Simultaneous Optimization of Mean andStandard DeviationN.docx

Simultaneous Optimization of Mean and
Standard Deviation
Nuno Ricardo Pais Costa
Polytechnic Institute of Setúbal,
Setúbal College of Technology,
Campus do IPS, Setúbal,
Portugal
ABSTRACT Checking whether process and product are
satisfying or
functioning according to the technical specification is not
enough to assure
competitiveness. Competition compels organizations to develop
efforts to
assure that product and process characteristics are on target
values and
the variability around those targets is minimal. This article
proposes an alter-
native method for optimizing both the mean and standard
deviation of a
quality characteristic of the process or product. The objective

function
accommodates all the response types, allowing the practitioner
to assign dis-
tinct weights to process mean and standard deviation and to find
trade-off
solutions between them, taking their relative magnitudes into
account.
Two classical examples from the literature are used to illustrate
the feasi-
bility of the proposed method and compare its results with those
of other
popular methods. A practical procedure for implementing the
proposed
method is also presented.
KEYWORDS dual response, GRG, optimization, response
surface, robust
design
INTRODUCTION
In the current market, rather than checking whether process and
product
are satisfying or functioning according to technical
specifications, it is essen-
tial to assure that their characteristics are on target value and

the variability
around those targets is minimal. With this aim, various
approaches have
been put forward and dissected in the literature.
Response surface methodology (RSM) since the 1950s and
robust
parameter design (RPD) since the mid-1980s have been
successfully used
in different application areas with various purposes, namely, to
get low
variability, low cost, high quality, and high reliability in the
process and pro-
duct. RPD has generated a great deal of interest among
researchers and
practitioners, although the statistical techniques used have been
the subject
of much discussion and controversy. A comprehensive overview
at a con-
ceptual level of the importance and usefulness of the principles
underlying
RPD, a methodology for implementing them, and proposals of
alternative
methods for data analysis is provided by Nair (1992). Recent
references

are Robinson et al. (2004), Park et al. (2006), and Arvidsson
and Gremyr
Address correspondence to Nuno
Ricardo Pais Costa, Polytechnic
Institute of Setúbal, Setúbal College
of Technology, Campus do IPS,
Estefanilha, 2910-761 Setúbal,
Portugal. E-mail:
[email protected]
Quality Engineering, 22:140–149, 2010
Copyright # Taylor & Francis Group, LLC
ISSN: 0898-2112 print=1532-4222 online
DOI: 10.1080/08982110903394205
140
(2008). Technical details on RPD can be found in the
extensive bibliographies of these references.
The integration of RPD principles with well-
established statistical techniques has been a much
researched subject. One research area that has
received considerable attention in the literature and
application in practice is the simultaneous optimiza-
tion of both the mean and standard deviation of a

characteristic of interest. Various approaches for
solving this problem in the RSM framework, which
is thoroughly discussed by Myers et al. (2004), have
been put forward in the literature, including the
popular priority-based, mean squared–based, and
goal programming–based approaches. However,
some drawbacks have been assigned to them;
namely, they may rule out better solutions, do not
consider the specification limits assigned to respon-
ses, and are not readily understood by practitioners
with limited mathematical and statistical skills. These
drawbacks must be a concern and highlight the need
for simple, yet effective, approaches to solve the
aforementioned problem as this is a current indus-
trial problem faced by practitioners, including non-
statisticians, who are using statistics than ever before
to solve problems in different application areas
(Hahn and Doganaksoy 2008; Meeker 2008).

This article introduces an alternative method for
simultaneous optimization of both the mean and
standard deviation of a characteristic of interest in
the RSM framework that has three major characteris-
tics: effectiveness, simplicity, and application easiness.
The remainder of the article is organized as fol-
lows: the following section presents a review on
methods for dual response optimization. Then an
alternative method is proposed. The next section
includes two examples to demonstrate the feasibility
of the proposed method and compares results with
various methods. The subsequent section discusses
the method results. In the final section conclusions
are presented. A procedure that allows mapping
steps for the application of the proposed method
can be found in the Appendix.
DUAL RESPONSE OPTIMIZATION—
METHODS REVIEW

Determination of the optimal mean value of
a quality characteristic of interest with the
consideration of variance reduction is a much
researched subject and a common practice in organi-
zations as this goal results in high-performance
process and high-quality product.
The optimization of both the mean and standard
deviation of a single quality characteristic is called
a dual response optimization problem, which is a
particular case of a multiple dual response optimiza-
tion problem where the objective is to optimize both
the mean and standard deviation of multiple quality
characteristics. Most literature focuses on these pro-
blems, so a review on approaches for dual response
optimization is relevant due to practical interest of
the subject and usefulness as adapting approaches
formulated for optimizing multiple dual response
problems to dual response problems may be a diffi-

cult and unnecessary task for practitioners who do
not have sufficient mathematical=statistical back-
ground. Moreover, those approaches are difficult to
find as they are published in various journals, which
may not be readily available.
This article reviews the most popular approaches
for optimizing dual response problems in the RSM
framework, namely, those based on priority, mean
square error, goal programming, and desirability
function. The emphasis is on ‘‘what to optimize’’:
defining an objective function, rather than ‘‘how to
optimize’’: determining the optimization algorithm.
Relevant references for this later issue are Del
Castillo and Montgomery (1993), Del Castillo et al.
(1997), Fan (2000), Köksoy and Doganaksoy
(2003), Pasandideh and Niaki (2006), and Köksoy
and Yalcinoz (2005, 2008).
Vining and Myers (1990) have the merit of revita-

lizing the optimization of dual response problems by
proposing to optimize a ‘‘primary response’’ subject
to an appropriate constraint on the value of a
‘‘secondary response.’’ The decision to make the
mean the primary or secondary response depends
upon the ultimate purpose of the experiment. Those
authors considered the following cases:
. Target is best: keeping the mean at a specified tar-
get value while minimizing the standard deviation.
. The larger the better: making the mean as large as
possible while keeping the standard deviation at a
specified target value.
. The smaller the better: making the mean as small
as possible while keeping the standard deviation
at a specified target value.
141 Simultaneous Optimization of Mean and Standard Deviation
An effective alternative to the priority-based

approach of Vining and Myers (1990) was suggested
by Copeland and Nelson (1996). These authors
proposed a method with more relaxed constraints
in the secondary response. Nevertheless, one must
take care if using priority-based approaches as
constraints on the secondary response may rule out
better solutions (Köksoy and Yalcinoz 2008).
Another popular approach to dual response
optimization consists of aggregating the mean and
standard deviation models into a single function.
Lin and Tu (1995) proposed objective functions
based on mean square error (MSE), which deals with
bias and variance and does not consider the con-
straint in the secondary response. Where tight
adherence to the target is necessary they suggest
weighting (to multiply by a parameter) each one of
the MSE terms. A data-driven approach and a
preference-based method to determine the weights

were proposed by Ding et al. (2004) and Jeong
et al. (2005), respectively. Shaibu and Cho (2009)
proposed a variant of Lin and Tu’s method that con-
siders the departure of standard deviation from its
target in the objective function for all response types.
Coetzer et al. (2008) used the exponential member-
ship functions of Kim and Lin (1998) in the weighted
MSE proposed by Lin and Tu (1995) instead of
the mean and variance models. An extension of Lin
and Tu’s method for multiple dual response
problems is proposed by Köksoy (2008).
Weighted square sum–based methods allow over-
taking the drawback of the priority-based methods
but may fail to capture appropriate solutions in the
concave part of the response surface defined by
objective function. As Messac et al. (2000) proved,
using exponents for assigning priorities to responses
is more effective than using a constant of propor-

tionality like in the weighted square sum–based
methods. Publications addressing this and other
limitations of weighted-sum linear and quadratic
objective functions include Koski (1985), Athan and
Papalambros (1996), Das and Dennis (1997), and
Mattson and Messac (2003).
Kim and Lin (1998) proposed a mathematical
programming method based on fuzzy optimization
methodology as alternative to previous methods.
Their approach consists of maximizing the minimum
satisfaction level with respect to the mean and
standard deviation within the feasible region, taking
explicitly into account their allowable ranges. They
claim that this is advantageous under the assumption
that all solutions within tolerance limits are not
necessarily equally desirable. As illustrated in their
paper, the membership function they proposed can
generate a rich variety of shapes by adjusting its

parameters and achieve a better balance between
bias and variance in the sense that the contribution
of both bias and variance is properly reflected in
the optimization process (Kim and Lin 1998). Some
variations in this approach are discussed by
Kim and Lin (2006) when multiple dual response
problems are considered.
Nonlinear goal programming techniques have
been also used to optimize dual response problems.
Goal programming is a technique that requires ordi-
nal and cardinal information for multiple objective
decision making (Tabucanon 1988), where deviation
variables (from goals) with assigned priorities and
weights are minimized instead of optimizing the
objective function directly as in previous appro-
aches. Tang and Xu (2002) proposed a goal
programming–based approach that is general
enough to include some of the existing methods as

special cases. The methods of Vining and Myers
(1990) and Lin and Tu (1995) are examples of those
that can be represented using that approach (see
Table 1 in Tang and Xu (2002) for details). Tang
and Xu’s approach consists of minimizing a convex
combination of two slack variables (unrestricted sca-
lar variables), one for the mean and the other for the
standard deviation, and adjusts weighting parameters
to those slack variables in the constraints so that the
trade-off between meeting the targets for the dual res-
ponses can be explicitly incorporated. Both the objec-
tive function and constraints are of quadratic form.
To consider asymmetric deviations of process
mean on both directions about the target and
reducing process variability, Kim and Cho (2002)
proposed a goal programming optimization model
whose objective function is expressed in terms of
deviational variables associated to the under- and

overachievement of process mean and variability.
The prioritization scheme for the optimization
procedure is established by assigning different
priorities=weights to the deviation variables. Variants
of this nonlinear goal programming technique for
multiple dual problems are illustrated by Kovach
and Cho (2008) and Bhamare et al. (2009).
N. R. P. Costa 142
Mathematical programming methods based on
fuzzy optimization methodology and goal program-
ming approaches have been often used for solving
dual response problems. Though these methodolo-
gies, among others reviewed by Jeyapaul et al.
(2005), have their own merits, the understandability
and application easiness of these approaches is
difficult for use by non-statistician practitioners. In
contrast, the desirability function approach proposed

by Derringer and Suich (1980) is very popular and
easy to understand and implement. Moreover, it is
available in most statistical software packages.
Köksoy (2005) showed that the approach proposed
by Derringer and Suich can be applied to dual
response optimization. In this type of problem, that
approach consists of converting the mean and stan-
dard deviation model into individual desirability func-
tions that are combined through geometric mean,
followed by optimization of that composite function.
A criticism to this approach was made by Chiao and
Hamada (2001), who affirm that the composite func-
tion value does not allow a clear interpretation, except
the principle that a higher value is preferred.
A summary of these methods is presented in
Table 1. The notation used hereafter is as follows:
ŷy: Estimated response (mean [l̂ l] and standard
deviation [r̂ r]),
(hl, hr): Target value for the mean and standard

deviation
ðLl; UlÞ: Lower and upper bounds assigned to mean
(Lr, Ur): Lower and upper bounds assigned to
standard deviation
D: Admissible deviation of mean from its target value
k: Membership score (degree of satisfaction, 0 � k � 1)
dj(z): Fuzzy membership function for the mean and
standard deviation
(dl, dr): Unrestricted scalar variables
x: Weight=priority assigned to estimated response
ðx�l ; xþl Þ: Priorities assigned to the under- and over-
achievement of the mean
ðg�l ; gþl Þ; ðg�r ; gþr Þ: Deviational variables associated
with the under and over achievement of the mean
and standard deviation, respectively
dj: Individual desirability function for the mean (dl)
and standard deviation (dr)
Table 1 Optimization Methods for Dual Response Problem
Reference The Smaller the Better Target is Best The Larger the
Better
Vining and Myers (1990) Minimize l̂ l Minimize r̂ r Maximize l̂ l
subject to r̂ r ¼ hr subject to l̂ l ¼ hl subject to r̂ r ¼ hr

Copeland and Nelson (1996) Minimize l̂ l Minimize r̂ r Maximize
l̂ l
subject to r̂ r � Ur subject to ðl̂ l � hlÞ2 � D2 subject to r̂ r � Ur
Lin and Tu (1995) Min (xl̂ l2 þ ð1 � xÞr̂ r2) Min (xðl̂ l � hlÞ2 þ
ð1 � xÞr̂ r2) Min (�xl̂ l2 þ ð1 � xÞr̂ r2)
Shaibu and Cho (2009) Min (l̂ l þ r̂ r � hrð Þ2) Min ððl̂ l � hlÞ2
þ ðr̂ r � hrÞ2Þ Min (�l̂ l þðr̂ r � hrÞ2)
Kim and Lin (1998) Maximize k
Subject to dj(z) � k, j ¼ l, r
with dj zð Þ ¼
exp tð Þ�exp t zj jð Þ
exp tð Þ�1 ; t 6¼ 0
1 � zj j; t ¼ 0
(
z ¼ ŷy�L
U�L z ¼
ŷy�h
h�L z ¼
U�ŷy
U�L
Tang and Xu (2002) Minimize d2l þ d
2
r
� �
subject to l̂ l � xldl ¼ hl and r̂ r � xrdr ¼ hr

Kim and Cho (2002) Min x�l g
�
l
� �2
þ xþl gþl
� �2
þ xr hr � g�r � gþr
� �2
subject to l̂ l þ g�l � gþl ¼ hl and r̂ r þ g�r � gþr ¼ hr
with g�l � gþl ¼ 0 , g�r � gþr ¼ 0 , and g�l ; gþl ; g�r ; gþr
� 0
Köksoy (2005)
Derringer and
Suich (1980)
Maximize dlxdr
� �1=2
dj ¼
ŷy � U
L � U
� �r
for L � ŷy � U
dj ¼
ŷy�L
h�L

� �s
; L � ŷy � h
ŷy�U
h�U
� �t
; h � ŷy � U
0 otherwise
8>><
>>:
dj ¼
ŷy � L
U � L
� �r
for L � ŷy � U
(r, s, t): Shape parameters assigned to individual
desirability functions (d).
PROPOSED METHOD
There are various analytical techniques in the
literature that allow practitioners to find a compro-

mise solution for dual response problems. However,
as Izarbe et al. (2008) noted, the practitioners who
undertake experimentation and need to optimize
process prefer to use less complicated techniques.
Assuming that an optimal solution for a dual
response problem may require a trade-off between
the process mean and the standard deviation, that
is, their simultaneous optimization, the following
method is proposed:
Minimize
X2
i¼1
ŷyi � hij j
Ui � Li
� �xi
½1�
where hi corresponds to the target value (hl, hr) of
the estimated response ŷyi ðl̂ l; r̂ rÞ, with Li � ŷyi � Ui.
The parameter xi (xl, xr > 0) represents the priority
assigned to responses, which may depend on either
economical and technical issues or practitioner’s
preference.

As Eq. [1] reflects the objective inherent to dual
response problems, to minimize both the mean and
standard deviation departure from their target values,
it becomes easily understood and attractive to practi-
tioners in contrast to other methods.
The proposed additive equation is built on the
so-called global criterion method such as presented
by Tabucanon (1988) but has features that make an
impact on the results. A feature shared by both meth-
ods is that they allow practitioner assigning of differ-
ent priorities to mean and standard deviation
through an exponent, which is effective to capture
solutions in convex and nonconvex response sur-
faces (Messac et al. 2000). The modulus in the
numerator and the denominator are the differentiat-
ing aspects of the proposed method.
The modulus set in the numerator allows simplify-
ing mathematical formulation of problems involving

different responses types, which is an appealing
characteristic to practitioners involved in process
and product optimization. In contrast with the
priority-based, weighted square sum–based, math-
ematical programming–based, and desirability-based
methods presented in Table 1, the proposed method
does not require response transformation and
accommodates in the same objective function differ-
ent response types, namely: nominal-the-best (NTB;
the value of the estimated response [ŷy] is expected
to achieve a particular target value), larger-the-best
(LTB; the value of ŷy is expected to be larger than a
lower bound ðŷy > LÞ), and smaller-the-best (STB;
the value of ŷy is expected to be smaller than an
upper bound ðŷy < UÞ).
The denominator set in Eq. [1], as alternative to the
objective value hi proposed by Tabucanon (1988),
has threefold purpose:
. It makes Eq. [1] dimensionless.

. The ratio 1=(Ui � Li) impacts on the optimal factor
settings noticeably as it can be considered a weight
assigned to each response that represents more
appropriately the existing differences in response
properties, such as the scale and allowable range.
This is especially advantageous for multiresponse
problems where simultaneous optimization of
multiple responses (means or means and standard
deviations) with different properties such as scale
and measurement unit is required.
. It provides flexibility for practitioners testing differ-
ent specification limits and exploring trade-offs
among responses since optimization results are
sensitive to the selection of bounds and targets
(Wurl and Albin 1999). As noted by Kovach et al.
(2008), flexibility is needed in order to consider
multiple responses simultaneously just as custo-
mers do when judging products and to capture
design preferences with a reasonable degree of

accuracy.
To use the proposed method, practitioners need
to define a target value, a lower and an upper bound
to each response. However, the author believes that
those involved in product and process optimization
have the necessary background for assigning appro-
priate values to targets and bounds of each response.
Even if practitioners have sufficient knowledge or
information on setting the appropriate values for
the bounds, it may be useful to study how sensitive
the optimization results are to changes in bounds
and targets in order to identify alternative solutions
of the practitioner’s preference. For example, the
standard deviation is an STB-type response and its
N. R. P. Costa 144
target value is typically set equal to zero (hr ¼ 0).
However, it may be reasonable to accept values for

variability up to a certain level and assign a positive
value to hr (hr > 0) for exploring alternative trade-
offs between the mean and standard deviation, such
as considered by Shaibu and Cho (2009). The target
value for the mean is typically set equal to the lower
and upper bounds assigned to STB- and LTB-type
responses, respectively. For NTB-type responses
the target value is set at equal distance of the lower
and upper bounds.
As concerns the implementation of the proposed
method, practitioners can easily do it in the popular
Excel. A built-in Microsoft Excel optimization routine
based on the generalized reduced gradient (GRG)
algorithm, the Solver tool, is used for optimizing
Eq. [1]. The GRG is considered one of the most
robust algorithms available and permits assigning
several forms of constraints to responses. Never-
theless, one must take care if using this algorithm
because it tends to converge to local optima

(compromise solutions where response values are
not as close to their target as possible). So, an optimi-
zation routine must be iterated using several random
starting points; for example, with all the experi-
mental design points. To use several random starting
points may not guarantee a global optimum. How-
ever, this practical procedure provides a global opti-
mal solution in many cases, in particular for the
multiresponse optimization problems encountered
in the RSM framework (Xu et al. 2004).
Regarding the results of Eq. [1], it is desirable to
obtain the minimum cumulative departure of
responses from their target value. Mean and standard
deviation on-target is the ideal solution. In these
conditions, the cumulative departure given by
Eq. [1] is equal to zero.
ILLUSTRATIVE EXAMPLES
To demonstrate the feasibility of the proposed

method and compare its results with those of other
methods, two classical examples from the literature
are used. The first one was adapted by Vining and
Myers (1990) and has appeared repeatedly in the
literature. Del Castillo and Montgomery (1993),
Copeland and Nelson (1996), Lin and Tu (1995),
Kim and Lin (1998), Tang and Xu (2002), and Köksoy
(2005) are authors who used that example. The
second one was reported by Luner (1994) and was
used by Kim and Cho (2002).
Example 1. With the purpose of improving a printing
machine’s ability to apply colored inks to package
labels, a three-level factorial design with three runs
at each design point was employed to generate data
and analyze the effect of speed (x1), pressure (x2),
and distance (x3) variables. The response surfaces
for the mean (l̂ l) and standard deviation (r̂ r) of the
quality characteristic were fitted to full second-order

models as follows:
l̂ l ¼ 327:6 þ 177x1 þ 109:4x2 þ 131:5x3 þ 32:0x21
� 22:4x22 � 29:1x
2
3 þ 66:0x1x2 þ 75:5x1x3
þ 43:6x2x3
r̂ r ¼ 34:9 þ 11:5x1 þ 15:3x2 þ 29:2x3 þ 4:2x21 � 1:3x
2
2
þ 16:8x23 þ 7:7x1x2 þ 5:1x1x3 þ 14:1x2x3
The NTB case is considered in this article because
various authors provide results for the following goals
of printing quality: l̂ l ¼ 500 and r̂ r < 60. The results are
taken directly from the original articles in order to pro-
vide a fair comparison basis to evaluate the perform-
ance of the proposed method. In particular, Köksoy
(2005) used the shape parameters s ¼ t ¼ 1 and
limits Ll ¼ 24; Ul ¼ 1010 and Lr ¼ 0; Ur ¼ 158:2.
Note that these limits correspond to the maximum
and minimum values of the mean and standard devi-
ation obtained from the experimental runs (see paper
for details on experimental design and results). Taking

into account these limits, in the proposed method are
considered the values of hl ¼ 500, hr ¼ 0, and
xi ¼ ð1; 1Þ.
The results presented in Table 2, based on the
cuboidal region �1 � xi � 1, are all in close agree-
ment. Though solutions found by Lin and Tu
(1995) and Kim and Lin (1998) show a slight bias
on the expected mean, the proposed method
especially supports the results of Del Castillo and
Montgomery (1993) and Köksoy (2005), where the
mean is on-target. Note that Del Castillo and Mon-
tgomery used an alternative approach to the dual
response method proposed by Vining and Myers
(1990) based on the GRG algorithm.
Example 2. With the purpose of predicting the dis-
tance to the point where a projectile lands from the
base of a catapult with minimal variability, a central

composite design with three replicates at each design
point was employed to generate the data (see Luner
(1994) for details) and analyze the effect of three
control factors: arm length (x1), stop angle (x2),
and pivot height (x3). The response surfaces for the
mean (l̂ l) and standard deviation (r̂ r) of the quality
characteristic were fitted to full second-order models
as follows:
l̂ l ¼ 84:88 þ 15:29x1 þ 0:24x2 þ 18:80x3 � 0:52x21
� 11:80x22 þ 0:39x
2
3 þ 0:22x1x2 þ 3:60x1x3
� 4:22x2x3
r̂ r ¼ 4:53 þ 1:84x1 þ 4:28x2 þ 3:73x3 þ 1:16x21
þ 4:40x22 þ 0:94x
2
3 þ 1:20x1x2 þ 0:73x1x3
þ 3:49x2x3
Kim and Cho (2002) used this example to demon-
strate the robust design model they proposed, con-

sidering that the target value for l̂ l and the
satisfactory level of the r̂ r are 80 and 3.0, respectively.
Taking these values into account, in this article it is
assumed that 78 � l̂ l � 82 and r̂ r < 5; that is,
Ll ¼ 78, Ul ¼ 82, hl ¼ 80, Lr ¼ hr ¼ 0, and Ur ¼ 5.
Results summarized in Table 3, in particular the
solution denoted by A, provide evidence that the
proposed model can yield similar results to the Kim
and Cho’s (2002) method, requiring values to xl
and xr, whereas values to x
�
l ; x
þ
l , and xr are
required in the Kim and Cho’s method.
Solution
s B
and C are similar to those presented in Kim and
Cho (2002) of Models I (Vining and Myers’s (1990),

Del Castillo and Montgomery’s (1993), and Copeland
and Nelson’s (1996) methods) and II (Lin and Tu’s
(1995) and Copeland and Nelson’s (1996) methods),
which incorporate a preemptive priority to process
bias and an equal priorities to process bias and varia-
bility, respectively.
A procedure that allows mapping steps for appli-
cation of the proposed method is presented in the
Appendix.
Sensitivity analysis with respect to the priorities
assigned to the deviational variables is performed
to observe the behavior of proposed method’s solu-

tions. The sensitivity of those solutions to xl is ana-
lyzed by varying xl from 1.0 to 5.6 while holding xr
at 1.0, using as starting point xi ¼ �1; 1; �1ð Þ . It can
be observed from Figure 1 that by increasing the
value of xl the process variability decreases with
an increased bias. This result provides evidence that
the proposed method can present compromise solu-
tions between the bias and the variance in close
agreement with those reported by Kim and Cho
(2002) when zero bias solution is not necessarily of
the practitioner’s interest.
TABLE 2 Results
Method Variable settings Mean Standard deviation
Del Castillo and Montgomery (1993) (1.0, 0.1184, �0.259)
500.00 45.097
Copeland and Nelson (1996) (0.9809, 0.0427, �0.1898) 499.00

45.200
Lin and Tu (1995) (1.0, 0.0700, �0.250) 494.44 44.429
Kim and Lin (1998) (1.0, 0.0860, �0.254) 496.08 44.628
Tang and Xu (2002) (1.0, 0.1157, �0.259) 499.66 45.057
Köksoy (2005) (1.0, 0.1200, �0.260) 499.99 45.111
Proposed (1.0, 0.2049, �0.3180) 500.00 45.132
TABLE 3 Results
Method Priorities=weights Variable settings Mean Standard
deviation
Kim and Cho (2002) (1.0, 3.0, 5.0) (0.115, �0.256, �0.369)
78.35 2.966
Proposed (A) (5.6, 1.0) (0.120, �0.256, �0.373) 78.34 2.965
(B) (2.3, 1.0) (0.129, �0.278, �0.303) 79.67 3.113
(C) (1.0, 1.0) (0.152, �0.321, �0.285) 80.00 3.158
N. R. P. Costa 146
RESULTS AND DISCUSSION
Results from different approaches cannot be com-

pared in a straightforward manner because the meth-
ods differ in terms of their optimization criteria. Each
method reviewed in this article has its own merits,
and how good their solutions are may depend on
either economical and technical issues or practi-
tioner’s preference. Nevertheless, Examples 1 and 2
show that the proposed method is effective, yielding
results in close agreement with those of the other
methods.
In Example 1 the method results were achieved
keeping priorities=weights and specification limits
unchanged. The possibility of changing these values

gives flexibility for practitioners exploring trade-offs
between on-target performance and low variance
while they learn more about the problem, as is poss-
ible in Kim and Lin’s (1998) and Derringer and
Suich’s (1980) methods. That flexibility is indispens-
able because the solution for a problem may be
sensitive to selection of bounds and targets. In this
example the bounds used are as set by Köksoy
(2005) to provide a fair methods comparison basis,
but studying how sensitive the optimization results
are to changes in bounds may be useful if the practi-
tioners do not have sufficient knowledge or infor-
mation on setting the appropriate values for the

bounds.
In Example 2 the solutions are achieved by chan-
ging the priority values. To date there is not a clear
and effective procedure of prioritization to achieve
a preferred solution. In practice, the greater the num-
ber of priority factors included in the method, the
more difficult the task of assigning values to them
will be, which definitely complicates the process of
achieving an effective compromise solution. In the
proposed method the information required from
the practitioner is minimal. The practitioners only
need to assign values to xl and xr, which is parti-
cularly advantageous in the optimization process

and when a sensitivity analysis is required. According
to Kim and Cho (2002), the results of the sensitivity
analysis may provide alternative solutions in situa-
tions when the practitioner does not feel comfortable
with or does not have sufficient information on
setting the precise values for the priorities.
Example 2 serves to illustrate that the proposed
method can yield alternative solutions if zero bias
solution is not of the practitioners’ interest by either
technical or economical reasons. In fact, as Cho et al.
(2000) demonstrated, zero bias solutions are not nec-
essarily optimal. In this example, by forcing process

mean to deviate from target it was possible to achieve
a collection of alternatives to zero bias solution, which
include the solutions of the other methods. This result
confirms that the proposed method can be used in
practice for dual response optimization.
CONCLUSIONS
Though the ideal univariate response is simul-
taneously on-target and of minimal variance, reality
usually forces the practitioner to make trade-offs
between on-target performance and low variance.
In this article an alternative method is proposed
for simultaneous optimization of process mean and
process variability that may compete with leading

schemes in the field. The proposed method consid-
ers the nature of quality characteristics, takes relative
magnitude of responses into account, and allows
practitioners to assign priorities separately for each
one of them. Moreover, it has a direct physical
interpretation, can be easily implemented by practi-
tioners, and provides appropriate trade-off solutions
between the two process parameters. These attri-
butes, and in particular the low number of priorities
(weights) the user has to input, justify using the pro-
posed method for optimizing dual response pro-
blems and are a stimulus to apply it to multiple

dual response problems, where trade-offs among
several means and standard deviations are required.
Future research will explore its potential with this
type of problem.
FIGURE 1 Process bias and standard deviation for various xl
values.
ACKNOWLEDGMENTS
The author is grateful for valuable comments and
suggestions provided by the reviewers.
ABOUT THE AUTHOR
Nuna Costa is a lecturer in Quality and Operations

Management at Setúbal College of Technology. He
has implemented Quality Management Systems (ISO
9000 systems) in industry and among his research
interests is the design of experiments. He has several
publications in journals and conference proceedings,
and is a member of UNIDEMI (Research and Develop-
ment Unit on Mechanical and Industrial Engineering).
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APPENDIX
To help practitioners in implementing the propo-
sed method, the spreadsheet for Example 1 is pre-
sented in Table A1. Under the assumption that
experiments were designed, data were collected
and suitable models were fitted to the mean (l̂ l)
and standard deviation (r̂ r) of the characteristic of
interest, the procedure may follow the next steps:
1. Enter the coefficients of the estimated models:
cells B3:L4.
2. Enter responses’ specification limits (Li, Ui) and
target (Ti): cells H7:J8.

3. Enter priorities=weights (xi): cells E7:E8.
4. Link model’s coefficients with variable settings:
cells D10:D11. Note that the variable settings
(the output of the Excel Solver tool) will be
displayed at cells B7:B9.
5. Enter objective function linked with estimated
response values (cells D10:D11), response’s target
values, specification limits, and weights: cell F11.
6. Open Excel Solver tool and select: Set Target Cell:
F11, Equal to: Min (Minimize), By Changing Cells:
B7:B9, Subject to the Constraints: �1 � B7:B9 � 1,
and H7: H8 � D10:D11 � J7:J8, click on Options
and select Use Automatic Scaling, Estimates:
Quadratic, Derivatives: Forward, Search: Conju-

gate. See Excel Help menu for technical details
on these options.
7. Click on Solve: If F11 ¼ 0, the global optimum is
found. If F11 > 0, employ the following heuristic
procedure:
. Use the same weight for each response; for
example, setting xi ¼ 1 and testing all the
design points.
. Select the design point that produces the sol-
ution with the lowest value for objective
function (cell F11) and assign different weights
to responses until a solution of preference is
achieved.

8. Repeat step 7 until a preferred solution is found.
TABLE A1 Proposed Method—Example 1
A B C D E F G H I J L
1 Response Model’s coefficients
2 a0 a1 a2 a3 a1
2 a2
2 a3
2 a12 a13 a23
3 l̂ l 327.6 177.0 109.4 131.5 32.0 �22.4 �29.1 66.0 75.5 43.6
4 r̂ r 34.9 11.5 15.3 29.2 4.2 �1.3 16.8 7.7 5.1 14.1
5
6 Variable Settings Weights L T U
7 x1 1.0000 x1 1 Y1 24 500 1010
8 x2 0.1365 x2 1 Y2 0 0 158.2
9 x3 �0.2722
10 l̂ l 500.00 Result

11 r̂ r 45.132 0.285
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