A Reliable Procedure on Performance Evaluation-- A Large Sample Approach Based on the Estimated Taguchi Capability Index

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ISSN: 2277-3754

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International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 5, November 2013

359 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.

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])  6 LSL USL Cp   , d m Ca   1  , (1)              3 3 min , LSL USL Cpk , (2)

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

obtained by substituting the sample mean x =



n_i1xi/n

for and the sample variance 2 ₁( )2/ 1 

  in i

n x x

s (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 2 2 ₍ ₎ 6 T LSL USL Cpm       . (3)

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 ---

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.ESTIMATINGCPMUNDERTHENORMALITY

ASSUMPTION

A. The Estimated Cpm Based on Single Sample

Assuming that the measurements of the characteristic

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

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ISSN: 2277-3754

ISO 9001:2008 Certified

360 investigated, (X1, X2, …, Xn), are chosen randomly from a stable process which follows a normal distribution N(, 2). Reference [9] recommended a MLE (maximum likelihood estimator) of Cpm by substituting the sample mean x =

n x

n i 1 i/



 for  and the MLE s x x n

n i i n 1( ) / 2 2

_

   for 2 in (3), which can be expressed as

n T x d T x s LSL USL C n i i n pm / ) ( 3 ) ( 6 ˆ 1 2 2 2



       , (4)

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)2 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 Cˆpm can be expressed in terms of a mixture of the

chi-square distribution and the standard normal distribution [12]: , )] ( ) ( [ } )] 9 /( ) {[( 1 ) ( ) 3 /( 0 2 2 2 dt n t n t t x n b U x F x n b s           

_

(5) , )] ( ) ( [ )] 9 /( ) 2 [( } )] 9 /( ) {[( ) ( 2 3 ) 3 /( 0 2 2 2 dt n t n t x n b t x n b u x f x n b s          

_

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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, …,

m, j = 1, 2, …, ni. Reference [14] considered the following estimator of Cpm based on m subsamples of size ni each:

 

  



  m i n j m i i ij pm i n T x d C 1 1 1 * / ) ( 3 ˆ . (7)

For 0 < x < ∞, the exact cdf Fm(x) and pdf fm(x) of Cˆ*pm can be expressed as a mixture of the chi-square distribution and the standard normal distribution:

, )] ( ) ( [ } )] 9 /( ) {[( 1 ) ( ) 3 /( 0 2 2 2 * dt N t N t t x N b U x F x N b m           

_

(8) , )] ( ) ( [ )] 9 /( ) 2 [( } )] 9 /( ) {[( ) ( 2 3 ) 3 /( 0 2 2 2 * dt N t N t x N b t x N b u x f x N b m          

_

(9) where b = d/,  = ( - T)/, N 



m_i_1ni, U *₍._{) and u}*₍.₎

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.THELIMITINGBEHAVIORSOF ˆ*

pm

C

A. Asymptotic Distribution and Large Sample Properties of ˆ*

pm

C

In this subsection, asymptotic properties of ˆ*

pm

C are investigated under general conditions. The limiting distribution of ˆ*

pm

C 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 ˆ*

pm

C 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 ,s2_N) is a MLE (maximum likelihood estimator) of (, 2) based on m subsamples, where x

 

_im1 n_ji1xij/N , and

N x x sN m_i n_ji ( ij ) / 2 1 1 2

 

  

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ISSN: 2277-3754

ISO 9001:2008 Certified

364 Proof: From Theorem, we know that (ˆ* )

pm pm C C N 





d

N(0,2pm). Under the normality assumption, 3 = 0 and 4 = 34 implies that N(Cˆ*pmCpm)





d

N(0, 2

N

 ),

where N2 = (Cp)2/2. The information matrix is I()

       ) 2 /( 1 0 0 / 1 ) , ( ₄ 2    

I . Since the Cramer-Rao lower bound

N C C N I C C _N pm pm pm pm 2 2 1 2 ) (                                 

is achieved, the proof is complete.

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.

REFERENCES

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AUTHOR BIOGRAPHY

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.