Weighted Principal Component

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DOI 10.1007/s00170-004-2248-7

O R I G I N A L A R T I C L E

Int J Adv Manuf Technol (2006) 27: 720–725

Hung-Chang Liao

Multi-response optimization using weighted principal component

Received: 7 November 2003 / Accepted: 15 May 2004 / Published online: 23 February 2005© Springer-Verlag London Limited 2005

Abstract Taguchi method is a very popular offline quality de-sign. However, it cannot solve the multi-response problem whichoccurs often in today’s society. Research shows that the multi-response problem is still an issue with the Taguchi method. Re-searchers have tried to find a series of theories and methods inseeking a combination of factors/levels to achieve the situationof optimal multi-response instead of using engineers’ judgementto make a decision in the Taguchi method. In 1997, Su et al. sub-mitted the multivariate method, and in 2000 Antony proposedprincipal component analysis (PCA), to solve this problem. Butwith the PCA method, there are still two main shortcomings. Inthis study, the weighted principal components (WPC) method isproposed to overcome these two shortcomings, and three cases intheir papers will be illustrated and compared in the applicationof WPC method. The result shows that the WPC method offerssignificant improvements in quality.

Keywords Multi-response · Principal component analysis(PCA) · Taguchi

1 Introduction

The Taguchi method is one of the optimal experimental designsin consideration of the lowest societal cost. Its objective is toobtain the best combination of factors/levels so that the qualityengineers can control the quality of industrial products in of-fline design. To achieve Taguchi’s objective, three stages mustbe applied-system design, parameter design, and tolerance de-sign [1, 2]. By now, the Taguchi method can not optimize themulti-response problem; engineers’ judgement is the only way to

solve the multi-response problem in the Taguchi method. How-ever, in most industrial products, the multi-response problemshould be considered in the quality characteristics. Many studies

H.-C. Liao (u)Department of Health Services Administration,Chung-Shan Medical University,Taichung (402), TaiwanE-mail: [email protected].: +886-4-24730022

have dealt with this problem in order to find a way to solve it andmake an important contribution to real manufacturing. In order tosolve the multi-response problem, a common approach is to as-sign each response with a weight, as submitted by [3–10]. How-ever, how to determine and define a weight for each responsein a real case still remains difficult. Pignatiello [11], Reddy etal. [12], and logothetis et al. [13] used regression-technique-based approaches to optimize the multi-response problem. How-ever, such an approach increases the complexity of the computa-tional process, and the possible correlation among the responsesmay still not be considered. Furthermore, the significant factor ina single-response case may not be considered significant whenconsidered in a multi-response case.

In 1997 and 2000, Su et al. [14] and Antony [15] proposedthe multi-response methods which are based on a principal com-ponent analysis (PCA) method to optimize the multi-responseproblem. They used a PCA method to transform the normal-

ized multi-response value into uncorrelated linear combinations.If n linear combinations are obtained, then n principal com-ponents will also be formed. Su et al. [14] used the Kasier’sstudy [16] to choose the components whose eigenvalue is biggerthan 1 to replace the original multi-response values. In Su’s pa-per [14], when the number of components is bigger than 1, hesuggested that “trade-off might be necessary to select a feasiblesolution”. Also, Antony [15] indicated that “the rule of thumb isto choose those components with an eigenvalue greater or equalto one”. Nevertheless, in Antony’s paper [15], there is no furtherdiscussion about how to deal with situations where the num-ber of components is bigger than 1. In the application of PCAmethod, this selected component is regarded as an index (multi-

response performance statistic) in order to conveniently optimizethe multi-response problem and to gain the best combination of factors/levels. However, there are still two obvious shortcom-ings in the PCA method. First, when more than one principalcomponent is selected whose eigenvalue is greater than 1, the re-quired trade-off for a feasible solution is unknown; and second,the multi-response performance index cannot replace the multi-response solution when the chosen principle component can onlybe explained by total variation.



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In order to overcome these two main shortcomings in thePCA method, this paper proposes a weighted principal compo-nents (WPC) method. In this WPC method, all components aretaken into consideration in order to completely explain variationin all responses. The WPC method uses the explained variationas the weight to combine all principal components in order toform a multi-response performance index. Then, the best com-bination of factors/levels will easily be obtained. The proposedWPC method will be demonstrated in detail in Sect. 2. The fur-ther illustrated examples will be presented in Sect. 3. Concludingremarks drawn from this study will be included in the last section.

2 The proposed weighted principal components

In the application of the PCA method [14, 15], the main pro-cesses of dealing with the multi-response problem are: (1) tocompute the quality loss of each response, (2) to normalize thequality loss of each response, (3) to transform these normal-ized quality loss into a multi-response index or a named multi-response performance statistic, (4) to obtain the best combinationof factors/levels, and (5) to perform a confirmation experiment.Above all, process (3) is the spirit of PCA method in solvingthe multi-response problem. Process (3) is based on Pearson andHotelling [17] to explain the structure of variance-covariance byway of the linear combinations of the normalized value of eachresponse. Let Y ibe the normalized value of the ith response, fori = 1, 2, . . . , p. To compute PCA, k (k ≤ p) components will beobtained to explain the variance in the p responses. Principalcomponents are independent (uncorrelated) of each other. Sim-ultaneously, the explained variance of each principal componentfor the total variance of responses is also gained. The formed j principal component is a linear combination Z j =

pi=1 a jiY i ,

for j = 1, 2, . . . , k subjecting to p

i=1 a2 ji = 1; also, the coeffi-

cient a ji

is called eigenvector.Now, this paper proposes the WPC method to overcome the

shortcomings of multi-response problem in the PCA method. Toachieve the object, first, all principal components will be used inthis WPC method; thus the explained variance can be completelyexplained in all responses. Second, because different principalcomponents have their own variance to account for the total vari-ance, the variance of each principal component is regarded as theweight. Because these principal components are independent toeach other (which means that these principal components are inan additive model), the multi-response performance index (MPI)is MPI =

k j=1 W j Z j , where W j is the weight of jth principal

components. The larger the MPI is, the higher the quality. Fi-

nally, with the application of ANOVA (Analysis of Variance),significant factors in this quality index and their contribution per-centage for total variation in MPI will be obtained.

3 Illustrations

In order to illustrate the proposed WPC method, three caseswill be used to confirm the feasible WPC method and to

compare the WPC method with other additive model-basedmethods.

Case study 1

Tarng et al. [7] attempted to improve the quality of the sub-merged arc welding process. Five controllable factors were con-sidered, as follows: (A) arc current, (B) arc voltage, (C) weldingspeed, (D) electrode protrusion, and (E) pre-heat temperature.

The desired responses were: (1) deposition rate (larger-the-better) and (2) dilution (smaller-the-better).

Each controllable factor had two levels. In addition, the stan-dard array L8 was selected for this example. The initial conditionwas set to A1B1C2D2E1. Tarng et al. utilized two weights to in-tegrate the two responses normalized loss functions into a totalloss function, and then the total loss function was further trans-formed into a multi-response signal-to-noise (SN) ratio in orderto obtain the optimal experimental setting condition. The optimalcombination of factors/levels for Tarng’s study was to set thisexperiment in A2B1C2D2E2. After two years, Antony [15] usedPCA to improve Tarng’s study. Also, the optimal combination of factors/levels was set in A2B1C2D2E2.

Now, with the proposed WPC method to revise the PCAmethod, it is shown that two principal components are obtained.Table 1 shows the explained variation in these two responses andthe eigenvector of each principal component.

So, the relations between principal components and re-sponses are:

Z 1 = 0.707×deposition rate+0.707×dilution, and

Z 2 =−0.707×deposition rate+0.707×dilution

The equation of weighted principal components, MPI, is

MPI = 0.65593× Z 1 +0.34407× Z 2

Table 2 shows the obtained MPI value in the standard array L8.In computing the main effect of each controllable factor’s levels,Table 3 shows the computation results.

It is evident that the best combination of factors/levels is alsoA2B1C2D2E2. In addition, in order to understand whether thesefactors’ effects have a significant impact on the MPI, Table 4 per-forms the ANOVA on MPI. From Table 4, factors D and E aresignificant in the multi-response performance. Factors D and Eaccount for approximately 91% contribution for the total vari-ation. The smallest contribution and no significant factor A isadded to the pooled error.

Table 1. The explained variation and eigenvector in case study 1

Principal Eigenvalue Explained Cumulative Eigenvectorcomponents variation variation [deposition rate,

(%) (%) dilution]

Z 1 1.312 65.593 65.593 [0.707, 0.707] Z 2 0.688 34.407 100.000 [−0.707, 0.707]



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Table 2. Data summary of the MPI in case study 1

Exp. No. L8 MPIA B C D E

1 1 1 1 1 1 0.24042 1 1 1 2 2 0.76173 1 2 2 1 1 0.14344 1 2 2 2 2 0.88335 2 1 2 1 2 0.4844

6 2 1 2 2 1 0.75387 2 2 1 1 2 0.37448 2 2 1 2 1 0.5404

Table 3. Main effects in case study 1

Factor Level 1 Level 2 Max-Min

A 0.5072 0.5383 0.0311B 0.5601 0.4854 0.0747C 0.4792 0.5662 0.0870D 0.3107 0.7348 0.4241E 0.4195 0.6260 0.2065

In order to predict the anticipated improvement under theselected optimum conditions in the parameter design stage, thequality contribution of SN ratios for these two responses arepredicted, using the additive model. Table 5 displays the re-sults deriving from Tarng’s study, the PCA method (Antony’sstudy), and the WPC method in the additive model. This tablereveals that as with the optimal condition in the WPC method,the quality of deposition rate is 19.30 dB, which is largerthan the 17.04 dB initial condition and of the same qualityas in both Tarng’s study and in the PCA method. Similarly,

Table 4. ANOVA on the MPI in case study 1

Source of Sum of Degree of Mean F test P-value Percentagevariation squares freedom square contri-(factor) bution

B 0.01160 1 0.01160 2.023 0.250 2.37%C 0.01514 1 0.01514 2.745 0.196 3.10%D 0.36000 1 0.36000 65.220 0.004 73.77%E 0.08524 1 0.08524 15.451 0.029 17.45%Pooled 0.01655 3 0.00552 3.40%errorTotal 0.48800 7 100.00%

Table 5. The result of the addition model in case study 1

Optimal condition (dB) Anticipated improvement (dB)Response Initial Tarng’s Antony’s Liao’s Tarng’s Antony’s Liao’s

condition (dB) study study (PCA) study (WPC) study study (PCA) study (WPC)

Deposition rate 17.04 19.30 19.30 19.30 2.26 2.26 2.26Dilution −21.64 −21.04 −21.04 −21.04 0.60 0.60 0.06

as for the optimal condition in the WPC method, the qualityof dilution is −21.04 dB, which has no significant differenceamong the quality in the initial condition, Tarng’s study, andthe PCA method. However, the application of the WPC methodnot only overcomes the difficulty of assigned weight (as inTarng’ study), but also overcomes the shortcomings in the PCAmethod. Therefore, the WPC method is much more appropri-ate and better than both Tarng’s study and the PCA method forsolving this multi-response problem of submerged arc weldingprocess.

Case study 2

Phadke [2] sought to improve the quality of a polysilicon depo-sition process, and Hey attempted this experiment in 1984 [14].In this case, controllable factor (A) had two levels, and othershad three levels. In addition, the standard array L18 was selectedfor this example. Six controllable factors included: (A) deposi-tion temperature, (B) deposition pressure, (C) nitrogen flow, (D)silane flow, (E) setting time, and (F) cleaning method.

Three desired multi-responses are considered: (1) surface de-fects (smaller the better), (2) thickness (nominal the better, targetis 3600 Å), and (3) deposition rate (larger the better).

The initial condition was set to: A2B2C1D3E1F1, the result of this experiment with the optimal combination of Phadke’s study(Taguchi method) was A1B2C1D3E2F2, and the result with thePCA method was A1B1C3D2E3F2.

Now, with the application of the proposed WPC method,Table 6 shows the explained variation as a result of these threeresponses and the eigenvector of each principal component.

So, the relations between principal components and re-sponses are:

Z 1 =0.616× surface defects+0.543× thickness

+0.571×deposition rate,

Z 1 =0.782× thickness−0.623×deposition rate, and

Z 1 =−0.780× surface defects+0.318× thickness

+0.539×deposition rate.

The equation of weighted principal components, MPI, is:

MPI = 0.79322× Z 1 +0.15616× Z 2 +0.05062× Z 3

Table 7 shows the obtained MPI value in the standard array L18.In computing the main effect of each controllable factor’s levels,Table 8 shows the computational results.



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(%) (%) dilution]

Z 1 2.380 79.322 79.322 [0.616, 0.543, 0.571] Z 2 0.468 15.616 94.938 [0.000, 0.782,−0.623] Z 3 0.152 5.062 100.000 [−0.780, 0.318, 0.539]

It is evident that the best combination of factors/levels is alsoA1B1C1D3E2F3. Additionally, in order to understand whetherthese factors’ effects have a significant impact on the MPI,Table 9 performs an ANOVA on the MPI. From Table 9, factorB is significant in the multi-response performance, and accountsfor approximately 65% contribution for the total variation. Fac-tor A is added to the pooled error, since it is non-significant andmakes the smallest contribution.

In order to predict the anticipated improvement under the se-lected optimum conditions in the parameter design stage, the SNratios for these three responses are predicted, using the additivemodel. Table 10 displays the results deriving from the Taguchimethod, the PCA method, and the WPC method in the additivemodel. This table reveals that with a significant anticipated im-provement in the WPC method, the anticipated improvement of surface defects is 56.74 dB, which is larger than the improvement36.85 dB in the Taguchi method and better than the improvement54.40 dB in the PCA method. Additionally, with the anticipatedimprovement in the WPC method, the anticipated improvementof thickness is 8.19 dB, which is better than the improvement6.84 dB in the Taguchi method and is less significant than thatin the PCA method. In the quality of deposition rate, a slightreduction occurs in the PCA method and in the WPC method.Therefore, in this multi-response problem, the WPC method per-forms better than the Taguchi method, and there are no similarresults between the WPC method and the PCA method.


Exp. No. L18 MPIA B C D E F G H

1 1 1 1 1 1 1 1 1 1.39072 1 1 2 2 2 2 2 2 1.35863 1 1 3 3 3 3 3 3 1.34754 1 2 1 1 2 2 3 3 1.36805 1 2 2 2 3 3 1 1 0.91556 1 2 3 3 1 1 2 2 1.26907 1 3 1 2 1 3 2 3 0.9205

8 1 3 2 3 2 1 3 1 0.44329 1 3 3 1 3 2 1 2 0.799910 2 1 1 3 3 2 2 1 1.308711 2 1 2 1 1 3 3 2 1.391812 2 1 3 2 2 1 1 3 1.349113 2 2 1 2 3 1 3 2 1.298614 2 2 2 3 1 2 1 3 1.345315 2 2 3 1 2 3 2 1 1.239316 2 3 1 3 2 3 1 2 1.209017 2 3 2 1 3 1 2 3 0.431418 2 3 3 2 1 2 3 1 0.1431


Factor Level 1 Level 2 Level 3 Max-Min

A 1.0903 1.0796 — 0.0107B 1.3577 1.2393 0.6579 0.6998C 1.2493 0.9809 1.0247 0.2684D 1.1035 0.9976 1.1538 0.1562E 1.0767 1.1612 1.0169 0.1443F 1.0303 1.0539 1.1706 0.1403



B 1.68400 2 0.84200 13.350 0.004 65.25%C 0.24900 2 0.12400 1.972 0.209 9.65%D 0.07629 2 0.03815 0.605 0.572 2.96%E 0.06305 2 0.03152 0.500 0.627 2.44%F 0.06769 2 0.03384 0.537 0.607 2.62%Pooled 0.44100 7 0.06306 17.09%error

Total 2.58100 17 100.00%

Case study 3

This case study considers Su et al.’s [14] attempt to improve thequalityofaharddiskdrive.Inthiscase,controllablefactor(A)hadtwolevels andthe othershad three levels. In addition, thestandardarray L18 was selected for this case. The controllable factors wereas follows: (A) disk writability, (B) magnetization width, (C) gaplength, (D) coercivity of media, and (E) rotational speed.

Four desired multi-responses were: (1) 50% pulse width(PW, smaller the better), (2) high-frequency amplitude (HFA,

larger the better), (3) over write (OW, larger the better), and (4)peak shift (PS, smaller the better).

The starting condition in this experiment was: A1B1C2D2E3,the optimal combination of factors/levels chosen in Taguchimethod was: A1B1C1D1E3, and that in the PCA method wasA2B1C1D3E3 (Su et al. [14] had wrong computation in hispaper and set the optimal combination of factors/levels toA1B1C1D3E3.)

Now, with the application of the WPC method, Table 11shows the explained variation resulting from these three re-sponses and the eigenvector of each principal component.

The relations between principal components and responseare:

Z 1 = 0.565×PW+0.514×HFA−0.399OW+0.508×PS,

Z 2 = 0.203×PW−0.001×HFA+0.866OW+0.457×PS,

Z 3 =−0.070×PW+0.806×HFA+0.287OW−0.512×PS,

and

Z 4 = 0.797×PW−0.292×HFA+0.088OW−0.521×PS.



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Table 10. The result of the addition model in case study 2

Optimal condition (dB) Anticipated improvement (dB)Response Initial Phadke’s Su et al.’s Liao’s Phadke’s Su et al.’s Liao’s

condition (dB) study (Taguchi) study (PCA) study (WPC) study (Taguchi) study (PCA) study (WPC)

Surface defects −56.69 −19.84 −2.29 0.05 36.85 54.40 56.74Thickness 29.95 36.79 41.23 38.14 6.84 11.28 8.19Deposition rate 34.97 29.6 27.21 27.17 −5.37 −7.76 −7.80

The equation of weighted principal components, MPI, is:

MPI =0.73777× Z 1 +0.16702× Z 2

+0.08350× Z 3 +0.01171× Z 4

Table 12 shows the obtained MPI value in the standard array L18.In computing the main effect of each controllable factor’s levels,Table 13 shows the computation results.

It is evident that the best combination of factors/levelsis A1B1C1D3E3. Additionally, in order to understand whetherthese factors’ effects have a significant impact on the MPI,Table 14 performs an ANOVA on the MPI. From this Table 14,factors C and E are significant in the multi-response perform-ance, and account for approximately 95% of the total variation.Factor A, B and D are added to the pooled error, since they arenon-significant and make the smallest contribution.

To predict the anticipated improvements under the chosenoptimum conditions, the SN ratios of these four responses canalso be predicted from the additive model. Table 15 displaysthe results from the additive model. This table reveals that witha significant anticipated improvement in the WPC method, thePW50 is 2.54 dB, which is equal to the results of the Taguchimethod and the PCA method. Additionally, with the antici-pated improvement in the WPC method, the HFA is 1.76 dB,which is better than the improvement 0.95 dB in the Taguchimethod and is less than that in the PCA method. In the qual-ity of OW, the anticipated improvement is −3.8 dB in the WPCmethod, which is less than that in the Taguchi method and bet-ter than that in the PCA method. In the quality of PS, the an-ticipated improvement is 2.11 dB in the WPC method, whichis equal to that in the Taguchi method and better than that inthe PCA method. Therefore, it is demonstrated that the WPC



(%) (%) dilution]

Z 1 2.951 73.777 73.777 [0.565, 0.514,

−0.399, 0.508] Z 2 0.668 16.702 90.479 [0.203,−0.001,

0.866, 0.457] Z 3 0.334 8.350 98.829 [−0.070, 0.806,

0.287,−0.512] Z 4 0.005 1.171 100.000 [0.797,−0.292,

0.088,−0.521]


Exp. No. L18 MPIA B C D E F G H

1 1 1 1 1 1 1 1 1 0.76422 1 1 2 2 2 2 2 2 0.85483 1 1 3 3 3 3 3 3 0.83724 1 2 1 1 2 2 3 3 0.99135 1 2 2 2 3 3 1 1 1.01916 1 2 3 3 1 1 2 2 −0.02247 1 3 1 2 1 3 2 3 0.78338 1 3 2 3 2 1 3 1 0.86819 1 3 3 1 3 2 1 2 0.6746

10 2 1 1 3 3 2 2 1 1.261011 2 1 2 1 1 3 3 2 0.436012 2 1 3 2 2 1 1 3 0.524413 2 2 1 2 3 1 3 2 1.212914 2 2 2 3 1 2 1 3 0.562615 2 2 3 1 2 3 2 1 0.299116 2 3 1 3 2 3 1 2 1.093617 2 3 2 1 3 1 2 3 0.980618 2 3 3 2 1 2 3 1 0.0363


Factor Level 1 Level 2 Level 3 Max-Min

A 0.7522 0.7118 — 0.0434B 0.7796 0.6771 0.7394 0.1025C 1.0177 0.7869 0.3915 0.6262D 0.6910 0.7385 0.7667 0.0757E 0.4267 0.7719 0.9976 0.5709

method not only performs better than the PCA method butalso reduces the complexity of decision-making in the Taguchimethod.



C 1.20300 2 0.60200 71.462 0.000 52.19%E 0.99200 2 0.49600 58.910 0.000 43.04%Pooled 0.10900 13 0.00842 4.77%errorTotal 2.30500 17 100.00%



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Table 15. The result of addition model in case study 3

Optimal condition (dB) Anticipated improvement (dB)Response Initial Taguchi’s Su et al.’s Liao’s Taguchi’s Su et al.’s Liao’s

condition (dB) method study (PCA) study (WPC) method Study (PCA) study (WPC)

PW50 −36.28 −33.73 −33.74 −33.74 2.55 2.54 2.54HFA 50.47 51.42 52.93 52.23 0.95 2.46 1.76OW 31.51 29.82 25.32 27.71 −1.69 −6.19 −3.80PS −21.48 −19.37 −21.03 −19.37 2.11 0.45 2.11

4 Conclusion

In this paper, the author submits a WPC method to improve themulti-response problem in the Taguchi method. Three case stud-ies from Su et al. [14] and Antony [15] are used to illustrate theWPC method and to compare this method with other methods.The results show that in case study 1, the results of the optimalcombination of factors/levels are the same in the WPC method,Tarng’s study, and the PCA method. In case study 2, the resultin the quality of anticipated improvement in the WPC method

is better than that in the Taguchi method, and WPC’s settingcondition of best combination of factors/levels is different fromthe PCA method’s setting condition. In case study 3, the qual-ity of the anticipated improvement in the WPC method is betterthan that in the PCA method, and the WPC’s optimal combina-tion of factors/levels is different from Taguchi method’s optimalcombination. Most importantly, the proposed WPC method notonly reduces the uncertainty and complexity of engineers’ judg-ment associated with the Taguchi method, but also overcomes theshortcomings of the PCA method.

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