ABSTRACT FOR ARTICLE DUE 5PM EASTERN TIME "PERFECT WRITER"

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 [l̂l] and standard deviation [r̂r]),

(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 l̂l Minimize r̂r Maximize l̂l subject to r̂r ¼ hr subject to l̂l ¼ hl subject to r̂r ¼ hr

Copeland and Nelson (1996) Minimize l̂l Minimize r̂r Maximize l̂l subject to r̂r � Ur subject to ðl̂l � hlÞ2 � D2 subject to r̂r � Ur

Lin and Tu (1995) Min (xl̂l2 þ ð1 � xÞr̂r2) Min (xðl̂l � hlÞ2 þ ð1 � xÞr̂r2) Min (�xl̂l2 þ ð1 � xÞr̂r2) Shaibu and Cho (2009) Min (l̂l þ r̂r � hrð Þ2) Min ððl̂l � hlÞ2 þ ðr̂r � hrÞ2Þ Min (�l̂l þðr̂r � 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 l̂l � xldl ¼ hl and r̂r � 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 l̂l þ g�l � gþl ¼ hl and r̂r 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

143 Simultaneous Optimization of Mean and Standard Deviation

(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 ðl̂l; r̂rÞ, 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 (l̂l) and standard deviation (r̂r) of the quality characteristic were fitted to full second-order

models as follows:

l̂l 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

r̂r 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: l̂l ¼ 500 and r̂r < 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

145 Simultaneous Optimization of Mean and Standard Deviation

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 (l̂l) and standard deviation (r̂r) of the quality characteristic were fitted to full second-order models

as follows:

l̂l 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

r̂r 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 l̂l and the satisfactory level of the r̂r are 80 and 3.0, respectively. Taking these values into account, in this article it is

assumed that 78 � l̂l � 82 and r̂r < 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. Solutions 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.

147 Simultaneous Optimization of Mean and Standard Deviation

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 (l̂l) and standard deviation (r̂r) 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 l̂l 327.6 177.0 109.4 131.5 32.0 �22.4 �29.1 66.0 75.5 43.6 4 r̂r 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 l̂l 500.00 Result 11 r̂r 45.132 0.285

149 Simultaneous Optimization of Mean and Standard Deviation

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