Measuring Process Capability for Bivariate Non-Normal Process Using the Bivariate Burr Distribution

Abstract

As is well known, process capability analysis for more than one quality variables is a complicated and sometimes contentious area with several quality measures vying for recognition. When these variables exhibit non-normal characteristics, the situation becomes even more complex. The aim of this paper is to measure Process Capability Indices (PCIs) for bivariate non-normal process using the bivariate Burr distribution. The univariate Burr distribution has been shown to improve the accuracy of estimates of PCIs for univariate non-normal distributions (see for example, [7] and [16]). Here, we will estimate the PCIs of bivariate non-normal distributions using the bivariate Burr distribution. The process of obtaining these PCIs will be accomplished in a series of steps involving estimating the unknown parameters of the process using maximum likelihood estimation coupled with simulated annealing. Finally, the Proportion of Non-Conformance (PNC) obtained using this method will be compared with those obtained from variables distributed under the bivariate Beta, Weibull, Gamma and Weibull-Gamma distributions.

Full text

Measuring Process Capability for Bivariate Non-Normal Process Using
the Bivariate Burr Distribution
B. ABBASI, S. AHMAD, M. ABDOLLAHIAN and P.ZEEPHONGSEKUL
School of Mathematical and Geospatial Sciences
RMIT University
Melbourne 3001
AUSTRALIA
abbasi.babak@rmit.edu.au
shafiq.ahmad@rmit.edu.au
mali.abdollahian@rmit.edu.au
panlopz@rmit.edu.au
Abstract: - As is well known, process capability analysis for more than one quality variables is a complicated
and sometimes contentious area with several quality measures vying for recognition. When these variables
exhibit non-normal characteristics, the situation becomes even more complex. The aim of this paper is to
measure Process Capability Indices (PCIs) for bivariate non-normal process using the bivariate Burr
distribution. The univariate Burr distribution has been shown to improve the accuracy of estimates of PCIs for
univariate non-normal distributions (see for example, [7] and [16]). Here, we will estimate the PCIs of bivariate
non-normal distributions using the bivariate Burr distribution. The process of obtaining these PCIs will be
accomplished in a series of steps involving estimating the unknown parameters of the process using maximum
likelihood estimation coupled with simulated annealing. Finally, the Proportion of Non-Conformance (PNC)
obtained using this method will be compared with those obtained from variables distributed under the bivariate
Beta, Weibull, Gamma and Weibull-Gamma distributions.
Key-Words: - Process Capability Index (PCI), bivariate Burr distribution, simulated annealing algorithm, non-
normal distribution, multivariate processes.
1 Introduction
In the field of statistical quality control, it is
generally assumed that the distributions of quality
characteristics are normal. But, in most practical
cases this assumption is not valid and the
distribution of the quality characteristics may follow
non-normal distributions such as Gamma, Beta, and
Weibull distributions.
Many industries are using a quantitative measure
called Process Capability Indices (PCIs) for the
purpose of process assessment and improvement.
The objective of these statistical measures is to
estimate process variability relative to process
specifications. Additionally, process capability
provides a common standard of product quality for
suppliers and customers. The standard Process
Capability Index is based on certain assumptions
which are as follow:
Data are collected from an in-control process.
Collected process data are independent and
identically distributed.
Collected process data are normally distributed.
For non-normal stable processes, capability ratio Cp
and process capability ratio for off center process
Cpk , defined by Kane (1986) (equation (1) and (2)),
are not appropriate.
σ
6
LSLUSL
Cp
−
= (1)
),min( plpupk CCC
=
(2)
where
σ
μ
3
LSL
Cpu
−
= (3)
and
σ
μ
3
−
=USL
Cpl (4)
USL is the upper specification limit, LSL is the
lower specification limit, µ is the process mean, and
σ is the process standard deviation. If µ and σ are
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not known, one can replace them by
X
and
respectively, where
S
X is the sample mean and is
the sample standard deviation.
S
In the past decade, several modifications of classical
PCIs have been proposed to resolve the issue of non-
normality of quality characteristics data. Castagliola
(1996) presented a new approach to compute
process capability. This approach is based on using
probability distribution to compute the proportion of
non-conforming items and then use these to estimate
the capability index. This approach is
straightforward, logical and easy to deploy by
engineers and managers, for normal as well as for
non-normal data. Castagliola et al. (2005) have also
extended univariate method in Castagliola (1996) to
bivariate distribution but again limited it to bivariate
normal data, and compared the results against
existing methods for multivariate normal processes.
In this paper, we will use method presented by
Castagliola et al. (2005) to evaluate PCI for bivariate
non-normal quality characteristics data. Preliminary
to this, we also use the bivariate Burr distribution
with three parameters (Durling (1975)) to fit our
bivariate non-normal data.
This paper is organized in the following manner. A
capability analysis for univariate non-normal data
and multivariate normal data is discussed in Section
2. A review of the bivariate Burr distribution is
discussed in Section 3. Section 4 explains our
proposed method to estimate the Burr parameters
using simulated annealing algorithm (SA).
Simulation studies for different bivariate non normal
distributions are presented in Section 5 and, finally,
we conclude the paper with suggestions for future
works.
2 PCI for Non-Normal Data and
Multivariate Normal Data
Many researchers have proposed several methods to
handle the issue of non-normality in the quality
characteristics data. Most of these efforts have been
devoted to estimate PCI for multivariate normal
data. In case of multivariate non-normal quality
data, this field is still wide open for researchers due
to the complex nature of the problem. In the
proceeding section; we will review research
literature related to the subject mater.
2.1 PCI for Univariate Non-Normal
Processes
One simple method to handle non normal data is to
transform the data into normal form using
mathematical functions and then use traditional
normal methods to estimate PCI. For transformation
purpose, Johnson (1949) built a system of
distributions based on moment method, called the
Johnson transformation system. Box & Cox (1964)
presented a useful family of power transformations
which transform non-normal data into normal ones.
Somerville & Montgomery (1996) also used a
square-root transformation to transform a skewed
distribution into a normal one. Niaki & Abbasi
(2007) presented the transformation called “root
transformation” to transform skewed discrete
multivariate data to multivariate normal data.
Another conceptually simple way to treat the non-
normal data is to use non-normal percentiles to
modify classical PCIs. Clements (1989) proposed
the method of non-normal percentiles to calculate
process capability Cp and process capability for off
center process Cpk indices for a distribution of any
shape, using the Pearson family of curves.
Clements’s method is widely used in industry. Pearn
& Kotz (1994) applied Clements’s method to
construct the second generation index Cpm and the
third generation Cpmk for non normal data. Pearn et
al. (1999) presented a generalization of Clements’
method with asymmetric tolerances. These quantile-
based indices, Cp and Cpk for non normal data are
defined as follow:
135.0865.99 XX
LSLUSL
p
C−
−
= (5)
),min( plpupk CCC
=
(6)
where
50865.99
50
XX
XUSL
Cpu −
−
= (7)
and
135.050
50
XX
LSLX
Cpl −
−
= (8)
In the above equations, percentile value
of the data. Although Clements’s method is
commonly used in industry, a research study by Wu
et al. (1998) indicated that the Clements’s method
can not accurately measure the PCI values,
especially when the underlying data distribution is
skewed. Tang & Than (1999) also did a
comprehensive review of the process capability
indices for non normal processes.
th
ppX 100* is
Liu & Chen (2006) proposed a modified Clements
method to evaluate PCI for non-normal data. They
suggested that accuracy of the estimated PCI for non
normal data can be improved by using Burr
distribution instead of the Pearson curves
percentiles. The parameters of a Burr probability
density distribution function can be set as to fit the
normal, Gamma, Beta, Weibull, log-normal,
extreme value type I distribution. Wang et al.
(1996), Zimmer et al. (1998), Kan and Yazici
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(2006) and Mousa & Jaheen (2002) have presented
a comprehensive review of Burr distribution and its
application to many non-normal situations.
Liu & Chen (2006) has used 3rd and 4th sample
moments of the data to get the standardized
moments and used Burr tables to fit Burr
distribution to process data.
Using simulation study, Liu & Chen (2006) showed
that Burr distribution is superior to Clements’s
method in estimation Cpu but both methods over
estimate the Cpu in cases of highly skewed
distributions (skewness ≥1.5).
Bai & Choi (1997) have developed a “weighted
variance” (WV) approach to measure PCIs for
skewed distributions. Pal (2005) evaluated PCI
using process capability of non normal to
generalized Lambda distribution. Parchami et. al
(2005) used a fuzzy approach to estimate PCI.
Castagliola (1996) defined the relationship between
process capability and proportion of non-
conforming items and presented a new approach to
evaluate PCI for non normal data. 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 used a
polynomial function to approximate a Burr
distribution and from this obtained the Process
Capability Indices.
For normal data, it is easily shown that
3
))(5.05.0(
1∫
+
−
Φ
=
usl
lsl dxxf
p
C (9)
),min( plpupk CCC = (10)
where
3
))(5.0(
1∫
+
−
Φ
=
T
lsl dxxf
pl
C (11)
and
3
))(5.0(
1∫
+
−
Φ
=
usl
Tdxxf
pu
C (12)
where f(x) represents the probability density
function of the process and T its mean. For non-
normal distribution, the above equations can still be
used to obtain PCIs but T now represents the
median.
2.2 PCI for Multivariate Normal Process
Data
Multivariate process capability indices, in general,
can be obtained from (a) the ratio of a tolerance
region to a process region, (b) the probability of
nonconforming product, and (c) other approaches
using loss functions. Hubele et al. (1991,) using
multivariate normal distribution, defined the PCI as
the ratio of the rectangular tolerance region to
modified process region which is the smallest
rectangle around the ellipse with 0027.0=
α
. The
number of quality characteristics in the process is
taken into account by taking the root of the ratio
where
th
υ
υ
present the number of quality
characteristics.
υ
1
region process gengineerin theof vol.
region tolerancegengineerin theof vol. ⎥
⎦
⎤
⎢
⎣
⎡
=
PM
C (13)
Here the modified tolerance region is the largest
ellipsoid centered at the target which falls
completely within the original tolerance region.
Another method for estimating PCI for multivariate
normal was proposed by Chen (1994). In that
paper, a tolerance zone is defined by
{
}
0
)
0
(: rXh
V
RXV ≤−∈=
μ
, where is a
positive number,
0
r
0
μ
is a target value and is a
positive function. The process is capable if
)(h x
α
−
≥
∈
1)( VXP .
Let
{
}
α
μ
−≥≤
−
=
1))
0
((:min cXhPcr . If the
cumulative distribution function of )( 0
μ
−
Xh is
increasing in a neighborhood of
r
, then
r
is simply
the unique root of equation
α
μ
−
=
≤
−(0
XhP 1))( c.
The process is deemed capable if . Here is
the half-width of the tolerance interval centered at
the target value,
0
rr ≤0
r
0
μ
. Here, , 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
r
α
−
1.
Wang et al. (2000) compared the above three
multivariate process capability indices and
presented some graphical examples to illustrate
them. Chen et al. (2006) extend Boyles’ work
(1994) for normal distribution and Liao et al.’s work
(2002) for non normal distribution. They have also
extended Huang et al.’s (2002) work for
multivariate data but they have not considered the
correlation between the variables. They computed
process capability for multivariate data (without
correlation) and for each individual variable.
Castagliola and Castellanos (2005) extended the
univariate method developed in Castagliola (1996)
to multivariate normal distribution by replacing the
univariate probability density function with
the multivariate normal probability density function.
They used equation (9) but replaced f(x) with a
multivariate normal pdf , i.e.
)(xf
),...,,( 21 p
xxxf
3
1
1
)...
21
),...,
2
,
1
(
2
2
... 5.05.0(
1∫∫ ∫
+
−
Φ
=
usl
lsl p
dxdxdx
p
xxxf
usl
lsl p
usl
p
lsl
p
C
(14)
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Keeping in view the above literature survey, there is
an opportunity for researchers to explore a suitable
PCI evaluation method that can address the complex
situation of multivariate non-normal data. In this
paper, we replace probability density function in
equation (14) with a Burr distribution. The efficacy
of the proposed method will be assessed by using
the Proportion of Non-Conformance (PNC)
criterion.
3 Review of the Bivariate Burr
Distributions
Durling (1975) introduced the bivariate Burr
distribution as follows:
0,
2
,
1
,0
2
,
1
,
)2(
)
2
2
1
1
1(*
1
2
2
1
1
121
)(
)2(
)
2
,
1
(
≥≥
+−
++
−−
Γ
+Γ
=
pbbxx
p
b
x
b
x
b
x
b
xbb
p
p
xxf
(15)
The cumulative distribution function has the form:
0,
2
,
1
,0
2
,
1
,)
2
2
1
1
1()
2
2
1(
)
1
1
1(1)
2
,
1
(
≥≥
−
+++
−
+
−
−
+−=
pbbxx
p
b
x
b
x
p
b
x
p
b
xxxF
(16)
In the bivariate Burr distribution there are three
parameters, , to be estimated. These
parameters can be estimated by maximizing the log
likelihood function based on a sample of size n
given by:
pbb and , 21
∑
∑∑
=
==
+++
−−+
−+++
++=
n
j
b
j
b
j
n
j
j
n
j
j
n
xxp
xbx
bpnpn
bnbnxxpbbL
1
21
1
2
1
21
1
21121
)1ln()2(
ln)1(ln
)1())1(ln()(ln
)(ln)(ln),...,:,,(
21
(17)
nj
j
x
j
x,...,2,1 ),
2
,
1
( =is an observed bivariate
sample. The first order condition for maximizing L
with respect to lead to the following
differential equations:
pbb and , 21
0
1
ln
)2(
ln
121
1
1
1
1
11
21
1
=
++
+
−+=
∂
∂
∑
∑
=
=
n
j
b
j
b
j
b
j
j
n
j
j
xx
xx
p
x
b
n
b
L
(18)
0
1
ln
)2(
ln
121
2
2
1
2
22
21
2
=
++
+
−+=
∂
∂
∑
∑
=
=
n
j
b
j
b
j
b
j
j
n
j
j
xx
xx
p
x
b
n
b
L
(19)
0)1ln(
1
1
21
21 =++
−
+
+=
∂
∂
∑
=
n
j
b
j
b
jxx
p
n
p
n
p
L
(20)
In this paper, instead of solving equations (18), (19)
and (20) for our maximum likelihood estimators of
, we will use, directly from equation
(17), a systematic random search algorithm called
“Simulated Annealing” (see Abbasi et al. (2006)) to
obtain the estimated parameters.
pbb and , 21
4. Computing CPI for Bivariate Non-
Normal Data Using Burr Distribution
To use equation (14), one needs to calculate the
probability of quality characteristics falling between
specification limits. In order to calculate this
probability we need to know the distribution of the
data. In this paper we use bivariate Burr distribution
to calculate the probability of non-conforming
products in a bivariate non-normal process.
Maximum likelihood estimation (MLE) method is
used to estimate its unknown parameters
.Since the maximum likelihood function
(MLF) of bivariate Burr is complex and may have
some local optima, and numerical method to solve
differential equations may also give local optima,
we will maximize likelihood function by using
Simulated Annealing algorithm (SA). Abbasi et al.
(2006) used simulated annealing algorithm to
estimate three parameters of Weibull distribution
through MLE method and observed that it was fast
and the results were very accurate. Having obtained
the Burr distribution, we will then use equation (14),
and replace in the numerator with the
bivariate Burr distribution (equation (15)) to
compute process capability (C
pbb and , 21
),...,,( 21 p
xxxf
p). Table 1 outlines
the procedure of the proposed procedure.
5. Simulation studies
The purpose of this section is to show the capacity
of the proposed method for estimating the Cp value
of non-normal bivariate processes. Simulation
studies have been conducted for bivariate non-
normal processes. For this simulation study,
bivariate non normal distributions such as Gamma,
Beta and Weibull and Weibull- Gamma are used.
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Table 2 depicts the simulation methodology for this
research study. (We used NORTA method to
generate simulation bivariate data. Refer to Cario &
Nelson (1997) and Niaki & Abbasi (2007) for a
discussion of the procedures used to generate these
data.)
Table (1) – the procedure for computation
p
C
Table (3) presents the parameters of the bivariate
non-normal distributions used in the simulation
study. The Cp value computed using the exact
bivariate distributions, for example Gamma, is
presented under heading Exact Distribution. We
have generated m=30 samples of size n=100 and
fitted a bivariate Burr distribution to each sample.
The parameters for the fitted Burr
distribution are estimated using SA algorithm. The
C
pbb and , 21
p’s of these 30 samples are then calculated using
equation (14). The mean and standard deviation for
30 computed Cp’s are presented in the last two
columns of Table (3). The results in Table (3) show
that the mean Cp values for different bivariate non-
normal distributions are very close to the exact Cp
value. Therefore one can conclude that the proposed
method enables one to estimate Cp value of the
bivariate non-normal data reasonably accurately. To
further assess the efficacy of the proposed method,
we have also calculated the Proportion of Non-
Conformance (PNC) data using in
Table (4). This table also indicates that the
proportions of non-conformance based on the
proposed method are close to the proportion of non-
conformance obtained using the exact distributions.
Cp) (-3 = PNC Φ
Table (2) – the procedure for simulation methodology
Step 1
Generate 100 vectors from bivariate non-
normal using one of above distributions.
(Compute expected proportion of non-
conformance (p*) by using 1,000,000 data
from the corresponding distribution e.g.,
Gamma and calculate the proportion of data
falling out side the given USL.)
Step 2-1 Fit MLF of Bivariate Burr distribution to
data.
Step 2-2 Estimate parameters of the fitted bivariate
Burr distribution using SA.
Step 3 Use Castagliola method to compute Cp for
Bivariate Burr distribution Eq (14)
Step 4 Compute proportion of non-conforming for
Cp say p**
)3( =PNC pu
CΦ−
Step 5
Compare p*and p** to evaluate the accuracy
of the proposed method
Table (3) – Simulation study for bivariate non-normal
distribution
Step 1 Select a sample from the process.
Step 2
Write down the maximum likelihood function
(MLF) for sample based on bivariate Burr
distribution.
Step 3 Maximize MLF by using Simulated Annealing
and obtain estimates of b1, b2 and p.
Step 4
From Eq (16) compute the difference between
cumulative density function at the upper
specification limits ( ) and the lower
specification limits ( ), i.e.
.
21,uslusl
21 ,lsllsl
),(),( 2121 lsllslFusluslFB −=
Step 5 3
)5.05.0(
1B
Cp
+Φ
=
−.
Step 6
Compute the corresponding PNC=
and compare it with the PNC obtained from
the exact distribution for example Gamma.
)3( cp−Φ
Distribution Parameters1
Σ
Gamma α=[3,4]
β=[2.5,3]
Gamma α =[3,6]
β=[8,10]
Beta α =[2,4]
β=[5,4]
Weibull α =[3,4]
β =[1,2]
Gamma,
Weibull
α =[5,2]
β=[3,5]
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Follow of Table (3)
1 Note that each value in the pair represents the corresponding
marginal distribution.
2 Specification limits are selected to represent almost natural
specification limits
Table (4) – proportion of non-conformance
Distribution Burr p**
Expected PNC In
the process p*
Gamma 0.0325 0.0199
Gamma 0.0025 0.0018
Beta 0.0151 0.0095
Weibull 0.0064 0.0073
Gamma, Weibull 0.0132 0.0104
6. Conclusion
In this paper, the method proposed in Castagliola et
al. is used to estimate the process capability index
for bivariate non-normal quality characteristic data.
We used the bivariate Burr distribution to fit the
probability density function of the data. The process
combines both MLE and simulated annealing
algorithm to estimate the parameters of the bivariate
Burr distribution. We have presented the results
using simulated data from bivariate distribution such
as Gamma, Beta and Weibull. Thirty samples of size
100 from each distribution are generated and used to
assess the accuracy of the proposed approach.
The simulation study results for different non-
normal bivariate distributions revealed that the
proposed method perform well. Using the expected
non-conformance proportion criterion, the results
also indicate that proportion of non-conformance
obtained using the proposed method is close to that
obtained under the exact distributions.
The extension of the proposed method to more than
two non-normal multivariate quality data is
straightforward and is recommended as a future
research area. Using other metaheuristic algorithms
such as Genetic Algorithm (GA) and Tabu Search
(TS) to estimate bivariate Burr parameters could be
another future research.
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Cp
Burr
Specification
Limits2
(n=100) (m=30)
LSL USL
Exact
Distribution Mean Std
[0,0] [20,45] 0.7761 0.7124 0.1117
[0,0] [100,160] 1.0405 1.0109 0.2371
[0.0025
,0.005] [0.9,0.92] 0.8645 0.8100 0.0403
[0,0] [15,12] 0.8943 0.9089 0.3830
[0,0] [45,6.8] 0.8541 0.8260 0.3381
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Issue 5, Volume 4, May 2007

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WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
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Issue 5, Volume 4, May 2007