O.R. Applications
Capability adjustment for gamma processes with mean
shift consideration in implementing Six Sigma program
Ya-Chen Hsu
a,*, W.L. Pearn
b, Pei-Ching Wu
b aDepartment of Business Administration, Yuanpei University of Science and Technology, Hsin Chu City 30015, Taiwan, ROC
b
Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC Received 23 November 2006; accepted 17 July 2007
Available online 25 July 2007
Abstract
In the 1980s, Motorola, Inc. introduced its Six Sigma quality program to the world. Some quality practitioners ques-tioned why the Six Sigma advocates claim it is necessary to add a 1.5r shift to the process mean when estimating process capability. Bothe [Bothe, D.R., 2002. Statistical reason for the 1.5r shift. Quality Engineering 14 (3), 479–487] provided a statistical reason for considering such a shift in the process mean for normal processes. In this paper, we consider gamma processes which cover a wide class of applications. For fixed sample size n, the detection power of the control chart can be computed. For small process mean shifts, it is beyond the control chart detection power, which results in overestimating process capability. To resolve the problem, we first examine Bothe’s approach and find the detection power is less than 0.5 when data comes from gamma distribution, showing that Bothe’s adjustments are inadequate when we have gamma pro-cesses. We then calculate adjustments under various sample sizes n and gamma parameter N, with power fixed to 0.5. At the end, we adjust the formula of process capability to accommodate those shifts which can not be detected. Consequently, our adjustments provide much more accurate capability calculation for gamma processes. For illustration purpose, an application example is presented.
Ó 2007 Elsevier B.V. All rights reserved.
Keywords: Quality management; Dynamic Cpk; Gamma distribution; Mean shift; Process capability index
1. Introduction
Process capability indices (PCIs), Cp, Cpk, Cpmand Cpmk have been proposed in the manufacturing industry
providing numerical measures on whether a process is capable of reproducing items within specification limits
preset in the factory (seeKane, 1986; Chan et al., 1988; Pearn et al., 1992; Kotz and Lovelace, 1998). These
indices have been defined as:
0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.07.023
* Corresponding author. Tel.: +886 03 573 1630; fax: +886 03 572 2392.
E-mail address:ychsu.iem93@nctu.edu.tw(Y.-C. Hsu).
European Journal of Operational Research 191 (2008) 517–529
Cp¼ USL LSL 6r ; Cpk ¼ min USL l 3r ; l LSL 3r ; Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q ; Cpmk ¼ min USL l 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q ; l LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q 8 > < > : 9 > = > ;:
where USL is the upper specification limit, LSL is the lower specification limit, l is the process mean, r is the
process standard deviation (overall process variation), and T is the target value. The index Cp considers the
overall process variability relative to the manufacturing tolerance, reflecting product quality consistency.
The index Cpk takes the magnitude of process variance as well as process departure from target value, and
has been regarded as a yield-based index since it providing lower bounds on process yield. The index Cpm
emphasizes on measuring the ability of the process to cluster around the target, which therefore reflects the
degrees of process targeting (centering). Since the design of Cpm is based on the average process loss relative
to the manufacturing tolerance, the index Cpmprovides an upper bound on the average process loss, which has
been alternatively called the Taguchi index. The index Cpmkis constructed from combining the modifications to
Cp that produced Cpk and Cpm, which inherits the merits of both indices.
Since Motorola, Inc. introduced its Six Sigma quality initiative in the 1980s, quality practitioners have ques-tioned why the followers of this initiative have added a 1.5r shift to the process mean when estimating process capability. The advocates of Six Sigma have claimed that such an adjustment is necessary, but they have offered
only personal experiences and three dated empirical studies as justification for this claim (seeBender, 1975;
Evans, 1975; Gilson, 1951). By examining the sensitivity of control charts to detect changes of various
magni-tudes,Bothe (2002)provided a statistically based reason for this claim. In his study, Bothe assumed that the
pro-cess data is approximately normally distributed. However, non-normal propro-cesses occur frequently, in particular,
in the semiconductor industry.Pyzdek (1992)mentioned that the distributions of certain chemical processes,
such as zinc plating in a hot-dip galvanizing process, are very often skewed.Choi et al. (1996)presented an
exam-ple of a skewed distribution in the ‘‘active area’’ shaping stage of the wafer’s production processes. Gamma dis-tribution (skewed), denoted as GammaðN ; hÞ, with various values of N and h, covers a wide class of non-normal applications, including the manufacturing of semiconductor products, head/gimbal assembly for memory stor-age systems, jet-turbine engine components, flip-chips and chip-on-board, audio-speaker drivers, wood prod-ucts, and many others. Therefore, it seems reasonable that we use gamma process for data analysis.
The control charts are commonly used in many industries for providing early warning for the shift in the process mean. For example, the cumulative sum chart is known to be effective on detecting sustained shifts in
the process mean (see e.g.Lucas and Crosier, 2000; Luceno and Puig-Pey, 2002; Lucas, 1976). If the control
chart detects a process mean shift, then the process is not under control. However, for momentary process mean shifts, it may be beyond the control chart detection power. Consequently, the undetected shifts may
result in overestimating process capability. If the process mean shifts are not detected, then unadjusted Cpk
would overestimate the actual process yield. Bothe (2002)provided a statistical reason for considering such
a shift in the process mean for normal processes. However, if the capability indices are based on the assump-tion of a normal distribuassump-tion of data but are used to deal with non-normal observaassump-tions, the values of the capability indices may, in the majority of situations, misrepresent actual product quality. This paper first examines Bothe’s approach and finds that the detection power of the control chart is less than 0.5 when data comes from gamma distribution. This shows that Bothe’s adjustments are inadequate when we have gamma processes. Then, the adjustments under various sample sizes (nÞ and gamma parameters (N Þ with a fixed detec-tion power of 0.5 are calculated. Finally, the process capability formula is adjusted to accommodate the unde-tected shifts. As a result, our adjustments provide significantly more accurate calculations of the capability of gamma processes. A real-world example taken from the manufacturing process of semiconductors is investi-gated to illustrate the applicability of the process capability index.
2. Gamma process
All of us know that the case of non-normal processes occurs frequently in practice, for example, in the
frequently in industry and gave several examples, such as a shearing process and a chemical dip process. The abundance of outputs from skewed distributions makes the normality assumption often unreasonable. A gamma distribution, with varied N and h values, covers a wide class of non-normal applications. Therefore, a gamma process for data analysis has been chosen for this study. The difference between normal and gamma
distributions is compared in Section 2.1. And the statistical property of gamma distribution is discussed in
Section2.2.
2.1. The gamma distribution
In this section, we investigate the gamma distribution to study the effect on the detection power of the con-trol chart. Observations from the gamma distribution are non-negative. The gamma distribution can be
denoted as GammaðN ; hÞ with the probability density function given byRoss (2005)to be as follows:
fðx; N ; hÞ ¼ 1
hNCðN Þx
N1expfx=hg; x >0; N > 0; h > 0
and the mean and variance are given, respectively, by
l¼ N h and r2¼ N h2
:
Denote the family of gamma distributions with mean N h by GammaðN ; hÞ. The gamma distributions are skewed. To see how this distribution are different from the standard normal distribution in terms of skewness
and kurtosis,Table 1presents the values of skewness and kurtosis (which are defined as the third and fourth
moments of the standardized distribution, respectively) of the gamma distributions under study. Note that the
case N ¼ 1 corresponds to the exponential distribution and the skewness and kurtosis of GammaðN ; 1Þ are
2=pffiffiffiffiN and 6=Nþ 3 respectively. We can find in Table 1when the N decreases, the corresponding values of
skewness and kurtosis will become large and far away from the values of the standard normal distribution. The result through these distributions, we can get some insights of the effects of non-normality in terms of skewness and kurtosis.
Fig. 1 presents several gamma distributions along with a normal distribution for the same mean and
variance. In this study, we let N ¼ 0:5, 1, 2, 3, 4, and 5, while (without loss of generality) fixing h ¼ 1. These
values of N and h correspond to the values used bySchilling and Nelson (1976). As can be seen fromFig. 1a–f,
as N increases, the gamma distribution appears more nearly normal distribution. In fact, we demonstrate this
convergence property inTable 1, by calculating the skewness and kurtosis. It can be seen that as N increases,
the skewness and kurtosis of gamma distribution are very close to those of normal distribution. Through these distributions, we wish to get some insights of the effects of non-normality on the detection power in terms of
skewness and kurtosis in Section3.
2.2. Statistical properties of gamma distribution
The gamma distribution has a reproductive property: If X1and X2are independent random variables and
each has a gamma distribution with possible different values of N1, N2 of N, but with common values of h,
then X1þ X2 also has a gamma distribution (seeRoss, 2005), with N ¼ N1þ N2, and with the same value
Table 1
Values of skewness and kurtosis of various gamma distributions
Distribution Skewness Kurtosis
Nð0; 1Þ 0 3 Gammað5; 1Þ 0.8944 4.2 Gammað4; 1Þ 1 4.5 Gammað3; 1Þ 1.1547 5 Gammað2; 1Þ 1.4142 6 Gammað1; 1Þ 2 9 Gammað0:5; 1Þ 2.8284 15
of h. Applying this property, let X1,X2; . . . ; Xnbe a sequence of independent distribution of GammaðN ; hÞ and
then the distribution of X1þ X2þ þ Xnis GammaðnN ; hÞ. Using simply statistical technique, we can
con-clude that Xn¼ ðX1þ X2þ þ XnÞ=n GammaðnN ; h=nÞ.
The standard deviation of the Xn distribution, rxn, is calculated from its relationship to the distribution
parameters and the subgroup size n as follows:
rxn ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nN h n 2 s ¼ ffiffiffiffi N n r h:
Let X1; X2; . . . ; Xnbe a sequence of independent distribution of Gamma(3, 1) and we plot the probability
den-sity function of the average Xnfor subgroup size n¼ 2ð1Þ5 inFigs. 2a–d. We can found that the variance of
-3 -2 -1 0 1 2 3 0 1 2 3 4 5 6 7 PDFs for GAM(0.5,1)and N(0.5,0.5) x density density density density -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PDFs for GAM(1,1)and N(1,1) x -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 density density 0 0.05 0.1 0.15 0.2 0.25 PDFs for GAM(2,1)and N(2,2) -10 -5 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 PDFs for GAM(3,1)and N(3,3) x x -5 0 5 10 15 PDFs for GAM(4,1)and N(4,4) x -5 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 PDFs for GAM(5,1)and N(5,5) x
a
b
d
c
e
f
Fig. 1. (a) Probability density functions for Gammað0:5; 1Þ and N ð0:5; 0:5Þ. (b) Probability density functions for Gammað1; 1Þ and N ð1; 1Þ. (c) Probability density functions for Gammað2; 1Þ and N(2, 2). (d) Probability density functions for Gammað3; 1Þ and N ð3; 3Þ. (e) Probability density functions for Gammað4; 1Þ and N ð4; 4Þ. (f) Probability density functions for Gammað5; 1Þ and N ð5; 5Þ.
average Xn will get smaller as subgroup size n increases. This situation means that the distribution of Xn is more centralized when n > 1.
3. The detection power of gamma process
The major purpose of individuals control chart is assisting on identifying shifts and drifts in processes and it is easily to be implemented. But, some assumptions should be satisfied before control charts are used. The assumptions include that the process characteristics must follow normal distributions. Actually, non-normal
processes occur frequently in practice. Due to above-mentioned statements, we replace the traditional, l 3r,
to be the upper or lower control limits by the quantile of cumulative distribution function for different
param-eters of GammaðN ; hÞ (F0:00135and F0:99865) and detect the power of gamma process under Bothe’s capability
adjustments.
Let X1; X2; . . . ; Xnbe a sequence observations of independent and identically distributed in GammaðN ; hÞ.
Using the reproductive property of gamma distribution, the mean of the observations is Xn ðXn¼1nPni¼1XiÞ
which is distributed in GammaðnN ; h=n), then we can obtain that lXi ¼ lXn¼ N h, rXi¼
ffiffiffiffi N p h, and 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 n= 1 n= 2 gamma (3,1)gamma (6,1/2) 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 n= 1 n= 3 gamma (3,1) gamma (9,1/3) 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 n= 1 n=4 gamma (3,1) gamma (12,1/4) 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 n= 1 n=5 gamma (3,1) gamma (15,1/5)
a
b
d
c
Fig. 2. (a) Probability density functions for Gammað3; 1Þ and the average Xnfor n¼ 2. (b). Probability density functions for Gammað3; 1Þ
and the average Xnfor n¼ 3. (c) Probability density function for Gammað3; 1Þ and the average Xnfor n¼ 4. (d) Probability density
function for Gammað3; 1Þ and the average Xnfor n¼ 5.
Table 2
Detection power of various gamma processes
Subgroup size n Shift d Distribution GammaðN ; 1Þ
N¼ 0:5 N¼ 1 N¼ 2 N¼ 3 N¼ 4 N¼ 5 N¼ 6 N¼ 7 N¼ 8 N¼ 9 N¼ 10 Nð0; 1Þ 2 2.12 0.027 0.054 0.100 0.136 0.164 0.187 0.206 0.222 0.235 0.247 0.257 0.5 3 1.73 0.040 0.078 0.136 0.176 0.205 0.228 0.246 0.262 0.274 0.285 0.294 0.5 4 1.50 0.054 0.100 0.164 0.206 0.236 0.258 0.275 0.289 0.301 0.311 0.320 0.5 5 1.34 0.066 0.119 0.187 0.228 0.257 0.278 0.294 0.308 0.319 0.328 0.336 0.5 6 1.22 0.077 0.134 0.203 0.244 0.272 0.292 0.307 0.320 0.330 0.339 0.346 0.5
rX n ¼ ffiffiffiffi N p h
=pffiffiffin. Consequently, we derived the power of gamma process as follows. Since the type II error
bis
b¼ P ðLCL 6 Xn6UCLjl1¼ l0þ krXiÞ ¼ P ðF0:001356Xn6F0:99865jl1¼ l0þ krXiÞ
¼ GXnðF0:99865Þ GXnðF0:00135Þ;
where 1 b is the detection power of the process, GXnðÞ is the cumulative distribution function of gamma
dis-tribution with that mean has shifted and l1is the mean after process shift (l0is the mean of the original
pro-cess). The control limits LCL and UCL are calculated as F0:00135 and F0:99865respectively.
Table 2presents the detection power when data comes from gamma distribution with N ¼ 0:5, 1(1)10 and
h¼ 1. The magnitude of shift in the second column on the left is Bothe’s capability adjustments determined
when data comes from normal distribution and the detection power is 0.5.
FromTable 2, we can find that the detection power is less than 0.5 when data comes from gamma distri-bution under Bothe’s capability adjustments. Our study shows that the detection power gets closer to 0.5 as N increases, which is reasonable since the corresponding distributions get closer to the standard normal
distri-bution. This is due toBothe’s (2002)approach is based on the normality assumption of the data and the
detec-tion power is 0.5. The skewness of GammaðN ; 1Þ is 2=pffiffiffiffiN. Therefore, as N decreases the gamma distribution is
more skewed and the detection power is poorer. For example, when N ¼ 0:5 and the subgroup size n ¼ 2, the
detection power is 0.027. It implies Bothe’s adjustments are inadequate when we have skewed processes. Con-sequently, in our study, we determined the capability adjustment and calculation when process data comes from gamma distribution.
Table 3
AS50values for several subgroup sizes n and various N values
n N 0.5 1 2 3 4 5 6 7 8 9 10 Nð0; 1Þ 2 4.182 3.611 3.185 2.992 2.876 2.797 2.738 2.692 2.655 2.625 2.599 2.12 3 3.127 2.732 2.443 2.313 2.236 2.182 2.143 2.113 2.088 2.067 2.050 1.73 4 2.553 2.252 2.034 1.936 1.878 1.838 1.808 1.785 1.767 1.752 1.738 1.50 5 2.188 1.944 1.769 1.690 1.644 1.612 1.588 1.570 1.555 1.543 1.532 1.34 6 1.932 1.727 1.581 1.515 1.476 1.450 1.430 1.415 1.403 1.392 1.384 1.22 7 1.741 1.565 1.439 1.383 1.350 1.327 1.310 1.297 1.286 1.278 1.270 1.13 8 1.592 1.438 1.328 1.279 1.249 1.229 1.215 1.203 1.194 1.186 1.180 1.06 9 1.473 1.336 1.237 1.194 1.168 1.150 1.137 1.127 1.118 1.112 1.106 1.00 10 1.375 1.251 1.162 1.123 1.100 1.084 1.072 1.063 1.055 1.049 1.044 0.95 11 1.292 1.179 1.099 1.063 1.042 1.027 1.016 1.008 1.001 0.996 0.991 0.90 12 1.222 1.118 1.044 1.011 0.992 0.978 0.969 0.961 0.955 0.950 0.945 0.87 13 1.160 1.064 0.996 0.966 0.948 0.936 0.927 0.920 0.914 0.909 0.905 0.83 14 1.107 1.018 0.954 0.926 0.910 0.898 0.890 0.883 0.878 0.874 0.870 0.80 15 1.059 0.976 0.917 0.891 0.875 0.864 0.857 0.850 0.846 0.842 0.838 0.77 16 1.017 0.939 0.883 0.859 0.844 0.834 0.827 0.821 0.817 0.813 0.810 0.75 17 0.979 0.905 0.853 0.830 0.816 0.807 0.800 0.795 0.790 0.787 0.784 0.73 18 0.944 0.875 0.826 0.804 0.791 0.782 0.775 0.770 0.766 0.763 0.760 0.71 19 0.913 0.847 0.801 0.780 0.768 0.759 0.753 0.748 0.744 0.741 0.738 0.69 20 0.884 0.822 0.778 0.758 0.746 0.738 0.732 0.728 0.724 0.721 0.718 0.67 21 0.858 0.798 0.756 0.738 0.726 0.719 0.713 0.709 0.705 0.702 0.700 0.65 22 0.834 0.777 0.737 0.719 0.708 0.701 0.695 0.691 0.688 0.685 0.683 0.64 23 0.811 0.757 0.718 0.701 0.691 0.684 0.679 0.675 0.672 0.669 0.667 0.63 24 0.790 0.738 0.701 0.685 0.675 0.669 0.664 0.660 0.657 0.654 0.652 0.61 25 0.771 0.721 0.685 0.670 0.660 0.654 0.649 0.646 0.643 0.640 0.638 0.60 26 0.753 0.704 0.670 0.655 0.646 0.640 0.636 0.632 0.629 0.627 0.625 0.59 27 0.736 0.689 0.656 0.642 0.633 0.627 0.623 0.619 0.617 0.615 0.613 0.58 28 0.720 0.675 0.643 0.629 0.621 0.615 0.611 0.608 0.605 0.603 0.601 0.57 29 0.704 0.661 0.631 0.617 0.609 0.604 0.599 0.596 0.594 0.592 0.590 0.56 30 0.690 0.648 0.619 0.606 0.598 0.593 0.589 0.586 0.583 0.581 0.579 0.55
4. Undetected mean shift under designated power
The undetected mean shift adjustment inTable 3is called AS50which is the magnitude of shift we need to
adjust based on designated detection power is 0.5 and process data comes from gamma distribution. We
develop a Matlab program (seeAppendix) to determine the adjustment AS50. The program reads the desired
detection power (set to be 0.5), the gamma parameter N and the subgroup size n. Table 3 displays the
magnitude of adjustments AS50based on the detection power is 0.5 and data comes from Gamma(N, 1) with
various values of N (= 0.5 and 1(1)10) and n¼ 2ð1Þ30. For example, if we set N ¼ 3 and n ¼ 5, then the
adjust-ment is AS50¼ 1:69. We conclude that the adjustment AS50 rð¼ 1:69rÞ is required based on the detection
power is 0.5 and data comes from Gamma(3, 1). It also shows fromTable 3that the adjustments AS50get
clo-ser to Bothe’s adjustments as N increases (when n¼ 2ð1Þ6), which is reasonable since the corresponding
dis-tributions get closer to the standard normal distribution. However, we should notice that when N is small (distribution is strongly skewed), the required adjustment in the capability index formula is much greater than those for normal processes. Using the adjusted process capability formula, the engineers can determine the actual process capability more accurately.
Fig. 3presents the power curves, these lines on the graph depict the probabilities of detecting a shift in l for
the commonly used subgroup size n¼ 3, 4, 5 (expressed in r units on the horizontal axis) when N ¼ 3. All
these lines are close to zero for small shifts in l. It can be found that the power of the chart with all three curves eventually leveling off close to 100% as the size of the shifts in excess of 3.5r. The dashed horizontal
line drawn inFig. 3shows that there is a 50% probability of missing a 1.69r shift in l when n is 5, while l
must move by 2.313r to have this same probability when n is only 3. The shift sizes that have a 50%
proba-bility of remaining undetected, called AS50values are listed inTable 3for subgroup sizes n¼ 2ð1Þ30.
Momen-tary movements in l smaller than AS50rare more than likely to be missed by a control chart. Therefore our
adjustment AS50takes into account those shifts that are not detected by the control chart.
5. Capability adjustment
5.1. Estimator of Cpk in the non-normal case
The index Cpkhas been viewed as an yield-based index since it provides bounds on the process yield for a
normally distributed process with a fixed value of Cpk. This index Cpk is defined as:
Cpk¼ min USL l 3r ; l LSL 3r ;
where as above USL is the upper specification limit, LSL is the lower specification limit, l is the process mean and r is the process standard deviation. The proper use of process capability indices, which are statistical mea-sures of process capability, is based on several assumptions. One of the most essential is that the process mon-itored is supposed to be stable and the output is approximately normally distributed. When the distribution of a process characteristic is non-normal, PCIs calculated using conventional methods could often lead to erro-neous and misleading interpretation of the process’s capability.
In the recent years, several approaches to the problems of PCIs for the non-normal populations have been
suggested (see e.g.Pal, 2005; Ding, 2004; Pearn and Chen, 1997; Kotz and Lovelace, 1998; Somerville and
Montgomery, 1996; Kocherlakota et al., 1992). Several authors used data transformation techniques such as the Box–Cox power transformation, Johnson’s transformations and quantile transform techniques to solve this problem. And some authors replaced the unknown distribution by a known three or four-parameter
dis-tribution. Examples include Clements (1989), Franklin and Wasserman (1992), Shore (1998) and Polansky
(1998). We did not consider the Box–Cox transformation because: (1) process characteristics might be lost after the transformation, and the transformed data is difficult to interpret. (2) In general, however, practitio-ners may feel uncomfortable working with transformed data and may have some difficulty in reversing the results of the calculations back to the original scale. Due to above-mentioned statements, we use the most common method for modifying PCIs in the non-normal case is the technique of quantile estimation. Analo-gous to the normal case, where the ‘‘natural’’ process width is between the 0.135 percentile and the 99.865 per-centile, PCIs can be redefined in terms of their quantiles for possible modification in the non-normal case. The
quantile definition for Cpu and Cpl are defined as:
Cpu¼
USL median
ðupper 0:135% pointÞ median¼
USL F0:5
F0:99865 F0:5 and
Cpl¼
median LSL
median ðlower 0:135% pointÞ¼
F0:5 LSL
F0:5 F0:00135 :
Then the index Cpk would be calculated as the minimum of Cpuand Cpl, namely:
Cpk ¼ minfCpu; Cplg ¼ min USL F0:5 F0:99865 F0:5 ; F0:5 LSL F0:5 F0:00135 ; ð1Þ
so that the normality assumption can be verified simultaneously.
We can obtain more accurate measures of these percentile points (F0:00135, F0:5 and F0:99865) under
consid-eration in the non-normal case, if we are able to find a better distributional form for the data, which provides a very satisfactory fit. This involves modeling the process data with alternative probability plot models, such
as the Weibull or gamma ones (see e.g.Dudewicz and Mishra, 1998; Kotz and Lovelace, 1998). Nevertheless,
an obvious disadvantage of probability plotting is that it is not a truly objective procedure. It is quite possible for two analysts to arrive at different conclusions using the same data. Accordingly, it is often desirable to supplement probability plots with goodness-of-fit tests, which possess more formal statistical foundations
(see, e.g.,Shapiro, 1995). Choosing proper distribution to fit the data is an important step in probability
plot-ting. Sometimes one can use the available knowledge of the physical phenomenon or the past experience to suggest a choice of the distribution.
5.2. Modifying the assessment of Cpk
Since a process will experience shifts in F0:5 (= median) of various magnitudes and not all of these will be
discovered, we must take them into account when estimating outgoing quality so customers are not
disap-pointed. Whereas the shifts of process mean ranging in size from 0 up to AS50rare the ones likely to remain
undetected (larger shifts should be detected by the control chart), a cautious method is to assume that every
missed shift is as large as AS50r.
Considering the undetected process mean shift as large as AS50r, we use F0:5 minus AS50rto evaluate how
estimating the index Cpk. Incorporating both of these adjustments into the Cpk formula (see Eq. (1)) we
obtained the ‘‘dynamic’’ Cpk index by making the following modifications:
Cpk¼ min USL ðF0:5þ AS50rÞ F0:99865 F0:5 ;ðF0:5 AS50rÞ LSL F0:5 F0:00135 ¼ min USL F0:5 F0:99865 F0:5 AS50r F0:99865 F0:5 ; F0:5 LSL F0:5 F0:00135 AS50r F0:5 F0:00135 : ð2Þ
By considering an adjustment AS50rin this assessment for undetected shifts in process median, the estimate of
dynamic index Cpkwill decrease and the expected total number of nonconforming parts will increase. It must
be noticed that this nonconforming level assumes that undetected shifts are happening almost constantly and
that every one is equal to AS50r. From Table 3, the practitioners can find the AS50to calculate the dynamic
index Cpk for determining whether their process meets the preset capability requirement, and make reliable
decisions to the process. 6. Application
The manufacture of integrated circuits (ICs) includes the front-end process of wafer and the back-end pro-cess of integrated circuit packaging. In an integrated circuit packaging factory, the manufacturing propro-cess gen-erally contains the following main steps: die sawing, die mounting, wire bonding, molding, trimming and
forming, marking, plating and testing (Fig. 4). Wire bonding is the most common means of providing an
elec-trical connection from the IC device to the lead-frame and it uses ultra-thin gold or aluminum wire to form the
electrical inter-connection between the chip and the package leads (Fig. 5). High-speed wire bonding
equip-ment consists of a handling system to feed the lead-frame into the work area. Image recognition systems ensure the die is orientated to match the bonding diagram for a particular device. Wires are bonded one at a time, and two wire bonds are formed at each interconnection: one at the die (first bond) and the other at the lead-frame (second bond). The first bond involves the formation of a ball which is placed within the bond
Fig. 4. Wire bonding process.
pad opening on the die, under load and ultrasonic energy within a few milliseconds and forms a ball bond at the bond pad metal.
In the wire bonding process, one of the most important factors which directly relates to its level of quality is the ball size. Since the process may easily shut down when the width between the two bond balls is too small, the size of the bond ball must be taken into consideration. Therefore, the proposed USL and LSL for the ball
size are 8 mil and 0.5 mil (1 mil = 1/1000 in. = 0.0254 mm), respectively. As shown inTable 4, a part of
his-torical data is collected.Fig. 6displays the histogram, andFig. 7displays the normal probability plot of these
historical data. From theFigs. 6 and 7, it is evident to conclude the data collected from the factory are not
normal distributed. The data analysis results justify that the process is significantly away from the normal dis-tribution. By the goodness-of-fit tests, the historical data indicates that the process pretty approximates to be
Table 4
The 100 observations are collected from the historical data
2.891 4.035 4.495 2.890 2.312 3.158 5.228 3.334 5.896 5.639 3.842 1.590 1.954 1.842 0.680 2.752 1.301 2.260 0.889 2.381 0.619 2.788 1.050 3.750 3.508 6.123 6.549 5.954 2.207 4.417 4.805 1.516 2.227 2.797 1.636 1.066 0.940 4.101 4.542 1.295 1.770 3.492 5.706 3.722 6.644 2.472 1.383 4.494 1.694 2.892 2.111 3.591 2.093 3.222 2.891 2.582 0.665 3.234 1.102 1.083 1.508 1.811 2.803 6.659 0.923 6.229 3.177 2.333 1.311 4.419 2.495 0.921 4.061 9.725 1.600 4.281 3.360 1.131 1.618 4.489 3.696 1.982 2.413 5.480 1.992 2.573 1.845 4.620 6.221 1.694 4.882 1.380 3.982 2.260 2.366 2.899 3.782 2.336 1.175 3.055 2 6 10 15 10 5 0 Historical data 4 8
Fig. 6. Histogram plot of the historical data.
-2 -1 0 1 2
Quantiles of Standard Normal 10 8 6 4 2 Historical data
distributed as gamma. The parameters N and h of this gamma process could be estimated from the historical
data, giving ^N ¼ 3 and ^h¼ 1.
Accordingly, it is appropriate to use this approach and we can obtain more accurate measures of the three
quantiles (F0:00135, F0:5 (= median), and F0:99865) for
r¼ ffiffiffiffi N n r h ¼ ffiffiffiffiffi 3 10 r ¼ 0:547
under consideration. Then ‘‘dynamic’’ Cpk index can be calculated as follows:
dynamic Cpk ¼ min USL F0:5 AS50r F0:99865 F0:5 ;F0:5 AS50r LSL F0:5 F0:00135 ¼ min 8 2:67 1:123ð0:547Þ 10:87 2:67 ; 2:67 1:123ð0:547Þ 0:5 2:67 0:211 ¼ minf0:58; 0:63g ¼ 0:58;
with AS50¼ 1:123 for n ¼ 10 from Table 3. Compared it to the value of the following conventional index:
Cpk¼ USL F0:5 F0:99865 F0:5 ; F0:5 LSL F0:5 F0:00135 ¼ f0:65; 0:88g ¼ 0:65
calculated by a traditional capability study (the shift of process mean is not considered), we can find that the
value of the modified Cpkis much smaller. This result indicates if the process mean shifts that are not detected
then unadjusted Cpk would overestimate the actual process yield which is not derisible. Our adjustment takes
into account those shifts that are not detected so that the practitioner would be able to keep its quality promise for this process. As the adjusted process capability drops below the desired quality level, the practitioner should stop the process because the process does not meet his preset capability requirement.
As the subgroup size n increases, the shift in process mean have a higher probability of detection. For
exam-ple, if n¼ 15, the AS50would be 0.891 for Gamma(3, 1) from Table 3, and then the ‘‘dynamic’’ Cpk index is
dynamic Cpk ¼ min USL F0:5 AS50r F0:99865 F0:5 ;F0:5 AS50r LSL F0:5 F0:00135 ¼ min 8 2:67 0:891ð0:547Þ 10:87 2:67 ; 2:67 0:891ð0:547Þ 0:5 2:67 0:211 ¼ minf0:6; 0:68g ¼ 0:6:
Changing n from 10 to 15 increases the dynamic Cpk index from 0.58 to 0.6, and the total number of
noncon-forming parts would be reduced. 7. Conclusion
In this paper, we considered the problem of how to determine the adjustments for process capability with mean shift when data follows the gamma distribution. We first examined Bothe’s approach and found the detection power is less than 0.5 when data comes from the gamma distribution, showing that Bothe’s adjust-ments are inadequate when we have gamma processes. For gamma processes, we calculated the adjustadjust-ments for various sample sizes (nÞ and gamma parameter (N Þ with detection power fixed to 0.5. For small value of N (distribution is strongly skewed), the required adjustment in the capability index formula is much greater than those for normal processes. Using the adjusted process capability formula, the engineers can determine the actual process capability more accurately. Tables are also provided for engineers/practitioners to use in their in-plant applications. A real-world semi-conductor production plant is investigated and presented to illustrate the applicability of the proposed approach.
Appendix. Matlab program for determining the adjustment AS50
% n is the sample size
% N and theta are the parameters of gamma distribution % power is the detection power of the control chart
clear all power = 0.5; for n=2:1:30; for N=[0.5 1 2 3 4 5 6 7 8 9 10]; theta = 1; AS_50_Upper = 5; AS_50_Lower = 0.5; Sigma = sqrt(N.*(thetaˆ2));
%The upper and lower limits of the control chart
F_99865 = gaminv(0.99865,n.*N,theta/n);
F_00135 = gaminv(0.00135,n.*N,theta/n);
%Bisection Method
B_AS_50_Upper = gamcdf(F_99865-AS_50_Upper.*Sigma,n.*N,theta/n)- gamcdf(F_00135-AS_50_Upper.
*Sigma,n.*N,theta/n);
P_AS_50_Upper = 1-B_AS_50_Upper;
B_AS_50_Lower = gamcdf(F_99865-AS_50_Lower.*Sigma,n.*N,theta/n)-
gamcdf(F_00135-AS_50_Lo-wer.*Sigma,n.*N,theta/n);
P_AS_50_Lower = 1-B_AS_50_Lower; AS_50 = (AS_50_Lower+AS_50_Upper)/2;
B_AS_50 = gamcdf(F_99865-AS_50.*Sigma,n.*N,theta/n)- gamcdf(F_00135- AS_50.*Sigma,n.*N,theta/
n); P_AS_50 = 1-B_AS_50; while (abs(P_AS_50-power) > 0.0001) if P_AS_50 > power AS_50_Upper = AS_50; AS_50 = (AS_50_Lower+AS_50_Upper)/2;
B_AS_50 = gamcdf(F_99865-AS_50.*Sigma,n.*N,theta/n)- gamcdf(F_00135- AS_50.*Sigma,n.*N,
theta/n);
P_AS_50 = 1-B_AS_50; else
AS_50_Lower = AS_50;
AS_50 = (AS_50_Lower+AS_50_Upper)/2;
B_AS_50 = gamcdf(F_99865-AS_50.*Sigma,n.*N,theta/n)- gamcdf(F_00135- AS_50.*Sigma,n.*N,
theta/n); P_AS_50 = 1-B_AS_50; end end fprintf (‘%g ’,AS_50); end fprintf (‘%gnn’, n); end References
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