Computers & Industrial Engineering 59 (2010) 976–985

Contents lists available at ScienceDirect

Computers & Industrial Engineering

journal homepage: www.elsevier .com/ locate/caie

Assessing effectiveness of the various performance metrics for multi-responseoptimization using multiple regression q

Surajit Pal a,⇑, Susanta Kumar Gauri b

a SQC & OR Unit, Indian Statistical Institute, 110, Nelson Manickam Road, Chennai 600 029, Indiab SQC & OR Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata 700 108, India

a r t i c l e i n f o

Article history:Received 6 August 2009Received in revised form 30 April 2010Accepted 12 September 2010Available online 17 September 2010

Keywords:Multiple responsesTaguchi methodMultiple regressionWeighted signal-to-noise ratioDesirability functionMultivariate loss function

0360-8352/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.cie.2010.09.009

q This manuscript was processed by Area Editor E.A⇑ Corresponding author. Tel.: +91 44 23740612; fax

E-mail address: surajitpal@hotmail.com (S. Pal).

a b s t r a c t

Several methods for optimization of multiple response problems using planned experimental data havebeen proposed in the literature. Among them, an integrated approach of multiple regression-basedoptimization using an overall performance criteria has become quite popular. In this article, we examinethe effectiveness of five performance metrics that are used for optimization of multiple responseproblems. The usefulness of these performance metrics are compared with respect to a utility measure,namely, the expected total non-conformance (NC), for three experimental datasets taken from theliterature. It is observed that multiple regression-based weighted signal-to-noise ratio as a performancemetric is the most effective in finding an optimal solution for multiple response problems.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

A common approach for process optimization is a plannedexperimentation. Taguchi methods (Taguchi, 1990) for designingan experiment using orthogonal arrays are extremely popularamong the practitioners. A large number of input process variablescan be assessed to find their contribution on the output responsevariable using a lesser number of experimental trials. Taguchi ap-plies the quality loss function to evaluate product quality andemploys the signal-to-noise (SN) ratio with simultaneous consid-eration of achieving the target and reducing variability aroundthe target value of the response variable. However, Taguchi meth-od focuses on optimization of a single response variable only.Whereas most of the modern manufacturing processes have sev-eral response variables and the process needs to be optimized forall response characteristics.

In a multiple response optimization problem, the main objec-tive is to find a setting combination of input process variables thatwould result in the optimum values of all response variables. Gen-erally, it is very difficult to obtain such a combination, becauseoptimum values of one response variable may lead to non-optimum values for the remaining response variables. Hence, it isdesirable to find a best combination of input variables that would

ll rights reserved.

. Elsayed.: +91 44 23740256.

result in an optimal compromise of response variables. Here opti-mal compromise means each response variable is as close as pos-sible to its target value and with minimum variability aroundthat target value.

Several methods for optimization of multiple response prob-lems using experimental data have been proposed in the literature.Some simpler methods (Pan, Wang, Wei, & Sher, 2007; Ramakrishnan& Karunamoorthy, 2006; Tai, Chen, & Wu, 1992; Tong, Chen, &Wang, 2007) use Taguchi’s quality loss function and signal-to-noise (SN) ratio as the starting point of analysis procedure.The basic approach is as follows: the quality loss or SN ratio of indi-vidual responses are computed from experimental data and theyare converted into an overall process performance index (PPI)and then, the factor-level combination, i.e. settings of the inputvariables that can optimize the PPI is determined by examiningthe level averages on the PPI. The advantage of the simpler PPIbased approach is that it can be easily comprehended and appliedeven by the engineers who do not have a strong background inmathematics/statistics. However, PPI based approach cannot en-sure that one or more responses will not move far away from theirtarget values at the optimal solution.

A few integrated methods are presented by Derringer and Suich(1980), Khuri and Conlon (1981), Logothetis and Haigh (1988),Pignatiello and Joseph (1993), Tsui (1999), Wu and Hamada(2010), Wu (2005), Ch’ng et al. (2005), and Pal and Gauri (2010).In these methods, a functional relationship of each response vari-able with the input decision variables is established using multiple

S. Pal, S.K. Gauri / Computers & Industrial Engineering 59 (2010) 976–985 977

regression techniques, and then, the optimal solution is deter-mined by considering an appropriately defined objective functionas the performance metric for optimization. Different researchershave defined different objective functions, e.g. (i) desirability func-tion (Derringer & Suich, 1980; Wu, 2005; Wu & Hamada, 2000), (ii)distance measure (Khuri & Conlon, 1981), (iii) multivariate (MV)loss function (Pignatiello & Joseph, 1993; Tsui, 1999), (iv) processcapability index C�pm (Ch’ng et al., 2005), and (v) multiple regres-sion-based weighted signal-to-noise (MRWSN) ratio (Pal & Gauri,2010).

The aim of the present work is to examine the effectiveness ofthe various performance metrics that are used in the multipleregression-based methods for optimization of multi-responseproblems. An ideal optimal solution to a multi-response problemshould result in that all the individual responses achieve theirrespective target value with the minimum variance around the tar-get value. In this context, expected non-conformance (NC) of theprocess can be an appropriate measure of the goodness of an opti-mal solution. So expected NC at the optimal solution is consideredhere as the utility measure of a performance metric. It may benoted that expected NC can be computed only if the specificationsfor the responses are known. For the comparison of the usefulnessof the various performance metrics, therefore, three experimentaldatasets with known specifications for the responses are analyzedusing various multiple regression-based multi-response optimiza-tion procedures.

The article is organized as follows: the second section providesa brief literature review on the existing methodologies for dealingwith the multi-response optimization problems. The third sectionoutlines the multiple regression-based optimization procedureusing different performance metrics. The fourth section describesthe utility functions, namely, expected process NC and the proce-dure for comparison of different performance metrics. In Section 5,we consider three sets of experimental data for analysis and com-parison. We conclude in Section 6.

2. Literature survey

The analysis of multiple response experiments has receivedhigh attention over the last three decades. Derringer and Suich(1980) and Khuri and Conlon (1981) proposed the multiple regres-sion-based approach to solve multiple response problems usingsome performance metric. Derringer and Suich (1980) used a desir-ability function, first defined by Harrington (1965), as a metric foroptimization of multiple response variables. Each response is con-verted into an individual desirability function and then an overalldesirability function is defined as the geometric mean of the indi-vidual response desirabilities. Khuri and Conlon (1981) developeda regression-based optimization method using a distance metric.This distance metric uses the squared deviations of the responsevariables from their targets and then standardizes these deviationsby the variance of the response variables. Logothetis and Haigh(1988) proposed an optimization technique by using multipleregression and the linear programming approach. In the linear pro-gramming model, one response variable is selected as primaryobjective function and is optimized by using other response vari-ables in the constraints criteria. Pignatiello and Joseph (1993) pres-ent a multiple regression technique based on the criteria ofminimizing the expected value of a multivariate loss function.Their procedure assumes that the responses follow a multivariatenormal distribution, are nominal-the-best (NTB) type and followan additive model. Tsui (1999) extended Pignatiello and Joseph’sprocedure to situations where responses may be smaller-the-better (STB) type or larger-the-better (LTB) type of qualitycharacteristics. Wu and Hamada (2000), and later Wu (2005)

suggested using double-exponential desirability function and pro-posed a regression-based approach for optimization of correlatedmultiple quality characteristics. Pal and Gauri (2010) proposedan integrated approach using multiple regression technique andweighted signal-to-noise ratio for multiple response optimization.In this method, the weighted SN ratio is maximized ensuring thatall the quality characteristics (responses) are very close to theirrespective target values and with minimum variability aroundthe target values.

A few simpler methods were developed using a metric knownas process performance index. These methods include weightedsignal-to-noise ratio (WSN) (Tai et al., 1992), multi-response sig-nal-to-noise ratio (MRSN) (Ramakrishnan & Karunamoorthy,2006), grey relational analysis (GRA) (Pan et al., 2007), and VIKOR(VlseKriterijumska Optimizacija I Kompromisno Resenje in Serbian)(Tong et al., 2007) methods. In these approaches, the quality lossor SN ratio of individual responses are converted first into an over-all process performance index (PPI) and then, the factor-level com-bination, i.e. settings of the input variables that can optimize thePPI is determined examining the level averages on the PPI. Thesemethods are well acceptable to the practitioners because of theirsimplicity. However, one limitation of those approaches, exceptthe VIKOR method (Tong et al., 2007), is that the optimal factor-level combination often result in one or more individual responsesto move far away from their target values which are not desirable.

Another group of researchers have attempted to integrate someof the advanced techniques such as principal component analysis(PCA) with the simpler approaches, i.e. PPI based approaches. Suand Tong (1997) and Liao (2006) have applied principal componentanalysis (PCA) and then, computed the PPI (named differently bythe authors) using one or more principal components instead ofthe original response variables. While Su and Tong (1997) haveused the first principal component only, Liao (2006) have utilizedall the principal components and his proposed approach is knownas weighted principal component (WPC) method. Tong et al. (2005)integrated PCA with the technique for order preference by similar-ity to ideal solution (TOPSIS).

Some researchers like Su and Hsieh (1998), Tong and Hsieh(2000), Hsi, Tsai, Wu, and Tzuang (1999) and Chiang and Su(2003) have found that the techniques of artificial intelligencecan be effectively used for process optimization. In these ap-proaches, parameters can be set optimally but nothing can belearned about the relationship between the control factors andthe responses, and so do not help engineers to learn efficient engi-neering experiences during process optimization.

A significant advancement has been made with the introductionof response surface methodology (RSM) for optimization of multi-ple response problems. RSM typically involves experimental de-sign, regression models and optimization. Regression models arebuilt based on the data collected in the experimental design andthen optimization is done using the regression models. Viningand Myers (1990) first introduced dual response surface optimiza-tion technique to solve joint optimization problem of mean andvariance of a single response variable in Taguchi’s experimentaldesign. Del Castillo and Montgomery (1993) and Lin and Tu(1995) have contributed significantly to the development of dualresponse surface optimization. In multiple response surface opti-mization, researchers seek to optimize simultaneously the meanand variance of several response variables. Ch’ng et al. (2005) haveproposed the usage of RSM and C�pm metric to optimize multiple re-sponse variables.

Multiple regression-based optimization methods are essentiallya part of response surface methodology. RSM techniques are basedon a series of experimentation which are different from Taguchi’sexperimental design using orthogonal arrays. RSM tries to obtainan optimal solution using the entire range of values of input vari-

978 S. Pal, S.K. Gauri / Computers & Industrial Engineering 59 (2010) 976–985

ables. Taguchi’s experimental design has set level values of inputcontrollable variables and hence optimal combination is obtainedusing these level values of input variables. A search procedure isfollowed in both RSM and multiple regression-based method usingdifferent types of metrics to obtain the optimal combination of in-put variables.

3. Multiple regression-based optimization methods

In this section, we describe the multiple regression-based ap-proaches for simultaneous optimization of multiple response vari-ables from designed experimental data. Five different types ofperformance metrics are considered: (1) total desirability function,(2) MV loss function, (3) normalized distance measure, (4) processcapability index C�pm, and (5) weighted signal-to-noise ratio. Opti-mization of a performance metric with respect to the inputcontrollable variables yields the best operating condition. It isimportant to note that one of the essential conditions for applica-tion of any multiple regression-based approaches for optimizationis that the relationship among the input variables and the responsevariables are well modeled by appropriate regression equations.The goodness of a fitted model is indicated by high values of R2

and adjusted R2 statistics. For an experimental dataset, if multipleregression equations with high value in R2 and adjusted R2 cannotbe found, then the solution obtained using any of these perfor-mance metrics may not be an optimal solution. So in such cases,multiple regression-based methods will not be a good choice foroptimization of multiple response variables.

3.1. Multiple regression

Let us assume that there is r number of response variables ob-served at each experimental run, i.e. at each set of conditions onk input variables (control factors) of a manufacturing process. Wedenote Yi, i = 1, 2, . . . , r as the response variable and Xj, j = 1, 2,. . . , k as the input variable. We assume that n (n P 2) number ofobservations is made for each response variable in each of the mexperimental runs (trials).

At first, mean and variance of each response variable for eachexperimental trial is computed. Then multiple polynomial regres-sion equations for both mean and variance of each response vari-able are developed. Let us denote the mean and standarddeviation of the response variable Yi by lyi and ryi respectively.These equations can be represented as follows:

lyi¼ h0 þ h1x1 þ h2x2 þ � � � þ hkxk þ hkþ1x2

1 þ � � � þ hp1xk�1xk ð1Þ

logðr2yiÞ ¼ b0 þ b1x1 þ b2x2 þ � � � þ bkxk þ bkþ1x2

1 þ � � � þ bp2xk�1xk

ð2Þ

where hi’s and bi’s are regression coefficients, and p1 and p2 givesthe number of regressor variables. Here, the logarithmic transfor-mation of variance is used to ensure positive values for variances.If necessary, the logarithmic transformation of mean also can beused to ensure feasible values for the mean. It is not necessary thatthe two regression equations of mean and variance should includethe same terms as regression variables. There may be some squareterms or cross terms which may appear in the regression equationfor mean but may not appear in the regression equation for varianceand vice versa. The number of regressor terms in any equationshould be smaller than the number of experimental runs in thedesigned experiment.

The multiple regression function for any response variable canbe easily developed using Microsoft Excel or any statistical pack-age, e.g. MINITAB and STATISTICA. The regression coefficients R2

and adjusted R2 are considered to drop unnecessary terms from

the model and to include only those terms that have some contri-butions on the response variable. Also, technical considerationsmust be used to include any term into the regression model. As athumb rule, if the value of R2 is more than 0.90 and value ofadjusted R2 is more than 0.85, the fitted model will be consideredadequate.

Diagnostic checks for validating the regression models must beperformed. Using analysis of variance (ANOVA) and an F-test for sig-nificance of regression, the adequacy of model is checked. A residualanalysis in terms of various plots, e.g. normality plot of residuals, plotof residual versus predicted values and plot of residual versus indi-vidual regression variable etc. are examined to detect possibleanomalies. If, after the diagnostic checks, no serious violations ofmodel assumptions are detected, then the regression equation is as-sumed to be adequate fit to predict the response variable.

3.2. Performance metrics

The performance metric is used to combine multiple responsesinto a single function or metric. In the following sub-sections, wedescribe five different performance metrics that are used as anobjective function for optimization.

3.2.1. Desirability function as performance metricThe utilization of desirability function is the most popular and

strongly suggested method for multiple response optimizationproblems (Carlyle, Montgomery, & Runger, 2000). Few statisticalpackages e.g. STATGRAPHICS and Design-Expert, use this desirabil-ity function approach for optimization of multiple responseproblems.

Each response variable is converted into an individual desirabil-ity function, proposed by Harrington (1965), whose value variesfrom 0 to 1. If the product characteristic is in an unacceptablerange, the desirability value is 0 and if the product characteristicis at the optimum value, the desirability value is 1. Derringer andSuich (1980) modified this desirability function for optimizationof multiple response variables. They defined three classes of desir-ability functions for three different types of response variables, viz.,NTB, STB and LTB. For the NTB type, the desirability function isdefined as

d ¼

y�LSLT�LSL

�� ��s; LSL � y � Ty�USLT�USL

�� ��t ; T � y � USL

0; y < LSL or y > USL

8>><>>: ð3Þ

where the exponents s and t are the shape constants of the desir-ability function, and LSL and USL are respectively the lower andupper specification limit for the NTB type response variable witha target value T. In general, the shape constants are chosen in therange from 0.01 to 10. We use both the shape constants are equalto 2 so as to look similar as Taguchi’s quadratic loss function.

For the STB type, the desirability function is defined as

d ¼y�USLa�USL

�� ��t ; a 6 y 6 USL0; y > USL

(ð4Þ

where a is a smallest possible value for response Y.For the LTB type, the desirability function is defined as

d ¼

y�LSLUSL�LSL

�� ��s; LSL 6 y 6 USL

0; y < LSL

1; y > USL

8>>><>>>:

ð5Þ

Once each response variable is converted to a desirability value,then the total desirability function D can be defined as the geomet-ric mean of the individual response di’s, i = 1, 2, . . ., r.

S. Pal, S.K. Gauri / Computers & Industrial Engineering 59 (2010) 976–985 979

D ¼ fd1 � d2 . . . drg1=r ð6Þ

It can be extended to

D ¼ dw11 � d

w22 . . . dwr

r ð7Þ

where wj (0 < wj < 1) is the weight value given for the importance ofjth response variable and

Pwj = 1. If all the response variables are

equally important for product quality, then wj is taken as 1/r, wherer is the number of response variables. The total desirability functionD is considered as the performance metric for optimization. Maxi-mizing the total desirability value, the optimal combination of inputvariables can be obtained.

The basic advantage of using desirability function as perfor-mance metric is that it is a simple unit-less measure and quiteappealing for easy implementation. However, if the specificationlimits and target values (for NTB type) of the response variablesare not provided, then desirability index cannot be computedand therefore this total desirability function cannot be used as aperformance metric for multiple response optimization. Anotherdisadvantage with this metric is that it does not consider theexpected variability and hence it may produce a solution wherethe expected mean is very close to its target value but with a highvariability around it. Thus the obtained solution may not yield anideal result.

3.2.2. Multivariate loss function as performance metricPignatiello and Joseph (1993) present an approach for multiple

response optimization based on the multiple regression techniqueand on the criteria of minimizing the expected value of a multivar-iate (MV) loss function. The MV loss function assumes additivity ofthe univariate loss functions.

The univariate loss function is based on Taguchi’s quality lossconcept that loss is incurred when a product’s functional qualitycharacteristic deviates from its target value regardless of theamount of deviations. The relationship between quality loss andthe amount of deviation from the target value is expressed by qua-dratic loss functions for different types of quality characteristics.The quality loss functions are computed as

For smaller the better; Lj ¼ c1n

Xn

k¼1

y2jk

!¼ c � �y2

j þ s2j

� �ð8Þ

For larger the better; Lj ¼ c1n

Xn

k¼1

1y2

jk

!

� c � 1�y2

j

1þ3s2

j

�y2j

! !ð9Þ

For nominal the best; Lj ¼ c1n

Xn

k¼1

ðyjk � TjÞ2 !

¼ c � ½ð�yj � TjÞ2 þ s2j � ð10Þ

where �yj ¼ 1n

Pnk¼1yjk denotes the mean, s2

j ¼ 1n�1

Pnk¼1ðyjk � �yjÞ2

denotes the variance, c is the loss coefficient and Tj is the targetvalue for jth response.

The MV loss function is the weighted sum of all univariate lossfunctions, where the weight is assigned according to the impor-tance of that particular response variable. The MV loss functionis computed as

L ¼Xr

j¼1

wjLj ð11Þ

where Lj is the univariate loss function of jth response variable, wj isthe weight of jth response chosen by the experimenter, andP

wj = 1.

Each loss function associated with a response variable has anunit of measurement. In any process/product, different responsevariables may have different units of measurement. Therefore, itmay be quite difficult to explain the unit of the MV loss functionwhich is a weighted sum of individual loss functions. To overcomethis problem, we may consider the MV loss function as the productof individual loss functions. We consider this multiplicative form ofthe MV loss function as the performance metric which is computedas

L ¼Yr

j¼1

Lwjj ð12Þ

where Lj is the univariate loss function of jth response variable, wj isthe weight of jth response chosen by the experimenter, andP

wj = 1.Minimizing the MV loss functional value, the optimal combina-

tion of input variables can be easily obtained. Most often, thismethod results in an optimal solution. This method works welleven if specification limits for the response variables are not pro-vided. This MV loss function metric is considered most acceptablefor multiple response optimization.

3.2.3. Distance measure as performance metricKhuri and Conlon (1981) developed an optimization method

using polynomial regression and a distance metric. This distancemetric uses the squared deviations of the response variables fromtheir targets, but then normalizes these deviations by the varianceof prediction of the response variables. This distance measure canbe viewed as the Euclidean distance between the target values andthe expected mean values of the response variables.

Khuri and Conlon’s method requires the same set of regressionterms (main factors, square terms and interaction terms) for all theresponse variables. It may not be possible to obtain a particular setof regressor terms that influences the mean and variance of eachresponse variable. Such a model would result in high residual er-rors. The predicted values of mean and variance of each responsevariable may become erroneous. Hence, the regression equationsdeveloped for the mean and variance of a response variable maynot include the same regressor terms.

Using the multiple regression equations, the means and vari-ances of all the response variables are computed for an arbitrarycombination of input variables. Then the distance metric iscomputed as

DM ¼Xr

j¼1

ðlYj� TjÞ2

r2Yj

ð13Þ

where lYj and r2Yj are the predicted mean and variance of jth

response variable, and Tj is the target value of jth response variable.For the STB type of quality characteristic, zero is considered as thetarget value and for the LTB type of quality characteristic, a largestpossible value is considered as the target value. This distance metricDM is considered as the performance metric for optimization. Theoptimal condition of input variables is obtained by minimizing thisdistance metric.

The basic disadvantage of this approach is that the distancemetric is minimized by maximizing the predicted variance valuesof the response variables. Hence, it is very difficult to obtain aworkable solution. By introducing a few constraints on the pre-dicted variances, we may get a reasonably good workable solution.Possibly after several attempts on optimization procedure, theanalyst may finally obtain the globally optimal solution.

3.2.4. Process capability index C�pm as performance metricCh’ng et al. (2005) proposed the use of process capability index

C�pm as the criteria for optimization of multiple response problems.

980 S. Pal, S.K. Gauri / Computers & Industrial Engineering 59 (2010) 976–985

This can be viewed as an extension of Lin and Tu’s (1995) proposedapproach of minimizing a mean square error function for optimiza-tion of dual response surface problem. The index C�pm is defined as

C�pm ¼minðUSL� T; T � LSLÞ

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðlY � TÞ2 þ r2

Y

q ð14Þ

where lY is the fitted value for the mean, r2Y is the fitted value for

the variance, T is the target value, and USL and LSL are the upperand lower specification limits for the response variable.

For a LTB type of response variable that specifies the target va-lue and LSL, the index C�pm is defined as

C�pm ¼T � LSL

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðlY � TÞ2 þ r2

Y

q ð15Þ

For the STB type of response variable that specifies the target valueand USL, the index C�pm is defined as

C�pm ¼USL� T

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðlY � TÞ2 þ r2

Y

q ð16Þ

In order to handle a process with r response variables, the overallindex is defined as

Total C�pm ¼Xr

j¼1

wjC�pmj ð17Þ

where C�pmj is the index for the jth response, wj is the weight of jthresponse chosen by the experimenter, and

Pwj = 1. This Total C�pm is

chosen as the performance metric. By maximizing this value, theoptimal combination of input decision variables can be easilyobtained.

The use of index C�pm works only when the target value and oneor both the specification limits are available. Generally, for STB andLTB type of quality characteristics, target values are not specified.In such cases, a smallest possible value can be considered as targetfor the STB type and a largest possible value can be considered asthe target for the LTB type of quality characteristic. Assuming thatspecification limits for all the response variables are known, TotalC�pm as a performance metric works quite well for optimization ofmultiple response variables.

3.2.5. MRWSN as performance metricTaguchi (1990) considers signal-to-noise (SN) ratio as a

measure of performance for optimization of single responsevariable. The most important aspect of the SN ratio is that itcombines location and dispersion of a response variable in a singleperformance measure. Taguchi defined the SN ratios for the threedifferent types of characteristics as:

For smaller-the-better type of characteristic

gj ¼ �10log101n

Xn

i¼1

y2i

!¼ �10log10 �y2

j þ s2j

� �ð18Þ

For larger-the-better type of characteristic

gj ¼ �10log101n

Xn

i¼1

1y2

i

!� �10log10

1�y2

j

1þ3s2

j

�y2j

! !ð19Þ

For nominal-the-best type of characteristic

gj ¼ 10log10

�y2j

s2j

!ð20Þ

where �yj ¼ 1n

Pnk¼1yjk denotes the mean, s2

j ¼ 1n�1

Pnk¼1ðyjk � �yjÞ2

denotes the variance. The SN ratio is always expressed in decibel

(dB) unit. A higher value of SN ratio implies a lower value of qualityloss and hence a better quality product.

Shiau (1990) and Tai et al. (1992) used weighted signal-to-noise(WSN) ratio as the combined process performance measure formultiple response optimization, where SN ratio for a response iscomputed from the observed experimental data. Pal and Gauri(2010) proposed to predict the means and variances of all the re-sponse variables using appropriately fitted multiple regressionequations and compute SN ratios of individual responses usingtheir predicted means and variances. Then they computed WSN va-lue, which is maximized to obtain the optimal solution. Since Paland Gauri (2010) computed the WSN value using multiple regres-sion-based prediction of means and variances of the individual re-sponses, their proposed performance metric for optimization is,therefore, called as multiple regression-based weighted signal-to-noise (MRWSN). The MRWSN value is computed using thefollowing equation:

MRWSN ¼Xr

j¼1

wjgj ð21Þ

where gj is the SN ratio for jth response variable computed based onthe predicted mean and variance from the appropriately fittedmultiple regressions, wj is the weight of jth response chosen bythe experimenter, and

Pwj = 1. If all the response variables are

equally important for product quality, then wj is taken as 1/r, wherer is the number of response variables. The optimal combination ofinput variables is obtained by maximizing this MRWSN value.

The advantage of using MRWSN value as performance metric isthat it tries to find the optimal condition by minimizing thevariability as low as possible. Therefore, the analyst has to searchcarefully to obtain the optimal solution where all the responsevariables are close to their respective target values. So, while max-imizing MRWSN value, the analyst may have to add constraintsrelated to the closeness of target of the NTB type of response vari-ables. This method works well even if specification limits for theresponse variables are not provided.

3.3. Optimization of performance metric

We now describe the procedure followed for optimization of aperformance metric using the fitted multiple regression equations.For any arbitrary setting (say, existing combination) of input vari-ables, we compute mean and variance for each response variableusing those regression equations. Depending on the performancemetric chosen, the value of the objective function is computedusing the expected mean and variance at that arbitrary setting.Then this objective function is optimized by changing level valuesof the input variables. If the objective function is based on desir-ability function, total C�pm or MRWSN, then the objective functionis maximized. On the contrary, if the objective function is basedon distance function or MV loss function, then the objectivefunction is minimized.

For optimization purpose, we use ‘‘Solver” tool of MicrosoftEXCEL package. The Solver tool employs the generalized reducedgradient (GRG) method, proposed by Del Castillo and Montgomery(1993), which is a popular and broadly acceptable method applica-ble to many mathematical optimization problems. However, it ispossible to arrive at any locally optimal solution, especially whenthere are a large number of decision variables. Therefore, it is ad-vised to select several starting points to ensure that the globallyoptimal solution is found.

While running the ‘‘Solver” tool it is necessary to specify therange of levels for the input variables. In certain cases, where theinput variable takes only discrete values, the integer restrictionfor that input variable need to be specified. Using Excel Solver,

Table 1Response variables and Specification limits (case study 1).

Response variable Targetvalue

Lower specificationlimit (LSL)

Upper specificationlimit (USL)

Deposition thickness(DT) (Y1)

1000 Å 900 Å 1100 Å

Refractive index(RI) (Y2)

2.0 1.9 2.1

S. Pal, S.K. Gauri / Computers & Industrial Engineering 59 (2010) 976–985 981

the optimal value of the objective function is obtained for a partic-ular distinct setting of the input variables. The expected means andvariances of all response variables at this optimal setting conditionare also obtained. The range of expected individual observationscan also be predicted with certain degree of confidence. These val-ues need to be verified when the confirmatory trial is made withthe optimal setting of input parameters.

4. Utility measure and comparison procedure

4.1. Total non-conformance as utility measure

At the optimal combination of input variables, the expectedmean and variance of each response variable can be computed. Gi-ven the specification limits of any response variable, the expectednon-conformance (NC) for that response variable can be easilycomputed. Assuming that the response variable follows a normaldistribution, the expected NC of that response variable is computedas

NCL ¼ PðY < LSLÞ ¼ U½ðLSL� lyÞ=ry�NCU ¼ PðY > USLÞ ¼ 1�U½ðUSL� lyÞ=ry�NCTotal ¼ NCL þ NCU

ð22Þ

where NCL and NCU gives the expected NCs below the lower speci-fication limit (LSL) and above the upper specification limit (USL)respectively. Here ly and ry denote the expected mean and stan-dard deviation of the response variable Y and U(�) denotes thecumulative distribution function of a standard normal variable.The NC of response Y is given by NCTotal. The expected NC of eachresponse variable at the optimal combination is computed. Andthen, the total process NC will be computed as the sum of individualNCs of all response variables. The expected total NC will be consid-ered as a utility measure for comparison.

It is known that the sum of individual NCs is not a proper mea-sure for the total expected NC, especially when the NC with respectto one response variable is dependent to NC with respect to otherresponse variables. Assuming that the NCs with respect to one re-sponse variable are independent to the other response variables,we use this approximate measure of total NC as our utility metricfor comparison.

4.2. Comparison procedure

We consider three sets of experimental data for multipleresponse optimization taken from the literature. In each dataset,we first develop multiple regression equations for the mean andvariance of each response variable. The adequacy of fitting aregression equation is verified through ANOVA and normality plotof residuals. We then consider a performance metric for optimiza-tion and optimize it using ‘‘Solver” tool of Microsoft Excel package.While optimizing, we have specified the integer restrictions for thelevel values of input variables, mainly to make the optimalsolutions comparable. The optimal setting combination of theinput variables is obtained and the expected means and variancesof all response variables under the optimum combination arepredicted. The expected NC for each response variable and the totalNC are also predicted. All the expected results are listed in a tableand then compared.

5. Experimental data and analysis

5.1. Case study 1

Tong and Su (1997) presented this case study that involves theimprovement of a plasma-enhanced chemical vapour deposition

(PECVD) process used in the fabrication of ICs. In this study anexperiment was performed to determine the effects of variousprocess parameters (input variables) on the deposition process inorder to improve quality for meeting specification requirements.The response variables are deposition thickness (DT) (NTB type)and refractive index (RI) (NTB type). The target value and lowerand upper specification limits for DT and RI are given in Table 1.In this experimentation, eight controllable factors were chosen.The standard orthogonal array L18 was used for the experiments.The input controllable factors, their levels and the experimentaldata can be obtained from Tong and Su (1997).

From the experimental data, we obtain multiple regressionequations for both mean and variance of each response variable.These regression equations are:

ly1¼ 855:77þ 15:08x1 � 263:24x2 � 110:36x3 � 103:75x4

þ 386:96x5 � 104:57x6 þ 89:44x7 � 116:35x8

þ 63:99x22 þ 56:52x2

6 � 133:28x3x5 þ 185:98x3x8

� 89:99x5x8

½R2 ¼ 0:975; adjusted R2 ¼ 0:893� ð23Þ

log10ðr2y1Þ ¼ 5:259þ 0:189x1 þ 0:574x2 � 2:346x3

� 0:696x4 þ 1:263x5 � 1:014x6 þ 0:307x7

� 0:696x8 þ 0:159x23 þ 0:315x2

6 þ 0:829x3x8

þ 0:187x4x8 � 0:714x5x8

½R2 ¼ 0:984; adjusted R2 ¼ 0:931� ð24Þ

ly2¼ 2:157þ 0:047x1 � 0:268x2 þ 0:416x3 þ 0:270x4

� 0:738x5 þ 0:062x6 � 0:113x7 þ 0:116x8 þ 0:121x3x5

� 0:357x3x8 þ 0:305x5x8

½R2 ¼ 0:971; adjusted R2 ¼ 0:919� ð25Þ

log10 r2y2

� �¼ 1:259þ 0:735x1 � 2:518x2 � 1:64x3 � 0:928x4

� 0:591x5 � 0:822x6 þ 0:575x7 � 0:033x8

þ 0:366x2x5 þ 0:598x2x6 þ 0:222x2x8

þ 0:626x3x4 þ 0:335x3x5 � 0:371x5x8

½R2 ¼ 0:99; adjusted R2 ¼ 0:942� ð26Þ

The R2 value and adjusted R2 value for each multiple regressionequation is given along with the equation. Using ANOVA and F-testfor significance of regression, the adequacy of the models arechecked. Residuals are checked for normality using normal proba-bility plot and are found satisfactory.

The relative weights for the two response variables were notknown. So, we assume equal weights for both the response vari-ables. Using the desirability function, distance function, MV lossfunction and total C�pm approaches, we obtain the same optimalcombination of input variables, which is A1B1C3D1E3F2G3H3. UsingMRWSN as performance metric, the optimal setting combination of

Table 2Expected mean and variance and performance measures at different optimal conditions (case study 1).

Performance metric Optimal condition Expected mean and variance at the optimal condition Expected values of the performance measures at the optimalcondition

DT RI Totaldesirability

MVLoss

Distancefunction

TotalC�pm

MRWSN Total NC(in PPM)

Mean Variance Mean Variance

Desirability function A1B1C3D1E3F2G3H3 998.25 1475.71 1.992 0.00022 0.904 0.653 0.287 1.329 35.385 9310MV loss functionDistance functionTotal C�pm

MRWSN A1B1C2D1E2F2G2H3 1010.64 638.26 1.978 0.00028 0.697 0.760 1.883 1.013 36.718 210

982 S. Pal, S.K. Gauri / Computers & Industrial Engineering 59 (2010) 976–985

input variables is obtained as A1B1C2D1E2F2G2H3, which is differentfrom the optimal solution obtained by other approaches. Theexpected means and variances of the response variables and theexpected values of the five performance measures, e.g. totaldesirability, MV loss, distance function, total C�pm and MRWSN,and the total NC (in PPM) at different optimal conditions are ob-tained. These values are shown in Table 2.

It is observed from Table 2 that the expected mean values ofboth response variables are very close to their target values atthe optimal condition A1B1C3D1E3F2G3H3 compared to the optimalcondition A1B1C2D1E2F2G2H3. But, the expected variance ofresponse variable DT at the optimal condition A1B1C3D1E3F2G3H3

is higher than the expected variance at the optimal conditionA1B1C2D1E2F2G2H3 obtained using MRWSN approach. This resultsin a higher value of expected total NC at the optimal conditionA1B1C3D1E3F2G3H3, which is obtained using desirability function,MV loss function, distance function and total C�pm approaches.Hence, for this dataset, we conclude that the optimal conditionobtained using MRWSN as performance metric is superior thanthe optimal condition obtained using the other four performancemetrics.

5.2. Case study 2

Phadke (1989) considered this case study to improve a polysill-icon deposition process. Six controllable factors were identified:deposition temperature (A), deposition pressure (B), nitrogen flow(C), silane flow (D), setting time (E) and cleaning method (F). Allthe factors were studied at three levels each. The L18 orthogonalarray was used and factors A–F were assigned to columns 2–6and 8 respectively. Three quality characteristics were consideredfor this process. The first one is surface defects (SD) (STB type),the second is thickness (TH) (NTB type) and the third is depositionrate (DR) (LTB type). The specification limits of the response vari-ables are listed in Table 3. For the characteristic DR, a larger valueis desirable. In each trial run, nine observations were taken on bothTH and SD and a single observation was made on the DR. The de-tails about the input variables and experimental data can be ob-tained from Phadke (1989).

We need to obtain multiple regression equations for both themean and variance of the response variables SD and TH. For that,we use logarithmic transformation for both mean and variance ofeach response variable and then develop multiple regression equa-tions. The logarithmic transformation is used to ensure positive

Table 3Response variables and specification limits (case study 2).

Response variable Targetvalue

Lower specificationlimit (LSL)

Upper specificationlimit (USL)

Surface defects (SD) (Y1) – 10 per sq cmThickness (TH) (Y2) 3600 3550 3650

values for means and variances. Since DR has a single observationat each trial run, we obtain multiple regression equation only forthe expected value of DR. The multiple regression equations are:

log10ðly1Þ ¼ � 5:239þ 3:986A� 0:476Bþ 2:08C þ 0:562D

þ 1:552E� 1:620F � 0:778A2 � 0:685C2 þ 0:307F2

þ 0:451BC þ 0:541BD� 0:253BE� 0:602DE

½R2 ¼ 0:974; adjusted R2 ¼ 0:889� ð27Þ

log10ðr2y1Þ ¼ � 11:567þ 7:646A� 0:996Bþ 4:333C þ 1:673D

þ 3:901E� 2:841F � 1:540A2 � 1:366C2 þ 0:485F2

þ 0:931BC þ 1:216BD� 0:630BE� 1:587DE

½R2 ¼ 0:966; adjusted R2 ¼ 0:856� ð28Þ

log10ðly2Þ ¼ 2:8565� 0:417Aþ 0:3368Bþ 0:459C þ 0:282D

� 0:056E� 0:126F þ 0:103A2 � 0:062B2 � 0:101C2

� 0:052D2 þ 0:0172E2 þ 0:0369F2

½R2 ¼ 0:976; adjusted R2 ¼ 0:919� ð29Þ

log10ðr2y2Þ ¼ 5:350� 3:653A� 0:377Bþ 3:082C þ 0:425D

þ 0:09E� 2:614F þ 0:810A2 � 0:880C2 þ 0:299F2

þ 0:511AF þ 0:388BC

½R2 ¼ 0:94; adjusted R2 ¼ 0:83�ð30Þ

ly3¼ � 43:95� 7:95Aþ 42:10Bþ 16:53C þ 9:58D� 2:02E

� 1:65F þ 10:20A2 � 8:25B2 � 4:18C2

½R2 ¼ 0:991; adjusted R2 ¼ 0:980� ð31Þ

The R2 and adjusted R2 values for each multiple regression equationis given along with the equation. Using ANOVA and F-test for signif-icance of regression, the adequacy of the models are checked. Resid-uals are checked for normality using normal probability plot and arefound satisfactory.

For computation of various performance metrics, only tworesponse variables SD and TH are considered. The response DR isnot considered because it is not a product characteristic as wellas there is no specification available for it. As there is no priorknowledge about the relative importance of SD and TH, we assumeequal weights of 0.5 for the two response variables while comput-ing various performance metrics. Using each approach, we obtainthe setting condition of input variables that will optimize theresponse variables SD and TH. At the optimal combination, wecompute the expected total NC and the expected DR. The expectedtotal NC is computed as the sum of individual expected NCs due toSD and TH.

Table 4Expected mean and variance and performance measures at different optimal conditions (case study 2).

Performance metric Optimal condition Expected mean and variance at the optimal condition Expected values of the performance measures at the optimalcondition

SD TH DR Totaldesirability

MVLoss

Distancefunction

TotalC�pm

MRWSN Total NC(in PPM)

Mean Variance Mean Variance Mean

Desirability function A1B1C3D2E2F3 0.166 0.0716 3582.6 109.45 14.32 0.642 6.389 3.147 5.473 30.364 912Distance function

MV loss function A1B1C3D2E1F3 0.133 0.0573 3619.1 88.96 16.33 0.610 5.835 4.409 6.244 31.463 525Total C�pm

MRWSN

S. Pal, S.K. Gauri / Computers & Industrial Engineering 59 (2010) 976–985 983

The optimal process conditions are determined separately withrespect to the five chosen performance metrics. It is found that thedesirability function and the distance function approaches lead tothe same optimal condition which is A1B1C3D2E2F3. On the otherhand, MV loss function, total C�pm and MRWSN approaches lead tothe same optimal condition for the input variables which isA1B1C3D2E1F3. The expected mean and variance of the responsevariables at these optimal combinations are obtained. Also theexpected total NC and values of the five performance measuresat these optimal conditions are estimated. These values are givenin Table 4.

It can be observed from Table 4 that, for the response variableSD, the expected mean and variance values obtained using MV lossfunction, total C�pm and MRWSN approaches are slightly better thanthe same obtained using desirability and distance functionapproaches. For the response variable TH, the expected varianceobtained using MV loss function, total C�pm and MRWSN approachis smaller than that obtained using the desirability and distancefunction approaches. The expected NC values show that theoptimal solution obtained using MV loss function, total C�pm andMRWSN approaches is superior than the optimal solutionobtained using the desirability function and distance functionapproaches.

5.3. Case study 3

Ramakrishnan and Karunamoorthy (2006) presented this casestudy for multiple response optimization of wire electrical dis-charge machining (WEDM) process. Three responses namely mate-rial removal rate (MRR), surface roughness (SR), and wire wearratio (WWR) were considered. Five input variables, each with fourlevels, were considered for the experiment. The orthogonal arrayL16 was chosen as the experimental design. The design layoutand the experimental data can be obtained from Ramakrishnanand Karunamoorthy (2006). The specification limits of the re-sponse variables, considered mainly for comparison of perfor-mance metrics, are listed in Table 5.

From the experimental data, we obtain the multiple regressionequations for both mean and variance of the response variablesMRR and SR, and regression equation for mean alone for the re-sponse variable WWR. These regression equations are given below:

Table 5Response variables and specification limits (case study 3).

Response variable Targetvalue

Lower specificationlimit (LSL)

Upper specificationlimit (USL)

Material removal rate(Y1) (LTB type)

90 60 90

Surface roughness (Y2)(STB type)

2.4 2.0 3.0

ly1¼ 23:055þ 7:7531Aþ 13:377B� 6:1781C þ 11:177D

� 9:03E� 1:8359B2 � 1:6484D2 þ 2:7578E2

½R2 ¼ 0:992; adjusted R2 ¼ 0:983� ð32Þ

log10ðr2y1Þ ¼ 1:33� 0:82A� 0:29Bþ 0:69D� 0:68Eþ 0:09A2

þ 0:08C2 � 0:16AC þ 0:31AEþ 0:10BC � 0:11CD

½R2 ¼ 0:978; adjusted R2 ¼ 0:933� ð33Þ

ly2¼ 4:1945þ 0:6996A� 0:5117B� 0:5738C � 0:3019D

þ 0:3029E� 0:1608AEþ 0:1376BC

½R2 ¼ 0:966; adjusted R2 ¼ 0:936� ð34Þ

log10ðr2y2Þ ¼ � 2:308þ 1:727Aþ 0:570B� 0:534C � 1:080E

� 0:368AB� 0:2ADþ 0:105BEþ 0:207DE

½R2 ¼ 0:85; adjusted R2 ¼ 0:69� ð35Þ

ly3¼ 0:054þ 0:009A� 0:016C þ 0:014Eþ 0:003AC � 0:004AE

þ 0:002BD� 0:002BEþ 0:001CE

½R2 ¼ 0:976; adjusted R2 ¼ 0:949� ð36Þ

Using ANOVA and F-test for significance of regression, the adequacyof the models are checked. Residuals are checked for normalityusing normal probability plot and are found satisfactory.

For computation of various performance metrics, only two re-sponse variables MRR and SR are considered. Using each approach,we obtain the setting condition of input variables that will opti-mize the response variables MRR and SR. Since relative weightsfor the two response variables are unknown, we assume equalweights of 0.5 for the response variables MRR and SR. It is foundthat desirability function, MV loss function and MRWSN ap-proaches lead to the same optimal combination of input variableswhich is A3B4C1D4E4. On the other hand, distance function and to-tal C�pm approaches result in the same optimal condition for the in-put variables and it is A4B3C3D4E4.

The expected total NCs at these optimal combinations are com-puted as the sum of individual expected NCs due to MRR and SR.The expected total NCs and the expected WWR values at theseoptimal combinations are given in Table 6. Table 6 also showsthe expected means and variances of the response variables as wellas values of the five performance measures at these optimal com-binations. Based on the expected total NC values given in Table 6, itcan be concluded that the optimal solution obtained using MRWSNor MV loss function or desirability function as performance metricis superior to the optimal solution obtained using other two perfor-mance metrics.

Although the desirability function approach is quite popularand strongly suggested method, the results of the three case stud-ies reveal that it often fails to minimize total NC, e.g. in case of case

Table 6Expected mean and variance and performance measures at different optimal conditions (case study 3).

Performance metric Optimal condition Expected mean and variance at the optimal condition Expected values of the performance measures at the optimalcondition

MRR SR WWR Totaldesirability

MVLoss

Distancefunction

TotalC�pm

MRWSN Total NC(in PPM)

Mean Variance Mean Variance Mean

Desirability function A3B4C1D4E4 90.60 69.18 2.30 0.0298 0.086 0.703 0.0257 0.362 1.096 15.894 140MV loss functionMRWSN

Distance function A4B3C3D4E4 85.47 100.00 2.405 0.0022 0.082 0.505 0.0287 0.217 2.571 15.419 5430Total C�pm

984 S. Pal, S.K. Gauri / Computers & Industrial Engineering 59 (2010) 976–985

studies 1 and 2. The problem with the desirability function ap-proach is that it does not consider the expected variability andtherefore, it gives a solution where all the response variables areclose to their target values, but not with minimum variability. Incomparison, multiplicative MV loss function and total C�pm as per-formance metric results in minimum total NC quite often. On theother hand, Pal and Gauri (2010) proposed MRWSN method, whichuses MRWSN as performance metric, tries to find the optimal solu-tion by minimizing the expected variability and therefore, thismethod most often results in the minimum total NC. For example,for all the three case studies, the optimal solutions obtained usingMRWSN method result in the minimum total NC, which impliesthat MRWSN method result in the best optimal solution in general.In some cases (mainly for NTB type of response variable), an addedconstraint on the closeness of target may help in finding the glob-ally optimal solution much faster.

It is important to note that the effectiveness of MRWSN and allother multiple regression-based methods depend highly on thepredictive accuracies of the means and variances of the responsevariables, i.e. goodness of fit of the regression equations, whichcan be assessed by examining the R2 and adjusted R2 values ofthe fitted regression equations. For all the three case studies, theR2 and adjusted R2 values of the fitted regression equations arefound to be very high. With the aim to assess the threshold valuesof the R2 and adjusted R2 values, we also analyzed all the three setsof experimental data using the fitted regression models obtained inthe earlier steps (which are having lesser R2 and adjusted R2 val-ues). We have found that the optimal solution remain unaffectedas long as R2 and adjusted R2 values for the fitted regression equa-tions are greater than equal to 0.90 and 0.85 respectively.

6. Conclusion

There are several multiple regression-based integrated ap-proaches for multi-response optimization. In these approaches, afunctional relationship of each response variable with the inputdecision variables is established first using multiple regressiontechniques, and then the optimal solution is determined by opti-mizing an appropriately defined objective function called as theperformance metric. In this paper, five different performance met-rics are considered and their effectiveness for optimization of mul-ti-response data is examined. The usefulness of these metrics iscompared with respect to expected total non-conformance (NC)analyzing three sets of planned experimental data obtained frompreviously published articles. It is found that the desirability func-tion approach, one of the most popular and strongly suggestedmethods, often fails to provide the optimal solution. The reasonis that the desirability function approach does not consider theexpected variability and therefore, it gives a solution where allthe response variables are close to their target values, but not withminimum variability. However, multiplicative MV loss function

and total C�pm as performance metric results in minimum total NCquite often. But, MRWSN method, which uses MRWSN as perfor-mance metric and tries to find the optimal solution by minimizingthe expected variability of individual responses, is found to be themost effective in finding the optimal solution.

Acknowledgement

The authors would like to thank the referees for their helpfulcomments and suggestions which have substantially improvedthe content and presentation of the article.

References

Carlyle, W. M., Montgomery, D. C., & Runger, G. C. (2000). Optimization problemsand methods in quality control and improvement. Journal of Quality Technology,31(1), 1–16.

Ch’ng, C. K., Quah, S. H., & Low, H. C. (2005). Index C�pm in multiple responseoptimization. Quality Engineering, 17, 165–171.

Chiang, T. L., & Su, C. T. (2003). Optimization of TQFP modelling process usingneuro-fuzzy-GA approach. European Journal of Operations Research, 147,156–164.

Del Castillo, E., & Montgomery, D. C. (1993). A nonlinear programming solution tothe dual response problem. Journal of Quality Technology, 25, 199–204.

Derringer, G., & Suich, R. (1980). Simultaneous optimization of several responsevariables. Journal of Quality Technology, 12, 214–219.

Harrington, E. C. Jr., (1965). The desirability function. Industrial Quality Control, 21,494–498.

Hsi, H. M., Tsai, S. P., Wu, M. C., & Tzuang, C. K. (1999). A genetic algorithm for theoptimal design of microwave filters. International Journal on IndustrialEngineering, 6, 282–288.

Khuri, A. I., & Conlon, M. (1981). Simultaneous optimization of multiple responsesrepresented by polynomial regression functions. Technometrics, 23, 363–375.

Liao, H. C. (2006). Multi-response optimization using weighted principalcomponent. International Journal on Advanced Manufacturing Technology, 27,720–725.

Lin, D. K. J., & Tu, W. (1995). Dual response surface optimization. Journal of QualityTechnology, 27(1), 34–39.

Logothetis, N., & Haigh, A. (1988). Characterizing and optimizing multi-responseprocesses by the Taguchi method. Quality Reliability Engineering International, 4,159–169.

Pal, S., & Gauri, S. K. (2010). Multi-response optimization using multiple regressionbased weighted signal-to-noise ratio (MRWSN). Quality Engineering, 22,336–350.

Pan, L. K., Wang, C. C., Wei, S. L., & Sher, H. F. (2007). Optimizing multiple qualitycharacteristics via Taguchi method-based grey analysis. Journal of MaterialsProcessing Technology, 182, 107–116.

Phadke, M. S. (1989). Quality engineering using robust design. Prentice-Hall:Englewood Cliffs.

Pignatiello, J. R., & Joseph, J. (1993). Strategies for robust multi-response qualityengineering. Industrial Engineering Research Development-IIET, 25, 5–15.

Ramakrishnan, R., & Karunamoorthy, L. (2006). Multi response optimization of wireEDM operations using robust design of experiments. International Journal onAdvanced Manufacturing Technology, 29, 105–112.

Shiau, G. H. (1990). A study of the sintering properties of iron ores using theTaguchi’s parameter design. Journal of Chinese Statistical Association, 28,253–275.

Su, C. T., & Hsieh, K. L. (1998). Applying neural networks to achieve robust design fordynamic quality characteristic. International Journal on Quality and ReliabilityManagement, 15, 509–519.

Su, C. T., & Tong, L. I. (1997). Multi-response robust design by principal componentanalysis. Total Quality Management, 8, 409–416.

Taguchi, G. (1990). Introduction to Quality Engineering. Tokyo Japan: AsianProductivity Organization.

S. Pal, S.K. Gauri / Computers & Industrial Engineering 59 (2010) 976–985 985

Tai, C. Y., Chen, T. S., & Wu, M. C. (1992). An enhanced Taguchi method foroptimizing SMT processes. Journal of Electronics Manufacturing, 2, 91–100.

Tong, L. I., Chen, C. C., & Wang, C. H. (2007). Optimization of multi-responseprocesses using the VIKOR method. International Journal on AdvancedManufacturing Technology, 31, 1049–1057.

Tong, L. I., & Hsieh, K. L. (2000). A novel means of applying artificial neural networksto optimize multi-response problem. Quality Engineering, 13, 11–18.

Tong, L. I., & Su, C. T. (1997). Optimizing multi-response problems in the Taguchimethod by fuzzy multiple attribute decision making. Quality ReliabilityEngineering International, 13, 25–34.

Tong, L. I., Wang, C. H., & Chen, H. C. (2005). Optimization of multiple responsesusing principal component analysis and technique for order preference by

similarity to ideal solution. International Journal of Advanced ManufacturingTechnology, 27, 407–414.

Tsui, K. (1999). Robust design optimization for multiple characteristic problems.International Journal on Production Research, 37, 433–445.

Vining, G. G., & Myers, R. H. (1990). Combining Taguchi and response surfacephilosophies: A dual response approach. Journal of Quality Technology, 22,38–45.

Wu, C. F. J. (2005). Optimization of correlated multiple quality characteristics usingdesirability function. Quality Engineering, 17, 119–126.

Wu, C. F. J., & Hamada, M. (2000). Experiments: Planning, analysis, and parameterdesign optimization. New York: Wiley Interscience.