Compound Noise - MIT Department of Mechanical Engineering

QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL
Qual. Reliab. Engng. Int. 2007; 23:387–398
Published online 12 October 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/qre.812
Research
Compound Noise: Evaluation as
a Robust Design Method
Jagmeet Singh 1,∗,† ,DanielD.Frey 1,2 , Nathan Soderborg 3 and Rajesh Jugulum 1
1 Massachusetts Institute of Technology, Department of Mechanical Engineering, 77 Massachusetts Avenue,
Cambridge, MA 02139, U.S.A.
2 Massachusetts Institute of Technology, Engineering Systems Division, 77 Massachusetts Avenue,
Cambridge, MA 02139, U.S.A.
3 Ford Motor Company, North American Product Creation, Six Sigma, MD 5029, Building 5,
20300 Rotunda Drive, Dearborn, MI 48124, U.S.A.
This paper evaluates compound noise as a robust design method. Application of
compound noise as a robust design method leads to a reduction in experimental effort.
The compound noise strategy was applied to two types of situation: the first type has
been described with active effects up to two-factor interactions and the second type
has been described with effects up to three-factor interactions. These two situations
are illustrated with help of case studies. The paper provides theoretical justification
for the effectiveness of the compound noise strategy as formulated by Taguchi and
Phadke. For example, we found that the compound noise strategy is very effective
for systems which exhibit effect sparsity. This paper gives an alternative procedure
to formulate a compound noise, distinctly different from Taguchi’s formulation.
The alternative method requires less information to formulate compound noise as
compared to Taguchi’s formulation. Overall, the paper studies the effectiveness of
such an alternative formulation, outlines scenarios where compound noise as a robust
design method can be effectively used and gives alternative strategies for the systems
on which compound noise cannot be effective. Copyright c○ 2006 John Wiley & Sons,
Ltd.
Received 7 February 2006; Revised 27 June 2006; Accepted 3 July 2006
KEY WORDS: robust design; crossed array; Taguchi methods; compound noise; robust setting
INTRODUCTION AND BACKGROUND
In robust parameter design, the adverse effects of noise factors on quality are reduced by exploiting controlby-noise
interactions. A methodology for efficiently improving robustness was developed and employed
widely in Japanese industry by Taguchi 1 starting in the 1950s. The methods became popular in the
United States and Europe, especially during the 1980s, partly as a response to Japanese competition. In the
past few decades, there have been theoretical advances and new robust design methods have been developed.
∗ Correspondence to: Jagmeet Singh, Massachusetts Institute of Technology, Department of Mechanical Engineering, 77 Massachusetts
Avenue, Cambridge, MA 02139, U.S.A.
† E-mail: jagmeet@mit.edu
Contract/grant sponsor: Ford–MIT Alliance
Copyright c○ 2006 John Wiley & Sons, Ltd.

388 J. SINGH ET AL.
As discussed in a recent review paper by Robinson et al. 2 , research efforts in robust design have led to novel
performance measures, new experimental designs, and alternatives to Taguchi’s methods building upon response
surface methodology. One of the principal goals of recent research has been to reduce run sizes while still
attaining good outcomes. The focus of this paper will be compound noise, which is also intended to reduce run
sizes.
Compound noise is a technique in which multiple noise factors are varied simultaneously as if they were a
single noise factor. In most documented applications, an outer array of noise factors is replaced by the compound
noise factor being varied between two levels. Taguchi 1 and Phadke 3 suggested forming compound noise factors
based on the directionality of the noise factor effects thereby creating two conditions that represent, in some
sense, opposite extremes. Du et al. 4 suggested forming compound noises based on the conditions of the two
‘most probable points of inverse reliability’. This approach to compounding is better able to account for skew
in the distribution of system performance. This new concept of compound noise, like Taguchi’s, is based on two
‘extreme settings’ of noise factors.
Hou 5 studied the conditions that will make compound noise yield robust settings for systems. Hou said
‘extreme settings should exist for compound noise to work’. We find below that compound noise can be effective
even when extreme settings do not exist. The conditions mentioned in ‘compound noise factor theory’ turn out
to be the sufficient conditions. In later sections we extend the analysis to determine conditions under which
compound noise will predict a robust setting. Hou’s formulation was limited to systems that had active effects
up to two-factor interactions. We extend the formulation to systems that can have active effects up to three-factor
interactions.
Compound noise can be considered as an extension of supersaturated designs (SSD). This concept initially
originated with a paper by Satterthwaite 6 . SSDs were assumed to offer a potentially useful way to investigate
many factors with few experiments. Holcomb and Carlyle 7 discussed the construction and evaluation of SSDs.
Holcomb et al. 8 outlined the analysis of SSDs. Compound noise is an unbalanced SSD. Allen and Bernshteyn 9
discussed the advantages of unbalanced SSDs in terms of performance and affordability. Heyden et al. 10 argued
that SSDs can be used to estimate variance of response, which can be used as a measure of robustness rather
than using it to find main effects. SSDs ‘do not allow estimation of the effects of the individual factors because
of confounding between the main effects’. However, ‘estimation of the separate factor effects is not necessarily
required’ in improving robustness. Using compound noise as a robust design method we try to estimate the
robustness of the system at a given control factor setting. The setting that improves this estimate is taken as the
predicted robust setting.
The aim of this study is to explore the effectiveness of compound noise as a robust design method. Engineering
systems show certain regularities, one of which is the hierarchical ordering principle (see Wu and Hamada 11
and Chipman et al. 12 ). We classified systems based on the hierarchical ordering principle. The classes are as
follows.
• Strong hierarchy systems are those that have only main effects and two-factor interactions active.
Some small three-factor interactions might be present in such systems, but they are not active.
• Weak hierarchy systems are those that also have active three-factor interactions.
We apply compound noise to three strong hierarchy systems and three weak hierarchy systems and analyze
robustness gain. There are three main questions we want to address in this paper.
• Why is compound noise effective in achieving a robust setting in certain cases, but ineffective in other
cases
• How can we measure the effectiveness of a compound noise strategy
• Do we need to know the directionality of noise factors to use compound noise
We analyze a compound noise factor strategy for six case studies and explore reasons for the effectiveness or
ineffectiveness of the strategy. We then present the conditions for compound noise to work. These conditions are
an extension of Hou 5 . First, we present conditions for strong and weak hierarchy systems. Second, we present
the effectiveness of the compound noise strategy in real scenarios using fractional factorial arrays for control
Copyright c○ 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398
DOI: 10.1002/qre

COMPOUND NOISE: EVALUATION AS A ROBUST DESIGN METHOD 389
Table I. Results from full factorial control factor array and two-level compound noise
Matching of
all control Matching of
Measure Existence of factors with control factors
of sparsity extreme Number of their robust with their Improvement
System Hierarchy of effects settings (Hou 5 ) replications setting (%) robust setting (%) ratio (%)
Op Amp Strong 0.105 Existing 100 96 99.2 99.2
PNM Strong 0.132 Non-existent 100 98 99 99.5
Journal bearing Strong 0.28 Existing 100 97.8 98.8 98.84
CSTR Weak 0.647 Non-existent 100 10 55.17 58.4
Temperature Weak 0.725 Non-existent 100 5 29.25 30.65
Control circuit
Slider crank Weak 0.119 Existing 100 95 98 98.5
and noise factors. Finally, we present a procedure to determine an outer array for robust design experiments,
which can be used by practitioners.
CASE STUDIES
We analyzed six case studies. Three of them exhibited strong hierarchy: passive neuron model (PNM; Tawfik and
Durand 13 ), operational amplifier (Op Amp; Phadke 3 ) and journal bearing: half Sommerfeld solution (Hamrock
et al. 14 ). Three case studies exhibited weak hierarchy: the continuous-stirred tank reactor (CSTR; Kalagnanam
and Diwekar 15 ), the temperature control circuit (Phadke 3 ) and the slider crank (Gao et al. 16 ). For each case study
we ran full factorial control and noise factor crossed array experiments. These runs were used to determine the
effect coefficients for the main effects, two-factor interactions and three-factor interactions. Lenth’s method
(Lenth 17 ) was employed to determine the active effects. The full factorial results were also used to determine
the most robust setting for all six systems.
In order to formulate compound noise for robust design experiments, we found the directionality of noise
factors on the system response for all six systems. For each system we ran a full factorial design in control
factors crossed with two-level compound noise. Pure experimental error was introduced into the system’s
response. The error was of the order of the active main effects. From Lenth’s method, we can estimate the
average magnitude of active main effects. Pure experimental error introduced into the system’s response was
normally distributed with zero mean and standard deviation as the average magnitude of active main effects.
The compound noise strategy performed extremely well for the PNM, Op Amp, journal bearing and slider
crank. Table I the shows results from the robust design experiments.
The third column of Table I shows the measure of sparsity of effects, which is the ratio of active effects in a
system to the total number of effects that can be present in the system. The sparsity of effects means that among
several experimental effects examined in any experiment, a small fraction will usually prove to be significant
(Box and Meyer 18 and Wu and Hamada 11,19 ). The sixth column of Table I shows how well the predicted robust
setting from compound noise experiments matches with the actual robust setting as found by carrying out full
factorial control and noise array experiments. The seventh column of Table I shows the percentage of control
factors whose robust setting matched for full factorial and compound noise experiments.
At the predicted robust setting of control factors from compound noise experiments, we found the
corresponding variance of the system’s response from full factorial experiments. Full factorial experiments
yield the optimal variance of the system’s response and the average variance of the response at all control
factor settings. With this knowledge, variance can be improved from the average value to the optimal value,
which is the maximum possible improvement. For compound noise experiments the variance of the response
can be improved from the average value to the variance at the predicted robust setting. The ratio of achieved
Copyright c○ 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398
DOI: 10.1002/qre

390 J. SINGH ET AL.
improvement to maximum possible improvement is called the improvement ratio. The average of this measure
for all replicates is shown in the eighth column of Table I.
From Table I we can see that compound noise strategy works reasonably well for the Op Amp, PNM, journal
bearing and slider crank, but not nearly as well for the CSTR and the temperature control circuit.
CONDITIONS FOR COMPOUND NOISE TO BE COMPLETELY EFFECTIVE
Hou 5 gave conditions under which a compound noise strategy will predict the robust setting for a system.
One of the conditions for compound noise to work is the existence of extreme settings. The extreme settings
of compound noise are those that maximize and minimize the system’s response to capture noise variations.
However, we found that in some systems (e.g., PNM; Tawfik and Durand 13 ) the compound noise strategy would
work even if extreme settings do not exist. So the conditions outlined in Theorem 4 of Hou 5 are sufficient but
not necessary conditions for compound noise to work.
For strong hierarchy systems the response y can be expressed as
m∑ m∑
y = f(x 1 ,x 2 ,...,x l ) + β j z j +
j=1
j=1 i=1
l∑
γ ij x i z j (1)
where f(x 1 ,x 2 ,...,x l ) is a general function in the control factors x i ,z j (j = 1,...,m)denotes the noise
factors affecting the system, β j denotes the effect intensity of noise factor j and γ ij denotes the effect intensities
of control-by-noise interactions present in the system. Control factors can have two settings, either −1 or1.
(Control factor variability can be represented as separate noise factors.) Noise factors are in the range −1 to1
and are independent, with zero mean and a variance of 1. Noise factors are symmetric about 0. The variance of
y with respect to z j termsisgivenby
( l∑ ( m∑ )
Var z (y) = C + 2 β j γ ij x i +
i=1 j=1
l∑
( m∑ )
γ ij γ kj
)x i x k
j=1
where C is a constant. The robust setting of the system is such that the variance of the response is minimized
with respect to noise factors. It can be represented as
[ ( l∑ ( m∑ ) l∑
( m∑ )]
X ≡ arg min β j γ ij x i + γ ij γ kj
)x i x k (3)
x i
i=1 j=1 j=1
where X is the setting of the x i terms that minimizes the variance.
If we follow Taguchi’s formulation of compound noise, based on the directionality of noise factors, then
responses at two levels of compound noise will be given by
m∑
( l∑
y + = f(x 1 ,x 2 ,...,x l ) + β j + γ ij x i
)sign(β j ) (4)
y − = f(x 1 ,x 2 ,...,x l ) −
j=1
k=1
i

COMPOUND NOISE: EVALUATION AS A ROBUST DESIGN METHOD 391
where C 1 is a constant. The estimated robust setting of the system is such that the estimated variance of the
response is minimized with respect to the compound noise levels. It can be represented as
[ ( l∑ ( m∑
X Compound ≡ arg min
x i
i=1 j=1
|β j |
m∑
sign(β j )γ ij
)x i +
j=1
l∑
k=1
i

392 J. SINGH ET AL.
System
Table II. Results from resolution III control and noise factor array
Hierarchy
Measure
of sparsity
of effects
Number of
replications
Matching of
all control
factors with
their robust
setting
(%)
Matching of
control factors
with their
robust setting
(%)
Improvement
ratio
(%)
Op Amp Strong 0.105 100 98 99.6 99.91
CSTR Weak 0.647 100 None 76.5 96.5
Temperature Weak 0.725 100 28 55 87
control circuit
Slider crank Weak 0.119 100 91 95.5 97.6
System
Table III. Results from resolution III control factor array and two-level compound noise
Hierarchy
Measure of
sparsity of
effects
Number of
replications
Matching of
all control
factors with
their robust
setting
(%)
Matching of
control factors
with their
robust setting
(%)
Improvement
ratio
(%)
Improvement
ratio (from
resolution III
noise array)
(%)
Op Amp Strong 0.105 100 86 97.2 99.37 99.91
CSTR Weak 0.647 100 None 41.17 39.8 96.5
Temperature Weak 0.725 100 None 43.5 25.2 87
control circuit
Slider crank Weak 0.119 100 89 94 96.5 97.6
System
Table IV. Average results from full factorial control factor array and two-level random compound noise
Hierarchy
Measure of
sparsity of
effects
Number of
replications
Matching of
all control
factors with
their robust
setting
(%)
Matching of
control factors
with their
robust setting
(%)
Improvement
ratio
(%)
Improvement
ratio (from
Taguchi’s
compound
noise)
(%)
Op Amp Strong 0.105 1000 30.8 73 80.2 99.2
PNM Strong 0.132 2 4 = 16 50 68.75 79.9 99.5
Journal bearing Strong 0.28 2 3 = 8 100 100 100 98.84
CSTR Weak 0.647 2 6 = 64 54.69 70.3 67.6 58.4
Temperature Weak 0.725 2 5 = 32 68.75 76.56 74.75 30.65
control circuit
Slider crank Weak 0.119 2 5 = 32 50 70 55.2 98.5
The compound noise strategy also does not perform well with a resolution III control factor array for the
temperature control circuit. The improvement ratio drops from 87% for a resolution III noise array to 25%
for two-level compound noise.
In the formulation of compound noise as suggested by Taguchi 1 and Phadke 3 , we need to know the
directionality of noise factors on the system’s response. We need to run some fractional factorial experiments
on the system to gather information about the directionality of noise factors. We tried a different formulation
Copyright c○ 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398
DOI: 10.1002/qre

COMPOUND NOISE: EVALUATION AS A ROBUST DESIGN METHOD 393
of compound noise. Noise factor settings can be picked randomly to form a two-level random compound noise.
We ran such a compound noise strategy with a full factorial control factor array for each of the six case studies.
Table IV presents the average results from these experiments.
The Op Amp has 21 noise factors, so there can be 2 21 combinations of random compound noise. Instead
we ran 1000 such combinations of random compound noise for the Op Amp. The PNM has four noise factors.
Hence, there are 2 4 combinations of random compound noise for the PNM. The journal bearing has three, the
CSTR has six and the temperature control circuit and slider crank have five noise factors each. From Table IV
we see that such a formulation of compound noise is very effective on average. Except for the slider crank
system, random compound noise can achieve improvement ratios greater than 68% on average for all systems.
For the slider crank system, half of the replications gave a robust setting of the system and half of them led to
poor settings. Hence, had we used random compound noise with care for the slider crank system we would have
attained a robust setting.
CONCLUSIONS
The use of compound noise in robust design experiments leads to a reduction in experimental effort. In this
paper we tried to gage the effectiveness of compound noise as a robust design method. We built upon the
conditions given by Hou 5 for the effectiveness of compound noise. Also we extended the conditions to include
the possibility of three-factor interactions.
Compound noise as a robust design strategy is very effective on the systems that exhibit effect sparsity. We ran
compound noise on six case studies. The compound noise strategy predicted robust settings for the systems that
exhibited effect sparsity.
The given formulation of compound noise was adopted from Taguchi 1 and Phadke 3 . For such a formulation
we need to know the directionality of noise factors on the system’s response. To know the directionality of noise
factors we need to run fractional factorial experiments on the system. We tried to measure the effectiveness of
random compound noise for which we do not need to know the directionality of noise factors. We found that
such a formulation of compound noise can be very effective in obtaining high robustness gains. The reason why
such a formulation of compound noise is very effective is that active effects in the system are sparse. Even if we
randomly compound noise there is a very low probability that two opposite acting interactions get compounded
with each other. The results outlined in Tables I–IV did not change significantly on removing experimental error
from observations. In this case we will obtain the same conclusions regarding the effectiveness of all robust
design methods on all six systems that we studied.
These results may be used in the overall approach to deploying compound noise as a robust design strategy.
The flowchart in Figure 1 presents our suggestion for implementing these findings in practice. First of all,
practicing engineers must define the scenario including the system to be improved, what objectives are being
sought and what design variables can be altered.
At this point, it may be possible to consider what assumptions can be made regarding effect sparsity. It should
be noted here that we do not argue that engineers need to make a factual determination of effect sparsity.
The experience on the system should be used to make decision.
If engineers decide that effects are dense, then they should follow the procedure on the right-hand side of
Figure 1. In this case, the results in this paper suggest that instead of using compound noise, a fractional factorial
noise factor array should be used as the outer array.
If engineers decide that effects are sparse, then they should follow the procedure on the left-hand side of
Figure 1. At this point, engineers should consider the directionality of noise factors. If the directionality of noise
factors is known, then they should use Taguchi and Phadke’s formulation of compound noise. Otherwise they
can formulate random compound noise as the outer array. From Table IV, we can say that random compound
noise is effective for the systems for which effect sparsity holds. However, experience, judgment and knowledge
of engineering and science are critical in formulating such a compound noise. It is hoped that the procedure
proposed here in Figure 1 will be of value to practitioners seeking to implement robust design efficiently and
using minimum resources.
Copyright c○ 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398
DOI: 10.1002/qre

394 J. SINGH ET AL.
Define the robust design scenario
Sparse
Assumptions about
effect sparsity
Dense
Yes
Is the directionality of
noise factors known
No
Use fractional/full
factorial array as
outer array (instead
of compound
noise)
Use compound noise
(as defined by Taguchi,
Phadke) as outer array
Use random
compound noise
as outer array
Carry out robust design using the
chosen outer array
Figure 1. Suggested procedure for compound noise in robust design
Acknowledgement
The support of Ford–MIT Alliance is gratefully acknowledged.
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COMPOUND NOISE: EVALUATION AS A ROBUST DESIGN METHOD 395
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APPENDIX A
For weak hierarchy systems the response can be represented by
m∑
m∑
y = f(x 1 ,x 2 ,...,x l ) + β j z j +
j=1
j=1 i=1
l∑
m∑
γ ij x i z j +
l∑
j=1 i=1
l∑
m∑
θ ikj x i x k z j +
k=1
i

396 J. SINGH ET AL.
+
+
l∑
m∑
i=1 j=1 n=1
j

COMPOUND NOISE: EVALUATION AS A ROBUST DESIGN METHOD 397
( l∑ ( m∑ ) )( l∑
+ sign(β j )γ ij x i
i=1 j=1
i=1
+
+
+
l∑
k=1
i

398 J. SINGH ET AL.
where X Compound is the setting of x i terms that minimize the estimated variance. For compound noise to be
completely effective X = X Compound .
Authors’ biographies
Jagmeet Singh received a PhD degree from the Department of Mechanical Engineering at MIT. He received a
SM degree in Mechanical Engineering from MIT in 2003 and a BTech degree in Mechanical Engineering from
the Indian Institute of Technology, Kanpur, India. He worked on this paper as part of his research towards his
PhD. His areas of expertise include assembly architecture, datum flow chains and noise strategies in large-scale
systems.
Daniel D. Frey is an assistant professor of the Mechanical Engineering and Engineering Systems Division
at MIT. He conducts research on system design methods including robust design, design of experiments,
probability, manufacturing and computational geometry. The overarching goal of his research is to identify
principles and practices that improve the process of engineering design. He teaches a range of courses related
to system design. He is member of ASME, ASEE, AIAA and ASA.
Nathan Soderborg is a Design for Six Sigma Master Black Belt in the Ford Motor Company’s North American
Product Development organization. He coaches product development teams in mathematical modeling and the
application of statistical engineering and robust design methods to improve product quality. He is a member of
the MAA, senior member of ASQ and ASQ Certified Reliability Engineer (CRE) and Six Sigma Black Belt
(CSSBB).
Rajesh Jugulum is a Researcher at MIT and a Vice President in the global wealth and investment management
group of Bank of America. He earned his Doctorate degree under the guidance of Dr. Genichi Taguchi. He
has published several articles in leading technical journals and magazines. He co-authored a book on pattern
information technology and a book on computer-based robust engineering. He is a senior member of the ASQ
and the Japanese Quality Engineering Society and a Fellow of the Royal Statistical Society and International
Technology Institute.
Copyright c○ 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398
DOI: 10.1002/qre