Sample size determination for production yield estimation with multiple independent process characteristics

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Production, Manufacturing and Logistics

Sample size determination for production yield estimation with multiple

independent process characteristics

Ya-Chen Hsu

a,*

, W.L. Pearn

b

, Ya-Fei Chuang

b a

Department of Business Administration, Yuanpei University, Taiwan, ROC b

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

a r t i c l e

i n f o

Article history: Received 20 July 2007 Accepted 24 April 2008 Available online 4 May 2008 Keywords:

Bootstrap resampling Lower conﬁdence bound Multiple characteristics Process capability indices Estimation accuracy Sample size determination

a b s t r a c t

Capability measure for processes yield with single characteristic has been investigated extensively, but is still comparatively neglected for processes with multiple characteristics. Wu and Pearn [Wu, C.W., Pearn, W.L., 2005. Measuring manufacturing capability for couplers and wavelength division multiplexers (WDM). International Journal of Advanced Manufacturing Technology 25(5/6), 533–541] proposed a capability index for multiple characteristics called CTPU, which provides an exact measure on process yield

for multiple characteristics with each characteristic normally distributed. However, the exact sampling distribution of CT

PU(multiple characteristics) is analytically intractable. In this paper, we apply the

boot-strap method for calculating the lower conﬁdence bounds of the index CTPU, and determine the sample size

for a speciﬁed estimation accuracy. In order to obtain a desired estimation quality assurance, the sample size determination is essential as it provides the accuracy of the lower bound obtained from the bootstrap method. For convenience of applications, we tabulate the sample size required for various designated accuracy for the engineers/practitioners to use. A real-world example from manufacturing process with multiple characteristics is investigated to illustrate the applicability of the proposed approach.

Ó 2008 Published by Elsevier B.V.

1. Introduction

Process capability indices (PCIS) are effective tools for quality assurance and process improvement. Numerous capability indices quantifying process potential and process performance are essen-tial to any successful quality improvement activities and quality program implementation. Several basic capability indices have been widely used in manufacturing industry as follows:

Cp¼ USL LSL 6

r

; ð1Þ CPU¼USL

l

3

r

; ð2Þ CPL¼

l

LSL 3

r

; ð3Þ Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

2_{þ ð}

_l

_TÞ2 q ; ð4Þ Cpk¼ min USL

l

3

r

;

l

LSL 3

r

; ð5Þ

where USL and LSL are the upper and the lower speciﬁcation limits,

l

is the process mean,

r

is the process standard deviation, and T is the target value.

In order to calculate the estimator, data must be collected. A great degree of uncertainty may be introduced into the capability assessments due to sampling errors. As the sampling errors have been ignored, the approach, simply by the calculated values of the estimated indices and then making a conclusion on whether the given process is capable, is highly unreliable. A reliable ap-proach for estimating the true value of process capability is to determine the sample size for desired estimation accuracy. The sample size is directly related to the estimation accuracy and the cost of the data collection plan. The capability measurements for processes with single characteristic have been investigated exten-sively (seeKane, 1986; Pearn et al., 1992; Chen, 1998; Chen and Hsu, 2004; Cheng et al., 2006; Flaig, 2006; Vännman, 2006; Vänn-man and Albing, 2007). However, the lacks of these studies associ-ated with analyzing the quality and efﬁciency of a process, are, so far, limited by discussing one single quality speciﬁcation. In this paper, we consider the process capability with multiple character-istics to determine the sample size for desired estimation accuracy. For process with multiple characteristics, several approaches have been suggested (see e.g.Bothe, 1992; Chen et al., 2003a,b; Castagliola and Castellanos, 2005; Huang et al., 2005; Wu and Pearn, 2005). For example, Bothe (1992) considered a simple

0377-2217/$ - see front matter Ó 2008 Published by Elsevier B.V. doi:10.1016/j.ejor.2008.04.029

*Corresponding author. Tel.: +886 968027645. E-mail address:[email protected](Y.-C. Hsu).

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measurement by taking the minimum measure of each single char-acteristic. For example, considering a v-characteristics product with v-yield measures P1, P2, . . . , Pv, the overall process yield is

mea-sured as P = min{P1, P2, . . . , Pv}. Furthermore, Chen et al. (2003a)

provided the process capability index with multi-characteristics as

STpk¼ 1 3

U

1 Y v j¼1 ð2

U

ð3SpkjÞ 1Þ þ 1 " #, 2 ( ) ; ð6Þ

whereU() is the cumulative distribution of the standard normal distribution N(0, 1),U1_{is the inverse function of}_U_{(), S}

pkjdenotes

the Spkvalue of the jth characteristic for j = 1, 2, . . . , v, and v is the

number of characteristics. This index, which provides an exact mea-surement on the process yield, establishes the relationship between the manufacturing speciﬁcation and the actual process performance (Pearn and Cheng, 2007).Wu and Pearn (2005)discussed the cess with multi-characteristics for one-sided speciﬁcation and pro-posed a capability index as

CT PU¼ 1 3

U

1 Y v j¼1

U

ð3CPUjÞ ( ) ; ð7Þ

where CPUj denotes the CPU value of the jth characteristic for

j = 1, 2, . . . , v, and v is the number of characteristics. A one-to-one correspondence relationship between the index CT

PUand the overall

process yield P can be established as

P ¼Y m j¼1 Pj¼ Ym j¼1

U

ð3CPUjÞ ¼

U

ð3CTPUÞ: ð8Þ

Bootstrap approach seems to be a reasonable method for tackling the problem that the sampling distribution of CT

PU(multiple

charac-teristics) is analytically intractable. Since lower confidence bound estimates the minimum process capability conveying critical infor-mation regarding product quality,Wu and Pearn (2005)estimated the confidence bound by percentile bootstrap (PB) method. How-ever, there are four types of bootstrap methods to estimate confi-dence bound, including the standard bootstrap conficonfi-dence interval (SB), the percentile bootstrap confidence interval (PB), the biased-corrected percentile bootstrap confidence interval (BCPB), and the bootstrap-t (BT) method. And the engineers/practitioners would want to know which one is recommended. In this paper, we com-pare the performance of confidence bound for the one-sided index CTPU with multiple characteristics by using these four bootstrap

methods. The modiﬁed index CT

PU proposed by Wu and Pearn

(2005)are calculated. Furthermore, we ﬁnd that the BCPB method would be the recommended method to estimate conﬁdence bound in the general cases. We also provide the tables about the sample sizes required for various designated estimation accuracy for the engineers/practitioners to use in their factory applications. A real-world example from manufacturing process with multiple charac-teristics is investigated to illustrate the applicability of the proposed approach.

2. Capability measures for multiple characteristics

Capability measure for processes with single characteristic has been investigated extensively. For normally distributed processes with a one-sided speciﬁcation limit, USL or LSL, the process yield

Table 1 Various CT

PUvalues and the corresponding process yield CT PU Process yield 1.00 0.9986501020 1.25 0.9999115827 1.33 0.9999669634 1.45 0.9999931931 1.50 0.9999966023 1.60 0.9999992067 1.67 0.9999997278 2.00 0.9999999990 Table 2

Minimal requirement for each single characteristic of various capability levels for multiple characteristics c0 _c L 1.000 1.33 1 1.000 1.330 2 1.068 1.383 3 1.107 1.414 4 1.133 1.436 5 1.153 1.452 Table 3

The total rank of the four bootstrap methods as CT

PU¼ 1; 1:33 and v = 2(1)5 n CT PU¼ 1 CTPU¼ 1:33 SB PB BCPB PT SB PB BCPB PT v = 2 30 3 2 1.006 3.994 2.996 1.996 1.008 4.000 40 2.996 2.004 1.004 3.996 2.982 2.008 1.034 3.974 50 2.992 2.002 1.024 3.982 2.99 2.012 1.030 3.968 60 2.982 2.014 1.042 3.962 2.984 2.010 1.056 3.950 70 2.994 2.004 1.018 3.984 2.984 2.020 1.030 3.966 80 2.988 2.012 1.026 3.974 2.972 2.028 1.072 3.928 90 2.994 2.016 1.022 3.968 2.970 2.026 1.084 3.920 100 2.980 2.020 1.026 3.974 2.992 2.002 1.072 3.934 125 3.000 2.010 1.018 3.972 2.968 2.040 1.104 3.888 150 2.988 2.012 1.040 3.960 2.974 2.056 1.094 3.876 200 3.004 2.018 1.036 3.942 2.972 2.018 1.148 3.858 v = 3 30 2.966 2.028 1.1 3.906 2.942 2.048 1.112 3.896 40 2.95 2.04 1.106 3.904 2.954 2.046 1.132 3.868 50 2.97 2.034 1.1 3.896 2.946 2.044 1.15 3.86 60 2.942 2.07 1.124 3.864 2.922 2.094 1.216 3.768 70 2.954 2.052 1.12 3.874 2.916 2.112 1.29 3.678 80 2.944 2.062 1.172 3.818 2.904 2.108 1.332 3.656 90 2.94 2.078 1.166 3.816 2.836 2.168 1.428 3.568 100 2.934 2.072 1.162 3.828 2.894 2.136 1.352 3.61 125 2.964 2.06 1.124 3.852 2.832 2.172 1.59 3.406 150 2.97 2.044 1.136 3.848 2.804 2.176 1.62 3.398 200 2.918 2.114 1.2 3.768 2.784 2.23 1.636 3.344 v = 4 30 2.97 2.052 1.116 3.856 2.94 2.076 1.17 3.814 40 2.912 2.084 1.208 3.796 2.882 2.11 1.3 3.708 50 2.922 2.094 1.224 3.76 2.826 2.186 1.508 3.478 60 2.91 2.096 1.222 3.772 2.798 2.188 1.53 3.478 70 2.882 2.126 1.254 3.736 2.822 2.186 1.664 3.328 80 2.916 2.108 1.26 3.716 2.79 2.274 1.666 3.268 90 2.892 2.12 1.318 3.668 2.742 2.272 1.784 3.196 100 2.886 2.124 1.28 3.706 2.73 2.308 1.85 3.11 125 2.922 2.126 1.308 3.644 2.634 2.4 2.054 2.912 150 2.92 2.1 1.328 3.648 2.684 2.412 2.034 2.868 200 2.88 2.152 1.394 3.57 2.632 2.434 2.322 2.606 v = 5 30 2.938 2.084 1.222 3.756 2.86 2.15 1.398 3.59 40 2.878 2.118 1.314 3.688 2.786 2.216 1.652 3.344 50 2.892 2.148 1.326 3.63 2.792 2.24 1.648 3.316 60 2.85 2.16 1.408 3.582 2.758 2.254 1.828 3.156 70 2.846 2.158 1.45 3.542 2.686 2.336 2.026 2.942 80 2.902 2.18 1.382 3.528 2.654 2.4 2.11 2.83 90 2.876 2.172 1.5 3.452 2.572 2.432 2.288 2.702 100 2.83 2.192 1.56 3.406 2.582 2.478 2.384 2.55 125 2.904 2.156 1.514 3.42 2.56 2.482 2.6 2.354 150 2.856 2.228 1.572 3.342 2.466 2.524 2.648 2.346 200 2.834 2.244 1.578 3.34 2.426 2.614 2.878 2.074

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is listed in the following, where Z follows the standard normal dis-tribution N(0, 1) PðX < USLÞ ¼ P X

l

r

< USL

l

r

¼ PðZ < 3CPUÞ ¼

U

ð3CPUÞ; ð9Þ PðX > LSLÞ ¼ P X

l

r

> LSL

l

r

¼ PðZ < 3CPLÞ ¼

U

ð3CPLÞ: ð10Þ

For easier presentation, we denote CIas either CPUor CPL. Thus,

pro-cess capability index CIprovides an exact measure of the potential

process yield for processes with a one-sided manufacturing speciﬁ-cation. The corresponding process yield for a well controlled nor-mally distributed process is easily calculated asU(3CI).

Considering processes with v-characteristics (assuming charac-teristics are mutually independent) and v yield measures

P1, P2, . . . , PV,Wu and Pearn (2005)suggested that the overall

pro-cess yield should be calculated as P = P1 P2 . . . PV which is

signiﬁcantly less than the calculated one. From the deﬁnition of one-sided yield index in(9), the process yield index with single characteristic can be rewritten as

CPU¼ 1 3

U

1

_U

_ðUSL

l

r

Þ ; ð11Þ

whereU() is the cumulative distribution of the standard normal distribution N(0, 1), and U1 _{is the inverse function of} _U_{(). For}

the process with multiple quality characteristics, a simple measure by taking the minimum of the measure of each single characteristic

30 40 50 60 70 80 90 100 125 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 sample size SB PB BCPB PT R

Fig. 1a. The total rank of the four bootstrap methods as CT

PU¼ 1, v = 2. 30 40 50 60 70 80 90 100 125 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 sample size SB PB BCPB PT R

Fig. 1d. The total rank of the four bootstrap methods as CT

Fig. 2a. The total rank of the four bootstrap methods as CT

PU¼ 1:33, v = 2. 30 40 50 60 70 80 90 100 125 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 sample size SB PB BCPB PT R

Fig. 1b. The total rank of the four bootstrap methods as CT

Fig. 2b. The total rank of the four bootstrap methods as CT

PU¼ 1:33, v = 3. 30 40 50 60 70 80 90 100 125 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 sample size SB PB BCPB PT R

Fig. 1c. The total rank of the four bootstrap methods as CT

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has been considered.Wu and Pearn (2005)proposed the following overall capability index is referred to as:

CTPU¼ 1 3

U

1 Y v j¼1

U

ð3CPUjÞ ( ) ; ð12Þ

where CPUj denotes the CPU value of the jth characteristic for

j = 1, 2, . . . , v, and v is the number of characteristics. The index, CTPU,

can be a generalization of the single characteristic yield index. Let CT_PU¼ c, we have Yv j¼1

U

ð3CPUjÞ ( ) ¼

U

ð3cÞ: ð13Þ

In fact,Wu and Pearn (2005) showed that the one-to-one corre-spondence relationship between the index CT

PUand the overall

pro-cess yields P can be established as follows:

P ¼Y v j¼1 Pj¼ Yv j¼1

U

ð3CPUjÞ ¼

U

ð3CTPUÞ: ð14Þ

Hence, the new index CTPUprovides an exact measure on the overall

process yield when the characteristics are mutually independent. For example, if CT

PU¼ 1:00, the entire process yield would be exactly

99.865%, and each single characteristic yield is no less than (0.9986501)1/5= 0.9997299 (equivalent to 270 NCPPM).Table 1 dis-plays various commonly used capability requirement and the corre-sponding overall process yield.

Wu and Pearn (2005)also showed that for process with v char-acteristics, if the requirement for the overall process capability is CT

PUPc0, a sufﬁcient condition (which is minimal) for the

require-ment to each single characteristic can be obtained by the following. Let c0_{be the minimum C}

PUjrequired for each single characteristic,

then 1 3

U

1 Y v j¼1

U

ð3CPUjÞ ( ) P1 3

U

1 Y v j¼1

U

ð3c0_Þ ( ) Pc0: ð15Þ

We can obtain the lower bound of each single characteristic to be

cL¼ 1 3

U

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

_U

ð3c0Þ v p : ð16Þ

Table 2displays the minimum cLof CPUjfor the required overall

pro-cess capability CTPUare 1.00 and 1.33 for

m

= 1(1)5 characteristics. For

example, if the overall capability requirement CT

PUP1.00 would be

satisﬁed, it means each single characteristic yield is no less than (0.9986501)1/5_{= 0.9997299 (equivalent to 270 NCPPM), and the}

capability for all the ﬁve characteristics is the following, for j = 1, 2, . . . , 5. 30 40 50 60 70 80 90 100 125 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 sample size SB PB BCPB PT R

Fig. 2c. The total rank of the four bootstrap methods as CT

PU¼ 1:33, v = 4. 30 50 60 70 80 90 100 125 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 sample size SB PB BCPB PT R

Fig. 2d. The total rank of the four bootstrap methods as CT

PU¼ 1:33, v = 5.

Table 4

Sample size n required for RcPRPU, with quality characteristics v = 3, RPU= 0.75(0.01)0.95,c= 0.9, 0.95, 0.975, 0.99, and CTPU¼ 1 RPU c= 0.90 c= 0.95 c= 0.975 c= 0.99 n Rc n Rc n Rc n Rc 0.75 – – – – – – 16 0.7518 0.76 – – – – 6 – 21 0.7609 0.77 – – – – 7 0.7776 24 0.7731 0.78 – – – – 14 0.7832 28 0.7807 0.79 – – – – 18 0.7904 31 0.7901 0.80 – – 6 – 22 0.8005 36 0.8012 0.81 – – 12 0.8119 26 0.8107 40 0.8103 0.82 – – 17 0.8222 32 0.8226 49 0.8201 0.83 – – 23 0.8316 38 0.8304 56 0.8305 0.84 – – 28 0.8414 44 0.8403 65 0.8414 0.85 6 – 35 0.8536 52 0.8502 75 0.8502 0.86 18 0.8608 41 0.8600 63 0.8613 88 0.8609 0.87 26 0.8708 51 0.8710 73 0.8700 105 0.8710 0.88 33 0.8805 60 0.8802 88 0.8800 124 0.8812 0.89 44 0.8909 76 0.8912 105 0.8908 146 0.8610 0.90 54 0.9005 93 0.9004 128 0.9000 176 0.9005 0.91 71 0.9107 115 0.9102 158 0.9103 213 0.9101 0.92 92 0.9205 146 0.9201 197 0.9204 268 0.9202 0.93 121 0.9306 188 0.9303 253 0.9300 339 0.9302 0.94 164 0.9400 251 0.9402 337 0.9400 451 0.9400 0.95 231 0.9500 350 0.9502 473 0.9505 634 0.9500 Table 5

Sample size n required for RcPRPU, with quality characteristics v = 3, RPU= 0.75(0.01)0.95,c= 0.9, 0.95, 0.975, 0.99, and CTPU¼ 1:33 RPU c= 0.90 c= 0.95 c= 0.975 c= 0.99 n Rc n Rc n Rc n Rc 0.75 – – – – 6 – 19 0.7533 0.76 – – – – 8 0.7608 22 0.7600 0.77 – – – – 14 0.7721 24 0.7719 0.78 – – – – 17 0.7820 28 0.7813 0.79 – – – – 21 0.7923 33 0.7914 0.80 – – 6 0.8074 24 0.8008 38 0.8019 0.81 – – 16 0.8119 28 0.8109 41 0.8101 0.82 – – 19 0.8200 33 0.8209 50 0.8219 0.83 – – 24 0.8310 40 0.8334 57 0.8320 0.84 6 – 31 0.8421 44 0.8406 65 0.8400 0.85 12 0.8518 35 0.8519 54 0.8506 77 0.8524 0.86 21 0.8615 42 0.8613 62 0.8602 88 0.8603 0.87 26 0.8700 51 0.8705 73 0.8704 102 0.8705 0.88 34 0.8807 62 0.8817 89 0.8814 122 0.8814 0.89 43 0.8910 74 0.8902 104 0.8904 145 0.8906 0.90 55 0.9010 91 0.9009 129 0.9017 174 0.9010 0.91 72 0.9118 113 0.9101 157 0.9109 213 0.9101 0.92 89 0.9202 144 0.9211 196 0.9201 269 0.9204 0.93 118 0.9303 184 0.9301 254 0.9301 348 0.9301 0.94 159 0.9400 249 0.9400 340 0.9400 501 0.9421 0.95 226 0.9501 353 0.9501 481 0.9500 658 0.9500

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CPUj¼

1 3

U

1p5ffiffiffiffiffiffiffiffiffiffiffi

_U

_ð3Þ

¼ 1:153 for j ¼ 1; 2; . . . ; 5: ð17Þ

3. Bootstrap methods for calculating the lower bounds of CT_PU 3.1. Lower conﬁdence bounds on CT

PU

For each single characteristic, the CPUj values can be estimated

by their natural estimators bCPUj¼ ðUSLj xjÞ=sj, j = 1, 2, . . . , v, where

xjand sjare the sample mean and the sample standard deviation of

the jth characteristic, respectively. Thus, the estimator of bCT PU are deﬁned as b CT PU¼ 1 3

U

1 Y m j¼1

U

ð3bCPUjÞ ( ) : ð18Þ

In order to calculate the estimator of CT

PU, however, sample data

must be collected. Therefore, due to sampling errors, a great degree of uncertainty may be introduced into capability assessments. It is highly unreliable simply by the calculated values of the estimated indices and then making a conclusion on whether the given process is capable. Since the sampling errors have been ignored, a reliable

approach for estimating the true value of process index is to con-struct the lower conﬁdence bound.

Determination of the lower conﬁdence bound on the actual pro-cess capability is essential for quality assurance. The lower conﬁ-dence bound can not only be essential to production yield assurance, but also be used in capability testing for decision mak-ing. Since the sample size provides the accuracy of the lower bound, for the given desired estimation accuracy RPU(RPU¼ CTðLBÞPU

=bCT

PU, where C TðLBÞ

PU is the lower conﬁdence bound on C

T

PUÞ and the

conﬁdence level

c

(ensures that the risk of making incorrect deci-sions will be no larger than the preset Type I error 1

c

), the approximate sample size must be obtained. Before estimating the sample size, it is necessary to determine a desired lower conﬁ-dence bound for CTPU, depending on the ratio of RPU¼ CTðLBÞPU = bCTPU.

Hence, we need to compute the lower conﬁdence bound to deter-mine sample sizes required for speciﬁed estimation accuracy of the CT_PU.

While the sampling distribution of the estimator bCT

PUfor

multi-ple sammulti-ples is unknown, we use the nonparametric bootstrap method and the following to estimate the lower conﬁdence bound CTðLBÞPU . Efron (1981) introduced a nonparametric, computational

intensive but effective estimation method, called the ‘‘Bootstrap”, which is a data-based simulation technique for statistical infer-ence. The merit of the nonparametric bootstrap approach is that it does not rely on any assumptions regarding the underlying dis-tribution. The bootstrap sampling is equivalent to sampling (with replacement) from the empirical probability distribution function. The essence of bootstrapping is that, without any knowledge about a population, the distribution found in a random sample of size n from the population is the best guide to the distribution in the pop-ulation. By resampling observations from the observed data, the population that consists of the n observed sample values is used to model the unknown real population.

In the bootstrap, B new samples, each of the same size as the ob-served data n, are drawn with replacement from the population.

Efron and Tibshirani (1986)developed four types of bootstrap con-fidence interval, including the standard bootstrap concon-fidence inter-val (SB), the percentile bootstrap confidence interinter-val (PB), the biased-corrected percentile bootstrap confidence interval (BCPB),

and the bootstrap-t (BT) method. Franklin and Wasserman

(1992)investigated the lower conﬁdence bounds for the capability indices, Cp, Cpkand Cpmusing these bootstrap methods. Some

sim-ulations results indicate that for normal processes the bootstrap confidence limits perform equally well (seeChou et al., 1990 and Bissell, 1990). In the following, we give an overview of four Boot-strap confidence intervals. These are employed to determine the lower confidence bounds of the index.

3.2. Bootstrap methods 3.2.1. Standard bootstrap (SB)

From the B bootstrap estimator bCT

PU, the sample average and the

sample standard deviation are calculated as follows:

bCT PU¼ 1 B XB i¼1 b CT PUðiÞ; ð19Þ SCT PU¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 B 1 XB I¼1 ½bCT PUðiÞ bCTPU 2 v u u t ; ð20Þ overlay 1'st deposited layer 2'nd deposited layer

Fig. 3. Deposited layers on TFT-LCD.

critical

dimension

panel

mask

Fig. 4. Exposure process on panel window.

Table 6

Speciﬁcations for thin ﬁlm transistor liquid crystal display

Parameter Speciﬁcations

Overlay 6_0.1lm

Critical dimension 60.3lm

Uniformality 60.03

Table 7

Calculations for process capability of the overlay, critical dimension and uniformality

Characteristics USL x s CbPUj LC

Overlay 0.1 0.0795 0.0065 1.0499 0.9394

Critical dimension 0.1 0.2693 0.0083 1.2298 1.1016 Uniformality 0.03 0.0267 0.00097 1.1404 1.0215

Table 8

Calculations for overall yield index Characteristic bCT

PU NCPPM CTðLBÞPU NCPPM

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where bCT

PUðiÞ is the ith bootstrap estimate. The quantity S CT

PUis

actu-ally an estimator of the standard deviation of bCT

PU, and if C _ T PU is

approximately normal distribution, the (1 2

a

) 100% SB conﬁdence interval can be obtained as

½bCT PU ZaS

CT

PU; ð21Þ

where Z

a

is the upper

a

quantile of the standard normal

distribution.

3.2.2. The percentile bootstrap (PB) From the ordered collection of bCT

PUðiÞ, select the

a

percent and

the (1

a

) percent points as the end points, and the PB conﬁdence interval is

½bCT

PUð

a

BÞ: ð22Þ

3.2.3. Biased-corrected percentile bootstrap (BCPB)

The bootstrap distribution may be biased while the percentile conﬁdence interval is possible due to sampling errors. In other words, that bootstrap distributions obtained using only a sample

of the complete bootstrap distribution may be shifted higher or lower than expected. Thus, a three steps procedure has been devel-oped to correct for this potential biasEfron (1982). First, using the ordered distribution of bCT

PU, calculate the probability of

P0¼ P½bCTPU6^cTPU: ð23Þ

Second, calculate

Z0¼

U

1ðP0Þ; ð24Þ

PL¼

U

ð2Z0 ZaÞ; ð25Þ

PU¼

U

ð2Z0þ ZaÞ; ð26Þ

whereU() is the standard normal cumulative distribution function. Finally, the BCPB conﬁdence is obtained as

½bCT

PUðPLBÞ: ð27Þ

3.2.4. Bootstrap-t (BT)

While the distribution of the statistic is skewed, the percen-tile bootstrap conﬁdence interval is probably lower. Thus, the

Table 9

The 150 sample observations for three quality characteristics Overlay (lm): USL = 0.1lm 0.0779 0.0697 0.0764 0.0763 0.0834 0.0860 0.0778 0.0849 0.0846 0.0649 0.0853 0.0801 0.0711 0.0847 0.0817 0.0747 0.0886 0.0777 0.0889 0.0716 0.0802 0.0776 0.0800 0.0811 0.0873 0.0804 0.0810 0.0729 0.0782 0.0794 0.0711 0.0712 0.0724 0.0839 0.0831 0.0846 0.0803 0.0851 0.0701 0.0741 0.0706 0.0826 0.0665 0.0843 0.0862 0.0824 0.0810 0.0804 0.0838 0.0693 0.0757 0.0842 0.0765 0.0742 0.0838 0.0832 0.0837 0.0745 0.0820 0.0911 0.0786 0.0751 0.0738 0.0801 0.0853 0.0667 0.0778 0.0888 0.0890 0.0638 0.0796 0.0859 0.0718 0.0799 0.0637 0.0789 0.0878 0.0926 0.0674 0.0745 0.0859 0.0913 0.0863 0.0695 0.0878 0.0753 0.0790 0.0798 0.0801 0.0736 0.0746 0.0885 0.0788 0.0746 0.0862 0.0787 0.0753 0.0793 0.0776 0.0945 0.0833 0.0709 0.0804 0.0780 0.0888 0.0842 0.0794 0.0793 0.0771 0.0835 0.0691 0.0806 0.0805 0.0735 0.0843 0.0837 0.0727 0.0834 0.0752 0.0877 0.0771 0.0850 0.0755 0.0826 0.0776 0.0833 0.0669 0.0740 0.0839 0.0743 0.0781 0.0754 0.0840 0.0840 0.0962 0.0780 0.0801 0.0742 0.0781 0.0908 0.0911 0.0849 0.0764 0.0932 0.0783 0.0732 0.0722 0.0775 0.0787 0.0715

Critical dimension (lm): USL = 0.1lm

0.2559 0.2627 0.2717 0.2656 0.2756 0.2747 0.2645 0.2671 0.2588 0.2703 0.2689 0.2633 0.2694 0.2573 0.2691 0.2776 0.2550 0.2632 0.2624 0.2605 0.2783 0.2623 0.2691 0.2571 0.2616 0.2759 0.2670 0.2688 0.2598 0.2620 0.2788 0.2507 0.2661 0.2726 0.2807 0.2735 0.2673 0.2478 0.2831 0.2653 0.2691 0.2792 0.2718 0.2791 0.2770 0.2581 0.2731 0.2660 0.2612 0.2718 0.2657 0.2711 0.2579 0.2649 0.2760 0.2707 0.2769 0.2605 0.2648 0.2723 0.2657 0.2650 0.2764 0.2827 0.2734 0.2676 0.2757 0.2662 0.2758 0.2753 0.2514 0.2654 0.2754 0.2842 0.2524 0.2734 0.2687 0.2743 0.2631 0.2719 0.2726 0.2828 0.2750 0.2721 0.2633 0.2608 0.2877 0.2628 0.2894 0.2638 0.2700 0.2654 0.2819 0.2728 0.2713 0.2670 0.2580 0.2730 0.2652 0.2794 0.2656 0.2850 0.2735 0.2774 0.2730 0.2757 0.2640 0.2707 0.2564 0.2634 0.2638 0.2727 0.2681 0.2647 0.2720 0.2687 0.2627 0.2828 0.2838 0.2700 0.2638 0.2640 0.2797 0.2708 0.2704 0.2475 0.2713 0.2710 0.2870 0.2610 0.2651 0.2729 0.2698 0.2702 0.2694 0.2586 0.2619 0.2790 0.2723 0.2833 0.2709 0.2592 0.2740 0.2598 0.2557 0.2790 0.2714 0.2874 0.2656 0.2789 Uniformality: USL = 0.03 0.0272 0.0264 0.0255 0.0267 0.0248 0.0272 0.0270 0.0267 0.0257 0.0265 0.0264 0.0265 0.0252 0.0278 0.0263 0.0272 0.0252 0.0264 0.0264 0.0247 0.0271 0.0276 0.0268 0.0293 0.0283 0.0265 0.0269 0.0275 0.0277 0.0257 0.0255 0.0269 0.0259 0.0271 0.0273 0.0256 0.0278 0.0283 0.0267 0.0277 0.0254 0.0265 0.0280 0.0283 0.0262 0.0269 0.0267 0.0266 0.0263 0.0261 0.0269 0.0270 0.0262 0.0279 0.0252 0.0255 0.0277 0.0254 0.0262 0.0279 0.0265 0.0271 0.0286 0.0252 0.0261 0.0266 0.0278 0.0270 0.0255 0.0274 0.0244 0.0272 0.0279 0.0259 0.0266 0.0265 0.0256 0.0274 0.0266 0.0282 0.0268 0.0260 0.0256 0.0253 0.0268 0.0287 0.0270 0.0294 0.0265 0.0258 0.0275 0.0265 0.0282 0.0270 0.0266 0.0267 0.0254 0.0270 0.0277 0.0257 0.0278 0.0255 0.0274 0.0260 0.0273 0.0269 0.0256 0.0293 0.0256 0.0274 0.0249 0.0265 0.0269 0.0269 0.0268 0.0267 0.0262 0.0266 0.0271 0.0269 0.0252 0.0257 0.0286 0.0267 0.0265 0.0270 0.0270 0.0261 0.0264 0.0263 0.0280 0.0271 0.0267 0.0274 0.0266 0.0277 0.0252 0.0268 0.0267 0.0257 0.0264 0.0273 0.0244 0.0263 0.0264 0.0258 0.0268 0.0260 0.0276 0.0256

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bootstrap-t is developed and that the generated distribution will mi-mic the distribution of T. First, approximate the distribution of a sta-tistic of T ¼ ðbCT

PU C T PUÞ=SCT

PUby using bootstrap. By taking bootstrap

samples from the original data values the bootstrap approximation in this case can be obtained, calculate the corresponding estimates b

CT

PUðiÞ and their standard error, and then ﬁnd the T-values

T ¼ ðbCT

PU bCTPUÞ=S CT

PU. The (1 2

a

) 100% BT conﬁdence interval can

be obtained as ½bCT PU t aS CT PU; ð28Þ where t

aand t1aare the upper

a

and 1

a

quantile of the

boot-strap T-distribution respectively.

4. Performance comparisons of bootstrap methods

We use bootstrap methods to calculate the lower conﬁdence

bound of CT

PU, used to demonstrate the estimation accuracy

RPU¼ CTðLBÞPU =bCTPU. Then we can determine the required sample sizes

for speciﬁed estimation accuracy on CT

PU. We also rank the four

bootstrap methods according to RPU for ascertaining their

performance.

Considering the data generated by MATLAB program with mul-tiple independent characteristics from normal distribution, the assumption of normality for each single characteristic is required for the process yield calculation. But the bootstrap approach, which does not require any assumption, is a general nonparametric meth-od. Let the capability CPUj of each single characteristic satisfy the

minimal value (seeTable 2) required for overall process capability CT

PU. For example, if a process has a capability requirement

CT_PUP1:00 with

m

= 5, i.e., the capability for all the ﬁve character-istics is the following CPUjP1.153 for j = 1, 2, . . . , 5. We repeated

500 simulations and then obtained each rank of the four bootstrap methods. The calculation of the total weighted average rank R of each bootstrap method is as follows:

R ¼ 1 500 X4 i¼1 Ni i; ð29Þ Table 10

The average rank of the four bootstrap methods as CPU= 1,1.33 and v = 2

n CT PU¼ 1 CTPU¼ 1:33 N1 N2 N3 N4 R N1 N2 N3 N4 R 30 SB 0 1 498 1 3 1 0 499 0 2.996 PB 2 497 0 1 2 2 498 0 0 1.996 BCPB 498 1 1 0 1.006 497 2 1 0 1.008 PT 0 1 1 498 3.994 0 0 0 500 4 40 SB 1 0 499 0 2.996 2 6 491 1 2.982 PB 0 499 0 1 2.004 3 491 5 1 2.008 BCPB 499 0 1 0 1.004 493 1 2 4 1.034 PT 0 1 0 499 3.996 3 1 2 494 3.974 50 SB 1 3 495 1 2.992 2 3 493 2 2.99 PB 2 495 3 0 2.002 2 493 2 3 2.012 BCPB 495 1 1 3 1.024 492 4 1 3 1.03 PT 2 1 1 496 3.982 4 0 4 492 3.968 60 SB 1 7 492 0 2.982 3 6 487 4 2.984 PB 1 492 6 1 2.014 4 488 7 1 2.01 BCPB 492 1 1 6 1.042 487 5 1 7 1.056 PT 6 0 1 493 3.962 6 1 5 488 3.95 70 SB 1 2 496 1 2.994 1 8 489 2 2.984 PB 0 498 2 0 2.004 2 488 8 2 2.02 BCPB 497 0 0 3 1.018 493 2 2 3 1.03 PT 2 0 2 496 3.984 4 2 1 493 3.966 80 SB 0 7 492 1 2.988 4 9 484 3 2.972 PB 1 492 7 0 2.012 3 482 13 2 2.028 BCPB 495 1 0 4 1.026 485 4 1 10 1.072 PT 4 0 1 495 3.974 8 5 2 485 3.928 90 SB 1 4 492 3 2.994 5 11 478 6 2.97 PB 0 494 4 2 2.016 4 482 11 3 2.026 BCPB 495 1 2 2 1.022 483 3 3 11 1.084 PT 4 1 2 493 3.968 8 4 8 480 3.92 100 SB 1 10 487 2 2.98 3 6 483 8 2.992 PB 2 488 8 2 2.02 8 484 7 1 2.002 BCPB 494 2 1 3 1.026 484 5 2 9 1.072 PT 3 0 4 493 3.974 5 5 8 482 3.934 125 SB 1 2 493 4 3 5 14 473 8 2.968 PB 1 494 4 1 2.01 3 478 15 4 2.04 BCPB 495 3 0 2 1.018 480 2 4 14 1.104 PT 3 1 3 493 3.972 12 6 8 474 3.888 150 SB 1 8 487 4 2.988 4 22 457 17 2.974 PB 2 490 8 0 2.012 6 465 24 5 2.056 BCPB 492 1 2 5 1.04 480 5 3 12 1.094 PT 5 1 3 491 3.96 10 8 16 466 3.876 200 SB 0 8 482 10 3.004 7 15 463 15 2.972 PB 3 487 8 2 2.018 12 469 17 2 2.018 BCPB 492 3 0 5 1.036 466 11 6 17 1.148 PT 5 2 10 483 3.942 16 5 13 466 3.858

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where Niis the total number of rank i (i = 1, 2, . . . , 5) during the

sim-ulations. For example, if a process has a capability requirement CT

PUP1:00 with

m

= 2 and sample size n = 30, we can obtained the

weighted average rank R of the SB method to be

R ¼ ð0 1 þ 1 2 þ 498 3 þ 1 4Þ=500 ¼ 3: ð30Þ

InTable 3, the weighted average rank R of four bootstrap method is illustrated with various sample size n = 30(10)100, 125, 150, 200, and v = 2(1)5 as CTPU¼ 1 and 1.33. For example, if the sample size

n is 60, v = 2 and the CT

PU is 1.33, the weighted average rank R of

the four bootstrap methods are 2.984, 2.010, 1.056, 3.950, respec-tively (seeTable 3).

FromTable 3, the BCPB (biased-corrected percentile bootstrap)

method performs better than other ones when CT

PU¼ 1. For the

ﬁxed values of CTPUand n, the weighted average rank of each

meth-od gets closer to each other as v increases. This fact indicates that the performances of four methods are not much different when v is large.

It is shown inFigs. 1a–1d, 2a–2dthat the BCPB method is dis-tinctly better when sample size n < 125. However, as the sample

size is greater than 125, the performances of four methods are not much different. Furthermore, as CT_PU¼ 1:33 and the quality characteristic v = 5, the weighted average rank R of BCPB method is larger than others. This indicates that BCPB method performs worse than the other ones when n > 125, v = 5 and CT

PU¼ 1:33

(seeFig. 2d). Actually, the estimation of the four methods is similar in this situation. As a result of this fact, we recommend the BCPB method is the best one to calculate bCT

PU when the sample size

n < 125. In the following section, we use BCPB method to evaluate the estimator bCT

PU.

5. Sample size required for designated estimation accuracy The sample size determination is important, as it directly re-lates to the cost of data collection plan. We develop a MATLAB pro-gram to compute the required sample size n. The BCPB method is the recommended one to calculate the estimator bCT

PU. In Section

5, we use the simulation data which is randomly generated from normal distribution to determine the required sample size n.

Table 11

The average rank of four bootstrap methods as CPU= 1,1.33 and v = 3

n CT PU¼ 1 CTPU¼ 1:33 N1 N2 N3 N4 R N1 N2 N3 N4 R 30 SB 5 11 480 4 2.966 5 19 476 0 2.942 PB 3 482 13 2 2.028 3 473 21 3 2.048 BCPB 480 4 2 14 1.1 479 3 1 17 1.112 PT 12 3 5 480 3.906 14 4 2 480 3.896 40 SB 7 12 480 1 2.95 4 20 471 5 2.954 PB 3 477 17 3 2.04 5 470 22 3 2.046 BCPB 479 4 2 15 1.106 473 6 3 18 1.132 PT 11 7 1 481 3.904 18 4 4 474 3.868 50 SB 2 15 479 4 2.97 2 25 471 2 2.946 PB 3 478 18 1 2.034 5 469 25 1 2.044 BCPB 481 3 1 15 1.1 472 4 1 23 1.15 PT 14 4 2 480 3.896 21 2 3 474 3.86 60 SB 9 16 470 5 2.942 8 34 447 11 2.922 PB 3 468 20 9 2.07 6 451 33 10 2.094 BCPB 473 6 7 14 1.124 456 9 6 29 1.216 PT 15 10 3 472 3.864 30 6 14 450 3.768 70 SB 5 19 470 6 2.954 5 47 433 15 2.916 PB 2 473 22 3 2.052 7 439 45 9 2.112 BCPB 475 6 3 16 1.12 444 9 5 42 1.29 PT 18 2 5 475 3.874 45 5 16 434 3.678 80 SB 8 24 456 12 2.944 12 42 428 18 2.904 PB 7 462 24 7 2.062 9 438 43 10 2.108 BCPB 463 8 9 20 1.172 436 8 10 46 1.332 PT 23 5 12 460 3.818 43 12 19 426 3.656 90 SB 8 24 458 10 2.94 15 69 399 17 2.836 PB 3 465 22 10 2.078 12 407 66 15 2.168 BCPB 466 6 7 21 1.166 415 9 23 53 1.428 PT 23 5 13 459 3.816 58 15 12 415 3.568 100 SB 9 23 460 8 2.934 15 49 410 26 2.894 PB 4 463 26 7 2.072 12 423 50 15 2.136 BCPB 465 7 10 18 1.162 428 13 14 45 1.352 PT 22 7 6 465 3.828 46 16 25 413 3.61 125 SB 6 18 464 12 2.964 19 79 369 33 2.832 PB 5 467 21 7 2.06 18 386 88 8 2.172 BCPB 471 11 3 15 1.124 386 19 9 86 1.59 PT 18 4 12 466 3.852 77 16 34 373 3.406 150 SB 3 22 462 13 2.97 25 85 353 37 2.804 PB 9 467 17 7 2.044 25 377 83 15 2.176 BCPB 470 7 8 15 1.136 378 18 20 84 1.62 PT 19 3 13 465 3.848 72 20 45 363 3.398 200 SB 15 42 412 31 2.918 26 97 336 41 2.784 PB 8 436 47 9 2.114 30 350 95 25 2.23 BCPB 456 10 12 22 1.2 364 34 22 80 1.636 PT 21 12 29 438 3.768 82 18 46 354 3.344

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Based on the procedure above, a Matlab algorithm for calculat-ing the required sample size is developed as follows:

Algorithm for the required sample size

Step 1. Input the value of characteristics v, the designated esti-mation accuracy RPU and the initial sample size values

of Lo and Hi. Compute RPU(Hi) and RPU(Lo) to ensure that

RPU(Hi) > RPUand RPU(Lo) < RPU.

Step 2. Let n = (Lo + Hi)/2. Compute RPU(n). If —Hi Lo— 6 1, stop

and choose n ¼ x MinjRf j PUðxÞ RPUj; x 2 ðHi; LoÞg, and

then return n (always rounding up if n is not an integer) as the required sample size.

Step 3. If RPU(n) > RPU, Hi n; otherwise, Lo n. Go back to Step

2.We implement the algorithm and develop a MATLAB program to compute the required sample size.Tables 4, 5tabulate the required sample size for RPU= 0.75(0.01)

0.95 and

c

= 0.9, 0.95, 0.975, 0.99.

Let the desired estimation accuracy be RPUand the conﬁdence

level be

c

, and then the minimum sample size n(always rounding up if n is not an integer) can be calculated. Tables 4, 5 display the sample size n required for R

c

PRPUwith quality characteristic

v = 3, RPU= 0.75(0.01)0.95 and

c

= 0.9, 0.95, 0.975, and 0.99 when

CT

PU¼ 1 and 1.33. For example, if RPUis set to 0.89, CTPU¼ 1, and

c

= 0.95, the sample size needed is n = 76. We conclude that a min-imum sample size of n = 76 is required to be 95% so that the true CPUis no less than R

c

= 89.12% of the sample estimate bCPU. Thus,

if the sample estimate bCT

PU¼ 1:2, the true value of C T

PU is no less

than 1.2 89.12% = 1.069, with 95% conﬁdence.

FromTables 4 and 5, we can ﬁnd that as RPUand

c

increase, the

required sample size n increases. However, some values of sample size can not be obtained (see the sign ‘‘” in the column of the sample size n) when the values of RPUand conﬁdence level

c

are

small. This is due to the problem of the bootstrap resampling procedure.

Table 12

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6. Application

We consider the following case from a manufacturing factory located on the Science-Based Industrial Park in Taiwan, making the thin ﬁlm transistor liquid crystal display (TFT-LCD). There are three major process groups in TFT-LCD manufacturing process, ar-ray process; cell process and module assemble process. The arar-ray process is similar to the semiconductor manufacturing process, ex-cept that transistors are fabricated on a glass substrate instead of a silicon wafer. Photolithography (one of the array process) is a crit-ical step within LCD manufacturing process since the panel quality depends on the entire pattern formation. Film deposition is done before photolithography. Overlay is a key parameter in deposition process and uniformality is a key parameter in coating and expo-sure, which are two processes in photolithography. We focus on these key parameters, such as overlay, critical dimension and uniformality.

InFig. 3, between one deposited layer and another, a distance called overlay may be existed. There are three steps in

photolithog-raphy process, coating, exposure, and development. It might result deviation as exposure on panel window, called critical dimension (seeFig. 4). In addition, coating photoresist on panel has to be uni-form. The speciﬁcations of these three key parameters are shown inTable 6. Since the assumption of normality for each single char-acteristic is required for the process yield calculation, the historical data of each key characteristic indicates the process being pretty approximate to a normal distribution. Thus, we can conclude that each characteristic data collected from the process is in control and normally distributed.

To obtain the sample size required n under the desired estima-tion accuracy RðPSÞ

pm, we can ﬁnd it inTable 4. If the practitioners set

RðPSÞ

pm to be 0.92, and

c

= 0.95 the sample size needed is n = 146. We

conclude that a minimum sample size of n = 146 is required to be 95% so that the true CT

PUis no less than R

c

= 92.11% of the sample

estimator bCT

PU. Thus, if the sample estimator bCTPU¼ 1:3, the true

va-lue of CTPUis no less than 1.3 92.11% = 1.197, with 95% conﬁdence.

Hence, sample data collected from 150 LCD is displayed inTable 9

of theAppendix. And the upper speciﬁcation limit, the calculated

Table 13

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sample mean, location departure, sample standard deviation, the estimated CPUjand the lower conﬁdence bound LCfor each

charac-teristic are summarized inTable 7. 6.1. Overall process yield analysis

The sample estimators of CT

PUand the BCPB method lower

con-ﬁdence bound of CT

PU for the single characteristic overlay, critical

dimension and uniformality coupler can be summarized inTable 8.

Table 8displays the manufacturing capability and its corre-sponding NCPPM for the LCD process using the estimated bCT

PU

val-ues (uncorrected) and the lower conﬁdence bounds CTðLBÞ_PU

(corrected). The CTðLBÞPU (the LCB of C T

PUÞ obtained using BCPB method

is certainly more reliable than the estimated bCT

PUindex values (an

approach widely used in current industrial applications), since the sampling errors are considered in the LCB approach. In fact, as the sample estimate bCT

PUmay overestimate the true capability (overall

process yield), it conveys unreliable and misleading information, which should be avoided in factory applications. Based on the va-lue of CTðLBÞ

PU , we thus can assure that the production yield is

99.7308%, and the number of the nonconformities is less than 2692 PPM.

7. Conclusions

In this paper, we considered the problem of finding the lower confidence bound and sample sizes required for specified estima-tion accuracy for the CTPU. Since the sampling distribution of C

T PU

is analytically intractable, we applied the bootstrap method to cal-culate the estimator of CT_PUand compared the estimation accuracy of the four bootstrap methods. The results indicated that the BCPB method has good performance when the sample size is smaller than 125. The lower confidence bounds present a measure on the minimum capability of the process based on the sample data. We also investigated the lower confidence bound values and sample sizes required for specified estimation accuracy using BCPB meth-od. The proposed approach ensures that the risk of making incor-rect decisions is no larger than the preset Type I error 1

c

. We also provided tables for the engineers/practitioners to use for their in-plant applications. A real-world example from TFT-LCD manu-facturing process is investigated to illustrate the applicability of our approach. In the future, we can consider same approach for indices with two-sided speciﬁcations, such as Cpk, Cpmand Cpmk.

We also consider multiple characteristics with some correlations.

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