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ISSN: 2277-3754
ISO 9001:2008 Certified
International Journal of Engineering and Innovative Technology (IJEIT)
Volume 3, Issue 5, November 2013
A Reliable Procedure on Performance
Evaluation - A Large Sample Approach Based on
the Estimated Taguchi Capability Index
Gu-Hong Lin
Professor, Department of Industrial Engineering and Management,
National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan
Abstract—the Taguchi capability index Cpm has been proposed
to the manufacturing industry for measuring manufacturing
capability. Contributions of the estimated Taguchi capability
index based on subsamples have been proposed and arrested
substantial research attention. In this paper, investigations based
on the proposed estimator are considered under general
conditions having fourth central moment exists. The limiting
distribution of the considered estimator is derived. A reliable
inferential procedure based on large samples is proposed. A
demonstrate example is also provided to illustrate how the
proposed approach may be applied for judging whether the
process runs under the desirable quality requirement.
Key words: asymptotic, capability, Taguchi.
obtained by substituting the sample mean x = in1 xi / n
for and the sample variance s n21  in1 ( xi  x ) 2 / (n - 1) for
 in (1) and (2). The statistical properties and the sampling
distributions of the natural estimators of Cp, Ca, and Cpk have
been widely investigated in literature (e.g., [2]-[8]).
The capability index Cpk is a yield-based index [9].
However, the design of Cpk is independent of the target value
T, which can fail to account for process targeting (the ability
to cluster around the target). For this reason, the index Cpm
was independently introduced (e.g., [10]-[11]) to take the
process targeting issue into consideration. The index Cpm is
defined as
USL  LSL
.
(3)
C pm 
6  2  (  T ) 2
I. INTRODUCTION
Process capability indices, whose purpose is to provide
numerical measures on whether or not a manufacturing
process is capable of reproducing items satisfying the quality
requirements preset by the customers or the product
designers, have received substantial research attention in the
quality control and statistical literature,. The three basic
capability indices Cp, Ca, and Cpk have been defined as (e.g.,
[1]-[4])
Cp 
USL  LSL
, Ca  1    m ,
6
d
USL     LSL 
C pk  min
,
 3 , 3 
(1)
(2)
We note that the index Cpm is not originally designed to
provide an exact measure on the number of non-conforming
items. But, Cpm includes the process departure ( - T)2 (rather
than 6 alone) in the denominator of the definition to reflect
the degree of process targeting. Some Cpm values commonly
used as quality requirements in most industry applications are
1.00, 1.33, 1.50, 1.67, and 2.00. A process is called
―inadequate‖ if Cpm < 1.00, called ―capable‖ if 1.00  Cpm <
1.33, called ―marginally capable‖ if 1.33  Cpm < 1.50, called
―satisfactory‖ if 1.50  Cpm < 1.67, called ―excellent‖ if 1.67 
Cpm < 2.00, and is called ―super‖ if Cpm  2.00. The above six
quality requirements and the corresponding Cpm values are
displayed in Table 1.
Table 1: Six quality conditions based on Cpm
------------------------------------------------------where USL and LSL are the upper and lower specification
limits preset by the customers, the product designers,  is the
process mean,  is the process standard deviation, m = (USL +
LSL)/2 and d = (USL - LSL)/2 are the mid-point and half
length of the specification interval, respectively. The index Cp
reflects only the magnitude of the process variation relative to
the specification tolerance and, therefore, is used to measure
process potential. The index Ca measures the degree of
process centering (the ability to cluster around the center) and
is referred as the process accuracy index. The index Cpk takes
into account process variation as well as the location of the
process mean. The natural estimators of Cp, Ca, and Cpk can be
359
Quality Condition
Cpm values
------------------------------------------------------Inadequate
Cpm < 1.00
Capable
1.00  Cpm < 1.33
Marginally Capable
1.33  Cpm < 1.50
Satisfactory
1.50  Cpm < 1.67
Excellent
1.67  Cpm < 2.00
Super
Cpm  2.00
-------------------------------------------------------
II. ESTIMATING CPM UNDER THE NORMALITY
ASSUMPTION
A. The Estimated Cpm Based on Single Sample
Assuming that the measurements of the characteristic
ISSN: 2277-3754
ISO 9001:2008 Certified
International Journal of Engineering and Innovative Technology (IJEIT)
Volume 3, Issue 5, November 2013
investigated, (X1, X2, …, Xn), are chosen randomly from a m, j = 1, 2, …, ni. Reference [14] considered the following
stable process which follows a normal distribution N( , 2). estimator of Cpm based on m subsamples of size ni each:
Reference [9] recommended a MLE (maximum likelihood
d
estimator) of Cpm by substituting the sample mean x =
.
(7)
Cˆ * 
pm
2
n
n
2
2
i 1 xi / n for  and the MLE sn  i 1 ( xi  x ) / n for  in
3 im1 nji1 ( xij  T ) / im1 ni
(3), which can be expressed as
Cˆ pm 
For 0 < x < ∞, the exact cdf Fm(x) and pdf fm(x) of Cˆ *pm can
USL  LSL
6 sn2  ( x  T ) 2

d
,
(4)
3 in1 ( xi  T ) 2 / n
be expressed as a mixture of the chi-square distribution and
the standard normal distribution:
where d = (USL – LSL)/2 is half the length of the specification
interval. As noted by Reference [12], the term
2
n
s + ( x - T)
Fm ( x)  1 
2
(8)
 [ (t   N )   (t   N )] dt ,
f m ( x) 
b N /(3 x )
*
2
2
2
2
3
 u {[(b N ) /(9 x )]  t }[( 2b N ) /(9 x )]
0
(9)
 [ (t   N )   (t   N )] dt ,
chi-square distribution and the standard normal distribution
[12]:
where b = d/,  = ( - T)/, N  im1 ni , U*(.) and u*(.)
b n /(3 x )
2
2
2
 U {[(b n) /(9 x )]  t }
0
(5)
 [ (t   n )   (t   n )] dt ,
f s ( x) 
*
2
2
2
 U {[(b N ) /(9 x )]  t }
0
in the denominator of (4) is the UMVUE (uniformly minimum
variance unbiased estimator) of the term 2 + ( - T)2 in the
denominator of Cpm, it is reasonable for reliability purpose.
Under the assumption of normality, the exact cdf Fs(x) and
pdf fs(x) of Cˆ pm can be expressed in terms of a mixture of the
Fs ( x)  1 
b N /(3 x )
b n /(3 x )
2
2
2
2
3
 u{[(b n) /(9 x )]  t }[( 2b n) /(9 x )]
0
(6)
 [ (t   n )   (t   n )] dt ,
for x > 0, where b = d/,  = ( - T)/, U(.) and u(.) are the cdf
and pdf of the chi-square distribution 2 with n – 1 degrees of
freedom respectively, and (.) is the pdf of the standard
normal distribution N(0, 1). Based on the suggested MLE of
Cpm, Reference [9] proposed two approximate 100(1 - )%
lower confidence bounds using the normal and chi-square
distributions for Cpm from the distribution frequency point of
view. Reference [2] investigated the statistical properties of
the MLE of Cpm. Reference [13] proposed a Bayesian
procedure based on the MLE of Cpm without the restriction 
= T on the process mean . Their results generalized those
discussed in Reference [11].
B. The Estimated Cpm Based on m Subsamples
In real-world practice, process information is often derived
from sub-samples rather than from one single sample. A
common practice of the process capability estimation in the
manufacturing industry is to first implement a daily-based
data collection program for monitoring the process stability,
then to analyze the past ―in control‖ data. Assuming that the
measurements of the i-th production line investigated, (xi1, xi2,
…, xini), are chosen randomly from a stable process which
follows a normal distribution N(, 2) for each i = 1, 2, …,
denote the cdf and pdf of the chi-square distribution 2 with N
– 1 degrees of freedom respectively, and ·(.) is the pdf of
the standard normal distribution N(0, 1). It is rather
complicated and computationally inefficient to make
statistical inference on Cpm from (8) or (9). Therefore, a
general asymptotic study on the distributional and inferential
properties of Cpm based on subsamples turns out to be most
desirable.
III. THE LIMITING BEHAVIORS OF Cˆ *pm
A. Asymptotic Distribution and Large Sample Properties
of Cˆ *pm
In this subsection, asymptotic properties of Cˆ *pm are
investigated under general conditions. The limiting
distribution of Cˆ *pm is derived for arbitrary populations
having fourth central moment 4 = E(X - )4 exists.
Consequently, approximate manufacturing capability can be
measured for processes under those described conditions,
particularly, for those with near-normal distributions.
Furthermore, consistent, asymptotically unbiased, and
asymptotically efficient properties of Cˆ *pm under large
samples are also presented. Let xi1, xi2, …, xini be a random
sample of measurements from a process which has
distribution G with mean  and variance 2 for each i = 1, 2,
…, m, j = 1, 2, …, ni. Under the normality assumption,
( x , s N2 ) is a MLE (maximum likelihood estimator) of (, 2)
based on m subsamples, where x  im1 nji1 xij / N , and
360
2
s N2  im1 nji1 ( xij  x ) / N . By substituting x for  and s N2
ISSN: 2277-3754
ISO 9001:2008 Certified
International Journal of Engineering and Innovative Technology (IJEIT)
Volume 3, Issue 5, November 2013
[12] P.C. Lin and W.L. Pearn, ―Testing process performance based
Proof: From Theorem, we know that N (Cˆ *pm  C pm )


d
N(0,  2pm ).
Under the normality assumption, 3 = 0
and 4 = 3 implies that
4
on the capability index Cpm,‖ The International Journal of
Advanced Manufacturing Technology, vol. 27, no. 3-4, pp.
351-358, Dec. 2005.
N (Cˆ *pm  C pm )
d
N(0,  N2 ),


where  N2 = (Cp)2/2. The information matrix is I ( ) 
1 /  2
0  . Since the Cramer-Rao lower bound
I ( ,  )  

1 /(2 4 )
 0
 C pm 
 C pm C pm  I 1 ( )     N2


 

 2  N  C pm  N

  2 
[13] J.H. Shiau, C.T. Chiang, and H.N. Hung, ―A Bayesian
procedure for process capability assessment,‖ Quality and
Reliability Engineering International, vol. 15, no. 4, pp.
369-378, Sep./Oct. 1999.
[14] C.W. Wu and W.L. Pearn, ―Capability testing based on Cpm
with multiple samples,‖ Quality and Reliability Engineering
International, vol. 21, no. 1, pp. 29-42, Feb. 2005.
[15] R.J. Serfling, ―Approximation theorems of mathematical
statistics,‖ John Wiley and Sons, New York. 1980.
is achieved, the proof is complete.
AUTHOR BIOGRAPHY
VI. ACKNOWLEDGMENT
This research was partially supported by National Science
Council of the Republic of China, under contract number
NSC 95–2221–E–151– 038-MY3.
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364
Gu-Hong Lin is a Professor at the Department of
Industrial Engineering and Management, National
Kaohsiung University of Applied Sciences, Kaohsiung,
Taiwan. He received his Ph.D. degree in quality
assurance from National Chiao-Tung University,
Taiwan. His research interests include applied statistics
and quality assurance.