A Note on "Capability Assessment for Processes with Multiple Characteristics: A Generalization of the Popular Index Cpk"

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A Note on

“Capability Assessment for

Processes with Multiple Characteristics: A

Generalization of the Popular Index

C

_pk

”

W. L. Pearn, C. H. Wu*

,†

and M. C. Tsai

The generalized yield indexCT

pkestablishes the relationship between the manufacturing speciﬁcations and the actual process

performance, which provides a lower bound on process yield for two-sided processes with multiple characteristics. The results attended are very practical for industrial application. In this article, we extended the results in cases with one-sided speciﬁcation and multiple characteristics. The generalized index CT

PUwas considered, and the asymptotic distribution of the

natural estimator ^C_PUT _{was developed. Then, we derived the lower con}_{ﬁdence bounds as well as the critical values of index C}T PU.

We not only provided some tables but also presented an application example. Copyright © 2012 John Wiley & Sons, Ltd. Keywords: critical values; lower conﬁdence bounds; multiple characteristics; one-sided speciﬁcation; process capability index

1. Introduction

P

rocess yield has been the most basic and common criterion used in the manufacturing industry as a base for measuring process performance. Recently, Pan and Lee2developed two novel indices to evaluate the performance of multivariate manufacturing process. The effects of the estimation of process capability index (PCI) on the nonconforming units in parts per million (NCPPM) estimates are analyzed by Ozkaya and Testik.3Later, Lin and Pearn4used the yield index Spkto deal with process selection problem.

Itay et al.5investigated an advanced multistage sampling plan based on Cpkindex.

Afterward, Yum and Kim6provided a bibliography of PCIs for 2000–2009. Spiring7then presented several method of process capability using Mathematica 7 software. Next, an applicable methodology to achieve the robustness of the multivariate process capability vector was proposed by Awad and Kovach.8Lately, more dissertations about PCI were published such as Pearn et al.,9 Hsu et al.,10Yen and Pearn,11Pearn and Cheng,12and Goethals and Cho.13It can be observed that the new investigations of PCI mainly focus on processes with multivariate or multiple characteristics.

Pearn et al.14proposed a generalization of the popular index Cpkfor evaluating the yield of a gold bumping manufacturing process

with multiple characteristics. The CT

pk index is deﬁned for a process with multiple characteristics and two-sided speciﬁcations.

However, the quality characteristics often have only one-sided speciﬁcation. At this time, the overall capability index CT

PUproposed

by Wu and Pearn1is considered for this purpose. The index CT

PUis deﬁned as follows: CT PU¼ 1 3Φ 1 Ym i¼1 Φ 3Cð PUiÞ ( ) (1)

where CPUi denotes the CPUvalue of the ith characteristic for i = 1, 2,. . ., m, and m is the number of characteristics. Φ() is the

cumulative distribution function of standard normal distribution. A one-to-one correspondence relationship between CT

PU and the

overall process NCPPM can be demonstrated as

NCPPM¼ 106_{1 Φ 3C}T PU

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

*Correspondence to: C. H. Wu, Department of Industrial Engineering and Management, National Chiao Tung University, 1001 University Road, Hsinchu, Taiwan 300, ROC.

†_{E-mail: hexjacal.iem96g@nctu.edu.tw}

(wileyonlinelibrary.com) DOI: 10.1002/qre.1295 Published online 8 February 2012 in Wiley Online Library

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2. Approximation distribution of the natural estimator

Consider the natural estimator of CT PUas ^C_PUT_¼ 1 3Φ 1 Ym i¼1 Φ 3^CPUi ( ) ¼1₃Φ1 Ym i¼1 Φ USL Xi Si ( ) ; i ¼ 1; . . . ; m (3) where Xi and Sidenote the sample mean and sample variance of ith characteristic. The exact distribution of ^CPUT is mathematically

intractable. By taking theﬁrst order of the Taylor expansion (see Appendix), the asymptotic distribution of ^C_PUT_is

^C_PUT_{N C}T PU; 1 9nf 3CT PU 2 Xm i¼1 a2_i þ b2_i ! (4) where ai¼ Ym j¼1;j6¼i Φ 3CPUj " # f 3CPUj and bi¼ 3CPUi ffiffiffi 2 p ai; i ¼ 1; . . . ; m

3. Estimation and testing on

C

T PU

From Equation (4), it can be seen that ^C_PUT _{is an asymptotic unbiased estimator of ^}_C

PUT. The determination of the lower conﬁdence

bound on the actual process capability is essential for quality assurance. An approximate 100(1 a)% lower conﬁdence bound for CT PUcan be expressed as CT PULB ^CPUT Za 1 9nf 3CT PU 2 Xm i¼1 a2 i þ b2i " #1=2 (5)

It is noted that aiand biparameters are unknown in Equation (5). To investigate the effects of CPUion C

T

PULB, the cases with processes

that have two independent characteristics are considered. Figure 1 displays the curves of CT

PULBfor various combinations of CPU1and

CPU2, with C

T

PU=1.0, 1.33, 1.5, 1.67. Then, we examined the results presented in Figure 1, which indicate that

(i) CT

PULBobtains its absolute maximum as CPU1 ¼ CPU2.

(ii) The minimum CT

PULB occurs when one of CPUi approaches inﬁnity, that is, another CPUi equals C

T

PU. Under this condition, the

minimum CT

PULBis the most reliable lower conﬁdence bound for a given CPUT .

By the discussion mentioned earlier, after doing some algebra, the asymptotic distribution of ^C_PUT _becomes

^C_PUT_{N C}T PU; 1 9nþ 1 2nC T PU2 (6) Therefore, an approximate 100(1 a)% lower conﬁdence bound for CT

PUcan be expressed as C_PUT LB¼2^C T PU ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4z2 a 9nþ 2z2 a n ^C T PU2 2z4 a 9n2 q 2 z2 a=n (7) Table I tabulates the 95% lower confidence bound CT LB

PU for ^CPUT =1.0(0.1)2.0, n = 10(10)400. For the convenience of hypothesis

testing, we also provide the critical value. From the asymptotic distribution listed in Equation (6), the critical value to hypothesis testing H0: CPUT ⩽C versus Ha: CTPU> C is expressed as

c0¼ C þ za ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 9nþ C2 2n r : (8)

Table II performs the critical values for type I error a = 0.05 with C=1.0(0.1)2.0, n = 10(10)400. Next, an application example is presented.

4. A case study

We applied the methodology to a set of real data (n = 100) presented in Wu and Pearn1for measuring manufacturing capability of a process making couplers and wavelength division multiplexers (WDM). Two quality characteristic including the polarization-dependent

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loss and the insertion loss, which are critical inﬁber-optic transmission quality, are considered. Table III (cited from Wu and Pearn1_{) displays}

the manufacturing capabilities and the corresponding NCPPM for coupler and WDM process using ^C_PUT _{values and C}T PULB.

In the coupler case, if the quality requirement is CT

PU⩾; 1:3, some statistical inferences can be made. First, because the 95% lower

conﬁdence bound CT

PULB¼1.35588 > 1.3, we say that the process satisﬁes the requirement. Moreover, CTPULB¼1.35588 means that there

are no more than 23.7 NCPPM, or, from Table III, the critical value 1.460835 (n = 100, C = 1.3) is less than the observation value ^C_PUT _=1.5261.

The two results are agreed. In WDM case, ^C_PUT _{= 0.7352}_{< 1 is obviously inadequate and incapable for high-tech product manufacturing.}

T PU C =1.0 T PU C =1.33 T PU C =1.5 T PU C =1.67 T LB PU C T LB PU C T LB PU C T LB PU C Figure 1. Curves of CT

PULBwitha = 0.05, n = 10(20)90 (bottom to top in plot) and CPUT =1.0, 1.33, 1.5, 1.67

Table I. 95% lower conﬁdence bounds of CT

PUfor ^CTPU=1.0(0.1)2.0, n = 10(10)400 n ^CT PU 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 10 0.6920 0.7684 0.8443 0.9198 0.9950 1.0700 1.1447 1.2192 1.2936 1.3679 1.4420 20 0.7661 0.8477 0.9291 1.0101 1.0909 1.1716 1.2521 1.3324 1.4127 1.4929 1.5729 30 0.8024 0.8867 0.9708 1.0547 1.1384 1.2219 1.3053 1.3886 1.4718 1.5550 1.6380 40 0.8252 0.9113 0.9972 1.0828 1.1683 1.2537 1.3390 1.4241 1.5092 1.5943 1.6792 50 0.8414 0.9287 1.0158 1.1027 1.1895 1.2762 1.3628 1.4493 1.5358 1.6221 1.7085 60 0.8536 0.9419 1.0299 1.1178 1.2056 1.2933 1.3809 1.4684 1.5559 1.6433 1.7307 70 0.8633 0.9523 1.0411 1.1298 1.2184 1.3069 1.3953 1.4836 1.5719 1.6601 1.7483 80 0.8712 0.9608 1.0503 1.1396 1.2288 1.3180 1.4070 1.4960 1.5850 1.6739 1.7627 90 0.8778 0.9680 1.0580 1.1479 1.2376 1.3273 1.4169 1.5065 1.5960 1.6854 1.7749 100 0.8835 0.9741 1.0646 1.1549 1.2451 1.3353 1.4254 1.5154 1.6054 1.6953 1.7852 120 0.8928 0.9841 1.0753 1.1664 1.2574 1.3483 1.4392 1.5300 1.6208 1.7115 1.8022 150 0.9032 0.9954 1.0874 1.1794 1.2712 1.3630 1.4548 1.5465 1.6381 1.7297 1.8213 180 0.9111 1.0039 1.0965 1.1891 1.2816 1.3741 1.4665 1.5588 1.6511 1.7434 1.8357 200 0.9153 1.0084 1.1015 1.1944 1.2873 1.3801 1.4728 1.5655 1.6582 1.7509 1.8435 250 0.9237 1.0175 1.1112 1.2048 1.2984 1.3919 1.4854 1.5788 1.6722 1.7656 1.8589 300 0.9300 1.0243 1.1185 1.2126 1.3067 1.4008 1.4948 1.5887 1.6827 1.7766 1.8705 350 0.9349 1.0296 1.1242 1.2188 1.3133 1.4077 1.5021 1.5965 1.6909 1.7852 1.8795 400 0.9389 1.0339 1.1289 1.2237 1.3186 1.4134 1.5081 1.6028 1.6975 1.7922 1.8869

161

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References

1. Wu CW, Pearn WL. Measuring manufacturing capability for couplers and wavelength division multiplexers. The International Journal of Advanced Manufacturing Technology 2005; 25:533–541. DOI: 10.1007/s00170-003-1793-9

2. Pan JN, Lee CY. New Capability Indices for Evaluating the Performance of Multivariate Manufacturing Process. Quality and Reliability Engineering International 2010; 26:3–15. DOI: 10.1002/qre.1024

3. Ozkaya BY, Testik MC. On the Expected Parts Per Million Nonconforming Levels Obtained from Estimated Process Capability Indices. Quality and Reliability Engineering International 2010; 26:817–829. DOI: 10.1002/qre.1156

4. Lin CJ, Pearn WL. Process Selection for Higher Production Yield Based on Capability Index Spk. Quality and Reliability Engineering International 2010;

26:247–258. DOI: 10.1002/qre.1051

5. Itay N, Yisrael P, Edna S. Developing a Sampling Plan Based on Cpk-Unknown Variance. Quality and Reliability Engineering International 2011;

27:3–14. DOI: 10.1002/qre.1094

6. Yum BJ, Kim KW. A Bibliography of the Literature on Process Capability Indices : 2000–2009. Quality and Reliability Engineering International 2011; 27:251–268. DOI: 10.1002/qre.1115

7. Spiring F. Exploring Process Capability with Mathematica. Quality and Reliability Engineering International 2011; 27:369–387. DOI: 10.1002/qre.1112 8. Awad MI, Kovach JV. Multiresponse Optimization using Multivariate Process Capability Index. Quality and Reliability Engineering International 2011;

27:465–477. DOI: 10.1002/qre.1141

9. Pearn WL, Kang HY, Lee AHI, Liao MY. Photolithography Control in Wafer Fabrication Based on Process Capability Indices with Multiple Characteristics. IEEE Transactions on Semiconductor Manufacturing 2009; 22(3):351–356. DOI: 10.1109/TSM.2009.2024851

10. Hsu YC, Pearn WL, Chuang YF. Sample size determination for production yield estimation with multiple independent process characteristics. European Journal of Operational Research 2009; 196:968–978. DOI: 10.1016/j.ejor.2008.04.029

11. Yen CH, Pearn WL. Select better suppliers based on manufacturing precision for processes with multivariate data. International Journal of Production Research 2009; 47(11):2961–2974. DOI: 10.1080/00207540701796985

12. Pearn WL, Cheng YC. Measuring production yield for processes with multiple characteristics. International Journal of Production Research 2010; 48(15):4519–4536. DOI: 10.1080/00207540701796985

13. Goethals PL, Cho BR. The Development of a Target-Focused Process Capability Index with Multiple Characteristics. Quality and Reliability Engineering International 2011; 27:297–311. DOI: 10.1002/qre.1120

14. Pearn WL, Shiau JJH, Tai YT, Li MY. Capability Assessment for Processes with Multiple Characteristics: A Generalization of the Popular Index Cpk.

Quality and Reliability Engineering International 2011; 27(8):1119–1129. DOI: 10.1002/qre.1200

Table II. Critical values c0for C =1.0(0.1)2.0, n = 10(10)400

n C 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 10 1.4066 1.5401 1.6741 1.8086 1.9433 2.0783 2.2134 2.3488 2.4843 2.6200 2.7557 20 1.2875 1.4112 1.5353 1.6596 1.7841 1.9089 2.0338 2.1588 2.2839 2.4091 2.5344 30 1.2347 1.3541 1.4737 1.5936 1.7136 1.8338 1.9541 2.0746 2.1951 2.3156 2.4363 40 1.2033 1.3200 1.4370 1.5543 1.6716 1.7891 1.9067 2.0244 2.1421 2.2600 2.3778 50 1.1818 1.2968 1.4120 1.5274 1.6429 1.7586 1.8743 1.9901 2.1060 2.2219 2.3379 60 1.1660 1.2796 1.3935 1.5076 1.6218 1.7360 1.8504 1.9648 2.0793 2.1939 2.3085 70 1.1536 1.2663 1.3792 1.4922 1.6053 1.7185 1.8318 1.9452 2.0586 2.1721 2.2856 80 1.1437 1.2556 1.3676 1.4798 1.5920 1.7044 1.8169 1.9294 2.0419 2.1545 2.2672 90 1.1355 1.2467 1.3580 1.4695 1.5811 1.6927 1.8044 1.9162 2.0281 2.1400 2.2519 100 1.1285 1.2391 1.3499 1.4608 1.5718 1.6828 1.7940 1.9051 2.0164 2.1276 2.2389 120 1.1173 1.2270 1.3368 1.4468 1.5568 1.6669 1.7771 1.8873 1.9975 2.1078 2.2181 150 1.1049 1.2136 1.3224 1.4313 1.5402 1.6493 1.7584 1.8675 1.9767 2.0859 2.1951 180 1.0958 1.2037 1.3117 1.4198 1.5280 1.6363 1.7446 1.8529 1.9613 2.0697 2.1781 200 1.0909 1.1984 1.3060 1.4137 1.5214 1.6293 1.7371 1.8450 1.9530 2.0609 2.1689 250 1.0813 1.1880 1.2948 1.4017 1.5086 1.6156 1.7226 1.8297 1.9368 2.0440 2.1511 300 1.0742 1.1803 1.2865 1.3928 1.4991 1.6055 1.7120 1.8184 1.9249 2.0314 2.1379 350 1.0687 1.1744 1.2801 1.3859 1.4918 1.5977 1.7036 1.8096 1.9156 2.0217 2.1277

Table III. Calculations for process capability of the coupler and WDMS

Characteristic ^C_PUT _NCPPM ^C

PUT LB NCPPM

Coupler 1.5261 2.3439 1.3588 22.86916

WDM 0.7352 13706.01 0.6425 26958.67

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Authors' biographies

Wen-Lea Pearn received the Ph.D. degree in operations research from the University of Maryland, College Park .He is a Professor of Operations Research and Quality Assurance at the National Chiao-Tung University (NCTU), Hsinchu, Taiwan. He was with Bell Labora-tories, Murray Hill, NJ, as a Quality Research Scientist before joining the NCTU, and others. His current research interests include pro-cess capability, network optimization, and production management. Dr. Pearn’s publications have appeared in the Journal of the Royal Statistical Society, Series C, Journal of Quality Technology, European Journal of Operational Research, Journal of the Operational Research Society, Operations Research Letters, Omega, Networks, and the International Journal Productions Research.

Chia-Huang Wu received his MS degree in Applied Mathematics from National Chung-Hsing University. Currently, he is a PhD candidate at the Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC.

Meng-Chun Tsai received her MS degree in Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC.