Optimization of multiple responses using principal component analysis and technique for order preference by similarity to ideal solution

DOI 10.1007/s00170-004-2157-9 O R I G I N A L A R T I C L E

Lee-Ing Tong · Chung-Ho Wang · Hung-Cheng Chen

Optimization of multiple responses using principal component analysis and

technique for order preference by similarity to ideal solution

Received: 3 December 2003 / Accepted: 25 February 2004 / Published online: 7 July 2004 ©Springer-Verlag London Limited 2004

Abstract Optimizing multi-response problems has become an increasingly relevant issue when more than one correlated product quality characteristic must be assessed simultaneously in a complicated manufacturing process. This study proposes a novel optimization procedure for multiple responses based on Taguchi’s parameter design. The signal-to-noise (SN) ratio is ini-tially used to assess the performance of each response. Principal component analysis (PCA) is then conducted on the SN values to obtain a set of uncorrelated components. The optimization direction for each component is determined based on the corres-ponding variation mode chart. Finally, the relative closeness to the ideal solution resulting from the technique for order prefer-ence by similarity to ideal solution (TOPSIS) is determined as an overall performance index (OPI) for multiple responses. En-gineers can easily employ the proposed procedure to obtain the optimal factor/level combination for multiple responses. A case study involving optimization of the chemical-mechanical polish-ing of copper (Cu-CMP) thin films from an integrated circuit manufacturer in Taiwan is also presented to demonstrate the ef-fectiveness of the proposed procedure.

Keywords Multi-response problems· Optimization · Principal component analysis· Taguchi method · TOPSIS

L.-I. Tong (u)

Department of Industrial Engineering and Management, National Chiao Tung University,

Hsinchu, Taiwan, R.O.C. 300 C.-H. Wang

Department of General courses, Chung Cheng Institute of Technology, National Defense University,

Taoyuan, Taiwan, R.O.C. H.-C. Chen

Enterprise Operation Planning Division, United Microelectronics Corporation, Hsinchu, Taiwan, R.O.C. 300

1 Introduction

The method that Taguchi developed in 1960 for enhancing product quality has been widely implemented throughout in-dustry to upgrade manufacturing products/processes [9]. The Taguchi method evaluates product quality by applying the signal-to-noise (SN) ratio and, in doing so, the optimal fac-tor/level combination obtained from the Taguchi method can be determined to reduce simultaneously the quality varia-tion and bring the mean close to the target value. Despite its widespread industrial applications, the Taguchi method can only be used for optimizing single-response problems. When the product design becomes increasingly complicated, more than one response must be optimized. Because these responses are usually moderately or highly correlated, developing pro-cedures capable of optimizing simultaneously multi-response problems has become increasingly important, particularly in high-tech industries. However, most available procedures do not consider the correlations among the responses and they are too mathematically complicated for engineers to use in practice. Additionally, the possible correlations among re-sponses may cause difficulty in optimizing multiple rere-sponses simultaneously.

This study develops a novel multi-response optimization pro-cedure capable of resolving the correlation problems among re-sponses, and reducing the computational complexity. The SN ratio is initially used to assess the performance of each response. The PCA is then conducted on SN values to obtain a set of un-correlated principle components, which are linear combinations of the original responses. The variation mode chart is plotted to interpret the variation mode (or principal component variation) resulting from PCA. Based on the engineering requirements, engineers can determine the optimization direction for each prin-cipal component using the variation mode chart. Finally, TOPSIS is employed to derive the OPI for multiple responses. The opti-mal factor/level combination is determined by the maximum OPI value. A case study involving the optimization of the chemical-mechanical polishing of copper (Cu-CMP) thin films, from an

integrated circuit manufacturer in Taiwan, is presented to demon-strate the effectiveness of the proposed procedure.

2 Literature review

Many Taguchi practitioners have used engineering knowledge to resolve multi-response optimization problems. For example, Phadke [9] combined engineering knowledge with relevant ex-perience to optimize three responses, i.e. surface, wafer thick-ness, and deposition rate, in a very-large-scale integrated (VLSI) circuit-manufacturing process. Other techniques, as proposed by Logothetis and Haigh [8], suggested that multi-response data must be transformed before determining the noise performance statistic (NPS) and the target performance statistic (TPS) for each response. The optimal factor/level combination and adjust-ment factors are determined based on the NPS and TPS values. Elsayed and Chen [4] distinguished between the control factor and adjustment factor by the performance measure independent of adjustment (PerMIA). The performance measure for quality (PerMQ) is calculated based on the selected adjustment fac-tor. Chang and Shivpuri [2] established a regression model for each response for control factors and applied the procedure of multiple-attribute decision making (MADM) to determine an op-timal factor/level. Tong and Su [10] used fuzzy theory and the MADM technique to resolve a multi-response problem. Ames et al. [1] adopted the response surface method (RSM) to re-solve a multi-response problem. Despite their contributions, the above multi-response optimization methods share the following limitations:

1. The optimal factor/level combination for multiple responses is determined based on pure engineering experience but the correlations among responses are not considered. Because the engineer’s judgment often leads to uncertainty during de-cision making, different engineers may produce conflicting results when addressing the same problem; and

2. These procedures are developed based on the linear pro-gramming technique or other complicated mathematical al-gorithms, thereby making them impractical for many engin-eering applications.

3 PCA and TOPSIS

Pearson and Hotelling [5] initially developed PCA to explain the variance-covariance structure of a set of variables by linearly combining the original variables. The PCA technique can ac-count for most of the variation of the original p variables via k uncorrelated principal components, where k≤ p. Restated, let x= x1, x2, . . . , xpbe a set of original variables with a

variance-covariance matrix
. Through the PCA, a set of uncorrelated linear combinations can be obtained in the following matrix:

Y= ATx (1)

where Y= (Y1, Y2, . . . , YP)T, Y1 is called the first principal component, Y2is called the second principal component and so on; A= (aij)P×Pand A is an orthogonal matrix with ATA= I.

Therefore, x can also be expressed as follows:

x= AY = p

j=1

AjYj (2)

where Aj= [a1 j, a2 j, . . . , ap j]Tis the jtheigenvector ofΣ.

3.2 Variation mode chart

The variation mode chart [11] is an effective means of ana-lysing the variation mode (or principal component variation) obtained from PCA. Analysing this chart can provide further in-sight into the different variation types for each variation mode. Therefore, the portion of variation contributed by the original variables (x1, x2, . . . , xp) in each mode can be obtained. The

calculation process for establishing a variation mode chart is given as follows: Let zj = A. jYj= [a1 j, a2 j, . . . , ap j]TYj=

[z1 j, z2 j, . . . , zp j]T, Eq. 2 can be rewritten as follows:

x= z1+ z2+ ... + zp (3)

where zj is a product of a random scalar Yjand a deterministic

vector A_{. j}; zj can be defined as a geometrical variation mode.

The mean, variance and standard deviation of zij are given as

follows:

E(zij) = E(aij× Yj) = aijE(Yj) = 0 (4)

Var(zij) = Var(aij× Yj) = aij2Var(Yj) = a2ijλj (5)

σ(zij) = |aij|



λj (6)

Figure 1 plots the variation mode chart based on a three-sigma zone (u± 3σ) that describes the pattern and magnitude for each variation mode. In this figure, the solid line denotes the variation extent limit (VE L1), which is equal to 3σ(zij) as shown in Eq. 7.

The dotted line denotes the variation extent limit (VE L2), which is equal to−3σ(zij) as shown in Eq. 8.

VE L1(zj) = (3a1 j  λj, . . . , 3ap j  λj) (7) VE L2(zj) = (−3a1 j  λj, . . . , −3ap j  λj) (8)

The following example, including four variables x= (x1, x2, x3, x4), illustrates how to use a variation mode chart to

charac-terize the exact pattern and magnitude of a variation mode. In this example, assume that the eigenvalueλ1= 33.62 and A.1=

(0.503, 0.332, −0.455, −0.656). The VEL1(z1) = (8.75, 5.77,

−7.91, −11.4) and the VEL2(z1) = (−8.75, −5.77, 7.91, 11.4)

are accordingly obtained using Eqs. 7 and 8. Thus, the variation mode chart for mode 1 is presented in Fig. 2.

Clearly, when x1 and x2 vary in the positive direction, then x3and x4vary in the negative direction. As x1moves in the

pos-itive direction up to 8.75, x2 moves in the positive direction up

to 5.77, and x3and x4 move in the negative direction up to 7.91

and 11.4, respectively. Therefore, analysis of the variation mode chart can provide further insight into the variation pattern of var-ious variables. Doing so can facilitate the reduction of the origins of variable variations.

3.3 TOPSIS

Hwang and Yoon [6] developed TOPSIS to assess the alterna-tives before multiple-attribute decision making. TOPSIS consid-ers simultaneously the distance to the ideal solution and negative ideal solution regarding each alternative, and also selects the most relative closeness to the ideal solution as the best alterna-tive. That is, the best alternative is the nearest one to the ideal solution and the farthest one from the negative ideal solution. The procedure of TOPSIS is summarized as follows:

1. Establish an alternative performance matrix. The structure of the alternative performance matrix is expressed as follows:

D= A1 A2 . Ai . Am X1 X2 Xj Xn ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x11x12. x1 j. x1n x21x22. x2 j. x2n . . . . xi1 xi2 . xij . xin . . . . xm1xm2. xm j. xmn ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (9)

where Ai denotes the possible alternatives, i= 1, . . ., m; Xj represents attributes relating to alternative performance,

Fig. 2. An example of variation mode chart

j= 1, . . ., n; and xij is the performance of Aiwith respect to

attribute Xj.

2. Normalize the performance matrix. The normalized perform-ance matrix can be obtained using the following transform-ation formula: rij= xij m i=1 x2_ij (10)

where rij represents the normalized performance of Ai with

respect to attributeXj. The matrix form of rij is given as

fol-lows:

R= [rij] (11)

where i= 1, 2, . . . , m and j = 1, 2, . . .n.

3. Multiply the performance matrix by its associated weights. Each column of matrix R is multiplied by weights associated with each attribute. The weighted performance matrix V is obtained as follows: V= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ w1r11w2r12. wjr1 j. wnr1n w1r21w2r22. wjr2 j. wnr2n . . . . . . w1ri1w2ri2 . wjrij . wnrin . . . . . . w1rm1w2rm2. wjrm j. wnrmn ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ v11v12. v1 j. v1n v21v22. v2 j. v2n . . . . vi1 vi2. vij . vin . . . . vm1vm2. vm j. vmn ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (12)

wherewj represents the weight of attribute Xj andvij

rep-resents the weighted normalized performance of Aiwith

re-spect to Xjfor i= 1, 2, . . ., m and j = 1, 2, . . ., n.

4. Determine the ideal and negative ideal solution. The ideal value set V+ and the negative ideal value set V− are deter-mined as follows: V+= {(max vij| j ∈ J) or (min vij| j ∈ J  ), i = 1, 2, . . . , m} = {v1+, v+2, . . . , v+n} V−= {(min vij| j ∈ J) or (max vij| j ∈ J  ), i = 1, 2, . . . , m} = {v1−, v−2, . . . , v−n} where

J= { j = 1, 2, . . . , n|vij, a larger response is desired} J= { j = 1, 2, . . . , n|vij, a smaller response is desired}

5. Calculate the separation measures. The separation of each al-ternative from the ideal solution (S+_i ) is given as follows:

S_i+=   
n j=1 (vij−v+_j )2 (13)

The separation of each alternative from the negative ideal so-lution (S_i−) is as follows: S−_i =   
n j=1 (vij− v−j )2 (14)

Calculate the relative closeness to the ideal solution and rank the preference order. The relative closeness Ci to the ideal

solution can be expressed as follows:

Ci= S−_i

S_i++ S−_i (15)

where Ci lies between 0 and 1. The closer Ci is to 1, the

higher the priority of the ithalternative.

4 Proposed procedure

This study proposes an optimization procedure for multiple re-sponses based on Taguchi’s parameter design. Because multi-ple responses always contain moderate or high correlations, the PCA is initially performed on the SN values obtained from each response to integrate the dimension of multiple responses to a smaller number of uncorrelated components. The variation mode charts for components obtained from PCA are then used to investigate the variation pattern of various integrated responses. Finally, TOPSIS is used to determine the optimal factor/level combination for multiple responses. The symbols used in this study are summarized in Table 1.

The proposed procedure for optimizing multi-response prob-lems includes the following seven steps:

Step 1. Calculate the SN ratio for each response.

Phadke [9] details the SN ratio formula.

Step 2. Conduct the PCA on normalized SN ratios.

The SN ratio for each response is normalized by the follow-ing formula:

SNij− SNj SSNj

(16)

where SNij denotes the SN ratio of the jth response in the ith

experimental run; SNj and SSNj represent the mean and

stan-dard deviation of the SN ratio for the jthresponse, respectively.

Table 1. List of symbols

Symbols Notation

SNij The signal-to-noise (SN) ratio of the jthresponse under the ith

experimental run.

SNj The mean of the SN ratio of the jthresponse.

S_SNj The standard deviation of the SN ratio for the jth_response. v+j The ideal solutions of the jthresponse.

v−j The negative ideal solutions of the jthresponse.

Ci The overall performance index (OPI) of multiple responses

under the ith_{experimental run.}

The eigenvalues and eigenvectors for each principal compon-ent are obtained after conducting PCA on the normalized SN ratios.

Step 3. Determine the number of principal components retained

and establish the variation mode charts.

Some principal components are selected for further analy-sis based on the significance of the linear correlation between the responses and principal components and the cumulative vari-ation of the responses accounted for by the selected principal components. The corresponding variation mode charts are also established using Eq. 7 and Eq. 8.

Step 4. Determine the optimization direction of the selected

prin-cipal components.

The optimization direction of each selected principal com-ponent is determined according to the variation mode chart. Ac-cording to Fig. 2, if responses x1and x2are more important than

responses x3 and x4, the first principal component score is

de-termined since a larger value is desired. In this case, optimizing (or maximizing) the first principal component increases response

x1 and x2 by 8.75 and 5.77, respectively, and decreases x3 and x4by 7.91 and 11.4, respectively. When responses x3and x4are

more important than responses x1and x2, the first principal

com-ponent score is determined, as a smaller value is desired. In this case, optimizing (or minimizing) the first principal component decreases responses x1and x2by 8.75 and 5.77, respectively and

increases x3 and x4by 7.91 and 11.4, respectively. When more

than one principal component is selected for further analysis, the first principal component initially determines the optimization direction. Thereafter, the optimization direction of the second principal component is determined, and so on for the remaining selected components.

Step 5. Conduct TOPSIS to obtain the OPI for multiple

re-sponses.

According to the optimization direction of the selected prin-cipal components obtained from Step 4, TOPSIS is used to deter-mine the OPI. The experimental runs are treated as alternatives; and the selected principal components are treated as attributes and a quality performance matrix is formed. The weighted qual-ity performance matrix can be obtained using Eqs. 9–12, where the weights are the eigenvalues associated with each principal component. If a larger value is desired, the ideal and negative ideal solutions representing the maximum and minimum princi-pal component scores of all experimental runs are expressed in Eq. 17 and Eq. 18, whereas if a smaller value is desired, the ideal and negative ideal solution representing the minimum and max-imum principal component scores of all experimental runs are expressed in Eq. 19 and Eq. 20. Correspondingly, the OPI values (or Ci values for i= 1, 2, . . ., m) for each experimental run are

derived using Eqs. 13–15.

v+j = max{v1 j, v2 j, . . . , vm j} (17)

v−

j = min{v1 j, v2 j, . . . , vm j} (18)

v+j = min{v1 j, v2 j, . . . , vm j} (19)

Step 6. Determine the optimal factor/level combination.

The main effects on OPI are determined based on the Ci

values. Thus, the corresponding diagram plots the factor effect on OPI. The optimal factor/level combination produces the max-imum OPI value.

Step 7. Conduct the confirmation experiment.

According to the optimal factor/level combination, confirma-tion experiments are performed to verify whether the experimen-tal results can be reproduced. The predicted SN value for each response is compared with the associated actual SN value ob-tained from the confirmation experiments. If the predicted SN values and the actual SN values differ only slightly then the ex-periment can be reproduced. If the predicted SN values and the actual SN values differ substantially, the experimental result can-not be reproduced. In this case, suitable quality characteristics, control factors, or signal factors must be reselected, and return to step 1 of the proposed procedure to start all over again.

5 Illustrative example

The effectiveness of the proposed procedure is demonstrated using a chemical-mechanical polishing process of copper thin-film provided by a Taiwanese integrated circuit (IC) manufac-turer. With the scaling down in size of integrated circuit devices and the increasing number of devices on the chip, on-chip in-terconnections play a dominant role in determining IC perform-ance. Although scaling down the size of the device enhances the device speed, the interconnection delays the signal propa-gation. Consequently, interconnect delay severely limits circuit performance. The interconnection RC-delay increases rapidly in sub-micron regions because the resistance of the metal line increases with the decreasing line width while the parasitic cap-acitance of metal lines increases with decreasing line spacing. Higher operating frequencies for IC chips increase current

den-Table 2. Experimental observations and SN ratio

Ex. no. Control factor Response SN ratio

L18 A B C D E RR NU TaN/Cu RR NU TaN/Cu 1 1 1 1 1 1 294 14.3 4.0 49.37 -23.11 12.04 2 1 2 2 2 2 289 15.7 4.3 49.22 -23.92 12.67 3 1 3 3 3 3 314 23.2 5.6 49.94 -27.31 14.96 4 2 1 1 2 2 375 12.1 3.7 51.48 -21.66 11.36 5 2 2 2 3 3 437 8.7 4.9 52.81 -18.79 13.80 6 2 3 3 1 1 498 6.5 6.1 53.94 -16.26 15.71 7 3 1 2 1 3 481 8.99 4.2 53.64 -19.08 12.46 8 3 2 3 2 1 588 11.8 4.3 55.39 -21.44 12.67 9 3 3 1 3 2 660 12.4 5.3 56.39 -21.87 14.49 10 1 1 3 3 2 242 16.2 4.6 47.68 -24.19 13.26 11 1 2 1 1 3 268 26.9 4.1 48.56 -28.60 12.26 12 1 3 2 2 1 340 10.5 5.3 50.63 -20.42 14.49 13 2 1 2 3 1 377 16.9 3.9 51.53 -24.56 11.82 14 2 2 3 1 2 434 5.06 4.7 52.75 -14.08 13.44 15 2 3 1 2 3 494 7.08 5.4 53.87 -17.00 14.65 16 3 1 3 2 3 483 8.76 5.2 53.68 -18.85 14.32 17 3 2 1 3 1 580 15.1 4.6 55.27 -23.58 13.26 18 3 3 2 1 2 651 5.0 5.8 56.27 -13.98 15.27

sities in smaller interconnection features. Consequently, a sta-ble metal like copper is highly promising for the metallization of advanced interconnection owing to its lower resistivity and enhanced electromigration resistance compared to current alu-minum alloy interconnects [3, 7]. A dual damascene process is initially developed to pattern the inter-layer dielectric (ILD) to form trenches and vias. These trenches and vias are then filled with copper, followed by chemical-mechanical polishing to flat-ten the wafer surface until the only remaining copper is in the trenches, and the vias have recessed into the ILD. Since CMP leaves the wafer surface globally planar, this sequence can be repeated to add multiple metal layers. Furthermore, Cu-CMP is essential in successfully implementing the dual damascene process for multilevel interconnects. Thus, multilevel Cu-interconnection using the dual damascene process is a viable alternative for ULSI manufacturing. The proposed procedure is verified via the optimization steps of removing the TaN barrier layer of the Cu-CMP process. Based on the engineering require-ments, the following quality characteristics are determined as the process responses:

1. Removal rate (RR): a larger value is desired. 2. Non-uniformity (NU): a smaller value is desired. 3. TaN/Cu selectivity: a larger value is desired.

Among these responses, NU is the most important quality char-acteristic, TaN/Cu is the next important one and RR is the least important one. Five control factors (A to E), each with three lev-els, are allocated sequentially to a L18orthogonal array. Owing to engineering confidentiality, the names of the factors and the contents of each level are omitted. The experiments are con-ducted randomly.

The experimental data was analysed by following the pro-posed procedure strictly. Table 2 displays the experimental observations and SN ratios for each response resulting from Taguchi’s SN ratio formula. Tables 3 and 4 display the

eigenval-Components Eigenvalue Difference Percentage Accumulative percentage

First component 1.99107 1.36858 0.663689 0.66369

Second component 0.62249 0.23604 0.207496 0.87118

Third component 0.38645 – 0.128816 1.00000

Table 3. The eigenvalues and explained

per-centage of variation for principal components

Response First component Second component Third component

Removal rate (RR) 0.589389 −0.487698 0.644028

Non-uniformity (NU) 0.610704 −0.252885 −0.750393

Selectivity (TaN/Cu) 0.528830 0.835583 0.148792

Table 4. The eigenvectors for principal

com-ponents

ues and eigenvectors arising from PCA conducted by employing the SAS statistical software package (many other statistical soft packages such as SPSS, Minitab or STATISTICA can also be used to conduct PCA).

Based on Tables 3 and 4, all three principal components are retained, since the first two principal components only account for 87% of the variation of the original variables. The three prin-cipal components are uncorrelated and can account for 100% of the variation of original variables. Fig. 3 displays the variation mode charts for each principal component. Clearly, the directions

of variation mode for responses RR, NU and TaN/Cu is consis-tent and their variation contributions do not significantly differ for the first principal component. Therefore, the first principal component is determined, as a larger value desired integrated response. The SN ratios of each response can be enhanced sim-ultaneously when optimizing the first principal component. The directions of variation mode for responses RR and TaN/Cu are opposite and the variation contributed by response NU is in-significant in the second principal component. Therefore, the second principal component is determined as a larger value de-sired integrated response since the response TaN/Cu is more important than the response RR. Similarly, the third principal component is determined as a smaller value desired integrated response since NU is the most important response and the direc-tions of variation mode for responses RR and NU are opposite in the third principal component. Thus, decreasing the SN ratio of response RR by 1.2 can increase the SN ratio of NU by 1.39 and the SN ratio of TaN/Cu is nearly unchanged.

Table 5 lists the OPI values, which are measures of relative closeness to the ideal solution resulting from TOPSIS. Accord-ingly, a response diagram on OPI values is established as shown in Fig. 4. According to this figure, the optimal factor/level com-bination is determined as A3B3C3D1E3.

Experimental runs OPI

1 0.23 2 0.26 3 0.39 4 0.26 5 0.57 6 0.83 7 0.47 8 0.48 9 0.64 10 0.28 11 0.15 12 0.53 13 0.21 14 0.65 15 0.72 16 0.64 17 0.47 18 0.83

Table 5. The overall

Table 6. Summary of the predicted SN value and the actual improvement

Response ∗Starting condition Optimal condition (prediction) Optimal condition (confirmation) Actual improvement

Removal rate (RR) 52.7071 55.9256 56.9391 4.2320

Non-uniformity (NU) −17.3205 −16.260 −12.8691 4.4514

Selectivity (TaN/Cu) 12.9393 16.0201 21.0076 8.0683

∗_{Starting condition are at level 2 for all factors.}

Fig. 4. Factor effects on OPI

The confirmation experiments are performed under the op-timal factor/level combination to verify whether the optimum condition is reproduced. Table 6 summarizes the computations of the predicted SN values for all three responses, revealing that the actual SN values slightly differ from the predicted SN values. The SN ratio for RR is improved by 4.2320 dB, NU is improved by 4.4514 dB and TaN/Cu is improved by 8.0683 dB. This find-ing confirms that the optimal factor/level combination can be reproduced and the proposed procedure for optimizing multiple responses can enhance the product/process quality efficiently.

6 Conclusion

This study utilizes PCA to simplify multi-response problems and determines the optimization direction by using a variation mode chart. The optimal factor/level combination is also determined based on the overall performance index for multiple responses obtained from TOPSIS. A case study in which the chemical-mechanical polishing process of copper thin films is optimized confirms the effectiveness of the proposed procedure.

The proposed procedure has the following merits:

1. The proposed procedure is relatively simple and does not in-volve much complicated mathematical processing, thus mak-ing it quite feasible for engineers to use without much statis-tical background.

2. Most of the multi-response optimization procedures de-veloped in previous studies provided non-inferior solutions. In addition, these procedures do not consider the relative

Fig. 5. The diagram of non-inferior solutions

importance of each response. The proposed procedure can identify the optimization direction via the variation mode chart, which reflects the relative importance of each re-sponse, to obtain a real optimal solution. To address this point, as illustrated in Fig. 5, assuming two responses, say y1and y2, with the larger values desired for both responses, the solid line represents the collection of all non-inferior solutions. The optimization solution for multiple responses obtained from previous studies falls anywhere in the interval between A and E. However, the optimization solution falls only in the interval between C and E if the response y2 is more important than y1. Whereas when the response y1 is more important than y2, the optimization solution falls in the interval between A and C.

3. The proposed procedure transforms the correlated multi-ple responses into uncorrelated components through PCA, thereby simplifying the optimization process.

4. The proposed procedure can also resolve the multi-response problems in a dynamic system with some modification.

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