the approach has some practical lim-
itations, because in general the mean and variance functions
are not known and are estimated from the experimental data.
M
only for nonnegative values of
, we must assume that
fˆƒ14a5 ¶
t
6a1 b7
.
2
0 to ensure unbiased adjustments for
D
f 4 5
ˆ
Typically
0
0, and hence this will be automatically
PM
satisŽ ed. From (10) and (11), the
in (7) becomes
³
´

Z
fˆƒ14t5
b
D
PM
‘ 24 5
dW 4t5
X
f4 1 M5 M
is a nonlinear function in , then data must be
If
X
‚4 5
X
a
M
collected at many levels of . Because in most experiments
the number of runs is a multiple of the signal factor levels,
the experiment becomes expensive to conduct. From a cost
standpoint, it is beneŽ cial to have few signal factor levels and
thus few parameters in the mean and variance function mod-
els. We show that this can be done and still achieve the stated
objectives.
Z
‘ 24 5
b
X
4fˆƒ14t55 dW 4t50

D
(12)

‚ 4 5
X
a
t
With the selected mean function in (11), the X and are sep-
PM
arated in the
, thus simplifying the procedure. From (12),
we deŽ ne the performance measure as

‚ 4 5
X
D
‡4 5
X
log
(13)
It is important to note that in our formulation, the perfor-
mance measure in (7) is obtained by integrating the variance
‘ 24 5
X
to be consistent with the form of the SN ratio in (2). Note
that we no longer need to specify the probability distribution
of customer intent to get the performance measure. Thus the
optimization problem is reduced to
t
over the distribution of . Tsui (1998) formulated the dynamic
parameter design problem by considering integration over sig-
nal values. In multiple-target systems, the distribution of
is induced from the distribution of and is dependent on the
control factor setting X. Therefore, in multiple-target systems
it is more appropriate to consider integrating over the distribu-
M
t
‡4 5
X
max
(14)
2
D
X
t
tion of . Integration over the signal values may be considered
‚4 5 ¶ ‚
subject to
X
L, where
for the problem of functional response, which can be treated
as a third type of dynamic parameter design problem. Here
the objective is to minimize the variations in the functional
response from an ideal function over a range of signal val-
ƒ
fˆ 14b5=MH 1
D
‚
(15)
L
which is obtained from the constraint in (8). The constraint (9)
is trivially satisŽ ed for
D
ML
0. The foregoing problem can
ues. Nair, Taam, and Ye (2002) have provided details on the be solved using a mathematical programming algorithm (see,
analysis of functional response.
e.g., Luenberger 1989). The two-step optimization procedure
TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4

ROBUST PARAMETER DESIGN
341
is found to work well in many practical situations. It will relationship. The signal-response relationship is usually
succeed if an adjustment parameter exists. In this case, an restricted to be a monotonic function. Therefore, if the LOF
cannot be completely eliminated with elaborate modeling,
then the cause behind it must be identiŽ ed and removed
adjustment parameter refers to a variable that has an effect on
‚
‡
but not on . The two-step optimization procedure in our
avoid the need of
formulation can be stated as follows:
from the system. This approach helps us
Ž tting elaborate signal-response models to each run, thereby
making the experiment less expensive
2
0
0
D
‡4 50
to maximize X
1 Find X
. In situations where
2 Use one or more adjustment parameters to
(16)
speciŽ c functional forms such as linearity are preferred in a
system, the LOF can be separately modeled and minimized.
A decomposition like (20) was used by Miller and Wu (1996)
in their analysis of an injection-molding experiment. Our
approach is more general, as we introduce the separation of
LOF from the residual variation as a strategy to deal with
nonlinear signal-response systems.
‚
0
adjust to the desired range
‚
Note that in the formulation, is not bounded above. A
very high sensitivity can become undesirable if there is error in
the signal factor settings. However, if the error is also treated
as a noise factor, then it will be mitigated by parameter design
technique.
We now consider the issue of estimation. Assume that the
data are obtained using a cross array design which is a prod-
uct of the control array, noise array, and signal factor levels.
A discussion on the choice of experimental plans for signal-
response systems has been given by Wu and Hamada (2000,
The steps in modeling and optimization are summarized as
follows:
2
‘

1. Estimate
and by Ž tting a log-linear model in (19)
i
s2
using a gamma GLM, where
is estimated from (18).
ij
‚
ˆ
yijk
fˆO4‚iMj 5
2. Estimate
and
from
by using
i
y
i
denote the response value at run , signal
sec. 11.6). Let
ijk
ƒ

j
2
i
1
O
4‘ M 5
weighted least squares with weights
.
j
k
level , and noise level . Then
O

O
2
O
O D
O
‡
‚ =‘
‚
3. Fit
log
and log in terms of X.
D
C
yijk f 4 1 M 5 …ijk1
Xi
(17)
j
4. Use a mathematical programming algorithm for (14) or
the two-step optimization procedure in (16) to Ž nd the opti-
mal X.

…
N4 1‘ 2M 5
where the ijk’s are independently distributed as
Suppose that we do not know the true functional form of the
0
.
i
j
5. Collect more data at the optimal X for different values
of the signal factor and Ž t a more elaborate signal-response
model.
N
yij¢
f 4 1 M 5
j
mean. Then
we can estimate
is an obvious estimate of Xi
. Hence
(18)

‘ 2M
from
i
j
K
X
1
ƒ
2
2
D
ƒ N
4yijk yij¢5 0
sij
The models used in steps 1 and 2 work only under
assumptions previously discussed. More general models could
be used, but they will complicate computation of the per-
formance measure and optimization. In many cases it would
D
K
1 kD
1
2
i
‘

We can estimate
and using a log-linear model,
2
ij
2
i
D
C
‘
‘

M 1
log
j
log
log
(19)
f 4‚M5 ‚M
sufŽ ce to choose
. It should be noted that even
ˆ
2
with this choice, our approach is different from Taguchi’s
approach. He used the linear signal-response relationship as
an ideal function and tried to minimizes the deviations from
it, whereas we separate the LOF term and use the linear
signal-response relationship only as an approximation to the
true signal-response relationship.
ƒ
4K
5s2 =‘ 2
•
where
1
1. The Ž tting can be done using a
ƒ
ij
ij
K
gamma generalized likelihood model (GLM) with a log-link
or using quasi-likelihood estimation (McCullagh and Nelder
1989). Thus we can compute the noise variability without
actually knowing the exact mean function. To Ž nd the variabil-
M
ity after adjustment, we have to know the value of to get the
t
f4 1 M5
response at on average. Instead of using
X
, we can
f 4‚4 5M5
to the
4.2 Response Modeling
M
obtain an approximate value of
fˆ 4‚M5
by Ž tting
X
ˆ
Y
data. The model
is to be Ž tted using weighted
When the noise factors have Ž xed settings in an experiment,
ƒ

4‘ 2M 5
the ijk’s in (17) are not independent, and thus therefore an
1. Note that
…
least squares with weights
XXX
approach that takes into account this structure is more appro-
priate (see Bérubé and Nair 1998). Tsui (1998, 1999) has
shown that mean-variance modeling can lead to bias in the
control factor effects. Modeling the response as a function of
the noise factors will avoid this problem.

2
2
ƒ
4yijk fˆ4‚iMj 55 =‘i Mj
i
j
k
X
XX

LOFi=‘i2 4K
5
si2j =‘i2Mj 1
D
C
ƒ
K
1
(20)
i
i
j
Let N be the set of observable noise factors. Then
where the lack of Ž t (LOF) sum of squares is
X
D
C
D
Y
f 4 1 1M5
X N
…
…
E4…5
for a given N, let
, where
0,

2
D
N ƒ
LOFi
4yij¢ fˆ 4‚iMj 55 =Mj 0
D
4…5 ”4 1 1 M5
var
X N
and is the random error caused by the
j
unobservable noise factors. As an approximation to the true
signal-response relationship, consider a polynomial model,
2
‘
In Taguchi’s approach, the
in model (1) includes both
the LOF and the noise variation. Here we separate these two
components and try to minimize only the noise variation. The
cause for LOF is not random, and thus the robustness should
not be sacriŽ ced to reduce the LOF. The minimization of LOF
‚ 4 1 5M ¢¢¢ ‚ 4 1 5Mr 0
D
C
C
C
f4 1 1M5 ‚ 4 1
X N X N
5
X N
1
X N
r
0
(21)
In the response function modeling suggested by Miller and
is not important, because in most cases it can be eliminated at Wu (1996), the parameters in (21) are estimated for each con-
the optimal setting by Ž tting a more elaborate signal-response trol and noise combination and are then modeled with respect
TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4

342
V. ROSHAN JOSEPH AND C. F. JEFF WU
to X and N. An alternative approach is to Ž t (21) directly from
the data (see Miller 1993; Tsui 1998, 1999). An important
issue not considered in the literature is that the signal-response
relationship should be monotonic. Polynomial models do not
have the monotonicity property; therefore, the estimation pro-
cedure should ensure that the Ž tted response is monotonic
Transformations on both the signal factor and response
should be used to enhance the modeling described in the
previous sections. There are several experiments reported
that use transformations based on engineering knowledge to
linearize the signal-response relationship (see, e.g., Fowlkes
and Creveling 1995). The robust setting obtained using the
transformed response can be considered to be an approxima-
tion to the robust setting for the original response under the
unbiased strategy. It is also possible to introduce unknown
parameters in the transformations. In such a case, estimation
of the parameters should be done using the iteratively
reweighted least squares method (see McCullagh and Nelder
1989; Engle and Huele 1996).
M
in
in the desired range. Most of the research in mono-
tonic regression is on nonparametric methods (see Robertson,
Wright, and Dykstra 1988). The estimation problem can be
avoided if the regression function is chosen from a monotone
class of functions.
As in Section 4.1, choose an appropriate model for the vari-
ance term and estimate it from replicates. Then (21) can be
O
Although the performance measure in (13) resembles the
=”4 1 1M5
Ž tted with weights 1
X N
using monotonic regression.
”4 1 1 M5
they are entirely different
ƒ
generalized SN ratio in (3),
. The in
It is not necessary to reestimate X N
from the resid-
ƒ , whereas we assume
/
4Y 5
‚
(3) is obtained by assuming var
uals if we decided to throw out the lack of Ž t component.
The mean and variance functions are then obtained using the
conditional expectation and variance formulas. This requires
explicit knowledge about the distribution of the noise factors,
and may need to use Monte Carlo methods. Most often the

/
4Y 5
var
M
ƒ

. Hence the value of will be different from
,
‚
4Y 5
. As
depending on how the control factors affect and var
we have seen, the minimization of variability demands a per-
formance measure depending on how the signal factor affects
the mean and variance and not on the overall slope-variance
relationship exhibited in the data. Because the static charac-
teristics (i.e., response with a single target value) represent
a special case of dynamic characteristics, similar comments
apply to the static parameter design optimization. As we show
PM
in (7) must be obtained using numerical integration for
PM
different control factor settings. The
with respect to X and optimized.
can then be modeled
D
f4 1 1M5
‚ 4 1 5M
X N and
For a special case with
X N
PM
1
”4 1 1 M5 ”4 1 5M2
D
X N
obtained as
X N
, the
in (7) can be explicitly

in the next section, the best noninformative choice of
is
2, and thus the static SN ratio analysis is justiŽ able in many
linear systems.
C
‚ 4 1
varN X N
5
E ”4 1 5
X N
1
N
PM ²
1
E2 ‚ 4 1
5
X N
1
N
5. SIGNAL-TO-NOISE RATIO
which can be viewed as the reciprocal of the SN ratio. Con-
sider the case of a single noise factor with mean 0 and variance
In this section we identify the underlying statistical model
for the SN ratio analysis. The explanation of the SN ratio
given here is different from that in the literature. Taguchi
(1993) assumed that when a scaling factor (same as the adjust-
2
D
C
‘
‚ 4 1
X N
5
‚ 4 5 ‚ 4 5N
X X
10 11 1
1 . If we further assume that
1
D
C
” 4 5 ” 4 5N
”4 1
and X N
lem becomes
5
X
X
1, then the optimization prob-
0
1
‚2 4 5
X
‚
ment parameter) is used to adjust the
‚
‘ 24‚ =‚52
I , the variance will change to
to its ideal value
, and therefore he
10
max
‚2 4 5‘ 2 ” 4 5
2
C
D
X
X
X
I
0
11
1
used the SN ratio in (2) for optimization. Phadke and Dehnad
(1988) extended the same idea in their derivation. Leon, Shoe-
maker, and Kacker (1987) justiŽ ed the SN ratio as a perfor-
mance measure independent of adjustment (PerMIA) under a
subject to
b
a
µ ‚ 4 5 µ
1
X
10
MH
ML
2
/
4Y 5
model with var
‚
(see also Lunani et al. 1997; Wu
which can be easily solved using a standard nonlinear pro-
gramming algorithm.
and Hamada 2000). We show that assuming the existence of
a scaling factor is not required and derive a version of the SN
ratio under some assumptions.
In contrast to Miller and Wu (1996) and Tsui (1998, 1999),
after
we focus on minimizing the variation in the response
Y
M
Assume that and are nonnegative variables and that the
signal-response relationship passes through the origin. Also
assume that the response is not the end result of an additive
process, to avoid situations like the electroplating example
given in Section 2. Let Z be the set of noise factors. Then
the relationship between the response and other factors can be
written as
adjusting for the mean. This is important, because the variance
is a function of the signal factor and can change while the
mean is being adjusted to a speciŽ ed target.
4.3 Discussion
The performance of the SN ratio in (2) is comparable to
D
‡

that of the
in (13) when
2 and with a linear signal-
D
Y
f4 1 1 M50
X Z
response relationship. Because of the importance of the SN
ratio, we give a more rigorous treatment in the next section.
Using Taylor’s theorem and series expansion,
2
‚
Taguchi (1993) used an unbiased estimate of
for estimating
µ
¶
µ
¶
¡f
¡f
the SN ratio. Because the performance measure is used only
for comparing different control factor settings, the bias is not
of much concern as long as it does not change greatly with X.
In the Appendix we show that the bias in maximum likelihood
estimate can be neglected.
D
D
Y
M
M
¡M
¡M
S
D
D
M
M
M
M0
µ
¶
¡2f
¡M2
S
M4M M05
C
ƒ
C
¢ ¢¢ 1
M0
D
M
TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4

ROBUST PARAMETER DESIGN
343
S
and
µ M µ M
M
is chosen so as to get a good linear
0
where 0
and
Ž t through the origin in the region of interest. Using only the
N 4
2 high
R
5 2 0 A= 0 B=0 C= 0 D1
95 1 05 95 1 05
ƒ
T
on
Ž rst term in the expansion, we get the approximate formulas
µ
¶
A1B1 C1
D
and are the nominal values of the four fac-
where
(22) tors. A 24 design is used for the control factors. The response
RT ƒ
¡f
D
EZ4Y 5 MEZ
¡M
D
M
M0
is generated from the transfer function in (4). For each
on
R
run, there are six values of
levels of the signal factor and two levels of the noise factor.
For comparison, we use both mean-variance modeling and
response modeling for analysis. The
against the signal factor for each control and noise combi-
nation in Figure 2. We see that the mean and variance of
corresponding to the three
and
ƒ
T
on
µ
¶
¡f
2VarZ
(23)
D
Z4Y 5
M
0
var
¡M
D
M
M0
R
values are plotted
ƒ
T
on
Thus an approximate statistical model under the stated
assumptions is
RT ƒ
M
increases with . Assume that the experimenter knows
on
D
C
Y
‚M …1
(24)
D
RT ƒ
M

0. Using a gamma GLM,
that the
is 0 when
on
2
D
D
4…5 ‘ M
E4…5
where
0 and var
2. Following the approach
is estimated from the log-linear model in (19) as 2.58. An
approximate 95% conŽ dence interval for
which clearly shows that a value greater than 2 should be used
in the analysis. Let
using nonlinear regression with weights
the value of as 1.111. The half-normal plot of the effects of
of Section 4.1, the performance measure to maximize can be

is (2.45, 2.72),
‚2=‘ 2
‚2=‘ 2 E 4Y 5= 4Y 5
2
D
derived as
. Now
var
, which can be
ˆ
R
4‚M5ˆ
D
f 4‚M5 4‚M5
viewed as a SN ratio. The performance measure from model
. Now Ž t
ƒ
T on
ˆ
2
O

ƒ
2
1. This gives
‘
O
4‘ M 5
(1) is
and is not the SN ratio. Thus the underlying statistical
model of the SN ratio is different from (1).
fˆ4‚M5
ˆ
O

2, given in Figure 3(a), shows that the four main
O
D
O
‚ =‘
In mean-variance modeling, using
for the mean
‡
log
function will improve the approximation in (22), whereas
CD
effects and the
interaction are signiŽ cant. The optimum

using instead of 2 and separating LOF from the residuals
A B C D
2
‚
1. The lower bound for can be calculated
levels are
1
2
were aimed at improving the approximation in (23) and
relaxing some of the assumptions. Also, the dependence of the
‚
from (15) as 1.06. The value of at the optimum combina-
tion (run 7) is 2.04, which satisŽ es the constraint, and hence
the adjustment step in (16) is not required. At the optimum
M
variance on
necessitates the use of weighted least squares
estimation. Hence we can expect an overall improvement
in the analysis by using the performance measure in (13)
compared to the SN ratio. We substantiate this claim with an
example in the next section.
M
R
setting, an approximate range of
to get
in (1, 5) is
ƒ
T
on
M
obtained as (.49, 2.08). So generate data by taking
.25 and 2.5 with an increment of .25. One model that Ž ts this
data well is
between
RT ƒ
0
M
0
M2
0447
0
M31
00099
D
C
C
2 24
on
6. AN EXAMPLE
which can be used to design the adjustment system in the
temperature controller.
For the response modeling, a quadratic polynomial with no
intercept term is Ž tted. The error in the model is solely due to
the LOF, and thus it is omitted from analysis. The performance
measure in (7) is obtained through numerical integration and
In this section we illustrate the proposed methodology using
the temperature controller example. For the purpose of illus-
R
R
so that
3
tration, we treat
as the signal factor instead of
2
we have a nonlinear signal-response relationship. We assume
no knowledge about the exact form of the transfer function in
(4) for the analysis, and then evaluate the performance of our
approach based on the transfer function. The factors and their
levels are given in Table 1.
PM
PM
given in
R
is denoted by
R. The half-normal plot of log
Figure 3(b) shows that the four main effects are signiŽ cant.
We now report the results of an exact analysis based on the
RT ƒ
Suppose that the customer requirement on
is uni-
on
‹
transfer function. Introduce so that
4 1 5
formly distributed in 1 5 and that the maximum value for
C
B4C AD5
M
that can be used to achieve this range is 4. Select three
D
ERT ƒ
‹
M0
(25)
on
ƒ
ƒ
A4C 4D
5M5
1
M
levels for
as 1, 2, and 3.5. Assume that the deviations in
D
ER
t
M
, the value of should be
To get
the factors from their nominal values are normally distributed
with standard deviation 5% of their nominal values. Two levels
of compounded noise factor are selected as
ƒ
T
on
ACt
D
M
0
(26)
ƒ
C
C
4D
5At ‹BC ‹ABD
1
N 4
1 low
R
5 2 0 A=0 B= 0 C=0 D
1 05 95 1 05 95
D
‹
ER
T
ƒ
Starting with
1 and using simulations to compute
,
T
on
ƒ
on
M
can be iteratively solved from (25) and (26). This value
M
4R on5
is then used to compute var
of
based on 10,000
ƒ
T
Table 1. Factors and Levels
D
t
simulations. The variance is computed for
and is numerically integrated using a Ž ve-point rule to get the
PM PM
1, 2, 3, 4, and 5
Levels
in (7). The half-normal plot of the effects of log
given
Sl.No.
Factors
Notation
1
2
in Figure 3(c) shows that only the main effects are important.
A B C D
The optimum combination is
setting as obtained earlier.
1, which is the same
1
2
2
1
2
3
4
R1
R3
R4
A
B
C
D
2
2
4
4
15
25
We also analyze this problem using the SN ratio in (2). For
=
E0 Ez
105
205
‚
‘
are estimated based on model (1) using
each run,
and
TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4

344
V. ROSHAN JOSEPH AND C. F. JEFF WU
0.5
1.0
1.5
2.0
M
2.5
3.0
3.5
Figure 2. Plot of RT
Against M.
-
on
(a)
(b)
D
A
A
D
B
B
C
C
CD
BD
BC
AD
AC
CD
BCD
ABD
AD
ACD
AC
ACD AB
ABC
ABCD
ABD BCD BC ABC ABCD BD
AB
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
half-normal quantiles
(c)
half-normal quantiles
(d)
B
D
A
D
C
C
AB
BD
AD
AC
CD
BC
CD
A
ACD
ABD ABC
ACD
AD
BCD
ABCD
BD ABD
B
ABC D BCD AB
ABC
BC
AC
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
half-normal quantiles
half-normal quantiles
Figure 3. Half-Normal Plots. (a) Mean-variance modeling; (b) response modeling; (c) exact analysis; (d) SN ratio analysis.
TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4

ROBUST PARAMETER DESIGN
345
(a)
(b)
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.0
1.1
1.2
1.3
1.4
1.5
1.6
log(1/PM)
log(1/PM)
‡
(
=
)
Figure 4. Plots of (a) and (b) SN Against log 1 PM .
‡
least squares, and the SN ratio is calculated. The
4 =PM5
and SN values are plotted against log 1
values model in (1). Many believe that the SN ratio is justiŽ ed
2
/
4Y 5
‚
in Figure 4. As only when var
is more correlated with the exact performance
values being 81.3% and
37.4%. The half-normal plot of the effects of SN is shown
in Figure 3(d). We see that the SN ratio analysis completely
, which requires the existence of a
scaling/adjustment factor. We have shown that this is not
true. Rather, it is how the signal factor affects the mean and
variance that leads to the SN ratio or a related performance
measure. Throughout the article we have emphasized the
importance of modeling variation in the response as a function
of the signal factor to get the proper performance measure for
robust design optimization.
‡
expected,
R2
measure than SN, with the
B
missed the most important effect of
and barely identiŽ es
PM
did not
R
A
‡
and log
as signiŽ cant. The analysis of
detect the effects in the exact order of importance [compare
Figs. 3(a), (b), and (c)], but did succeed in capturing the
important effects.
ACKNOWLEDGMENTS
The authors thank the associate editor, two referees, and
Vijay Nair for helpful comments. The research was supported
by National Science Foundation grant DMS-0072489.
7. CONCLUSIONS
In this article we have formulated the robust parameter
design of multiple-target systems as a mathematical program-
ming problem. We have deŽ ned the performance measure to
evaluate a control factor setting as the average variability in
the response after adjusting for the mean through the sig-
nal factor. We have explained two practical approaches for
estimation of the performance measure. The signal-response
relationship in many multiple-target systems can be nonlinear.
We have proposed some methods to address nonlinearity and
have demonstrated the usefulness of our approach using an
example.
APPENDIX: BIAS IN ESTIMATION
Consider the SN ratio model
D
C
yijkl ‚iMj
…ijkl1
y
i
j
where
is the observation at control level , signal level ,
ijkl
k
l
…
N4 1‘ 2M25
0 . We want
i j
noise level , and replicate and
ijkl
2
i
‚
D
‡
to estimate
(MLEs) of
log ‘ 2 . The maximum likelihood estimators
i
i
2
i
‚
‘
and
are
i
J
K
L
XXX
yijkl
Mj
1
O
The statistical model underlying the SN ratio analysis
is given in (24), which is different from the widely used
D
D N
ui : : :
‚
i
JKL jD
D
D
l
1
k
1
1
TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4

346
V. ROSHAN JOSEPH AND C. F. JEFF WU
Leon, R. V., Shoemaker, A. C., and Kacker, R. N. (1987), “Performance
and
Measures Independent of Adjustment: An Explanation and Extension of
Taguchi’s Signal-to-Noise Ratios,” Technometrics 29, 253–285.
Luenberger, D. G. (1989), Linear and Nonlinear Programming, Reading, MA:
Addison-Wesley.
J
K
L
XXX
1
2
O D
ƒ N
‘
4uijkl ui : : : 521
i
JKL jD
D
D
l
1
k
1
1
Lunani, M., Nair, V. N., and Wasserman, G. S. (1997), “Graphical Meth-
ods for Robust Design With Dynamic Characteristics,” Journal of Quality
Technology, 29, 327–338.
D
O D
‡
is given by
i
u
yijkl=M
‡
where
j . Then the MLE of
ijkl
i
log ‘‚22 . The bias in the estimate can be obtained using Taylor’s
O
Oi
i
McCaskey, S. D., and Tsui, K. L. (1997), “Analysis of Dynamic Robust
Design Experiments,” International Journal of Production Research, 35,
1561–1574.
McCullagh, P., and Nelder, J. A. (1989), Generalized Linear Models, New
York: Chapman and Hall.
Miller, A. (1993), “Improving Signal-Response Systems Through Designed
Experiments,” unpublished doctoral thesis, University of Waterloo.
Miller, A., and Wu, C. F. J. (1996), “Parameter Design for Signal-Response
Systems: A Different Look at Taguchi’s Dynamic Parameter Design,” Sta-
tistical Science, 11, 122–136.
series expansion as
O ƒ
E4‡i ‡i5
D ƒ
C
B4‡i5 c1
where
ˆ
X
ƒ
WW
4 n
2
5
1
n‡i
D
B4‡i5
eƒ
n4JKL5n
D
n
1
and
Nair, V. N., Taam, W., and Ye, K. Q. (2002), “Analysis of Functional
Responses From Robust Design Studies with Location and Dispersion
Effects,” Journal of Quality Technology, to appear.
Phadke, M. S. (1989), Quality Engineering Using Robust Design, Englewood
Cliffs, NJ: Prentice-Hall.
³ ´
ˆ
n
XX
C
ƒ
WW
JKL
n
4JKL
m
5
3
ƒ
2
WW
c
5m
1
0
D
C
ƒ
4
log
m n4JKL 5 4JKL 5m
ƒ
ƒ
JKL
1
3
1
D
D
n
m
1
0
Phadke, M. S., and Dehnad, K. (1988), “Optimization of Product and Process
Design for Quality and Cost,” Quality and Reliability Engineering Interna-
tional, 4, 105–112.
WW
€ €
¢ ¢¢n
n
n
when is odd
The notation
€ €
means that 1
3
5
¢¢ ¢n
n
c
and 2
4
6
when is even. Because is a constant,
B4‡ 5
i
Robertson, T., Wright, F. T., and Dykstra, R. L. (1988), Order Restricted
Statistical Inference, New York: Wiley.
Taguchi, G. (1987), System of Experimental Design, White Plains, NY:
Unipub/Kraus International.
we need consider only the term
. In typical experiments,
for the normal-
decreases with increasing
J ¶ 1K ¶ 1
L ¶
‚
‘
2
2
2
and
1. Also,
i
i
B4‡i5
ity assumption to be valid.
JKL
‚ =‘
i
B4‡ 5
i. Hence the maximum value of
i
and
can be ——— (1990), “Experimental Design for Product Design,” Statistical Design
and Analysis of Industrial Experiments, ed. S. Ghosh, New York: Marcel
D
D
JKL
‚ =‘
i i
obtained by setting
accounts for only 5% of the i, and therefore the bias can be
neglected.
4 and
‡
2. We obtain that it
Dekker.
——— (1993), Taguchi Methods: Design of Experiments, Quality Engineering
Series, Vol. 4, Dearborn, MI: ASI Press.
Taguchi, G., Chowdhury, S., and Taguchi, S. (2000), Robust Engineering,
New York: McGraw-Hill.
[Received May 2001. Revised December 2001.]
Tsui, K. L. (1998), “Alternatives of Taguchi’s Approach for Dynamic Robust
Design Problems,” International Journal of Reliability, Quality, and Safety
Engineering, 5, 115–131.
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TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4