Measuring production yield for processes with multiple characteristics

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Measuring production yield for

processes with multiple characteristics

a

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

Published online: 02 Jul 2009.

To cite this article: W.L. Pearn & Ya-Ching Cheng (2010) Measuring production yield for processes with multiple characteristics, International Journal of Production Research, 48:15, 4519-4536, DOI: 10.1080/00207540903036313

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Vol. 48, No. 15, 1 August 2010, 4519–4536

Measuring production yield for processes with multiple characteristics

W.L. Pearn and Ya-Ching Cheng*

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

(Received 18 September 2008; final version received 20 April 2009) Numerous capability indices have been proposed to measure the performance of processes with multiple characteristics. The index ST

pkprovides an exact measure

on the production yield of multinormal processes in which the characteristics are mutually independent. In this paper, we thoroughly investigate the relationship between process parameters and the sampling distribution of ST

pk. Our

investi-gation shows that for a fixed ST

pk, the variance of sample estimator of STpk

is restricted in an interval. For reliability consideration, the maximal variance is used in the estimation and testing of the production yield to ensure the level of confidence. Also, information about sample sizes required for specified precision of estimation and for convergence is determined. At last, we implement a trivariate process with data collected from a plastics manufacturing industrial to demonstrate the practicability of the proposed method in measuring the production yield.

Keywords: capability indices; manufacturing processes; production yield; reliability

1. Introduction

Numerous capability indices have been proposed to measure the production yield, which is an important concern for manufacturing factories. The production yield is defined as the percentage of processed product units passing the inspection. For a measured character-istic X, the production yield can be defined as follows, where F(x) is the cumulative distribution function of X, USL and LSL are the upper and lower specification limits of the product characteristic, respectively:

Yield ¼ ZUSL

LSL

dFðxÞ

¼PrðLSL  X  USLÞ:

For normal processes with a single characteristic, the production yield can be measured by some well-known process capability indices. For example, Boyles (1991) found that for an on-centre process, Yield ¼ 2ð3CpÞ 1, and for any process, 2ð3CpkÞ

1  Yield 5 ð3CpkÞ. Ruczinski (1996) obtained a lower bound on the production yield

as Yield  2ð3CpmÞ 1. Furthermore, Boyles (1994) proposed the yield index Spk,

*Corresponding author. Email: [email protected]

ISSN 0020–7543 print/ISSN 1366–588X online ß 2010 Taylor & Francis

DOI: 10.1080/00207540903036313 http://www.informaworld.com

which provides an exact measure on the production yield, Yield ¼ 2ð3SpkÞ 1.

These indices have been explicitly defined as follows: Cp¼ USL  LSL 6 , Cpk¼min USL   3 ,   LSL 3   , Cpm¼ USL  LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2_{þ ð  TÞ}2 q , Spk¼ 1 3 1 1 2 USL       þ1 2   LSL    ,

where  is the process mean,  is the process standard deviation, T is the target value, ðÞ is the cumulative distribution function of the standard normal distribution N(0, 1), and 1ðÞis the inverse of ðÞ.

The above indices have been designed for measuring the capability of processes with a single characteristic. However, commonly the manufactured product involves more than one characteristic. That is, manufactured items require values of several different characteristics for adequate description of their quality. The assessed quality of a product depends on the combined effects of those characteristics, rather than on their individual values. Normally each of those characteristics must satisfy certain specifications, and the corresponding production yield can be expressed as:

Yield ¼PrðLSL1X1USL1\    \LSLvXvUSLvÞ,

where X1, . . . , Xvrepresents v characteristics of the investigated product.

To measure the performance of processes with multiple characteristics, various capability indices are proposed for assessing process capability. For example, Pearn et al. (1992) proposed the indices vCpm, vCp which require the assumption of multivariate normality and thus the tolerance region is ellipsoidal. Taam et al. (1993) proposed an index MCpm defined as a ratio of a modified tolerance region and a scaled 99.73% process region. Chen (1994) proposed the MCp index given by a ratio of the width of the tolerance interval centred at the target value and the width of the interval that the probability of a process falling within this interval is 1  . The MCp index can be easily interpreted for the univariate normal cases. Shahriari et al. (1995) proposed a multivariate capability vector denoted as [CpM, PV, LI ] representing, respectively, a ratio of regions (similar to MCp and MCpm), locations of centres, and whether or not the modified process region is contained within the tolerance region. Wang and Du (2000) presented a comparison of three methods proposed by Taam et al. (1993), Chen (1994), and Shahriari et al. (1995). They applied the principal component analysis (PCA) to these process capability indices to handle normal and non-normal data. The capability indices proposed by Pearn et al. (1992), Taam et al. (1993) and Shahriari et al. (1995) require the assumption of multivariate normality, while those proposed by Chen (1994) and Wang and Du (2000) make no assumptions.

However, the issues between the production yield of processes with multiple characteristics and the capability indices have received little attention. In this paper, we focus on a more newly proposed index ST

pk, which can provide an exact measure on the

production yield for the multinormal processes in which the characteristics are mutually independent. The sampling distribution of ST

pk is investigated based on various

combinations of the production parameters. A statistical method for estimating the true production yield is presented by the lower confidence bound of ST

pk. Hypothesis testing on

the production yield is also conducted. For the engineers’ usefulness, critical values of the hypothesis testing for some commonly used capability requirements are tabulated. Finally, a real world example is presented to demonstrate the applicability of the proposed approach.

2. Production yield of multivariate processes 2.1 General consideration

Let X

 ¼ ðX1, . . . , XvÞ T

be the vector of the v interested variables of a specific product which is used for statistical process control. Assume that X

 has a multinormal distribution,

NvðX

j, Þ, where  is the mean vector and  represents the covariance matrix of X.

Under the assumptions mentioned, the production yield can be represented as: Yield ¼ ZUSLv LSLv    Z USL1 LSL1 NvðX j, Þ dX1  dXv,

where LSLi and USLi, i ¼ 1 , . . . , v, are the v lower and upper specification limits, respectively. If the v variables are mutually independent, the production yield can be alternatively represented as:

Yield ¼Y v i¼1  USLii i    LSLii i     ,

where i and i are the mean and standard deviation of the ith characteristic Xi,

respectively.

2.2 Yield index ST_pk

Chen et al. (2003) proposed the following capability index, referred to as ST pk: ST_pk¼1 3 1 1 2 Yv i¼1 ð2ð3SpkiÞ 1Þ þ 1 " # ( ) ,

where Spkidenotes the Spkvalue of the ith characteristic for i ¼ 1, 2, . . . , v, and v is the number of characteristics. The index ST_pk may be viewed as a generalisation of the yield index Spk proposed by Boyles (1994) for processes with a single characteristic. For multinormal processes in which the characteristics are mutually independent, there is a one-to-one correspondence between the index ST_pk and the overall production yield, Yield ¼2ð3ST

pkÞ 1. For example, if STpk¼1:50, then the overall production yield is

roughly 99.9993%, or equally, the fraction of defectives is roughly 7 parts per million (ppm). Table 1 displays various commonly used capability requirements and the corresponding production yields as well as non-conformities in ppm.

2.3 Sampling distribution of ST pk

The natural estimator of ST

pk is defined as: ^ ST_pk¼1 3 1 1 2 Yv i¼1 ð2ð3 ^SpkiÞ 1Þ þ 1 " # ( ) ,

where ^Spki denotes the estimator of Spki. For univariate normal processes, the index Spk

provides an exact measure on the production yield. Lee et al. (2002) derived the sampling distribution of the univariate capability index Spk, which is an asymptotic normal distribution with mean Spkand variance ða2þb2Þ=36n2ð3SpkÞ, where  is the probability

density function (PDF) of the standard normal distribution N(0, 1). a ¼ 1ffiffiffi 2 p 1  Cdr Cdp  1  Cdr Cdp   þ1 þ Cdr Cdp  1 þ Cdr Cdp     , b ¼  1  Cdr Cdp    1 þ Cdr Cdp   , Cdr ¼  m d , Cdp ¼  d, m ¼ ðUSL þ LSLÞ=2, d ¼ ðUSL  LSLÞ=2: Table 1. Various ST

pk values and the corresponding production yields as

well as non-conformities in parts per million (ppm). ST pk Yield ppm 1.00 0.997300204 2699.796 1.05 0.998367295 1632.705 1.10 0.999033152 966.848 1.15 0.999439413 560.587 1.20 0.999681783 318.217 1.25 0.999823165 176.835 1.30 0.999903807 96.193 1.35 0.999948782 51.218 1.40 0.999973309 26.691 1.45 0.999986386 13.614 1.50 0.999993205 6.795 1.55 0.999996681 3.319 1.60 0.999998413 1.587 1.65 0.999999258 0.742 1.70 0.999999660 0.340 1.75 0.999999848 0.152 1.80 0.999999933 0.067 1.85 0.999999971 0.029 1.90 0.999999988 0.012 1.95 0.999999995 0.005 2.00 0.999999998 0.002

Inheriting the single variate result of Lee et al. (2002) and applying the Taylor expansion of v-variate, Pearn et al. (2006) showed that the sampling distribution of ST

pk is an asymptotic normal distribution with mean STpkand variance:

1 36n2_ð3ST pkÞ Xv j¼1 a2_j þb2_j Qv i¼1ð2ð3SpkiÞ 1Þ2 ð2ð3SpkjÞ 1Þ2 " # ( ) ,

where aj, bj, Spkjare the corresponding parameters of the jth characteristic. That is, ^ ST_pk N ST_pk, 1 36n2_ð3ST pkÞ Xv j¼1 a2_j þb2_j Qv i¼1ð2ð3SpkiÞ 1Þ2 ð2ð3SpkjÞ 1Þ2 " # ( )! :

It should be noted that the production yield index ST_pkprovides an exact yield measure based on the assumption of multinormal processes with independent characteristics. With dependent characteristics, our proposed procedure will use the principal component analysis (PCA) to project the original set of dependent variables Y

 into a set of

independent ones X

, which is used to estimate the index value of S T

pk. Furthermore, if the

original characteristics is not multinormal but any particular distribution, we will assume that Y

 is some transformation Y ¼hðWÞ, where W is the original non-normal variables, so

that Y

 is normally distributed. In practice, some Box-and-Cox type transformations can

help improve the normality of the data. In the rest of the article, to avoid further notation we will assume that the original observations follow a multinormal distribution with v independent characteristics, that is X

 NvðXj, Þ.

3. Estimation of the production yield 3.1 Lower confidence bound

In practice, to estimate the true production yield in a conservative way, the engineers will take the lower confidence bound as the true yield estimation. Since the sampling distribution of ST

pk is presented in 3v þ 1 parameters, Spkj, aj, bj, j ¼ 1, . . . , v and STpk, the

lower confidence bound is also a function of those parameters. Thus, we have to consider the effect of all the parameters in the calculation of lower confidence bound to ensure that the lower bounds obtained are reliable. The word ‘reliable’ here means that the probability that the obtained lower bound (subject to the sample estimate) is smaller than the actual capability index ST

pk is greater than the desired confidence level. In the following, we

present how to obtain the reliable lower bound in the two- and three-dimensional processes. We also execute cases in higher (greater than three) dimensions. To save the capacity of the article, the high dimensional cases are dropped. Fortunately, in the high dimensional cases, the effects of all the parameters in the calculation of lower confidence bound come out in a similar pattern as in the two- or three-dimensional cases.

3.1.1 Two-dimensional processes

For processes with two interested characteristics, i.e., v ¼ 2, we perform extensive calculations to obtain the variance of ^ST

pk, Varð ^STpkÞ, which directly affects the value of the

lower confidence bound. We note that for an identical ST

pk, there are numerous

combinations of Spk1and Spk2, and when one of them becomes larger, the other one will converge to ST

pk. Figure 1 shows the relationship between Spk1and Spk2under various STpk.

Furthermore, for a fixed Spkj, there are also numerous combinations of aj and bj. Table 2 displays a few combinations of the parameters Spk1, Spk2, a1, a2, b1, and b2under ST

pk¼1, and the corresponding variance of ^STpk.

Our extensive calculation results show that: (i) the variance of ^ST

pkobtains its minimum

at Spk1¼Spk2, and maximum at Spk12.5 and Spk2¼ST

pk (or Spk1¼STpk and Spk22.5);

and (ii) for fixed Spk1and Spk2, the variance of ^ST

pkreaches its maximum at b1¼b2¼0, i.e.,

the mean vector is on-centre,  ¼ m. Figure 2 shows the nVar( ^ST

pk) of two-dimensional

processes for ST

pk¼1.00, 1.33, 1.50, 1.67, 2.00 with combinations of Spk1and Spk2 as in

Figure 1. It is clear that the nVar( ^ST

pk) reaches its maximum when Spk12.5 and Spk2¼STpk.

Thus, in the calculations of lower confidence bound of ST

pk for the two-dimensional

processes, we will set Spk1¼2.5, Spk2¼ST

pk (or Spk1¼STpk and Spk2¼2.5), a1 ¼ ffiffiffi 2 p ð3Spk1Þð3Spk1Þ, a2¼ ffiffiffi 2 p

ð3Spk2Þð3Spk2Þ, and b1¼b2¼0 to make a reliable yield

estimation.

3.1.2 Three- and high-dimensional processes

For three-dimensional processes, we perform similar calculations as in the two-dimensional cases. Figure 3 shows the combinations of Spk1, Spk2 and Spk3 of three-dimensional processes for ST

pk¼1.00. Note that the plots in Figure 3 are identical, just with

opposite angles. The top point in Figure 3 represents the combination of (Spk1, Spk2, Spk3) ¼ (1.1066, 1.1066, 1.1066) for ST

pk¼1.00, which can be shown with the smallest

corresponding variance (as in Figure 5). Furthermore, we can see that Spk3 converges to ST

pk as Spk1and Spk2become larger. In such a condition, the variance of ^STpk can be

shown to be the largest (see Figure 5). Similar to Figure 3, Figure 4 presents the combinations of ^ST

pk for three-dimensional processes for STpk¼1.00, 1.33, 1.50, 1.67 and

2.00. It can be seen that the patterns of combinations of Spk1, Spk2and Spk3for a fixed ST pk

are the same in three-dimensional cases. There is a peak point with Spk1¼Spk2¼Spk3, and a convergence with Spk3¼ST

pk and Spk1¼Spk22.5.

Figure 1. Contour of combinations of Spk1 and Spk2 for ST_pk¼1.00, 1.33, 1.50, 1.67, 2.00 (from

bottom to top).

Table 2. Variance of ^ST

pkversus combinations of the parameters under STpk¼1.

Spk1 Spk2 a1 a2 b1 b2 nVar( ^ST_pk) 1.06832 1.06832 0.01064 0.01064 0.00000 0.00000 0.319084 1.06832 1.06832 0.01063 0.01063 0.00015 0.00015 0.319084 1.06832 1.06832 0.01063 0.01063 0.00031 0.00031 0.319084 1.06832 1.06832 0.01063 0.01063 0.00047 0.00047 0.319084 1.06832 1.06832 0.01062 0.01062 0.00064 0.00064 0.319082 1.06832 1.06832 0.01060 0.01060 0.00082 0.00082 0.319077 1.06832 1.06832 0.01059 0.01059 0.00100 0.00100 0.319069 1.06832 1.06832 0.01057 0.01057 0.00118 0.00118 0.319054 1.06832 1.06832 0.01055 0.01055 0.00137 0.00137 0.319028 1.06832 1.06832 0.01052 0.01052 0.00157 0.00157 0.318987 1.66832 1.00002 1.02652E-05 0.01880 0.00000 0.00000 0.499832 1.66832 1.00002 1.02641E-05 0.01880 1.48866E-07 0.00027 0.499832 1.66832 1.00002 1.02607E-05 0.01879 3.05375E-07 0.00055 0.499832 1.66832 1.00002 1.02544E-05 0.01878 4.69921E-07 0.00083 0.499831 1.66832 1.00002 1.02449E-05 0.01877 6.42929E-07 0.00113 0.499829 1.66832 1.00002 1.02316E-05 0.01874 8.24847E-07 0.00143 0.499823 1.66832 1.00002 1.02138E-05 0.01872 1.01616E-06 0.00175 0.499813 1.66832 1.00002 1.01907E-05 0.01868 1.21739E-06 0.00207 0.499794 1.66832 1.00002 1.01612E-05 0.01864 1.42911E-06 0.00240 0.499763 1.66832 1.00002 1.01239E-05 0.01860 1.65195E-06 0.00275 0.499712 2.26832 1.00000 3.37837E-10 0.01880 0.00000 0.00000 0.500000 2.26832 1.00000 3.37801E-10 0.01880 4.9441E-12 0.00027 0.500000 2.26832 1.00000 3.37682E-10 0.01879 1.02364E-11 0.00055 0.500000 2.26832 1.00000 3.37459E-10 0.01878 1.59017E-11 0.00083 0.499999 2.26832 1.00000 3.37109E-10 0.01877 2.19671E-11 0.00113 0.499997 2.26832 1.00000 3.36602E-10 0.01875 2.84607E-11 0.00143 0.499991 2.26832 1.00000 3.35893E-10 0.01872 3.54154E-11 0.00175 0.499981 2.26832 1.00000 3.34914E-10 0.01869 4.28673E-11 0.00207 0.499962 2.26832 1.00000 3.33572E-10 0.01865 5.08564E-11 0.00240 0.499931 2.26832 1.00000 3.31662E-10 0.01860 5.94371E-11 0.00275 0.499880

Figure 2. Variance plots of two-dimensional processes for ST

pk¼1.00, 1.33, 1.50, 1.67, 2.00 (from

bottom to top).

Table 3 displays a few combinations of the parameters Spk1, Spk2, Spk3, a1, a2, a3, b1, b2, and b3under ST

pk¼1 as well as the corresponding variance of ^STpk. Similar to the

two-dimensional cases, the calculation results show that: (i) the variance of ^ST

pk obtains its

minimum at Spk1¼Spk2¼Spk3, and maximum at Spki¼ST

pkand Spkj2.5, where j 6¼ i; and

Figure 4. Surfaces of combinations of Spk1, Spk2and Spk3for STpk¼1.00, 1.33, 1.50, 1.67, 2.00 (from

bottom to top).

Figure 5. Variance plots of three-dimensional processes for ST

pk¼1.0 and 2.0 (bottom to top).

Figure 3. Surface of combinations of Spk1, Spk2and Spk3for ST_pk¼1.00.

(ii) for fixed Spk1, Spk2, and Spk3, the variance of S^T_pk reaches its maximum at b1¼b2¼b3¼0, that is, the mean vector is on-centre.

Figure 5 shows the corresponding variance of ^ST_pk with parameters Spk1, Spk2, Spk3 identical to those as in Figure 4 for ST

pk¼1.00 and 2.00. Again, the plots in Figure 5 are the

same plot with opposite angles. It can be seen that the larger the Spk3, the smaller the nVar( ^ST

pk), and the larger the STpk, the greater the variation of the nVar( ^STpk). We

should remark that to make a reliable decision, the maximal nVar( ^ST

pk) for a fixed STpkis the

only concern, and the nVar( ^ST

pk) reaches its maximum at Spk12.5, Spk22.5 and

Spk3¼ST pk.

For processes with high (greater than three) dimensions, the relationship between variance of ^ST

pk and the 3v þ 1 parameters is in a similar pattern as in the two- or

three-dimensional processes. Figure 6 shows the plots for four-dimensional cases. Figures 6(a)–6(b) present the combinations of Spk2, Spk3 and Spk4 for Spk1¼2.5 and ST

pk¼1.0, 1.5, 2.0 from bottom to top. Figures 6(c)–6(d) present the nVar( ^STpk) versus Spk2,

Spk3with parameter combinations as in Figures 6(a)–6(b). Again, Figures 6(a) and 6(b) are the same plots with opposite angle, and so are Figures 6(c) and 6(d).

Table 3. Combinations of the parameters and the corresponding nVar( ^ST

pk) for STpk¼1. Spk1 Spk2 Spk3 a1 a2 a3 b1 b2 b3 nVar( ^ST_pk) 1.10661 1.10661 1.10661 0.00757 0.00757 0.00757 0.00000 0.00000 0.00000 0.242496 1.10661 1.10661 1.10661 0.00757 0.00757 0.00757 0.00011 0.00011 0.00011 0.242496 1.10661 1.10661 1.10661 0.00757 0.00757 0.00757 0.00022 0.00022 0.00022 0.242496 1.10661 1.10661 1.10661 0.00757 0.00757 0.00757 0.00034 0.00034 0.00034 0.242495 1.10661 1.10661 1.10661 0.00756 0.00756 0.00756 0.00046 0.00046 0.00046 0.242494 1.10661 1.10661 1.10661 0.00755 0.00755 0.00755 0.00058 0.00058 0.00058 0.242490 1.10661 1.10661 1.10661 0.00754 0.00754 0.00754 0.00071 0.00071 0.00071 0.242483 1.10661 1.10661 1.10661 0.00753 0.00753 0.00753 0.00084 0.00084 0.00084 0.242470 1.10661 1.10661 1.10661 0.00751 0.00751 0.00751 0.00098 0.00098 0.00098 0.242448 1.10661 1.10661 1.10661 0.00749 0.00749 0.00749 0.00112 0.00112 0.00112 0.242413 2.10661 1.10661 1.04043 0.00000 0.00757 0.01350 0.00000 0.00000 0.00000 0.337989 2.10661 1.10661 1.04043 0.00000 0.00757 0.01350 0.00000 0.00011 0.00019 0.337989 2.10661 1.10661 1.04043 0.00000 0.00757 0.01349 0.00000 0.00022 0.00039 0.337988 2.10661 1.10661 1.04043 0.00000 0.00757 0.01348 0.00000 0.00034 0.00060 0.337988 2.10661 1.10661 1.04043 0.00000 0.00756 0.01347 0.00000 0.00046 0.00081 0.337986 2.10661 1.10661 1.04043 0.00000 0.00755 0.01346 0.00000 0.00058 0.00103 0.337982 2.10661 1.10661 1.04043 0.00000 0.00754 0.01344 0.00000 0.00071 0.00126 0.337973 2.10661 1.10661 1.04043 0.00000 0.00753 0.01341 0.00000 0.00084 0.00149 0.337958 2.10661 1.10661 1.04043 0.00000 0.00751 0.01338 0.00000 0.00098 0.00174 0.337932 2.10661 1.10661 1.04043 0.00000 0.00749 0.01335 0.00000 0.00112 0.00198 0.337890 2.10661 2.10661 1.00000 0.00000 0.00000 0.01880 0.00000 0.00000 0.00000 0.500000 2.10661 2.10661 1.00000 0.00000 0.00000 0.01880 0.00000 0.00000 0.00027 0.500000 2.10661 2.10661 1.00000 0.00000 0.00000 0.01879 0.00000 0.00000 0.00055 0.500000 2.10661 2.10661 1.00000 0.00000 0.00000 0.01878 0.00000 0.00000 0.00083 0.499999 2.10661 2.10661 1.00000 0.00000 0.00000 0.01877 0.00000 0.00000 0.00113 0.499997 2.10661 2.10661 1.00000 0.00000 0.00000 0.01875 0.00000 0.00000 0.00143 0.499991 2.10661 2.10661 1.00000 0.00000 0.00000 0.01872 0.00000 0.00000 0.00175 0.499981 2.10661 2.10661 1.00000 0.00000 0.00000 0.01869 0.00000 0.00000 0.00207 0.499962 2.10661 2.10661 1.00000 0.00000 0.00000 0.01865 0.00000 0.00000 0.00240 0.499930 2.10661 2.10661 1.00000 0.00000 0.00000 0.01860 0.00000 0.00000 0.00275 0.499880

In brief: (i) for an identical ST

pk, variance of ^STpkis maximal at Spki¼STpkand Spkj=2.5,

where j 6¼ i, (also variance of ^ST_pkis minimal while all v Spkjare equal); and (ii) for fixed Spkj, where j ¼ 1, . . . , v, the variance of ^ST

pkreaches its maximum at bj¼0, i.e., the mean vector is

on-centre. Hence, in the calculation of lower confidence bound of ST

pk, we will set Spki¼STpk

and Spkj¼2.5, for all j 6¼ i, aj¼

ffiffiffi 2 p

ð3SpkjÞð3SpkjÞ, and bj¼0 for all j ¼ 1, . . . , v. In this way,

the level of confidence can be ensured, and the decisions (lower confidence bounds) made based on such an approach are indeed more reliable.

We note that with the above parameter setting, the sampling distribution of ST pkcan be

rewritten in a shorter and simpler form, that is:

^ ST_pk N ST_pk, ST_pk 2 2n 0 B @ 1 C A: Thus, given an estimate ^ST

pk, a sample size n and a confidence level 1  , the lower

confidence bound of ST pk(denoted as S TðLÞ pk ) can be obtained: STðLÞ_pk ¼ ^ ST pk 1 þ Z= ffiffiffiffiffi 2n p

Figure 6. Plots of four-dimensional processes with Spk1¼2.5 and STpk¼1.0, 1.5, 2.0 (from bottom

to top): (a)–(b) surfaces of combinations of Spk2, Spk3and Spk4; (c) –(d) nVar( ^STpk) versus Spk2and

Spk3with parameter combinations as in (a) and (b).

where Zis the upper 100 percentile of the standard normal distribution. Table 4 lists the

lower confidence bounds of ST

pk for the estimates ^STpk¼1.0(0.1)2.0, n ¼ 5(5)100, and the

confidence level 1   ¼ 95%.

We remark that the lower confidence bound obtained according to our methodology is identical for processes of various dimensions, since the reliable lower confidence bound (based on the maximum Varð ^ST

pkÞfor a fixed STpk) is a function of ^STpk and n only.

3.2 Sample size determination

The sample size determination is important, as it directly relates to the cost of the data collection plan. Applying the parameter setting as in computing the lower confidence bound, the sample size required for a given estimation precision R can be expressed as:

n 1 2 Z 1 R1 !2 , where R ¼ STðLÞ_pk = ^ST

pk. Note that no matter what value of ^STpkis, the ratio of the lower bound

STðLÞ_pk to the estimate (value of ^ST

pk) is identical, because of the parameters setting in our

methodology. Thus, given a desired estimation precision R, a significance level  and the parameters setting as previously, the required sample size n can be obtained. Table 5 shows the sample size n required for desired estimation precisions R ¼ 0.75(0.01)0.95 and significant levels  ¼ 0.1, 0.05, 0.025, 0.01. For example, if the desired estimation precision

Table 4. Lower confidence bound of ST

pkfor various ^STpk, n ¼ 5(5)100, and  ¼ 0.05.

^ ST pk n 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 5 0.6578 0.7236 0.7893 0.8551 0.9209 0.9867 1.0525 1.1183 1.1840 1.2498 1.3156 10 0.7311 0.8042 0.8773 0.9504 1.0235 1.0966 1.1697 1.2428 1.3159 1.3890 1.4622 15 0.7690 0.8459 0.9228 0.9997 1.0766 1.1535 1.2304 1.3073 1.3842 1.4611 1.5380 20 0.7936 0.8729 0.9523 1.0316 1.1110 1.1904 1.2697 1.3491 1.4284 1.5078 1.5872 25 0.8112 0.8924 0.9735 1.0546 1.1357 1.2169 1.2980 1.3791 1.4603 1.5414 1.6225 30 0.8248 0.9073 0.9898 1.0722 1.1547 1.2372 1.3197 1.4022 1.4847 1.5672 1.6496 35 0.8357 0.9192 1.0028 1.0864 1.1699 1.2535 1.3371 1.4206 1.5042 1.5878 1.6714 40 0.8446 0.9291 1.0135 1.0980 1.1825 1.2669 1.3514 1.4359 1.5203 1.6048 1.6893 45 0.8522 0.9374 1.0226 1.1079 1.1931 1.2783 1.3635 1.4488 1.5340 1.6192 1.7044 50 0.8587 0.9446 1.0304 1.1163 1.2022 1.2881 1.3739 1.4598 1.5457 1.6316 1.7174 55 0.8644 0.9508 1.0373 1.1237 1.2102 1.2966 1.3830 1.4695 1.5559 1.6424 1.7288 60 0.8694 0.9563 1.0433 1.1302 1.2172 1.3041 1.3911 1.4780 1.5650 1.6519 1.7388 65 0.8739 0.9613 1.0487 1.1361 1.2234 1.3108 1.3982 1.4856 1.5730 1.6604 1.7478 70 0.8779 0.9657 1.0535 1.1413 1.2291 1.3169 1.4047 1.4925 1.5803 1.6681 1.7559 75 0.8815 0.9697 1.0579 1.1460 1.2342 1.3223 1.4105 1.4987 1.5868 1.6750 1.7631 80 0.8849 0.9734 1.0619 1.1504 1.2388 1.3273 1.4158 1.5043 1.5928 1.6813 1.7698 85 0.8879 0.9767 1.0655 1.1543 1.2431 1.3319 1.4207 1.5095 1.5983 1.6871 1.7759 90 0.8907 0.9798 1.0689 1.1580 1.2471 1.3361 1.4252 1.5143 1.6034 1.6924 1.7815 95 0.8933 0.9827 1.0720 1.1614 1.2507 1.3400 1.4294 1.5187 1.6081 1.6974 1.7867 100 0.8958 0.9853 1.0749 1.1645 1.2541 1.3437 1.4332 1.5228 1.6124 1.7020 1.7916

R is set to 0.80, then with  ¼ 0.05 the sample size required is 22. Thus, if a sample estimate ^

ST_pk¼1.5 is obtained, then we can conclude that the actual value of ST_pkwould be no less than 1.5  0.80 ¼ 1.2, or equally, be 95% confident that the production yield would be greater 0.999681783.

We further consider how large a sample size, n, should be collected to ensure that the sample estimator ^ST_pkis close to the actual capability performance ST

pkwithin a designated

accuracy ". The word ‘close’ here means that the occurring probability is greater than a desired level 1  , say 0.95. That is:

Pr S^T pkSTpk     " n o 1   )Pr ^ ST pkSTpk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var ^ST pk r 8 > > < > > :  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" Var ^ST pk r 9 > > = > > ; 1  2 ) _{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi}" Var ^ST pk r 1 1  2 )Var ^ ST_pk " 2 1 _{1 } 2   2 )n ðS T pkÞ 2 2  1 _{1 } 2   2 "2 :

Table 5. Sample size required for specific estimation precision.

R  ¼0.10  ¼0.05  ¼0.025  ¼0.01 0.75 8 13 18 25 0.76 9 14 20 28 0.77 10 16 22 31 0.78 11 18 25 35 0.79 12 20 28 39 0.80 14 22 31 44 0.81 15 25 35 50 0.82 18 29 40 57 0.83 20 33 46 65 0.84 23 38 53 75 0.85 27 44 62 87 0.86 31 52 73 103 0.87 37 61 87 122 0.88 45 73 104 146 0.89 54 89 126 178 0.90 67 110 156 220 0.91 84 139 197 277 0.92 109 179 255 358 0.93 145 239 340 478 0.94 202 333 472 665 0.95 297 489 694 977

Table 6 displays the required sample sizes n for ^ST_pkto converge to the actual ST pkwithin

a designated accuracy ", with " ¼ 0.01(0.01)0.10. For example, for ST_pk¼1.0 and  ¼ 0.05, a sample size of n  193 ensures that the difference between the sample estimator ^ST_pkand the actual performance ST_pk is smaller than 0.1. This convergence investigated is not for practical purpose, but to illustrate the behaviour and the rate of convergence for the normal approximation.

4. Testing on the production yield

From a customer’s view point, to determine whether or not a production process is capable is the main work. To deal with this, we consider the following hypothesis testing:

H0: Yield  C (process is incapable),

H1: Yield 4 C (process is capable), where C is a designated constant. Since the production yield for a process with multiple characteristics has a one-to-one relationship with the yield index ST

pk, Yield ¼ 2ð3STpkÞ 1, testing the above hypotheses is

equal to testing: H0: ST pkS, H1: ST pk4 S, where S ¼13 1_½1 2ðC þ1Þ.

Based on the sampling distribution of ST

pkand a desired confidence level 1  , the decision

rule for this hypothesis testing should be to reject H0if the testing statistic ^ST

pk4 c0, where

c0is the critical value that satisfies:

Pr ^ST_pkc0jH0: STpkS

n o

:

Table 6. Sample sizes required to converge.

Designated accuracy, " ST pk  0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 1.00 0.01 33175 8294 3687 2074 1327 922 678 519 410 332 0.025 25120 6280 2792 1570 1005 698 513 393 311 252 0.05 19208 4802 2135 1201 769 534 392 301 238 193 1.33 0.01 58683 14671 6521 3668 2348 1631 1198 917 725 587 0.025 44434 11109 4938 2778 1778 1235 907 695 549 445 0.05 33976 8494 3776 2124 1360 944 694 531 420 340 1.50 0.01 74643 18661 8294 4666 2986 2074 1524 1167 922 747 0.025 56519 14130 6280 3533 2261 1570 1154 884 698 566 0.05 43217 10805 4802 2702 1729 1201 882 676 534 433 1.67 0.01 92521 23131 10281 5783 3701 2571 1889 1446 1143 926 0.025 70056 17514 7784 4379 2803 1946 1430 1095 865 701 0.05 53568 13392 5952 3348 2143 1488 1094 837 662 536 2.00 0.01 132698 33175 14745 8294 5308 3687 2709 2074 1639 1327 0.025 100478 25120 11165 6280 4020 2792 2051 1570 1241 1005 0.05 76830 19208 8537 4802 3074 2135 1568 1201 949 769

Note that the smaller the value of ST

pk, the larger the probability of ^STpk4 c0. Thus,

the above probability should be calculated with H0: ST

pk¼S. Also, since S^Tpk is

asymptotically normal distributed with mean ST

pk and for a given fixed STpk the largest

Var( ^ST pk), ðSTpkÞ

2_{=2n, is identical, the critical value c0is also a function of S}T

pk only, which

can be expressed as:

c0¼STpkþZ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varð ^ST pkÞ q ¼S þZ S ffiffiffiffiffi 2n p ,

where Z is the upper 100 percentile of the standard normal distribution. Table 7 shows

the critical values for testing the production yield for a process with multiple characteristics by the yield index ST

pk, covering the most commonly used performance

requirements ST

pk¼1.00, 1.33, 1.50, 1.67 and 2.00.

5. A real world application

To illustrate how to apply the proposed method to estimate and test production yield of a multivariate process, we implement a real-world application presented in the article of Wang and Chen (1998). The data collected from a plastics manufacturing industrial has three interesting characteristics: depth (D), length (L), and width (W). Specification limits and target value (LSL, T, USL) for D, L, W are set to (2.1, 2.2, 2.3), (304.5, 304.8, 305.1), and (304.5, 304.8, 305.1), respectively. The sample mean vector and sample covariance matrix calculated from 50 collected observations were:

^  ¼ 2:16 304:72 304:77 0 B @ 1 C A and ^ ¼ 0:0021 0:0008 0:0007 0:0008 0:0071 0:0012 0:0007 0:0012 0:0020 0 B @ 1 C A:

By performing the principal component analysis (PCA), the eigenvectors u1, u2, u3and eigenvalues 1, 2, 3of the sample covariance matrix can be obtained as in Table 8. In the

article of Wang and Chen (1998), the principal components (PCs) were calculated to estimate the multivariate capability indices MCp, MCpk, MCpm, and MCpmk. However, none of the indices can provide an adequate connection to the production yield. Next, we would employ the proposed methodology by the yield index ST

pk to the estimation and

testing on the production yield.

Let U be the matrix of (u1, u2, u3), Y

¼ ðD, L, WÞ T_{, and X} ¼U T_Y ¼ ðX1, X2, X3Þ T_.

Then, according to PCA, Xiis the ith principal component with variance i, and all the Xi’s

are independent. Based on such a transformation, the specification limits and target value (LSL, T, USL) for X1, X2, X3 become (368.14092, 368.55476, 368.9686), (216.82815, 216.6969, 216.56565), and (55.00047, 55.0394, 55.07833), respectively. Consequently, the sample mean vector of X

 becomes:  X ¼ 0:5222 0:5824 0:6230 0:8385 0:2172 0:4998 0:1558 0:7834 0:6017 0 B @ 1 C A 2:16 304:72 304:77 0 B @ 1 C A ¼ 368:46859 216:69807 55:001011 0 B @ 1 C A:

Table 7. Critical values for testing production yield with  ¼ 0.05, 0.025, 0.01. S T pk 1.00 1.33 1.50 1.67 2.00 n 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01 5 1.5202 1.6198 1.7357 2.0218 2.1543 2.3084 2.2802 2.4297 2.6035 2.5387 2.7051 3.0403 3.0403 3.2396 3.4713 10 1.3678 1.4383 1.5202 1.8192 1.9129 2.0219 2.0517 2.1574 2.2803 2.2842 2.4019 2.7356 2.7356 2.8765 3.0404 15 1.3003 1.3578 1.4247 1.7294 1.8059 1.8949 1.9505 2.0368 2.1371 2.1715 2.2676 2.6006 2.6006 2.7157 2.8495 20 1.2601 1.3099 1.3678 1.6759 1.7422 1.8192 1.8901 1.9649 2.0517 2.1043 2.1875 2.5202 2.5202 2.6198 2.7357 25 1.2326 1.2772 1.3290 1.6394 1.6987 1.7676 1.8489 1.9158 1.9935 2.0585 2.1329 2.4652 2.4652 2.5544 2.6580 30 1.2124 1.2530 1.3003 1.6124 1.6665 1.7294 1.8185 1.8796 1.9505 2.0246 2.0926 2.4247 2.4247 2.5061 2.6007 35 1.1966 1.2343 1.2781 1.5915 1.6416 1.6998 1.7949 1.8514 1.9171 1.9983 2.0612 2.3932 2.3932 2.4685 2.5561 40 1.1839 1.2191 1.2601 1.5746 1.6214 1.6759 1.7759 1.8287 1.8901 1.9771 2.0360 2.3678 2.3678 2.4383 2.5202 45 1.1734 1.2066 1.2452 1.5606 1.6048 1.6561 1.7601 1.8099 1.8678 1.9596 2.0150 2.3468 2.3468 2.4132 2.4904 50 1.1645 1.1960 1.2326 1.5488 1.5907 1.6394 1.7467 1.7940 1.8490 1.9447 1.9973 2.3290 2.3290 2.3920 2.4653 55 1.1568 1.1869 1.2218 1.5386 1.5785 1.6250 1.7353 1.7803 1.8327 1.9319 1.9821 2.3137 2.3137 2.3738 2.4436 60 1.1502 1.1789 1.2124 1.5297 1.5680 1.6125 1.7252 1.7684 1.8186 1.9208 1.9688 2.3003 2.3003 2.3578 2.4247 65 1.1443 1.1719 1.2040 1.5219 1.5586 1.6014 1.7164 1.7579 1.8061 1.9109 1.9571 2.2885 2.2885 2.3438 2.4081 70 1.1390 1.1657 1.1966 1.5149 1.5503 1.5915 1.7085 1.7485 1.7949 1.9022 1.9466 2.2780 2.2780 2.3313 2.3932 75 1.1343 1.1600 1.1900 1.5086 1.5428 1.5826 1.7015 1.7401 1.7849 1.8943 1.9373 2.2686 2.2686 2.3201 2.3799 80 1.1300 1.1550 1.1839 1.5030 1.5361 1.5746 1.6951 1.7324 1.7759 1.8872 1.9288 2.2601 2.2601 2.3099 2.3678 85 1.1262 1.1503 1.1784 1.4978 1.5299 1.5673 1.6892 1.7255 1.7676 1.8807 1.9210 2.2523 2.2523 2.3006 2.3569 90 1.1226 1.1461 1.1734 1.4931 1.5243 1.5606 1.6839 1.7191 1.7601 1.8747 1.9140 2.2452 2.2452 2.2922 2.3468 95 1.1193 1.1422 1.1688 1.4887 1.5191 1.5545 1.6790 1.7133 1.7532 1.8693 1.9075 2.2387 2.2387 2.2844 2.3375 100 1.1163 1.1386 1.1645 1.4847 1.5143 1.5488 1.6745 1.7079 1.7468 1.8642 1.9015 2.2326 2.2326 2.2772 2.3290

The correlation between the ith characteristic and the jth principal component is given by: ij¼uij ffiffiffiffi j sii r ,

where uijdenotes the coefficient for the ith characteristic in the jth principal component, j

denotes the eigenvalue associated with the principal component, and siiis the variance of the ith characteristic. The correlations between characteristics and principal components are tabulated in Table 9. As we can see, at least one of the absolute correlation coefficients between the first two principal components and the original characteristics is greater than 0.7, which is generally identified as highly correlated. The absolute value of correlations between the third principal component and the three characteristics are all less than 0.33, which indicates that the third principal component does not correlate well with the three original characteristics. Thus, the first two principal components are used to evaluate the capability performance of the three-dimensional process.

Applying the formulae of ^Spki and ^ST_pk:

^ Spki¼ 1 3 1 1 2 USL  Xi si    þ1 2  XiLSL si   ^ ST_pk¼1 3 1 1 2 Yv i¼1 2ð3 ^SpkiÞ 1 þ1 " # ( ) , we can calculate that:

^ Spk1¼ 1 3 1 1 2 368:9686  368:46859 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:0037 p    þ1 2 368:46859  368:14092 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:0037 p   ¼1:8367

Table 8. Eigenvectors and eigenvalues of the sample covariance.

Characteristics Eigenvector u1 u2 u3 D 0.5222 0.8385 0.1558 L 0.5824 0.2172 0.7834 W 0.6230 0.4998 0.6017 Eigenvalue, i 0.0037 0.0015 0.0006 % explained 63.93 25.11 10.96

Table 9. Correlation coefficients between the characteristics and principal components.

Characteristics Principal components X1 X2 X3 D 0.6932 0.7087 0.0833 L 0.4204 0.0998 0.2277 W 0.8474 0.4328 0.3296

^ Spk2¼ 1 3 1 1 2 216:56565 þ 216:69807 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:0015 p    þ1 2 216:69807 þ 216:82815 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:0015 p   ¼1:1291 ^ ST_pk¼1 3 1 1 2 Yv i¼1 2ð3 ^SpkiÞ 1 þ1 " # ( ) ¼1 3 1 1 2½ð2ð3  1:8367Þ  1Þ ð2ð3  1:1291Þ  1Þ þ1   ¼1:1291:

Thus, the 95% lower confidence bound of ST

pkcan be calculated as:

STðLÞ_pk ¼ ^ ST_pk 1 þ Z= ffiffiffiffiffi 2n p ¼ 1:1291 1 þ 1:645=pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2  50 ¼0:9696:

Then, we can conclude that the true value of the yield index ST

pkwould be no less than 0.9696,

or equally, be 95% confident that the production yield would be greater than 0.9964. To test the yield performance, H0: ST

pk1:0 versus H1: STpk4 1:0, one could calculate

the critical value c0:

c0¼S þZ S ffiffiffiffiffi 2n p ¼1:0 þ 1:645 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:0 2  50 p ¼1:1645: Since the estimated ST

pk, ^STpk¼1.1291, is smaller than the critical value, c0¼1.1645, one

could conclude that the sample does not provide sufficient evidence to support that the process capability performance ST

pkis larger than 1.0, or equally, be 95% confident that the

production yield is greater than 0.9973.

6. Conclusion

Production yield is the most common and standard criterion for evaluating the performance of products manufactured. Numerous multivariate capability indices have been proposed to measure the performance of processes with multiple characteristics. However, few of them can be applied to measure the production yield for processes with multiple characteristics. The capability index ST_pkproposed by Chen et al. (2003) provides an exact yield measure for multinormal processes with independent characteristics. With the aids of principal component analysis (PCA) and some normalising methods, e.g., Box-and-Cox transformation, the yield index ST

pk can be applied to measure the production

yield of multivariate processes.

Assuring the production yield for processes with multiple characteristics to meet the requirement is important. So, the proposition of a technique of assuring production yield is necessary in this field. In this paper, statistical inferences on the capability index ST

pkhave

been considered. To make reliable decisions, we investigated the effect of all the parameters on the sampling distribution of ST

pk, and obtained the lower confidence bounds

and critical values which can ensure that the risk of making incorrect decisions will be smaller than the significant level . A real-world application, with data investigated by Wang and Chen (1998), was executed to illustrate the applicability of the proposed approach. The proposed methodology can be used to make a reliable decision to determine whether the production meets the yield requirement, and bridges the gap between the theoretical development and factory application.

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