Estimating process capability indices of multivariate nonnormal processes

Abstract

The capability analysis of production processes where there are more than one correlated quality variables is a complicated task. The problem becomes even more difficult when these variables exhibit nonnormal characteristics. In this paper, a new methodology is proposed to estimate process capability indices (PCIs) of multivariate nonnormal processes. In the proposed methodology, the skewness of the marginal probability distributions of the variables is first diminished by a root transformation technique. Then, a Monte Carlo simulation method is employed to estimate the process proportion of nonconformities (PNC). Next, the relationship between PNC and PCI is found, and finally, PCI is estimated using PNC. Several multivariate nonnormal distributions such as Beta, Weibull, and Gamma are taken into account in simulation experiments. A real-world problem is also given to demonstrate the application of the proposed procedure. The results obtained from both the simulation studies and the real-world problem show that the proposed method performs well and is able to estimate PCI properly.

Full text

1
Estimating Process Capability Indices of Multivariate Non-Normal
Processes
Babak Abbasi, Ph.D.
Department of Industrial Engineering, Sharif University of Technology
Email: B_Abbasi@Mehr.Sharif.edu, B.Abbasi@Gmail.com
Seyed Taghi Akhavan Niaki, Ph.D.1
Professor of Industrial Engineering, Sharif University of Technology
P.O. Box 11155-9414, Azadi Ave., Tehran 1458889694 Iran
Phone: (+9821) 66165740, Fax: (+9821) 66022702, Email: Niaki@Sharif.edu
Abstract
The capability analysis of production processes where there are more than one
correlated quality variables is a complicated task. The problem becomes even more
difficult when these variables exhibit non-normal characteristics. In this paper, a
new methodology is proposed to estimate process capability indices (PCIs) of
multivariate non-normal processes. In the proposed methodology, the skewness of
the marginal probability distributions of the variables is first diminished by a root
transformation technique. Then, a Monte Carlo simulation method is employed to
estimate the process proportion of non-conformities (PNC). Next, the relationship
between PNC and PCI is found, and finally PCI is estimated using PNC. Several
multivariate non-normal distributions such as Beta, Weibull, and Gamma are taken
into account in simulation experiments. A real-world problem is also given to
demonstrate the application of the proposed procedure. The results obtained from
both the simulation studies and the real-world problem show that the proposed
method performs well and is able to estimate PCI properly.
Key Words
Process Capability Index (PCI); Transformation Techniques; Skewness Reduction; Non-
Normal Distribution; Multivariate Processes; Monte Carlo Simulation
1 Corresponding Author

2
1. Introduction
As a quantitative measure, process capability indices (PCIs) are widely used to
determine whether a process is capable of producing items within customer specification
limits. The objective of these statistical measures is to estimate process variability relative
to process specifications. Furthermore, a process capability index provides a common
standard of product quality to suppliers and customers.
In the classical methods, while normality of the quality characteristics is usually
assumed in the estimation process of PCI, in many practical cases there are some non-
normal distributions involved. For a stable univariate normal process with mean

and
standard deviation

if we define and USL LSL to be the upper and lower specification
limits respectively, then the capability ratio ( p
C) and the process capability ratio for off-
center processes ( pk
C), defined by Kane (1986) in equation (1) and (2), are the classical
measures that are inappropriate for non-normal processes.
6
pUSL LSL
C


 (1)
( , ), where and
33
pk pu pl pu pl
USL LSL
CMinCC C C






(2)
In which, one can replace unknown

and

by their estimates defined as the sample
mean
X
and sample standard deviation S, respectively.
Although various methods are available in the literature to estimate PCI of
univariate non-normal processes, in many production processes there are more than one
quality characteristics involved. These characteristics are generally correlated and hence
some multivariate techniques should be employed. While many researchers have

3
discussed estimating PCI of multivariate normal processes, there is a limited research for
estimating PCI of multivariate non-normal processes.
2. Literature Review and Background
In the past decade, several modifications of the classical PCIs have been proposed
to resolve the issue of non-normality of quality characteristic data. These modifications
can be generally classified into two basic approaches. The first conceptually simple
approach is to transform the original non-normal data to normal and then use the classical
method to estimate PCI. To name a few, Johnson (1949), based upon the method of
moments, built a system of distributions called the Johnson transformation system to
transform non-normal data. Box & Cox (1964) presented a useful family of power
transformations that transform non-normal data into normal ones. Somerville &
Montgomery (1996) used a square-root transformation to transform a skewed distribution
into a normal one. Niaki & Abbasi (2007) proposed a root transformation method to
transform skewed discrete multivariate data to multivariate normal data. Hosseinifard et
al. (2009) used the root transformation technique to estimate PCI of univariate non-
normal processes.
The alternative approach to deal with non-normal data is to fit a distribution to the
data and then estimating PCI using non-normal percentiles. In this approach, the term
6

in equation (1) is replaced by the lengths of interval between the upper (99.865) and
the lower (0.135) percentage points of the fitted distribution. To name a few researches in
this approach, Clements (1989) proposed the method of non-normal percentiles to
estimate PCI of a distribution of any shape using the Pearson family of curves, which is

4
widely used in industry. Pearn & Kotz (1994) applied Clements’s method (1989) to
construct the second and the third generation indices ( pm
Cand pmk
C, respectively) for
non-normal data. Pearn et al. (1999) presented a generalization of Clements’ method with
asymmetric tolerances. Although Clements’s method is commonly used in industry, a
research study by Wu et al. (1998) indicated that this method could not accurately
measure the PCI values, especially when the underlying data distribution is skewed.
Liu & Chen (2006) proposed a modified Clements method to evaluate PCI of
non-normal data. They suggested that the accuracy of the estimated PCI for non-normal
data could be improved using Burr XII distribution instead of the Pearson curves
percentiles. The parameters of a Burr XII probability density function can be set to fit the
normal, Gamma, Beta, Weibull, log-normal, and extreme value type-I distributions.
Wang and Zimmer (1996), Zimmer et al. (1998), and Mousa & Jaheen (2002) have
presented a comprehensive review of the Burr XII distribution and its application to
many non-normal situations.
Tang and Than (1999) carried out simulation studies for different non-normal
distributions such as Lognormal, Gamma, and Weibull and concluded that the Box-Cox
method performed better than the Johnson transformation procedures of Johnson (1949),
Clements (1989), Choi and Bai (1996), and Wright (1995). Bai & Choi (1997) developed
a weighted variance approach to measure the PCIs of processes with skewed
distributions. Pal (2005) evaluated PCI using process capability of non-normal cases by a
generalized Lambda distribution.
Castagliola (1996) used the relation between process capability and proportion of
non-conforming items to evaluate PCI of non-normal data. This relation is:

5
10.5 0.5 ( )
3
USL
LSL
p
f
xdx
C





 (3)
(,)
pk pu pl
CMinCC (4)
Where pl
Cand pu
Care given in equations (5) and (6), respectively.
10.5 ( )
3
T
LSL
pl
f
xdx
C





 (5)
10.5 ( )
3
USL
T
pu
f
xdx
C





 (6)
In equations (3), (5), and (6)  is the cumulative probability distribution function (cdf)
of normal distribution, ()
f
xrepresents the probability density function of the process and
T
is the process median. We note that in the Castagliola method, ()
f
xneeds to be either
known or estimated. Castagliola used a method based on the generalized Burr distribution
to assess the capability of the process data. Through the sample empirical distribution
function, he first used a polynomial function to approximate a Burr distribution and then
estimated the PCI. Requiring ()
f
xis a big request and this is one of the most
shortcomings of his method.
In a different approach, Abbasi (2009) proposed an artificial neural network to
estimate PCI for right skewed distributions without appeal to pdf of the process. The
proposed neural network estimated PCI using skewness, kurtosis and upper specification
limit as input variables. Then, the performance of his proposed method was validated by
simulation study for different non-normal distributions.

6
Multivariate process capability indices, in general, can be obtained in three ways;
a) from the ratio of a tolerance region to a process region, b) by calculating the
probability of nonconforming products, and c) by approaches in which loss functions are
employed.
Hubele et al. (1991) using a multivariate normal distribution, defined PCI as the
ratio of the rectangular tolerance region to a modified process region which is the
smallest rectangle around the ellipse with specified type-I error ( 0.0027

). The
number of quality characteristics in the process is taken into account by taking the
th

root of the ratio where

represents the number of quality characteristics.
vol. of the engineering tolerance region
vol. of the engineering process region
PM
C

 (7)
Taam et al. (1993) defined a multivariate capability index ( PM
M
C) as the ratio of
the volume of a modified tolerance region to the volume of the scaled 99.73 percent
process region. The modified tolerance region was defined as the largest ellipsoid that is
centered at the target completely located inside the original tolerance region. Then, in
order to adjust the estimated multivariate normal capability index they used


PM
PM MC
MC D
, where

11
n
Dn


T-1
oo
(X-μ)S (X-μ), X is the sample mean
vector, Sis the sample variance-covariance matrix, and 0

is the target mean of the
process.
Another method to estimate
PCI of multivariate normal processes is proposed by
Chen (1994). In this method, a tolerance zone is defined by


00
:( )
V
VXRhX r

 , where 0
r is a positive number, 0

is a target value, and

7
)(xh is a positive function. The process is capable if inequality



1)( VXP holds. In
other words, let


0
(( ) ) 1
c
rMinPhX c



. If the cumulative distribution
function of 0
()hX

is increasing in a neighbourhood of r, then r is simply the unique
root of equation



 1))(( 0cXhP . The process is deemed capable if 0
rr. Here 0
r
is the half-width of the tolerance interval centered at the target value 0

and r is the half
width of an interval centered on the target value such that the probability of a process
realization falling within this interval is 1


.
Wang et al. (2000) compared the aforementioned multivariate process capability
indices and presented some graphical examples to illustrate them. Chen et al. (2006)
extended Boyles’ method (1994) for normal distribution, Liao et al.’s work (2002) for
non-normal distributions, and Huang et al.’s procedure (2002) for multivariate data.
However, the correlation between the variables was not taken into account. They
estimated both the overall and the individual process capability indices for multivariate
data (without correlation). Castagliola & Castellanos (2005) extended the univariate
method of Castagliola (1996) to multivariate distribution by replacing the univariate
probability density function ()
f
x with its multivariate version.
While Wang and Hubele (1999) and Wang and Du (2000) applied geometric
distance approach to estimate process capability of a univariate processes, Wang (2006)
used geometric distance in multivariate normal and non-normal processes as well. Yeh
and Chen (2001) applied nonparametric methods to estimate proportion of non-
conformities and process capability index for multivariate non-normal processes. For
examples of multivariate non-normal processes, Wang (2006) presented the capability

8
analysis of a real-life data set contained seven non-normal quality characteristics called
the connecter obtained from a manufacturer.
In this paper, using a transformation technique called root transformation, the
multivariate non-normal data is first transformed such that the inherent skewness of the
data diminishes and the marginal distributions of transformed data are almost normal.
Then, the Monte Carlo simulation method is employed to estimate the process proportion
of non-conformities (PNC). Next, the relationship between PNC and PCI is found and
finally the PCI is estimated using PNC.
The rest of the paper is organized as follows: Section 3 involves the proposed
method to estimate PCI. Some simulation experiments for different multivariate non-
normal distributions are presented in Section 4. A real world example is given in Section
5, and finally, we conclude the paper in Section 6.
3. Skewness reduction by transformation methods
Skewness is the most serious issue involved in using non-normal data. In some
research in the area of quality engineering and risk analysis where analyzing skewed data
is important, researchers and engineers prefer to transform data in a way that their
distribution is not skewed. To do this, some normalizing transformations, such as the
square root, inverse, arcsin, and parabolic inverse have been proposed in the literature
(see, for example, Box and Cox 1964 and Ryan 1989). Quesenberry (1995) proposed Q-
transformation, and Xie et al. (2000) proposed double square root transformation for
Geometric distribution. Niaki & Abbasi (2007) used the root transformation technique to
design multi-attribute control chart while there were skewed multivariate data.

9
In the proposed root transformation technique, we search for a proper root ( r) of
the right-skewed non-normal data in a way that if we raise the data to the power r (i.e.
r
X
) the skewness in the distribution of the transformed data becomes almost zero. The
bisection method is employed to find the r value.
The bisection method is based on the point that a function changes sign when it
passes through zero. By evaluating the function in the middle of an interval and replacing
whichever limit has the same sign, the bisection method can halve the size of the interval
in different iterations and eventually finds the root. For example, to find a root of
() 0fxin the interval of 00
(,)ab with which 00
()()0fa fb

, a tolerance

is chosen and
then the following algorithm is applied:
0k
While 1
()
k
fx


12
kk
kab
x


If

1
()()0
kk
fx fa


Then
111
and
kk k k
aa bx


Else
111
and
kk kk
bb ax


End If
1kk
End While
*k
x
x

10
In the application of the root transformation technique for skewness reduction
problem, let ( ) ; 1,2,...,
i
Xi
f
ri p be the amount of skewness on the th
i
rroot transformed
quality characteristic i
X
, i.e., i
r
i
X
. Then, finding an appropriate i
r is desired such that
()
i
Xi
f
r becomes zero. Therefore, by applying the bisection method we try to find a root
(i
r) for () 0
i
Xi
fr in the interval )1,0( . Note that for multivariate case one needs to find
different i
r values for different quality characteristics.
3.1 Using root transformation to estimate PCI of multivariate non-normal processes
In univariate processes when data does not follow a normal distribution,
transformation techniques such as Box-Cox and root transformation are commonly used
(See for example Box and Cox 1964, Tang and Than 1999, and Hosseinifard et al. 2009).
To estimate PCI of multivariate non-normal processes, the root transformation technique
can be first applied to diminish skewness in the marginal distributions of i
X
s. We stop
the iteration in the bisection procedure when skewness is either less than 0.05 or 200
iterations are used.
Let i
Y and i
Z
be the transformed and its standardized variable, respectively. The
skewness in the i
Z
s is almost zero such that they approach to standard normal variables
with the correlation matrix z
Σ. In other words, we may assume that the new vector Z
follows a multivariate normal with zero mean vector and covariance z
Σ. Then, the
correlation matrix of i
Z
s, denoted by ˆz
Σ, is estimated in the next step.
The specification limits of the original variables are transformed in the same
manner, i.e., we power each specification limit to the root obtained for its corresponding

11
variable and then standardize it in the same way as its corresponding variable. By this
procedure, a new specification limit for each i
Z
is obtained. As an example, assume that
the upper specification limit of the th
ioriginal quality characteristic ( i
X
) is i
X
USL . Then,
the upper specification limit of the th
istandardized-transformed variable ( i
Z
) will
be i
Z
USL which is calculated using equation (8)
ˆ
()
ˆ
i
ii
i
i
r
XY
ZY
USL
USL



 (8)
where ˆi
Y

and ˆi
Y

are the estimated mean and standard deviation of the ith transformed
variable, and i
r is the root obtained for the th
ioriginal variable such that the skewness
of i
r
i
Xwould be almost zero.
In the next step, a large sample from multivariate normal distribution with mean
zero and covariance ˆz
Σis generated. Then, based on the transformed specification limits
given in equation (8) the PNC of the generated sample is estimated. Finally, equation (9)
is used to estimate the process PCI.
1(1 )
3
pu PNC
C

 (9)
It is worth noting that in most non-normal processes, the marginal distributions
are right-skewed, resulting a need to define only USL (See Tang and Than 1999).
Consequently, the proposed step-by-step algorithm to estimate pu
Cwith the defined upper
specification limits is presented in Table (1).
In Table (1), the second step refers employing the root transformation technique
to find a specific root for each variable using the bisection method. In the third and fourth

12
steps, the data is transformed and the mean and variance of the new variables are
estimated using the method of moments. In step 5, new variables are standardized and in
step 6, the correlation matrix of the new variables is estimated. In step 7, the specification
limits are transformed in a similar fashion as their corresponding variables. In step 8, a
large number of data from the distribution of the new variables (the multivariate standard
normal distribution with correlation estimated in step 6), is generated. In step 9, the
generated data is compared with the transformed specification limits of step 7 and PNC is
estimated accordingly. Finally, based on the relationship between PNC and PCI
(Equation (9)), in step 10 the PCI of the process is estimated.
In case where both the upper and the lower specification limits are given, based
on equation (3) we simply change equation (8) and (9) by equation (10) and (11),
respectively.
ˆˆ
() ()
and
ˆˆ
ii
ii ii
ii
ii
rr
XY XY
ZZ
YY
LSL USL
LSL USL




 (10)

10.5 0.5(1 )
3
pPNC
C

 (11)
Insert Table (1) about here
4. Simulation experiments
In these experiments, multivariate non-normal processes of Beta, Weibull, and
Gamma are employed to estimate PCIs with pre-determined upper specification limits.
Given a sample of the process, the upper specification limits, and the algorithm of Table

13
(1) we first estimate the process PNC and use equation (9) to estimate pu
C. In order to
find PNC, a large sample is generated from a multivariate normal distribution with zero
mean and covariance ˆz
Σ.
In each of 1000 replications of simulation experiments, different sample sizes of
50, 100, 500, and 1000 are taken from the process. Furthermore, the number of generated
vectors of multivariate normal with zero mean and covariance ˆz
Σ in step 8 of the
algorithm is 1,000,000. The estimated mean and standard deviation of the estimated PCI
of the proposed method, pu
C
ˆ, are presented in Table (2). In order to evaluate the
performance of the proposed method, the right-most column of Table (2) shows the exact
pu
Cthat is computed by assuming that the distribution of the process is known. This
value is obtained using numerical integration methods on a modified version of equation
(6) for multivariate cases that is given in equation (12).
1
1
12 12
10.5 ... ( , ,..., ) ...
3
X
Xp
XX
p
USL
USL
pp
TT
pu
f
xx x dxdx dx
C







(12)
In which i
X
Tare the median of i
Xand 12
( , ,..., )
p
f
xx x is the joint probability density
function of the process.
The results in Table (2) indicate that the proposed method performs well, even for
small sample sizes. Note that the process distribution was assumed known solely for the
sake of performance evaluation. Having the process distribution is not a requirement to
employ the proposed method.

14
Insert Table (2) about here
5. A Case study
In a part manufacturing process of a sensitive plate in cars with one hole there are
two quality characteristics. These quality characteristics are shown in Figure (1). The
upper specification limit for 1
X
(maximum distance between the center of the hole and
the edge of the plate) is 5.85 mm, with no determined lower specification limit. The
upper and the lower specification limits for 2
X
(distance between the center of the hole
and the middle of breakdown on the plate) is 67 mm and 64 mm, respectively.
Insert Figure (1) about here
In 10 samples of size 100 each taken from the process, the proposed method was
employed and p
Cwas estimated accordingly. Based on all observations, i.e. 1000, Figure
(2) and (3) show the histogram of the observations on the first and second quality
characteristics, respectively. These figures indicate that although the marginal
distribution of the first quality characteristic seems to be normal (with a estimated
skewness of 0.06), for the second quality characteristic the marginal distribution is far
from normality (the estimated skewness is 1.95). Hence, the multivariate distribution of
the quality characteristics is not normal.
Insert Figure (2) about here

15
Insert Figure (3) about here
The estimated pu
C of the proposed method along with the estimated means,
standard deviations, and correlation of the quality characteristics for each sample are
given in Table (3). The mean of p
C
ˆ is 1.2777 and its standard deviation is 0.1033. The
corresponding value of the estimated mean of the estimated proportion of non-
conforming is 0.00008. We applied the proposed method for samples of 1000 products
and reached 1.315 as the estimated mean of pu
C.
Insert Table (3) about here
6. Conclusion
In this paper, the root transformation technique was first employed to diminish the
skewness involved in multivariate non-normal processes. Then, based on the transformed
specification limits and Monte Carlo simulation, the proportion of nonconforming (PNC)
products was estimated. Next, applying the relation between PNC and PCI the estimated
value of PCI for multivariate non-normal processes was obtained. Multivariate Gamma,
multivariate Beta, and multivariate Weibull distributions were used in some simulation
experiments to compare the results of the proposed method with the exact PCI values.
Finally, to demonstrate the application of the proposed method we used it in a real case

16
problem. The results of both simulation experiments and the real case problem were
encouraging.
Acknowledgement
The authors are thankful for constructive comments of the reviewers that
improved the presentation of the manuscript. This research was supported by the Iranian
National Science Foundation under grant number 86121120, which is greatly appreciated.
References
Abbasi, B. (2009). A neural network applied to estimate process capability of non-normal
processes. Expert Systems with Applications 36: 3093-3100.
Bai, D.S., Choi, S.S. (1997). Process capability indices for skewed populations. Master
Thesis, Department of Industrial Engineering, Advanced Institute of Science and
Technology, Taejon, South Korea.
Box, G., Cox, D.R. (1964). An analysis of transformation. Journal of the Royal Statistical
Society, Series B 26: 211–243.
Boyles, R.A. (1994). Process capability with asymmetric tolerances. Communications in
Statistics–Simulation and Computation 23: 615-643.
Castagliola, P. (1996). Evaluation of non-normal process capability indices using Burr’s
distributions. Quality Engineering 8: 587-593.
Castagliola, P., Castellanos, J.V.G. (2005). Capability indices dedicated to the two quality
characteristic cases. Quality Technology and Quantitative Management 2: 201-
220.

17
Chen, H. (1994). A multivariate process capability index over a rectangular solid
tolerance zone. Statistica Sinica 4: 749-758.
Chen, K.S., Hsu, C.H., Wu, C.C. (2006). Process capability analysis for a multi-process
product. International Journal of Advanced Manufacturing Technology 27: 1235-
1241.
Choi, I.S., Bai, D.S. (1996). Process capability indices for skewed populations. In
Proceedings of 20th International Conference on Computers and Industrial
Engineering, 1211-1214.
Clements, J.A. (1989). Process capability calculations for non-normal distributions.
Quality Progress 22: 95–100.
Hosseinifard, S.Z., Abbasi, B., Ahmad, S., Abdollahian, M. (2009). A transformation
technique to estimate the process capability index for non-normal processes.
International Journal of Advanced Manufacturing Technology 40: 512-517.
Huang, M.L., Chen, K.S., Hung, Y.H. (2002). Integrated process capability analysis with
an application in backlight module. Microelectronic Reliability 42: 2009–2014.
Hubele, N., Shahriari, H., Cheng, C. (1991). A bivariate capability vector. In Statistics
and Design in Process Control: Keeping Pace with Automated Manufacturing,
Keats and Montgomery, eds., Marcel Dekker, 299-310.
Johnson, N.L. (1949). System of frequency curves generated by methods of translation.
Biometrika 36: 149–176.
Kane, V.E. (1986). Process capability indices. Journal of Quality Technology 18: 41-52.

18
Liao, S.J., Chen, K.S., Li, R.K. (2002). Capability evaluation for process of the larger-
the-better type for non-normal populations. Advances and Applications in
Statistics 2: 189–198.
Liu, P.H., Chen, F.L. (2006). Process capability analysis of non-normal process data
using the Burr XII distribution. International Journal of Advanced Manufacturing
Technology 27: 975-984.
Mousa, M.A., Jaheen, Z.F. (2002). Statistical inference for the Burr model based on
progressively censored data. Computers & Mathematics with Applications 43:
1441–1449.
Niaki, S.T.A., Abbasi, B. (2007). Skewness reduction approach in multi-attribute process
monitoring. Communications in Statistics-Theory and Methods 36: 2313–2325.
Pal, S. (2005). Evaluation of non normal process capability indices using generalized
lambda distribution. Quality Engineering 17: 77-85.
Pearn, W.L., Kotz, S. (1994). Application of Clements’ method for calculating second
and third generation process capability indices for non-normal Pearsonian
populations. Quality Engineering 7: 139–145.
Pearn, W.L., Chen, K.S., Lin, G.H. (1999). A generalization of Clements’ method for
non-normal Pearsonian processes with asymmetric tolerances. Quality and
Reliability Management 16: 507-521.
Quesenberry, C.P. (1995). Geometric Q-charts for high quality processes. Journal of
Quality Technology 27: 304–315.
Ryan, T.P. (1989). Statistical methods for quality improvement. John Wiley, New York.

19
Somerville, S., Montgomery, D. (1996). Process capability indices and non-normal
distributions. Quality Engineering 19: 305–316.
Taam, W., Subbaiah, P., Liddym J.W. (1993). A note on multivariate capability indices.
Journal of Applied Statistics 20: 339-351.
Tang, L.C., Than, S.E. (1999). Computing process capability indices for non-normal data:
a review and comparative study. Quality and Reliability Engineering
International 15: 339-353.
Wang, F.K. (2006). Quality evaluation of a manufactured product with multiple
characteristics. Qual and Reliability Engineering International 22: 225–236
Wang, F.K., Du, T.C. (2000). Using principal component analysis in process performance
for multivariate data. Omega 28:185–194.
Wang, F.K., Hubele, N.F. (1999). Quality evaluation using geometric distance approach.
International Journal of Reliability, Quality, and Safety Engineering 6:139–153.
Wang, F.K., Hubele, N.F., Lawrence, F.P., Miskulin, J.D., Shahriari, H. (2000).
Comparison of three multivariate process capability indices. Journal of Quality
Technology 32: 263-275.
Wang, K.J.B., Zimmer, W.J. (1996). The maximum likelihood estimation of the Burr XII
parameters with censored and uncensored data. Microelectronic Reliability 36:
359–362.
Wright, P.A. (1995). A process capability index sensitive to skewness. Statistical
Computations and Simulation 52:195-302.

20
Wu, H.H., Wang, J.S., Liu, T.L. (1998). Discussions of the Clements-based process
capability indices. In: Proceedings of the 1998 CIIE National Conference, 561–
566.
Xie, M., Goh, T.N., Tang, Y. (2000). Data transformation for geometrically distributed
quality characteristics. Quality and Reliability Engineering International 16: 9–
15.
Yeh, A., Chen, H. (2001). A nonparametric multivariate process capability index.
International Journal of Modeling and Simulation 21: 218–223.
Zimmer, W.J., Keats, J.B., Wang, F.K. (1998). The Burr XII distribution in reliability
analysis. Journal of Quality Technology 30: 2-9.

21
Table (1): The Proposed Algorithm to Estimate pu
C
Steps Descriptions Comments
Step 1 Given a sample of the process p is number of quality characteristics
and n is the sample size
Step 2 Compute a specific root ( i
r) for each variable
i
Xby the root transformation technique ( ) 0 ; 1,2,...,
i
r
i
Skewness X i p
Step 3 Compute new variable ; 1,2,...,
i
r
ii
YX i p 12
12
... pT
r
rr p
XX X





Y
Step 4 Estimate the mean and the standard deviation
of the new variable i
Y say ˆi
Y

and ˆi
Y

ˆ
ˆˆˆ
(), ()
ii
YiY i
E
YVarY


Step 5
Standardize the variable i
Y as
ˆ ; 1,2,...,
ˆi
i
iY
iY
Y
Z
ip




12
12
12 ˆ
ˆˆ
.
ˆˆ ˆ
p
p
T
pY
YY
YY Y
Y
YY


 











Z
Step 6 Estimate the correlation matrix of Z, say ˆz
Σ
Step 7
Transform the USLs
ˆ
() ; 1,2,...,
ˆ
i
ii
i
i
r
XY
ZY
USL
USL i p




1
11
1
ˆ
()
ˆ
.
.
.
ˆ
()
ˆ
p
pp
p
r
XY
Y
r
XY
Y
USL
USL


















Z
USL
Step 8
Generate large sample vectors from
multivariate normal with mean zero and
covariance ˆz
Σ
Step 9
Estimate the proportion of new generated
vector (say PNC) that are out of the new USL
(Z
USL )
otal number of generated vectors
Number of vectors that are out of
NT
NNC USL
NNC
PNC N



Step 10 Estimate PCI using PNC 1(1 )
3PNC
Cpu




22
Table (2): The results of simulation experiments
Sample
size Distribution USL


ρ
ˆpu
C Exact
p
C
Mean Std
50
Gamma 26
13 2
1 3
2 





10.49
0.491
0.862 0.122
0.89
100 0.873 0.087
500 0.873 0.039
1000 0.873 0.029
50
Gamma 150
58
130
2
6
5 8
3
7
1 0.37 0.58
-0.37 1 0.28
0.58 -0.28 1












26
13
1.161 0.134
1.18
100 1.151 0.116
500 1.163 0.065
1000 1.172 0.050
50
Beta 99.0
99.0 4
2 4
5 





10.79
0.791
0.939 0.100
1.12
100 0.949 0.070
500 0.982 0.034
1000 0.995 0.026
50
Weibull 10
9
5.6
6
4
2
5
3
2










10.490.588
0.4910.29
0.580.291
1.114 0.117
1.28
100 1.147 0.086
500 1.199 0.039
1000 1.213 0.029

23
Figure (1): The part and its quality characteristics
2
X
1
X
2
X

24
Figure (2): The histogram of 1000 observations on the first quality characteristic
64.5 65 65.5 66 66.5 67
0
50
100
150
200
250
300
f
X
1

25
Figure (3): The histogram of 1000 observations on the second quality characteristic
4.2 4.4 4.6 4.8 55.2 5.4 5.6 5.8 6
0
50
100
150
200
250
300
350
400
450
500
f
X
2

26
Table (3): Estimated pu
C for 10 samples of size 100 for the real case
Sample 1 2 3 4 5 6 7 8 9 10
pu
C
ˆ 1.300 1.300 1.060 1.260 1.440 1.300 1.390 1.200 1.250 1.220
1
ˆX

65.91
2 65.90
7 65.91
0 65.80
7 65.82
8 65.93
5 65.91
5 65.92
2 65.95
4 65.89
2
2
ˆX

4.455 4.457 4.449 4.430 4.447 4.437 4.449 4.460 4.510 4.468
1
ˆX

0.289 0.293 0.364 0.328 0.291 0.282 0.275 0.307 0.289 0.319
2
ˆX

0.207 0.275 0.218 0.254 0.298 0.253 0.267 0.247 0.271 0.285
ˆ

0.350 0.474 0.429 0.425 0.443 0.456 0.337 0.398 0.287 0.457