This article was downloaded by: [National Chiao Tung University 國立交通大學] On: 26 April 2014, At: 01:29
Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Production
Research
Publication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/tprs20
Measuring production yield for
processes with multiple quality
characteristics
W. L. Pearn a , F. K. Wang b & C. H. Yen a a
Department of Industrial Engineering and Management , National Chiao Tung University , Taiwan, ROC
b
Department of Industrial Management , National Taiwan University of Science and Technology , Taiwan, ROC Published online: 22 Feb 2007.
To cite this article: W. L. Pearn , F. K. Wang & C. H. Yen (2006) Measuring production yield for
processes with multiple quality characteristics, International Journal of Production Research, 44:21, 4649-4661, DOI: 10.1080/00207540600589119
To link to this article: http://dx.doi.org/10.1080/00207540600589119
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &
Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
International Journal of Production Research, Vol. 44, No. 21, 1 November 2006, 4649–4661
Measuring production yield for processes with
multiple quality characteristics
W. L. PEARNy, F. K. WANG*z and C. H. YENy
yDepartment of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC
zDepartment of Industrial Management,
National Taiwan University of Science and Technology, Taiwan, ROC
(Revision received January 2006)
Process yield is an important criterion used in the manufacturing industry for measuring process performance. Methods for measuring yield for processes with single characteristic have been investigated extensively. However, methods for measuring yield for processes with multiple characteristics have been compara-tively neglected. In this paper, we develop a generalized yield index, called TSpk,PC, based on the index Spk introduced by Boyles (Journal of Quality
Technology, 23, 17–26, 1991) using the principal component analysis (PCA) technique. We obtained a lower confidence bound (LCB) for the true process yield. The proposed method can be used to determine whether a process meets the preset yield requirement, and make reliable decisions. Examples are provided to demonstrate the proposed methodology.
Keywords: Process yield; Process capability indices; Lower confidence bound; Principal component analysis
1. Introduction
Process capability indices, which establish the relationship between the actual process performance and the manufacturing specifications, have been the focus in quality assurance and capability analysis for the past 15 years. Those capability indices quantifying process performance are essential to any successful quality improvement activities and quality program implementation. The capability indices, Cp, Cpkand Cpm, are widely used in the manufacturing industry to evaluate process
performance for cases with a single quality characteristic. The index Cpmeasures the
overall process variation relative to the specification tolerance. The index Cpktakes
into account the magnitude of process variation as well as the degree of process centering. The index Cpm emphasizes measuring the ability of process to cluster
around the target, which reflects the degrees of process targeting. On the other hand, the index Spk (Boyles 1991) is introduced to establish the relationship between the
manufacturing specification and the actual process performance, which provides an exact measure on the process yield. Capability calculations for processes with single
*Corresponding author. Email: [email protected]
International Journal of Production Research
ISSN 0020–7543 print/ISSN 1366–588X onlineß 2006 Taylor & Francis http://www.tandf.co.uk/journals
DOI: 10.1080/00207540600589119
characteristic have been investigated extensively. Kotz and Johnson (2002) presented a thorough review for the development of process capability indices from 1992 to 2000.
Often, a manufactured product is described in multiple characteristics. That is, manufactured items require values of several different characteristics for adequate description of their quality. Each of those characteristics must satisfy certain specifications. The assessed quality of a product depends on the combined effects of those characteristics, rather than on their individual values. For example, automobile paint needs a range of light reflective abilities and a range of adhesion abilities. A paint that satisfies one criterion but not the other is undesirable. Those characteristics are related through the compositions of the paint. It is therefore natural to consider a bivariate characterization of this paint. As for the tolerance region of multiple characteristics, we often take an ellipsoidal region or a rectangular region. In the two-dimension cases, those tolerance ranges compose a rectangular tolerance region. In higher dimensions, they form a hypercube. For more complex engineering specifications, the tolerance region is very complicated. For instance, a drawing of a connecting rod in a combustion engine consists of crank-bore inner diameter, pin-bore inner diameter, rod length, bore true-location and so on.
In order to handle the issue for cases with multiple quality characteristics, multivariate methods for assessing process capability are proposed. These relevant multivariate capability indices can be found in Chan et al. (1991); Pearn et al. (1992); Taam et al. (1993); Chen (1994); Shahriari et al. (1995); Wang and Du (2000), etc. A brief summary for multivariate capability indices is given in table 1. The multivariate capability indices proposed by Chan et al. (1991), Pearn et al. (1992), Taam et al. (1993) and Shahriari et al. (1995), respectively, require the assumption of multivariate normality while those proposed by Chen (1994) and Wang and Du (2000) make no assumption on multivariate normality. The tolerance regions of those methods using multivariate normality assumption are ellipsoidal except for Shahriari et al. (1995). Relatively, Chen (1994) and Wang and Du (2000) provide more flexible methods to assess the capability for multivariate data. Chen (1994) proposed this over a general tolerance zone which includes ellipsoidal and rectangular solid ones, and this manner does not rely on a particular distribution. Wang et al. (2000) presented a comparison of three methods proposed by Taam et al. (1993), Shahriari et al. (1995) and Chen (1994). Also, Wang and Du (2000) applied the principal component analysis (PCA) to process capability indices to handle normal and non-normal data. However, the issues between process yield and the multivariate capability indices have received little attention. In this paper, we focus
Table 1. The summary of multivariate capability indices.
Authors Index Distribution application Tolerance form Chan et al. (1991) Cpm Multivariate normal Elliptical
Pearn et al. (1992) vCpm,vCp Multivariate normal Elliptical
Taam et al. (1993) MCpm, MCp Multivariate normal Elliptical
Chen (1994) MCp No specific No specific
Shahriari et al. (1995) [CpM, PV, LI ] Multivariate normal Elliptical
Wang and Du (2000) MCpm, MCpk, MCp No specific No specific
on the process yield for the correlated multiple quality characteristics. We calculate the process yield using Spk through PCA for processes with correlated multiple
quality characteristics. We present the PCA method and the procedure of obtaining the lower confidence bound (LCB) for the true process yield using Spkthrough the
principal component analysis (PCA). Illustrative examples are given to demonstrate the applicability of the proposed approach.
2. Principal Component Analysis
PCA is a useful statistical technique that has been widely applied to face recognition and image compression, which is a common technique for finding patterns in high dimensional data. It is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences. In many cases the patterns in data can be difficult to find in high-dimensional applications, particularly when graphical representation is not available, and PCA is a powerful tool in such situations. The other main advantage of PCA is that after finding patterns in the data, one could compress the data by reducing the number of dimensions without losing much information. PCA is a multivariate technique in which a number of related variables are transformed to a set of uncorrelated linear functions of the original measurements. The first principal component linearly combines all of the original variables in which the maximum variation among the objects is displayed. The second, third, and further components are, similarly, the linear combinations representing the next largest variation, irrespective of those represented by previous ones. In most practical applications, analysing the major components can retain most of the information regarding the variability of the process. In general, multivariate methods often assume the data satisfy multivariate normal distribution. But in applying the PCA technique one does not require such assumption.
Assume that X is a n sample data matrix, where is the number of product quality characteristic from one part and n is the sample size of part measured. Also, X is the sample mean vector ( 1) of observations and S is a symmetric matrix representing the covariance between observations. Engineering specifications are given for each quality characteristic, where LSL and USL are their -vectors of the lower specification limits and upper specification limits, respectively. The vector T( 1) represents the target values of the quality characteristics. In addition, the spectral decomposition can be used to obtain D ¼ UTSU, where D is a diagonal matrix. The diagonal elements of D, 1, 2, . . . , v, are the eigenvalues of S and the columns of U, u1, u2, . . . , uv are the eigenvectors of S. Consequently, the ith principal component (PCi) is expressed as
PCi¼uTix, 8 i ¼ 1, 2, . . . ,v ð1Þ where x is 1 vectors on the original variables. The engineering specifications and target values of PCisare as follows:
LSLPCi ¼u T iLSL USLPCi¼u T iUSL 8i ¼1, 2, . . . , TPCi ¼u T iT 8 > < > : ð2Þ
Measuring production yield 4651
Similarly, the relevant sample estimators, S2and X of PCiscan defined as S2 PCi ¼i XPCi ¼u T iX 8i ¼1, 2, . . . , ð3Þ The ratio of each eigenvalue to the summation of the eigenvalues is the proportion of variability associated with each principal component variable. That is,
i . Xv
i¼1
i, 8i ¼1, 2, . . . , v ð4Þ However, only a few principal components can explain most of the total variability (about 80–90%). Anderson (1963) proposed a 2test for identifying the significant components. It is 2¼ ðn 1Þ X v j¼kþ1 ln jþ ðn 1Þðv kÞ ln Pv j¼kþ1j v k ð5Þ
where 2 has r ¼ ð1=2Þðv kÞðv k þ 1Þ 1 degrees of freedom. Jackson (1980) further applied the test to the hypothesis H0: kþ1¼ ¼vagainst the alternatives with at least one different eigenvalue. Referring to this method, we can choose the suitable number of PCisrightly.
3. Process yield
Process yield has been the most basic and common criterion used in the manufacturing industry for measuring process performance. It is closely related to the production cost as well as customer satisfaction. Process yield is currently defined as the percentage of processed product unit passing inspection. That is, the product characteristic must fall within the manufacturing tolerance. For product units rejected (non-conformities), additional costs would be incurred to the factory for scrapping or repairing the product. All passed product units are equally accepted by the producer, which incurs the factory no additional cost. For processes with high yield, it produces few percentages of non-conforming products. That is, most of the products produced in this process satisfy the requirement of specifications. In many cases, a benchmark of minimum 99.73% for assessing the process is suggested. Enterprises get more profit and cost down with high process yield, hence companies make their efforts to increase the process yield. The relationships between the process yield and the process capability indices have been discussed extensively for processes with single characteristics, but comparatively neglected for processes with multiple characteristics.
Consider a production process in which, possibly dependent, quality character-istics determine the quality of the product. In other words, the product has multiple correlated characteristics. We are concerned with the probability of producing a good product satisfying all its specifications. Assume that the observations X have a multivariate normal distribution, Nvð,PÞ, where v is the dimension of variables, is the mean vector and represents the variance–covariance matrix of X. The components of the vectors LSL and USL are the v lower and upper specification
limits, respectively. Under the assumptions mentioned, the probability that a production process produces a good product is
p ¼ Z
½LSL,USL
NvðXj, ÞdX ð6Þ
It is also called the true process yield.
For normally distributed processes, the index Spk is used to establish the
relationship between the manufacturing specification and the actual process performance, which provides an exact measure on the process yield, defined as
Spk¼ 1 3 1 1 2 USL þ1 2 LSL ¼1 3 1 1 2 1 Cdr Cdp þ1 2 1 þ Cdr Cdp ð7Þ
where Cdr ¼ ð mÞ=d, Cdp ¼=d, m ¼ ðUSL þ LSLÞ=2, d ¼ ðUSL LSLÞ=2. It provides an exact measure of process yield. If Spk¼c, then the process yield
can be expressed as Yield ¼ 2ð3cÞ 1. Obviously, there is a one-to-one correspon-dence between Spk and the process yield. Considering processes with multiple
characteristics, Chen et al. (2003) defined the yield index as STpk¼1 3 1 Yv j¼1 2ð3SpkjÞ 1 þ1 " # 2 ( ) , ð8Þ
where Spkidenote the Spkvalue of the jth characteristic for j ¼ 1, 2, . . . , v and v is the
number of characteristics. The asymptotic distribution for an estimate ^ST
pk can be found from Theorem 1 (see Appendix). This index provides an exact measure of the overall process yield when the characteristics are mutually independent. Also the overall process yield P can be established as
P ¼Y v j¼1 2ð3SpkjÞ 1 ¼2ð3STpkÞ 1: ð9Þ
Assume that the multivariate processes data are from a multivariate normal distribution. In this case, the principal components can be applied to the capability study. Consequently, the new variables (principal components) are mutually independent and normal distributed (see Theorem 2 in the Appendix). Applying equation (8), the combined yield index for the multivariate processes data can be determined by TSpk,PC¼ 1 3 1 Y v j¼1 2ð3Spkj;PCÞ 1 þ1 " # 2 ( ) ð10Þ where Spkj;PCi represents the univariate measure of process yield index for the ith principal component. By analogy to equation (9), the overall process yield can be established as Yield¼2ð3TSpk;PCÞ 1.
Measuring production yield 4653
Lee et al. (2002) inferred the asymptotic distribution for an estimate ^Spk of the process yield index Spk. An approximate 100(1 )% confidence interval for Spkis
expressed as ^ Spk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ a2þ ^b2 p 6pffiffiffinð3 ^SpkÞ Z=2, ^Spkþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ a2þ ^b2 p 6pffiffiffinð3 ^SpkÞ Z=2 ! ð11Þ where ^ Spk¼ 1 3 1 1 2 USL X S þ1 2 X LSL S ¼1 3 1 1 2 1 ^Cdr ^ Cdp ! þ1 2 1 þ ^Cdr ^ Cdp ! ( ) , ^ a ¼ d. ffiffiffip2S ð1 ^CdrÞ 1 ^Cdr ^ Cdp ! þ ð1 þ ^CdrÞ 1 þ ^Cdr ^ Cdp ! ( ) , ^ b ¼ 1 ^Cdr ^ Cdp ! 1 þ ^Cdr ^ Cdp ! ,
where Z/2is the upper 100(/2)% point of the standard normal distribution, and
is the probability density function of the standard normal distribution. Applying the above formula, we can obtain an approximate 100(1 )% lower confidence bound for Spk, then an approximate 100(1 )% lower confidence bound for the process
yield can be obtained.
Equation (11) can be used to establish approximate lower confidence bound for the process yield in the case of single characteristic. However, how to estimate the process yield is more difficult for processes with multiple characteristics. By using the PCA method, such difficulty can be overcome. Applying the PCA method to equations (10) and (11), an approximate 100(1 )% confidence interval for the combined index, TSpk,PC, is expressed as
1 3 1 Y j¼1 ð2ð3kjÞ 1Þ þ 1 " # 2 ( ) TSpk,PC 1 3 1 Y 1 ð2ð3‘jÞ 1Þ þ 1 " # 2 ( ) , ð12Þ where kj¼ ^Spkj;PC ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ a2 j;PCþ ^b2j;PC q 6pffiffiffinð3 ^Spkj;PCÞ Z=2m, ‘j¼ ^Spkj;PCþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ a2 j;PCþ ^b2j;PC q 6pffiffiffinð3 ^Spkj;PCÞ Z=2: Thus, an approximate 100(1 )% lower confidence bound for TSpk,PC can be
obtained, and an approximate 100(1 )% lower confidence bound for the true process yield can also be obtained by using the one-to-one correspondence between TSpk,PCand the process yield (Yield ¼ 2ð3TSpk;PCÞ 1).
4. Application examples
Three examples are given below to illustrate the proposed methodology. We show how to calculate the lower confidence bound for process yield. In the first example, the case involves two quality characteristics. For the other two examples, the cases involve more than three quality characteristics.
4.1 Example 1
Chen (1994) discussed a bivariate normal example and employed Sultan (1986) bivariate process data (n ¼ 25). Of particular interest were the brinell hardness (H ) and the tensile strength (S) of a process. The specification limits for H and S were set at [112.7, 241.3] and [32.7, 73.3], respectively. The centre of the specifications was TT¼[177, 53]. The sample mean vector and sample covariance matrix were
XT¼ ½177:2, 52:32 and S ¼ 338 88:75 88:75 33:47414
: The process points and tolerance region is illustrated in figure 1.
By performing the principal components analysis, the eigenvecters and eigen-values can be obtained. Table 2 shows the loading and eigenvalue of PCs using the principal component analysis. Testing the hypothesis H0: 1¼2yields a value of 2
2¼55:35, which is quite significant at the 95% confidence level. Thus, we only used the first PC to evaluate the capability at 97% total variability. The prin-cipal components are USLPC1¼252.0660, LSLPC1¼117.3279, XPC1¼184.7172, TPC1¼184.6970, and SPC1¼19.0257. Referring to equation (11), ^Spk1;PC can be calculated as 1.1803. Applying equation (12), the approximate 95% lower confidence bound for the combined index, TSpk,PC, is 0.9058. Using the one-to-one
Hardness Strength 100 120 140 160 180 200 220 240 20 30 40 50 60 70 80 Tolerance region
Figure 1. Process points and tolerance region for example 1.
Measuring production yield 4655
correspondence between TSpk,PC and the process yield (Yield ¼ 2ð3TSpk;PCÞ 1), the approximate 95% lower confidence bound for the true process yield is 0.993418. Notably, this process does not meet the process yield requirement.
4.2 Example 2
The previous study (Wang and Chen 1998) presented a trivariate quality control involving the joint control of the depth (D), the length (L) and the width (W) of a plastic product. Fifty observations are collected from a plastic production line. The specified limits for D, L, and W are set at [2.1, 2.3], [304.5, 305.1] and [304.5, 305.1], respectively. The specification of the target value is TT¼[2.2, 304.8, 304.8]. The p-value for Mardia’s SW statistic is 0.32. Thus, the assumption of multivariate normality can not be rejected at 95% confidence level. The sample mean vector and sample covariance matrix were
XT¼ ½2:1616, 304:7182, 3:4:7678 and S ¼ 0:002051 0:000875 0:000656 0:000785 0:001717 0:001204 0:000656 0:001204 0:002034 2 6 4 3 7 5: Figure 2 illustrate the process points and tolerance region.
By performing the principal components analysis, the eigenvecters and eigen-values can be obtained. Table 3 shows the loading and eigenvalue of PCs using the principal component analysis. First, the test of the hypothesis H0: 1¼2¼3, produced a value of 2
5¼36:47, which is significant at the 95% confidence level. That is, the hypothesis is rejected. Then, testing the hypothesis H0: 2¼3 produced a value of 2
2 ¼8:19, which is also significant at the 95% confidence level. Thus, we used the first two PCs to evaluate the capability at 89.04% total variability. The principal components are USLPC1¼ 368.1421,
LSLPC1¼ 368.9698, XPC1¼ 368.4682, TPC1¼ 368.5560, SPC1¼0.0609,
USLPC2¼216.8123, LSLPC2¼216.5499, XPC2¼216.6794, TPC2¼216.6811 and
SPC2¼0.0382. Referring to equation (11), ^Spk1;PC and ^Spk2;PC2can be calculated as 1.7331 and 1.1450, respectively. Applying equation (12), the approximate 95% lower confidence bound for the combined index, TSpk,PC, is 0.9566. Using the one-to-one
correspondence between TSpk,PC and the process yield (Yield ¼ 2ð3TSpk;PCÞ 1), the approximate 95% lower confidence bound for the true process yield is 0.995894. Notably, this process does not meet the process yield requirement.
Table 2. The results of PCA for example 1.
Variable PC1 loading PC2 loading
H 0.965389 0.260814
S 0.260814 0.965389
Eigenvalue 361.9771 9.4970 % Explained of total variability 97.4434 2.5566
4.3 Example 3
Let we consider a real production process of multivariate normal distribution from an electronic thermos manufacturer located in Taiwan. One special type of thermos investigated has five target-the-best quality characteristics with unequal manufac-turing specifications. Forty observations are generated from a multivariate normal distribution. The specification, target value, and the statistics of sample data are summarized as follows: Variable LSL T USL X1 5.598 6.220 6.842 X2 606.5 680 753.5 X3 0.215 0.354 0.493 X4 31.5 35.0 38.5 X5 30 40 50 Tolerance region
Figure 2. Process points and tolerance region for example 2.
Table 3. The results of the first two PCs for example 2. Variable PC1 loading PC2 loading
D 0.522185 0.838481
L 0.582407 0.217154
W 0.622997 0.499794
Eigenvalue 0.003709 0.001457 % Explained of total variability 63.9253 25.1129
Measuring production yield 4657
XT¼ ½6:4104, 681:1656, 0:3597, 35:2364, 41:0645 , S ¼ 0:082811 0:286253 0:018106 0:043890 0:042138 0:286253 19:08823 0:101615 0:591470 0:93861 0:018106 0:101616 0:006509 0:030978 0:098379 0:043890 0:591470 0:030978 0:497112 0:446935 0:042138 0:93861 0:098379 0:446395 13:65237 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 :
By performing the PCA, the eigenvectors and eigenvalues can be obtained. Table 4 shows the loading and eigenvalue of PCs using the principal component analysis. First, the test of the hypothesis H0: 1¼2¼3¼4¼5, produced a value of 2
14 ¼545.85, which is significant at the 95% confidence level. Second, the test of the hypothesis H0: 1¼2¼3¼4, produced a value of 29¼242.72, which is significant at the 95% confidence level. Third, the test of the hypothesis H0: 1¼2¼3, produced a value of 25¼94:03, which is significant at the 95% confidence level. That is, the hypothesis is rejected. Then, testing the hypothesis H0: 2¼3 produced a value of 22¼79:60, which is also significant at the 95% confidence level. Thus, we used the first two PCs to evaluate the capability at 98.35% total variability. The principal components are USLPC1¼ 591.2569,
LSLPC1¼ 732.4531, XPC1¼ 662.8138, TPC1¼ 661.8550, SPC1¼4.3902,
USLPC2¼177.2339, LSLPC2¼132.5744, XPC2¼156.1566, TPC2¼154.9042 and
SPC2¼3.6748. Referring to equation (11), ^Spk1;PC and ^Spk2;PC can be calculated as 1.7331 and 1.7331, respectively. Applying equation (12), the approximate 95% lower confidence bound for the combined index, TSpk,PC, is 1.6844. Using the one-to-one
correspondence between TSpk,PC and the process yield (Yield ¼ 2ð3TSpk;PCÞ 1), the approximate 95% lower confidence bound for the true process yield is 0.999999. Notably, this process meets the process yield requirement.
5. Conclusions
Process yield is the most common and standard criteria for evaluating the quality of products manufactured. Process yield measure for processes with a single charac-teristic has been investigated extensively. However, process yield measure for processes with multiple quality characteristics is comparatively neglected. Assuring the process yield in processes with multiple characteristics to meet the requirement
Table 4. The results of the first two PCs for example 3. Variable PC1 loading PC2 loading
1 0.015148 0.000406 2 0.985115 0.1684891 3 0.006121 0.005973 4 0.035063 0.026208 5 0.167490 0.985336 Eigenvalue 19.27390 13.50434 % Explained of total variability 57.8326 40.5207
is important. So the proposition of a technique assuring the process yield is necessary in this field. In this paper, we proposed a generalized yield index, called TSpk,PC,
based on the yield index Spkproposed by Boyles (1991), by using the PCA method.
We also developed an approximate lower confidence bound (LCB) for the true process yield by using the Spkthrough the principal component analysis (PCA). This
methodology is easy to be understood and used. The proposed procedure can be used to determine whether their production meets the present yield requirement, and make a reliable decision.
Appendix Theorem 1: S^T pkis defined as ^ STpk¼1 3 1 Y v j¼1 2ð3 ^SpkjÞ 1 þ1 " # 2 ( ) ,
where ^Spkj denotes the estimator of Spkj, ^SpkjNðSpkj, ða2j þb2jÞ=36nðð3SpkjÞÞ2Þ, and all ^S0
pkjs are mutually independent, then ^STpkhas the asymptotic normal distribution with the mean ST
pk and variance 1 36nðð3ST pkÞÞ 2 Xv j¼1 ða2j þb2jÞ Qv i¼1 2ð3SpkiÞ 1 2 2ð3SpkjÞ 1 2 " # ( )! : That is, ^ STpkN STpk, 1 36nðð3ST pkÞÞ 2 Xv j¼1 ða2j þb2jÞ Qv i¼1 2ð3SpkiÞ 1 2 2ð3SpkjÞ 1 2 " # ( )!! :
Proof: Applying the first-order expansion of v-variate Taylor,
) f ðX Þ ¼ fðX0Þ þ Xv j¼1 @fðX0Þ @xi ðxixi0Þ, where X ¼ ðx1, x2, . . . , xvÞ.
We take ¼ 2 for example to derive the asymptotic distribution of ^ST pk. Here Eð ^SpkjÞ ¼Spkj, Varð ^SpkjÞ ¼ a2 j þg2j 36nðð3SpkjÞÞ2 , 8j ¼1, 2: From the definition, we have
^ STpk¼f ðSpk1, Spk2Þ þ @f ðSpk1,Spk2Þ @ ^Spk1 ð ^Spk1Spk1Þ þ @f ðSpk1,Spk2Þ @ ^Spk2 ð ^Spk2Spk2Þ:
Measuring production yield 4659
Then, we have Eð ^STpkÞ ¼Eð f ðSpk1,Spk2ÞÞ þE @f ðSpk1, Spk2Þ @ ^Spk1 ð ^Spk1Spk1Þ ! þE @f ðSpk1,Spk2Þ @ ^Spk2 ð ^Spk2Spk2Þ ! ¼f ðSpk1, Spk2Þ ¼STpk¼ 1 3 1 ð2ð3S pk1Þ 1Þð2ð3Spk2Þ 1Þ þ 1 =2 : Varð ^STpkÞ ¼ @f ðSpk1,Spk2Þ @ ^Spk1 !2 Varð ^Spk1Þ þ @f ðSpk1,Spk2Þ @ ^Spk2 !2 Varð ^Spk2Þ: ,f ð ^Spk1, ^Spk2Þ ¼ 1 3 1 ð2ð3 ^S pk1Þ 1Þð2ð3 ^Spk2Þ 1Þ þ 1 h i 2 n o : We have @f ð ^Spk1, ^Spk2Þ @ ^Spk1 ¼ ð2ð3 ^Spk2Þ 1Þð3 ^Spk1Þ 1 ð2ð3 ^S pk1Þ 1Þð2ð3 ^Spk2Þ 1Þ þ 1 h i 2 n o n o ;@f ðSpk1, Spk2Þ @ ^Spk1 ¼ ð2ð3Spk2Þ 1Þð3Spk1Þ 1 ð2ð3S pk1Þ 1Þð2ð3Spk2Þ 1Þ þ 1 2 : Similarly, we have @f ðSpk1, Spk2Þ @ ^Spk2 ¼ ð2ð3Spk1Þ 1Þð3Spk2Þ 1 ð2ð3S pk1Þ 1Þð2ð3Spk2Þ 1Þ þ 1 =2 : So, Varð ^STpkÞ ¼ 1 36nðð3ST pkÞÞ 2 ða 2 1þb 2 1Þð2ð3Spk2Þ 1Þ2þ ða22þb 2 2Þð2ð3Spk1Þ 1Þ2 :
By Central Limit Theorem ¼> ^ST
pkhas the asymptotic normal distribution with the mean ST pk and variance 1 36nðð3ST pkÞÞ 2 ða 2 1þb21Þð2ð3Spk2Þ 1Þ2þ ða22þb22Þð2ð3Spk1Þ 1Þ2 :
Similarly, consider v variables, the asymptotic distribution of ^ST
pk can be derived as ^ STpkN STpk, 1 36nðð3ST pkÞÞ 2 Xv j¼1 ða2j þb2jÞ Qv i¼1 2ð3SpkiÞ 1 2 2ð3SpkjÞ 1 2 " # ( )!! :
Theorem 2: Let be the covariance matrix associated with the random vector X ¼ x1 .. . xv 2 6 4 3 7 5:
Let have the eigenvalues 1 v & eigenvectors e1, . . . , ev. Then the ith principal component variable is given by yi¼e
0
iX ¼ ei1x1þ þeivxv, i ¼ 1, 2, . . . ,v. With these choices, varðyiÞ ¼e
0
iei¼i, i ¼ 1, 2, . . . , v and covðyi, ykÞ ¼e
0
iek¼0, i 6¼ k.
Proof: The proof can be found in Johnson and Wichern (2002) on page 428.
Acknowledgements
The authors wish to gratefully acknowledge the referees of this paper who helped to clarify and improve the presentation.
References
Anderson, T.W., Asymptotic theory for principal component analysis. Ann. Math. Stat., 1963, 34, 122–148.
Boyles, R.A., The Taguchi capability index. J. Qual. Technol., 1991, 23, 17–26.
Chan, L.K., Cheng, S.W. and Spiring, F.A., A multivariate measure of process capability. J. Model. Simul., 1991, 11, 1–6.
Chen, H., A multivariate process capability index over a rectangular solid tolerance zone. Stat. Sin., 1994, 4, 749–758.
Chen, K.S., Pearn, W.L. and Lin, P.C., Capability measures for processes with multiple characteristics. Qual. Reliab. Engng Int., 2003, 19, 101–110.
Jackson, J.E., Principal component and factor analysis: part I—principal components. J. Qual. Technol., 1980, 12, 201–213.
Johnson, R.A. and Wichern, D.E., edition. in Applied Multivariate Statistical Analysis, fifth edn, pp. 428, 2002 (Prentice Hall: New Jersey).
Kotz, S. and Johnson, N.L., Process capability indices—a review, 1992–2000. J. Qual. Technol., 2002, 34, 2–19.
Lee, J.C., Hung, H.N., Pearn, W.L. and Kueng, T.L., On the distribution of the estimated process yield index Spk. Qual. Reliab. Engng Int., 2002, 18, 111–116.
Pearn, W.L., Kotz, S. and Johnson, N.L., Distributional and inferential properties of process capability indices. J. Qual. Technol., 1992, 24, 216–231.
Shahriari, H., Hubele, N.F. and Lawrence, F.P, A multivariate process capability vector. in Proceedings of the 4th Industrial Engineering Research Conference. Institute of Industrial Engineers, Norcross, Georgia, USA, pp. 304–309/24–25 May, 1995. Sultan, T.I., An acceptance chart for raw materials of two correlated properties. Quality
Assurance, 1986, 12, 70–72.
Taam, W., Subbaiah, P. and Liddy, J.W., A note on multivariate capability indices. J. Appl. Stat., 1993, 20, 339–351.
Wang, F.K., Capability index using principal component analysis. Quality Engineering, 1998, 11, 21–27.
Wang, F.K. and Du, T.C.T., Using principal component analysis in process performance for multivariate data. Omega, Int. J. Manag. Sci., 2000, 28, 185–194.
Measuring production yield 4661
