Testing process performance based on the yield: an application to the liquid-crystal display module

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Testing process performance based on the yield: an

application to the liquid-crystal display module

Jann-Pygn Chen

a,*

, W.L. Pearn

b

a

Department of Industrial Engineering and Management, National Chin-Yi Institute of Technology, No. 35, Lane 215, Section 1, Chung Shan Road, Taiping, Taichung, 411 Taiwan, ROC

b

Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC Received 17 January 2002; received in revised form 18 March 2002

Abstract

Process capability indices have been introduced to provide numerical measures on whether a manufacturing process is capable of reproducing items meeting the speciﬁcations predetermined by the product designers or the consumers. Process yield is one of the most common criteria used in the manufacturing industry for measuring process perfor-mance. The formula Spkhas been proposed to calculate the process yield for normal processes. The formula Spkprovides

an exact measure on the process yield. Unfortunately, the statistical properties of the estimated ^SSpkare mathematically

intractable. In this paper, we apply the bootstrap simulation method to construct the lower conﬁdence bound of Spk. We

then present a real-world application to the liquid-crystal display module process, to illustrate how we may apply the formula Spkto actual data collected from the factories.

1. Introduction

Process capability indices (PCIs) have been proposed to the manufacturing industry to provide numerical measures on whether a process is capable of producing product items within the specification limits predeter-mined by the product designers or the consumers. The PCIs were first applied to the automatic industry in Japan and America. In a purchasing contract, a mini-mum value of the PCI is usually specified. If the pre-scribed minimum value of the PCI fails to be met, the process is determined to be incapable. Otherwise, the process will be determined to be capable.

Process yield has longtime been the most common and standard criteria used in various manufacturing industries to characterize the process performance. Process yield is currently deﬁned as the percentage of the processed product units passing the inspections. That is, the product characteristic must fall within the

manu-facturing tolerance. For product units rejected during the inspection (nonconforming), additional costs would be incurred to the factory for scrapping or reworking the product. All passed product units are treated equally and accepted by the producer, which requires no addi-tional cost to the factory.

For processes involving two-sided manufacturing speciﬁcations, the process yield can be calculated as Yield¼ F ðUSLÞ F ðLSLÞ, where USL and LSL are the upper and the lower speciﬁcation limits, respectively, and FðxÞ is the cumulative distribution function of the process characteristic. If the process characteristic is normally distributed, then the process yield can be ex-pressed as Yield¼ U½ðUSL lÞ=r U½ðLSL lÞ=r, where the parameter l is the process mean, parameter r is the process standard deviation, and UðxÞ is the cu-mulative distribution function of the standard normal distribution Nð0; 1Þ. Based on the above expression of the process yield, Boyles [1] considered the index Spkto

calculate the yield for normal processes, deﬁned as

Spk¼ 1 3U 1 1 2U USL l r þ1 2U l LSL r : *

Corresponding author. Tel.: +886-4-392-4505x7642; fax: +886-4-393-4620.

E-mail address:[email protected](J.-P. Chen).

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The index Spkestablishes the relationship between the

manufacturing speciﬁcations and the actual process performance, which provides an exact measure on the process yield. If Spk¼ c, then the process yield can be

expressed as Yield¼ 2Uð3cÞ 1. The fraction of non-conforming is deﬁned as the percentage of product items fall outside the speciﬁcation limits, which can be repre-sented as 1 Yield ¼ 2ð1 Uð3SpkÞÞ. Obviously, there

is a one-to-one correspondence between Spk and the

process yield. Thus, Spkprovides an exact measure of the

process yield. Note that other existing methods only provide approximate (instead of exact) yield measures. Table 1 displays the fraction of nonconforming (in ppm) as a function of the index Spk, (calculated by SAS

computer programs).

2. Estimating the measure Spk

A natural estimator of the index Spkmay be obtained

by replacing the unknown process mean l by the sample mean X ¼ ðX1þ X2þ þ XnÞ=n, and the unknown

process standard deviation r by the sample standard deviation, S¼ ½PðXi X Þ

2

=ðn 1Þ1=2, respectively, which may be obtained from a stable (statistically in control) process. Thus, the natural estimator ^SSpkmay be

expressed as ^ S Spk¼ 1 3U 1 1 2U USL X S þ1 2U X LSL S : Unfortunately, the exact distribution of ^SSpkis

math-ematically intractable. Lee et al. [2] derived an approx-imate distribution of the estimator ^SSpk using Taylor

expansion technique. They showed that the estimator ^

S

Spkis approximately distributed as the normal

distribu-tion. The calculation of the approximation is, however, rather messy, and cumbersome to deal with. Further, the accuracy of the approximation has not been investi-gated. Thus, the approximation would not be practically useful until the difficulty is overcome. For practical purpose, in this paper we use the bootstrap resampling technique to find the lower confidence bound on Spk, so

that practitioners/engineers can use them to perform quality testing and determine whether their process meets the preset quality requirement. We also present a case study on a liquid-crystal display module (LCM) manufacturing process to illustrate how the bootstrap lower conﬁdence limit of Spkmay be applied to actual

process data collected from the factory.

3. The bootstrap methodology

Efron [3,4] introduced a nonparametric, computa-tional intensive but eﬀective estimation method called ‘‘the bootstrap’’, which is a data based simulation tech-nique for statistical inference. In particular, one can use the nonparametric bootstrap method to estimate the sampling distribution of a statistic, while assuming only that the sample is a representative of the population from which it is drawn, and that the observations are independent and identically distributed. The merit, in its simplest form, is that the nonparametric bootstrap does not rely on any distributional assumptions about the underlying population.

Suppose the set of observations fx1; x2; . . . ; xng is a

random sample of size n taken from a process. A bootstrap sample,fx

1; x2; . . . ; xng, is a sample of size n

drawn (with replacement) from the original sample fx1; x2; . . . ; xng. Hence, there are a total of nn possible

Nomenclature

FðxÞ the cumulative distribution function of the process characteristic

UðxÞ the cumulative distribution function of the standard normal distribution Nð0; 1Þ fx1; x2; . . . ; xng a random sample of size n taken from

a process fx

1; x2; . . . ; xng a bootstrap sample of size n drawn

(with replacement) from the original sample fx1; x2; . . . ; xng

^ h

h an estimate of h can be computed from the bootstrap sample

za=2 the upper a=2 percentile of the Nð0; 1Þ

dis-tribution ^

h

hðiÞ the ith ordered ^hh_i, i¼ 1; 2; . . . ; B

Table 1

Various Spkvalues and the fraction of nonconformities (in ppm)

Spk 0.25 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 2.00

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resamples. Thus, the bootstrap sampling is equivalent to sampling with replacement from the empirical distribu-tion funcdistribu-tion, and the bootstrap distribudistribu-tion is an esti-mator of the distribution of ^SSpk. In practice, usually only

a small number of random samples out of the nn_possible

resamples is drawn, the estimate is calculated for each of these, and the subsequent empirical distribution is re-ferred to as the statistic’s bootstrap distribution. Efron and Tibshirani [5] indicated that a rough minimum of 1000 bootstrap resamples is usually suﬃcient for com-puting conﬁdence interval estimates with reasonably accuracy.

Suppose the random variable X measures process performance with respective to certain quality charac-teristic. The distribution of X is generally unknown. Suppose we wish to estimate some parameter, h, that characterizes the performance of the process. An esti-mate of h can be computed from the bootstrap sample, denoted as ^hh, which is called the bootstrap estimate. The resampling procedure can be performed repeatedly to obtain a certain number of bootstrap samples, for example, B times. Then, the B bootstrap estimates ^

h

h₁; ^hh₂; . . . ; ^hh_B, can be computed from the resamples. Research papers discussing the bootstrap methods in-clude Efron and Gong [6], Gunter [7,8], Mooney and Duval [9], Young [10], and many others. In particular, Efron and Tibshirani [5] developed three types of boot-strap confidence interval. Those include the standard bootstrap (SB) confidence interval, the percentile boot-strap (PB) confidence interval, and the biased-corrected percentile bootstrap (BCPB) confidence interval, which are defined as follows:

(1) Standard bootstrap: From the B bootstrap esti-matesf ^hh_ig, i ¼ 1; 2; . . . ; B, we calculate the sample av-erage, and the sample standard deviation as

h h¼1 B XB i¼1 ^ h h_i; S ^ h h ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 B 1 XB i¼1 ð ^hh_i hhÞ2 v u u t :

If the distribution of the estimate ^hh is approximately normal, theð1 aÞ100% level SB conﬁdence interval for his ^hh za=2S_h_h^, where za=2 is the upper a=2 percentile of

the Nð0; 1Þ distribution.

(2) Percentile bootstrap: From the ordered collection of f ^hh_ig, i ¼ 1; 2; . . . ; B, the a=2 percentage and the ð1 a=2Þ percentage points are used to obtain the ð1 aÞ100% level PB conﬁdence interval for h, where ^

h

hðiÞ is the ith ordered ^hh_i, i¼ 1; 2; . . . ; B.

½ ^hhða=2 BÞ; ^hhðð1 a=2Þ BÞ:

(3) Biased-corrected PB: The bootstrap distribution obtained using only a sample of the complete bootstrap distribution may be shifted higher or lower than would be expected (i.e, a biased distribution). Thus, a modi-ﬁed method is developed to correct for this potential bias (see Efron [4] for a complete justiﬁcation of this method). First, we use the (ordered) distribution of

^ h

h_i to calculate the probability p0¼ Pr½ ^hhðiÞ 6 ^hh, i ¼ 1;

2; . . . ; B, where ^hh is the estimated value of h using a random samplefx1; x2; . . . ; xng. Second, we calculate the

percentile point z0¼ U1ðp0Þ , and PL¼ Uð2z0 za=2Þ,

PU¼ Uð2z0þ za=2Þ, where Uð:Þ is the standard normal

cumulative distribution function. Finally, the BCPB conﬁdence interval is found as½ ^hhðPLBÞ; ^hhðPUBÞ.

Franklin and Wasserman [11] investigated the lower conﬁdence bounds for the capability indices, Cp, Cpkand

Cpm using the three bootstrap methods. Some

simula-tions were conducted, and a comparison was made among the three bootstrap methods based on the para-metric estimates. The simulation results indicate that for normal processes the bootstrap confidence limits per-form equally well (see Chou et al. [12], Bissell [13], and Boyles [14]). And for nonnormal processes the bootstrap estimates performed significantly better than other methods. Franklin and Wasserman [11] also found that in all cases investigated the SB method performs better than the PB and the BCPB methods. Pearn and Chen [15] also applied the three bootstrap methods to find the lower confidence bounds of the yield measure Spk, and

compared their performance based on the coverage percentage. Their results showed that the SB performs better than the other two methods. In particular, as the sample size n exceeds 45, the coverage percentage of the 95% lower conﬁdence bound for the SB method, is greater than 90%.

4. LCMmanufacturing process

To illustrate how the bootstrap lower conﬁdence bound of Spkmay be established and applied to the

ac-tual data collected from the factories, we present a case study on the liquid-crystal module manufacturing pro-cess. The case we investigated was taken from a manu-facturing factory located on the Science-Based Industrial Park, Taiwan, making the LCMs. The LCM is one of the key components used in many high-tech electronic commercial devices, such as the cellular phone, the PDA (personal digital assistant), the pocket calculator, digital watch, automobile accessory visual displays, and many others. Three key components make the LCM functions properly. Those include the liquid-crystal display, the back lighting, and the peripheral (interface) system. A typical assembly drawing for the LCM product is de-picted in Fig. 1.

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The mounting technology for the chip on glass (COG) makes the exposed particle overturned, with the side of circuits facing downward. Then, the electricity conduction is joined between the IC and the panel of the liquid-crystal display through the mounting material. Currently, the mounting technology of the COG is the best manufacturing technology for the LCM in terms of the mounting density. It is important to note that dif-ferent mounting material requires diﬀerent mounting technology of the COG.

In the factory, the manufacturing process control ﬂow-chart of the COG is illustrated in Fig. 2. For the main bonding process, the bonding precision is an

es-sential process parameter we focused on in our study. We investigated a particular model of the LCM product with the upper and the lower speciﬁcation limits set to USL¼ 15 lm, LSL ¼ 15 lm, and the target value is set to T ¼ 0. If the characteristic data does not fall within the tolerance (LSL, USL), the lifetime or reliability of the LCM will be discounted. To ensure the production quality, and to satisfy the customers’ requirement, the company has set the yield index SpkP

1:50, which implies that no more than 7 ppm fraction of nonconforming for the product. If the capability re-quirement fails to be met, the LCM product would be seriously aﬀected on its reliability or lifetime.

Fig. 1. The assembly drawing of the LCM product.

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4.1. The lower conﬁdence bound

In order to evaluate the process capability based on the 95% lower conﬁdence limit using the SB method. According to the process control plan of the COG, the random sample data in consecutive two day are col-lected, which are displayed in Table 2. The sample ob-servations are obtained through the inspection, using microscope by visual, which were collected eight pieces per every 2 h. These observations were justiﬁed taken from a stable process, and the characteristic distribution is shown to be approximate normal.

Then, B¼ 10 000 bootstrap resamples (each is of size 64) are drawn randomly from the original sample. A 95% bootstrap lower conﬁdence limit of the SB method for Spkis constructed. If the calculated bootstrap lower

conﬁdence limit is found to be smaller than the speciﬁed 1.50 index value, we would judge that the process is incapable. Quality improvement activities will be initi-ated. Otherwise, the process is considered to be capable. From the 10 000 bootstrap estimates ^SS_pk ðiÞ, the sample average can be calculated as

Spk¼ 1 10 000 X 10 000 i¼1 ^ S SpkðiÞ;

and the sample standard deviation can be obtained as

S ^ S Spk ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 9999 X 10 000 i¼1 ð ^SS pkðiÞ S pkÞ 2 v u u t :

Thus, the 95% SB lower conﬁdence limit for Spkcan be

constructed as

LCB¼ SS_pk Z0:05 SSS^pk:

The bootstrap resampling computation is rather com-plicated. In order to accelerate the computation, an in-tegrated SAS computer program is developed (see Appendix A). The practitioners only need to input the manufacturing speciﬁcations, USL, LSL, a speciﬁed quality level of Spkand a collected sample data of size n.

The estimated value ^SSpk and the SB lower conﬁdence

bound of Spkmay be easily obtained. Thus, whether or

not the process is capable may be determined.

For the LCM real example we investigated, Table 2 shows the simulation result with ^SSpk¼ 1:72588 using the

SB bootstrap method, the lower conﬁdence bound on Spk, was found to be 1.44244, which is less than the

speciﬁed 1.50 (the minimal value of the measure to reach a positive judgment). Thus, we may not conclude that the bonding precision of the bonding process is capable. Quality improvement team must immediately initiate some improvement activities to ensure the minimal re-quirement met.

5. Conclusions

Bootstrap resampling method is a nonparametric, computational intensive but eﬀective estimation method, which is a data based simulation technique for statistical inference. In particular, one can use the nonparametric bootstrap method to estimate the sampling distribution of a statistic. Bootstrap method has been widely applied to statistical process control. In this paper, we apply the bootstrap method to the process yield measure Spk to

obtain the conﬁdence bounds. The proposed approach makes it feasible for the engineers to perform approxi-mate process performance testing using the calculated Spk. We also provide an eﬃcient SAS computer program

for the engineers to use. The program only requires an input of the manufacturing speciﬁcation limits, and the sample data, then the estimated value ^SSpkand SB lower

conﬁdence limit of Spk will be outputted quickly. In

summary, bootstrap lower conﬁdence limit of Spk may

be used to evaluate process capability in statistical pro-cess control.

Appendix A.

/* – – – – – – – – – – – – – – – – -input the speciﬁcation limits, USL, LSL, and the target value T – – – – – – – – – – – – – – – – –*/ data para; input usl lsl T; cards; 1515 0 ; /* – – – – – – – – – – – – – – – – -Table 2

The collected sample data with 64 observations (unit: lm) 0.98 4.63 0.78 1.67 2.34 1.97 0.07 2.32 2.18 3.79 0.35 1.20 3.13 5.58 1.91 1.18 0.66 3.16 5.01 1.42 2.20 2.20 2.16 2.87 0.31 0.83 0.86 2.04 1.24 5.83 0.50 0.81 3.07 1.42 2.42 4.75 1.02 6.51 1.34 1.06 0.10 5.02 0.76 4.84 1.28 2.19 0.16 2.66 0.87 3.52 0.05 1.11 1.47 0.28 1.02 9.23 2.01 2.30 4.26 3.41 3.34 2.85 2.43 1.84

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store the input of the original sample data of size n¼ 64 – – – – – – – – – – – – – – – – –*/ data size; set para; array x{64} x1–x64; array y{64} y1–y64; retain seed 123; do i¼ 1 to 64;

input x{i} @; end;

/* – – – – – – – – – – – – – – – – -calculate the estimated Spk based on the original

sample of n¼ 64 – – – – – – – – – – – – – – – – –*/ omean¼ mean(of x1–x64); ostd¼ std(of x1–x64); c¼ (uslomean)/ostd; d¼ (omeanlsl)/ostd; ESpk¼ 1/3*probit(0.5*probnorm(c)+0.5*prob-norm(d)); /* – – – – – – – – – – – – – – – – -generate 10000 bootstrap samples from the original sample of n¼ 64 – – – – – – – – – – – – – – – – –*/ do m¼ 1 to 10000; do n¼ 1 to 64; L¼ int(ranuni(seed)*64+1); y{n}¼ x{L}; end; /* – – – – – – – – – – – – – – – – -Compute the 10000 bootstrap estimates ^SS_pkðiÞ – – – – – – – – – – – – – – – – –*/

mean¼ mean(of y1–y64); std¼ std(of y1–y64); e¼ (uslmean)/std; f¼ (meanlsl)/std; Spk¼ 1/3*probit(0.5*probnorm(e)+0.5*prob-norm(f)); output; end; /* – – – – – – – – – – – – – – – – -Input the original sample data of n¼ 64 sample observations – – – – – – – – – – – – – – – – –*/ cards; 0.98 4.63 0.78 1.67 2.34 1.97 0.07 2.32 2.17 3.78 0.351.20 3.12 5.58 1.91 1.18 0.66 3.16 5.01 1.42 2.20 2.20 2.162.87 0.30 0.830.86 2.04 1.24 5.83 0.500.81 3.07 1.42 2.42 4.75 1.02 6.50 1.34 1.06 0.10 5.020.76 4.84 1.28 2.19 0.162.66 0.87 3.52 0.05 1.11 1.47 0.28 1.02 9.24 2.01 2.30 4.26 3.41 3.34 2.852.43 1.84 ; /* – – – – – – – – – – – – – – – – -Compute SS_pkand S^SSpk from the 10000 bootstrap esti-mates ^SS_pkðiÞ

– – – – – – – – – – – – – – – – –*/ proc univariate data¼ size normal plot;

var Spk;

output out¼ out1 mean ¼ mean std¼ stddev;

/* – – – – – – – – – – – – – – – – -Calculate the lower conﬁdence bound on the Spk

– – – – – – – – – – – – – – – – –*/ data sbcl; set out1; SBLB¼ mean-probit(0.95)*stddev; proc print; var SBLB; run; References

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