O R I G I N A L A R T I C L E
Measuring manufacturing capability based on lower confidence
bounds of
C
pmk
applied to current transmitter process
Received: 5 December 2002 / Accepted: 24 February 2003 / Published online: 21 August 2003 Springer-Verlag London Limited 2003
Abstract Several process capability indices, including Cp,
Cpk, and Cpm, have been proposed to provide numerical
measures on manufacturing potential and actual perfor-mance. Combining the advantages of those indices, a
more advanced index Cpmkis proposed, taking the
pro-cess variation, centre of the specification tolerance, and the proximity to the target value into account, which has been shown to be a useful capability index for manufac-turing processes with two-sided specification limits. In
this paper, we consider the estimation of Cpmk, and we
develop an efficient algorithm to compute the lower
confidence bounds on Cpmk based on the estimation,
which presents a measure on the minimum manufacturing capability of the process based on the sample data. We also provide tables for practitioners to use in measuring their processes. A real-world example of current trans-mitters taken from a microelectronics device manufac-turing process is investigated to illustrate the applicability of the proposed approach. Our implementation of the existing statistical theory for manufacturing capability assessment bridges the gap between the theoretical development and the in-plant applications.
Keywords Process capability index Æ Lower confidence bound
1 Introduction
Process capability indices, including Cp, Cpk, and Cpm
[1, 4], have been proposed in the manufacturing industry to provide numerical measures on whether a process is capable of reproducing items meeting the manufacturing
quality requirement preset in the factory. Combining the advantages of those indices, Pearn et al. [5] proposed a
more advanced capability index called Cpmk, which has
been shown to be a useful capability index for processes with two-sided specification limits. These indices are defined as: Cp¼ USL LSL 6r ; Cpk¼ min USL l 3r ; l LSL 3r ; Cpm¼ USL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q ; Cpmk¼ min USL l 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q ; l LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q 9 > = > ; 8 > < > :
where USL is the upper specification limit, LSL is the lower specification limit, l is the process mean, r is the process standard deviation, and T is the target value predetermined by the product designer or the manu-facturing engineer.
Criteria that have been considered for measuring manufacturing capability include: process variation (product quality consistency), process departure, process
yield, and process loss. The index Cp considers the
overall process variability relative to the manufacturing tolerance; therefore, it only reflects the consistency of the
product quality characteristic. The index Cpktakes the
process mean into consideration but it can fail to dis-tinguish between on-target processes and off-target processes. It is a yield-based index providing lower
bounds on process yield. The index Cpm takes the
proximity of process mean from the target value into account and is more sensitive to process departure than
Cpk. Since the design of Cpm is based on the average
process loss relative to the manufacturing tolerance, the DOI 10.1007/s00170-003-1693-z
W. L. Pearn Æ Ming-Hung Shu
W. L. Pearn (&)
Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan
E-mail: roller@cc.nctu.edu.tw M.-H. Shu
Department of Commerce Automation and Management, National Pingung Institute of Commerce, Taiwan, ROC
index Cpm provides an upper bound on the average
process loss; Cpm has alternatively been called the
Taguchi index. The index Cpmk is constructed from
combining the modifications to Cp that produced Cpk
and Cpm, inheriting the merits of both indices. The
ranking of the four basic indices in terms of sensitivity to the departure of process mean from the target value, from the most sensitive one to the least sensitive, are (1) Cpmk, (2) Cpm, (3) Cpk, and (4) Cp. For semiconductor
manufacturing, the index is appropriate for capability measures due to high standards and stringent require-ments on product quality and reliability.
We note that a manufacturing process satisfying the
capability requirement ‘‘Cpk‡ c0’’ may not satisfy the
capability condition ‘‘Cpm‡ c0’’. On the other hand, a
process satisfying the capability requirement ‘‘Cpm‡ c0’’
may not satisfy the capability requirement ‘‘Cpk‡ c0’’
either. But, a manufacturing process does satisfy both capability requirements ‘‘Cpk‡ c0’’ and ‘‘Cpm‡ c0’’ if the
process satisfies the capability requirement ‘‘Cpmk‡ c0’’
since Cpmk£ Cpk and Cpmk£ Cpm. Thus, the index
Cpmkdoes provide more capability assurance with respect
to process yield and process loss to the customers than
the other two indices Cpk and Cpm. This is a desired
property according to today’s modern quality improve-ment theory, as reduction of process loss (variation from the target) is just as important as increasing the process
yield (meeting the specifications). While Cpk is still the
more popular and widely used index, the index Cpmkis
considered to be the most useful index to date for pro-cesses with two-sided manufacturing specifications. Chen and Hsu [2] investigated the asymptotic sampling
distri-bution of the estimated Cpmk. Wright [14] derived an
explicit but rather complicated expression for the
prob-ability density function of the estimated Cpmk. Pearn
et al. [7] considered an extension of Cpmk for handling
process with asymmetric tolerances. Jessenberger and
Weihs [3] studied the behaviour of Cpmk, looking for
processes with asymmetric tolerances. Pearn et al. [8] obtained an alternative, simpler form of the probability
density function of the estimated Cpmk and considered
capability testing based on Cpmk. Pearn et al. [9]
inves-tigated the statistical properties of the estimated Cpmk.
Pearn and Lin [10] focused on a Bayesian-like estimator
of Cpmk under a different manufacturing condition, in
which the probability p(l>m) is available. Pearn and Lin [11] developed efficient SAS/Maple computer programs to calculate the critical values and the p-value for testing
manufacturing capability based on Cpmk.
2 Manufacturing capability of a current transmitter process
In practice, a manufacturing process is said to be
inadequate if Cpmk<1.00; this indicates that the
process is not adequate with respect to the
manu-facturing tolerances, and the process variation r2needs
to be reduced (often by changing the design of the
experiments). The fraction of nonconformities for such a process exceeds 2700 ppm (parts per million). A manufacturing process is said to be marginally capable
if 1.00£ Cpmk<1.33; this indicates that caution needs
to be taken regarding the process consistency and some process control is required (usually using R or S con-trol charts). The fraction of nonconformities for such a process is within 66–2700 ppm. A manufacturing pro-cess is said to be satisfactory if 1.33£ Cpmk<1.67; this
indicates that process consistency is satisfactory, ma-terial substitution may be allowed, and no stringent precision control is required. The fraction of non-conformities for such a process is within 0.54–66 ppm. A manufacturing process is said to be excellent if 1.67£ Cpmk<2.00; this indicates that process precision
exceeds satisfactory. The fraction of nonconformities for such a process is within 0.002–0.54 ppm. Finally, a
manufacturing process is said to be super if
Cpmk‡ 2.00. The fraction of nonconformities for such a
process is less than 0.002 ppm.
Table 1 summarizes the above five capability
require-ments and the corresponding Cpmk values. Some
mini-mum capability requirements have been recommended in the manufacturing industry [15] for specific process types, which must run under more designated stringent quality conditions. For existing manufacturing processes, the capability must be no less than 1.33, and for new manu-facturing processes, the capability must be no less than 1.50. For existing manufacturing processes on safety, strength, or critical parameters (such as manufacturing soft drinks or chemical solutions bottled with glass con-tainers), the capability must be no less than 1.50, and for new manufacturing processes on safety, strength, or crit-ical parameters, the capability must be no less than 1.67. We consider the following case taken from a microelectronic manufacturing factory making various types of microelectronic devices. There is one pro-duction line, controlled and monitored in the factory, that makes current transmitters. The process investi-gated is the making of a monolithic 4–20 mA, two-wire current transmitter integrated circuit (2WCT IC) designed for bridge input signals. This device provides complete bridge excitation, instrumentation amplifier, linearization, and the current output circuitry neces-sary for high impedance strain gage sensors. The instrumentation amplifier can be used over a wide range of gain, accommodating a variety of input sig-nals and sensors. Linearization circuitry consists of a
Table 1 Some commonly used capability requirement and the corresponding precision conditions
Precision Condition Cpmkvalues
Inadequate Cpmk< 1.00
Marginally capable 1.00 £ Cpmk<1.33
Satisfactory 1.33 £ Cpmk<1.67
Excellent 1.67 £ Cpmk<2.00
second, fully independent instrumentation amplifier that controls the bridge excitation voltage. It provides the second-order correction to the transfer function, typically achieving a 20:1 improvement in nonlinearity, even with low cost transducers. Total unadjusted error of the complete current transmitter, including the lin-earized bridge, is low enough to permit use without adjustment in many applications such as industrial process control, factory automation, SCADA remote data acquisition, weighting systems, and
accelerome-ters. This 2WCT IC product is available in 16-pin
plastic DIP and SOL-16 surface-mount packages, as depicted in Fig. 1.
The total unadjusted error of the 2WCT IC is an essential product characteristic, which has significant impact on product quality. Because the total unadjusted error is a two-sided specification, the upper specification limit, USL, is set to 5 lA, and the lower specification
limit, LSL, is set to )5 lA,; therefore, the factory
engi-neers recommend using Cpmk for determining whether
products meet specifications and taking action to improve the process if necessary. In practice, we never
know the true values of l and r2nor Cpmk. Hence, these
parameters need to be estimated and sampling error needs to be considered.
For a normally distributed process that is demon-strably stable (under statistical control), Pearn et al. [5] considered the maximum likelihood estimator (MLE) of
Cpmkas defined below: ^ Cpmk¼ min USL X 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 nþ ðX T Þ 2 q ; X LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 nþ ðX T Þ 2 q 8 > < > : 9 > = > ;
where X ¼Pni¼1Xi=n and S2n ¼
Pn
i¼1ðXi XÞ2=n are
the MLEs of l and r2, respectively. We note that
S2n þ ðX TÞ2¼Pni¼1ðXi T Þ2=n, which is in the
denominator of ^Cpmk, is the uniformly minimum variance
unbiased estimator (UMVUE) of r2þ l Tð Þ2¼
EðX TÞ2in the denominator of Cpmk.
3 Sampling distribution ofCpmk
For symmetric manufacturing tolerance (T=m), Pearn et al. [5] expressed the natural estimator ^Cpmk,
alterna-tively, as the following: ^ Cpmk¼ d jX mj 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 nþ ð X T Þ 2 q
and showed that the distribution of the natural
estima-tor ^Cpmk is a mixture of the chi-square distribution and
the non-central chi-square distribution, as expressed in the following: ^ Cpmk~ dpffiffin r v01ðkÞ 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 n1þ v02n1ðkÞ q where v2
n1 is the chi-square distribution with n–1
de-grees of freedom, v01ðkÞ is the non-central chi-square
distribution with one degree of freedom and non-cen-trality parameter k, and v02n1ðkÞ is the non-central chi-square distribution with n–1 degrees of freedom and
non-centrality parameter k, where k¼ nðl T Þ2=r2.
The cumulative distribution of ^Cpmk, therefore, can be
found as the following [7, 8, 9]:
Fig. 1 A current transmitter with bridge excitation and linearization FCpmkðxÞ ¼ 0; x\1 3 ; P1 j¼0 pjRb1FK Dpffiffiffiy 2. 9x2 h i y fYjðyÞdy n o ; 13 6x\0; 1P1 j¼0 pj RD2 0 fYjðyÞdy n o ; x¼ 0; 1P1 j¼0 pjR0bFK Dpffiffiffiy 2. 9x2 h i y fYjðyÞdy n o ; x >0; 8 > > > > > > > > > < > > > > > > > > > :
where b¼ D= 1 þ 3x½ ð Þ2, D¼ n1=2ðd=rÞ, K ¼ nS2 n=r2, FK
is the cumulative distribution function of K, and the probability density function (PDF) is:
fCpmkð Þ ¼x BP 1 j¼0 pjBj R1 0 Ijðx; zÞ dz ; 1 3\x\0; BP 1 j¼0 pjBj R1=x 0 Ijðx; zÞ dz n o ; x >0; 0; otherwise; 8 > > > > < > > > > : where pj¼ ðk=2Þjek=2=j!; B¼ 12ðn1=2dÞn= ð18r2Þn=2 C ½ðn 1Þ=2 , Bj¼ ðn1=2dÞ2j= ð2r2Þj C1=2þ j , and Ijðx; zÞ ¼ ð1xzÞ 2jðz3Þ2 zðn3Þ=2½zþ3ð2xzÞðn3Þ=2 ð1þ3xÞ2jþðnþ3Þ=2 exp D2ðzþ3Þ2 18ð1þ3xÞ2 n o :
Using variable transformation and integration tech-nique, for x>0, the cumulative distribution function
(CDF) of the estimated Cpmk may be alternatively
ex-pressed as the following, which can be used for calcu-lating the critical values c0, the p-values, and the lower
confidence bounds C on Cpmk. FC^pmkðxÞ ¼ 1 Z bpffiffin=ð1þ3xÞ 0 G ðb ffiffiffi n p tÞ2 9x2 t 2 ! /ðt þ n pffiffiffinÞ þ /ðt npffiffiffinÞdt;
where b¼ d=r; n ¼ l Tð Þ=r; G ð Þ is the cumulative
distribution function of the chi-square distribution v2
n1, and /( .
) is the probability density function of the
standard normal distribution N(0,1). Note that for
l>USL or l<LSL, the capability Cpmk<0.0, and for
l=USL or l=LSL, the capability Cpmk=0.0. The
requirement of LSL <l<USL is a minimum
cap-ability requirement applying to most start-up
engineering applications or new processes. Figure 2a, b
displays the PDF plots of the MLE estimator ^Cpmk
with n=0.5 and 1, b=3, d=2, and n=10, 20, 50. Figure 2c, d displays the CDF plots of the natural
estimator ^Cpmk with n=0.5 and 1, b=3, d=2, and
n=10, 20, 50.
4 Calculating manufacturing capability
Critical values are used for making decisions in man-ufacturing capability testing with designated type-I error a, which is the risk of misjudging an incapable
process (H0: Cpmk£ c0) as a capable one (H1:
Cpmk>c0). The p-values are used for making decisions
in manufacturing capability testing, which presents the
actual risk of misjudging an incapable process (H0:
Cpmk£ c0) as a capable one (H1: Cpmk>c0). Thus, if
p<a, we reject the null hypothesis and conclude that the process is capable with actual type-I error p (rather than a). Both approaches, the critical values and the p-values, do not convey any information regarding the minimal value of the actual manufacturing capability (lower confidence bound). The development of the lower confidence bound on the actual manufacturing capability is essential. The lower confidence bound not
Fig. 2 a PDF plots of ^Cpmk with n=1, b=3, d=2, and n=10, 20, 50 (bottom to top). b PDF plots of ^Cpmkwith n=0.5, b=3, d=2, and n=10, 20, 50 (bottom to top). c CDF plots of ^Cpmk with n=1, b=3, d=2, and n=10, 20, 50 (bottom to top). d CDF plots of ^Cpmk with n=0.5, b=3, d=2, and n=10, 20, 50 (bottom to top)
only gives us a clue about the minimal level of the actual manufacturing performance, which is closely related to the fraction of nonconforming units (defec-tives), but is also useful in making decisions for manufacturing capability testing. For processes with a target value set to the mid-point of the manufacturing
specifications (T=m), the index Cpmk may be rewritten
as the following. When Cpmk=C, b=d/r can be
expressed as b¼ 3Cpffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2þ nj j. Cpmk¼ Cpmk ¼ d l Tj j 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ l Tð Þ2 q ¼ d=r nj j 3pffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2; where n¼ l Tð Þ=r.
Hence, given the sample of size n, the confidence level
c, the estimated value, ^Cpmk, and the parameter n, the
lower confidence bounds C can be obtained using the numerical integration technique with iterations to solve the following equation. In practice, the parameter n=(l–T)/r is unknown, but it can be calculated from the sample data as ^n¼ ðX TÞ=Sn. It should be noted
that the equation is an even function of n.Thus, for both
n=n0 and n=)n0 we have the same lower confidence
bounds. Z bpffiffin=ð1þ3C^pmkÞ 0 G ðb ffiffiffi n p tÞ2 9C^ 2pmk t2 0 B @ 1 C A /ðt þ n pffiffiffinÞ þ /ðt npffiffiffinÞdt¼ 1 c ð1Þ
4.1 Algorithm for the LCB
Using Eq. 1, we may compute the lower confidence bounds, C, and a Matlab algorithm called the LCB is developed. Three auxiliary functions for evaluating C are included here: (a) the cumulative distribution
func-tion of the chi-square v2
n1, G(Æ), (b) the probability
density function of the standard normal distribution N(0,1), /(Æ), and (c) the function of numerical evaluate
integration using the recursive adaptive Simpson
quardrature – ‘‘quad’’. The algorithm used commonly is known as the direct search.
1. Read the sample data (X1, X2, ..., Xn), LSL, USL, T,
and c.
2. Calculate X, Sn, ^n, and ^Cpmk.
3. Compute an initial guess for C.
4. Find the lower confidence bound C on Cpmk.
5. Output the conclusive message, ‘‘The true value of
the manufacturing capability Cpmkis no less than the
Cwith 100c% level of confidence.’’
We implement the algorithm and develop the fol-lowing Matlab computer program to compute the min-imal manufacturing capability.
4.2 Matlab program for LCB
%---% Input the sample data (X1, X2,..., Xn),
LSL, USL, T, and c.
%--- ---[n1 usl lsl T r1]=read(‘Enter values of sample size, lower specification limit,
upper specification limit, target value, confidence level:’);
global b n epsilon ecpmk n=n1;r=r1; [data(1:n,1)]= textread (‘eeprom.dat’,‘%f’,n); %---% Compute X, Sn, n, and Cpmk. %---mdata=mean(data); stddata=std(data)*sqrt((n-1)/n); epsilon=(mdata-T)/stddata; ecpmk=(min(usl-mdata,mdata-lsl))/ (3*sqrt(stddata^2+(mdata-T)^2));
fprintf(‘The Sample Mean is %g.\n’,
mdata);
fprintf(‘The Sample Standard Deviation is %g.\n’,stddata)
fprintf(‘The Epsilon is %g.\n‘,epsilon)
fprintf(‘The Estimate of Cpmk is
%g.\n’,ecpmk) %
---% Compute a good initial value of C. % ---c=0.2:0.025:3; for i=1:1:113 b=0;d=0;y=0;b=3*c(i)*sqrt(1+epsilon^2) +abs(epsilon); d=b*sqrt(n)/(1+3*ecpmk); y=quad(‘cpmk’,0,d); if (y-(1-r))>0 break end; end %---%Evaluate the lower confidence bound C on Cpmk. %---c=0.2+0.025*(i-1):-0.001:0.2; for k=1:(0.025*(i-1)*1000)+1 b=0;d=0;y=0;b=3*c(k)*sqrt(1+epsilon^2) +abs(epsilon); d=b*sqrt(n)/(1+3*ecpmk); y=quad(‘cpmk’,0,d); if ((1-r)-y)>0.0001 break end; end %---% Output the conclusive message, ‘‘The true value of the process
% capability Cpmk is no less than C with
100c% level of confidence.’’
%---fprintf(‘The true value of the manufacturing capability Cpmk is no less than %g’, c(k)) fprintf(‘with %g’,r) fprintf(‘level of confidence.’) %---%Two function files includedread.m and cpmk.m
%---function Q1=cpmk(t)
global n b epsilon ecpmk
Q1=chi2cdf(((((b*sqrt(n)-t).^2)./ (9*ecpmk^2))-t.^2),n-1).*…
(normpdf((t+epsilon*sqrt(n))) +normpdf((t-epsilon*sqrt(n))));
function [a1, a2, a3, a4, a5]=read(labl) if nargin==0, labl=‘?’; end
n=nargout;str=input(labl,‘s’); str=[‘[’,str,‘]’];v=eval(str); L=length(v);
if L>=n, v=v(1:n);
else, v=[v,zeros(1,n-L)]; end for j=1:nargout
eval([‘a’, int2str(j),‘=v(j);’]); end %---%The End
%---5 Manufacturing capability and process parameter n
Since the process parameters l and r are unknown, the process characteristic parameter n=(l–T)/r is also un-known. Thus, it has to be estimated further in real applications, normally by substituting l and r with its sample mean and sample standard deviation. Such an approach certainly would make our approach less reli-able as the level of the confidence c cannot be ensured. To eliminate the need to further estimate the parameter n, we examine the behaviour of the lower confidence bounds C as the function of the process characteristic n.
We perform extensive computations to calculate the
lower confidence bounds C for n=0(0.05) 3.00,
^
Cpmk¼ 0:7ð0:1Þ3:0; and n=5(5)200. We note that
n=(l–T)/r=0(0.05)3.00 covers a wide range of
appli-cations with process capability Cpmk‡ 0. The result
indicates that the lower confidence bound C obtains its minimum either at n=0.50 (for most cases), or at 0.45 (in a few cases), and the difference between the two
lower confidence bounds is less than 5·10)4 (possibly
due to computational precision errors). In fact, the lower confidence bound value C first decreases as n increases, obtains its minimum value at n=0.45 or 0.5, then increases again within the range of n2[0.5, 3.0]. Hence, for practical purposes we may solve Eq. 1 for n=0.50 to obtain the required lower confidence bounds
for a given ^Cpmk, n and c, without having to further
estimate the parameter n. Thus, the level of confidence c can be ensured, and the decisions made based on such approach must be reliable.
We note the above result is almost impossible to prove mathematically. It is important to recognize that the lower confidence bounds obtained from solving Eq. 1 with n=0.50 takes the minimal value among all possible C values, and therefore the capability estima-tion is conservative. But, given that the process param-eter n is always unknown in real applications (as the process mean l and the process variation r are unknown), C must be the maximum value for which the confidence level c can be assured. Any other value of C¢ that is greater than the minimal value C (C¢>C) would certainly result in a confidence level c¢ less than the preset c (c¢<c). Therefore, our approach provides the most accurate solution among other existing estimations for the manufacturing capability.
Figure 3a–d plots the curves of the lower confidence bound, C, versus the parameter n for ^Cpmk=0.7, 1.2, 2.0,
3.0, respectively, with c=0.95. For bottom curve 1, sample size n=30. For bottom curve 2, sample size n=50; for bottom curve 3, sample size n=70; for top curve 2, sample size n=100; for top curve 1, sample size
n=200. We note that for all ^Cpmkvalues we investigated,
as n increases the discrepancy between those C values with different n values decreases. Tables 2 and 3 tabulate
the lower confidence bound, C, for ^Cpmk=0.7(0.1)3.0,
n=5(5)200, and c=0.95 with the process parameter n set
to n=0.50. For example, if ^Cpmk=1.4, then with n=100
we find the lower confidence bound C=1.208, and so the minimal manufacturing capability is no less than 1.297, Fig. 3 a Plots of C versus |n| for ^Cpmk=0.7, c=0.95, n=30, 50, 70,
100, 200 (bottom to top). b Plots of C versus |n| for ^Cpmk=1.2,
c=0.95, n= 30, 50, 70, 100, 200 (bottom to top). c Plots of C versus |n| for ^Cpmk=2.0, c=0.95, n= 30, 50, 70, 100, 200 (bottom to top).
d Plots of C versus |n| for ^Cpmk=3.0, c=0.95, n= 30, 50, 70, 100,
i.e. Cpmk>1.208. Consequently, the manufacturing yield
is no less than 99.97% and the fraction of nonconfo-rmities is no greater than 290.08 ppm.
6 Data analysis and manufacturing capability computation
We collected sample data of the total unadjusted error from 150 current transmitters (displayed in Table 4). Figures 4 and 5 display the histogram and normal probability plot of the 150 observations. The sample data appears to be normal. The Shapiro-Wilk test is also used to check whether the sample data is normal. The statistic W is found to be 0.9934 with p-value 0.7283. Thus, we conclude that the sample data can be regarded as taken from a normal process.
In order to measure manufacturing capability of the current transmitter process, we execute the Matlab
program to obtain the lower confidence bound on Cpmk
(with process characteristic set to n=(l)T)/r=0.5).
The program reads the sample data file, and the input
of sample size n=150, LSL=)5, USL=5, target value
T=0, and confidence level c=0.95, then outputs with the sample mean, X=0.187, the sample standard
devi-ation Sn=1.081, the estimator ^Cpmk=1.463, and the
lower confidence bound C=1.299. The actual program execution output is listed below. We therefore conclude
that the true value of the process capability Cpmkis no
less than 1.299 with a 95% level of confidence. The 2WCT manufacturing process is capable of reproduc-ing products with a yield no less than 99.9902% and the fraction of nonconformities is no greater than 97.39ppm. We note that the conclusions made here have used the particular value of n=0.5 in finding the lower confidence bound; thus, the confidence level is ensured to be no less than 0.95 (or the Type I error is no greater than 0.05).
Table 2 Lower confidence bounds C of Cpmkfor ^Cpmk=0.7(0.1)1.8, n=5(5)200, c=0.95
n 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 5 0.247 0.297 0.346 0.395 0.444 0.493 0.542 0.591 0.637 0.682 0.729 0.791 10 0.358 0.423 0.488 0.551 0.616 0.679 0.741 0.803 0.867 0.930 0.993 1.056 15 0.413 0.484 0.556 0.627 0.698 0.770 0.841 0.912 0.983 1.054 1.125 1.195 20 0.447 0.523 0.598 0.673 0.747 0.822 0.897 0.972 1.047 1.121 1.196 1.270 25 0.466 0.545 0.623 0.700 0.778 0.855 0.933 1.010 1.087 1.164 1.241 1.318 30 0.488 0.568 0.648 0.728 0.808 0.887 0.966 1.046 1.125 1.204 1.284 1.363 35 0.505 0.586 0.668 0.749 0.830 0.911 0.992 1.073 1.154 1.235 1.316 1.397 40 0.518 0.601 0.683 0.766 0.848 0.931 1.013 1.095 1.177 1.259 1.342 1.424 45 0.529 0.612 0.696 0.780 0.863 0.946 1.030 1.113 1.196 1.279 1.363 1.446 50 0.537 0.622 0.707 0.791 0.875 0.960 1044 1128 1.212 1.296 1.380 1.464 55 0.545 0.631 0.716 0.801 0.886 0.971 1.056 1.141 1.226 1.310 1.395 1.480 60 0.552 0.638 0.724 0.809 0.895 0.981 1.066 1.152 1.237 1.323 1.408 1.494 65 0.558 0.644 0.731 0.817 0.903 0.989 1.076 1.162 1.248 1.334 1.420 1.506 70 0.563 0.650 0.737 0.824 0.910 0.997 1.084 1.170 1.257 1.344 1.430 1.517 75 0.567 0.655 0.742 0.830 0.917 1.004 1.091 1.178 1.265 1.352 1.439 1.526 80 0.572 0.660 0.747 0.835 0.923 1.010 1.098 1.185 1.273 1.360 1.448 1.535 85 0.575 0.664 0.752 0.840 0.928 1.016 1.104 1.192 1.279 1.367 1.455 1.543 90 0.579 0.668 0.756 0.844 0.933 1.021 1.109 1.198 1.286 1.374 1.462 1.550 95 0.582 0.671 0.760 0.849 0.937 1.026 1.114 1.203 1.291 1.380 1.468 1.557 100 0.585 0.675 0.763 0.852 0.941 1.030 1.119 1.208 1.297 1.385 1.474 1.563 105 0.588 0.677 0.767 0.856 0.945 1.034 1.123 1.213 1.302 1.391 1.480 1.569 110 0.590 0.680 0.770 0.859 0.949 1.038 1.128 1.217 1.306 1.395 1.485 1.574 115 0.593 0.683 0.773 0.862 0.952 1.042 1.131 1.221 1.310 1.400 1.489 1.579 120 0.595 0.685 0.775 0.865 0.955 1.045 1.135 1.225 1.314 1.404 1.494 1.584 125 0.597 0.697 0.778 0.868 0.958 1.048 1.138 1.228 1.318 1.408 1.498 1.588 130 0.599 0.690 0.780 0.870 0.961 1.051 1.141 1.232 1.322 1.412 1.502 1.592 135 0.601 0.692 0.782 0.873 0.963 1.054 1.144 1.235 1.325 1.415 1.506 1.596 140 0.603 0.694 0.784 0.875 0.966 1.056 1.147 1.238 1.328 1.419 1.509 1.600 145 0.604 0.695 0.786 0.877 0.968 1.059 1.150 1.240 1.331 1.422 1.513 1.603 150 0.606 0.697 0.788 0.879 0.970 1.061 1.152 1.243 1.334 1.425 1.516 1.606 155 0.608 0.699 0.790 0.881 0.972 1.064 1.155 1.246 1.337 1.428 1.519 1.610 160 0.609 0.701 0.792 0.883 0.974 1.066 1.157 1.248 1.339 1.430 1.522 1.613 165 0.610 0.702 0.794 0.885 0.976 1.068 1.159 1.250 1.342 1.433 1.524 1.616 170 0.612 0.704 0.795 0.887 0.978 1.070 1.161 1.253 1.344 1.435 1.527 1.618 175 0.613 0.705 0.797 0.888 0.980 1.072 1.163 1.255 1.346 1.438 1.529 1.621 180 0.614 0.706 0.798 0.890 0.982 1.073 1.165 1.257 1.348 1.440 1.532 1.623 185 0.615 0.707 0.799 0.891 0.983 1.075 1.167 1.259 1.351 1.442 1.534 1.626 190 0.616 0.709 0.801 0.893 0.985 1.077 1.169 1.261 1.352 1.444 1.536 1.628 195 0.617 0.710 0.802 0.894 0.986 1.078 1.170 1.262 1.354 1.446 1.538 1.630 200 0.619 0.711 0.803 0.895 0.988 1.080 1.172 1.264 1.356 1.448 1.540 1.632
Table 3 Lower confidence bounds C of Cpmkfor ^Cpmk=1.9(0.1)3.0, n=5(5)200, c=0.95 n 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 5 0.870 0.916 0.961 0.1007 1.057 1.107 1.157 1.218 1.274 1.329 1.402 1.492 10 1.133 1.195 1.257 1.319 1.381 1.453 1.525 1.587 1.649 1.700 1.780 1.850 15 1.268 1.338 1.417 1.487 1.557 1.626 1.696 1.765 1.840 1.909 1.984 2.050 20 1.342 1.421 1.495 1.569 1.643 1.720 1.796 1.870 1.944 2.020 2.094 2.168 25 1.396 1.473 1.550 1.626 1.703 1.780 1.857 1.934 2.011 2.088 2.165 2.242 30 1.442 1.521 1.600 1.679 1.758 1.837 1.916 1.995 2.074 2.153 2.232 2.311 35 1.477 1.558 1.639 1.720 1.800 1.881 1.962 2.042 2.123 2.203 2.284 2.365 40 1.506 1.588 1.670 1.752 1.834 1.916 1.998 2.079 2.161 2.243 2.325 2.407 45 1.529 1.612 1.695 1.778 1.861 1.944 2.027 2.110 2.193 2.276 2.359 2.442 50 1.548 1.632 1.716 1.800 1.884 1.968 2.052 2.136 2.219 2.303 2.387 2.471 55 1.565 1.649 1.734 1.819 1.903 1.988 2.073 2.157 2.242 2.327 2.411 2.496 60 1.579 1.664 1.750 1.835 1.921 2.006 2.091 2.176 2.262 2.347 2.432 2.518 65 1.592 1.678 1.764 1.850 1.936 2.021 2.107 2.193 2.279 2.365 2.451 2.537 70 1.603 1.690 1.776 1.862 1.949 2.035 2.122 2.208 2.295 2.381 2.467 2.554 75 1.613 1.700 1.787 1.874 1.961 2.048 2.135 2.221 2.308 2.395 2.482 2.569 80 1.622 1.710 1.797 1.884 1.972 2.059 2.146 2.234 2.321 2.408 2.495 2.583 85 1.631 1.718 1.806 1.894 1.981 2.069 2.157 2.245 2.332 2.420 2.508 2.595 90 1.638 1.726 1.814 1.902 1.990 2.079 2.167 2.255 2.343 2.431 2.519 2.607 95 1.645 1.734 1.822 1.910 1.999 2.087 2.175 2.264 2.352 2.441 2.529 2.617 100 1.652 1.740 1.829 1.918 2.006 2.095 2.184 2.272 2.361 2.450 2.538 2.627 105 1.658 1.747 1.836 1.925 2.014 2.102 2.191 2.280 2.369 2.458 2.547 2.636 110 1.663 1.752 1.842 1.931 2.020 2.109 2.198 2.288 2.377 2.466 2.555 2.644 115 1.668 1.758 1.847 1.937 2.026 2.116 2.205 2.295 2.384 2.473 2.563 2.652 120 1.673 1.763 1.853 1.942 2.032 2.122 2.211 2.300 2.391 2.480 2.570 2.660 125 1.678 1.768 1.858 1.948 2.037 2.127 2.217 2.307 2.397 2.487 2.577 2.666 130 1.682 1.772 1.862 1.953 2.043 2.133 2.223 2.313 2.403 2.493 2.583 2.673 135 1.686 1.777 1.867 1.957 2.047 2.138 2.228 2.318 2.408 2.499 2.589 2.679 140 1.690 1.781 1.871 1.962 2.052 2.142 2.233 2.323 2.414 2.504 2.594 2.685 145 1.694 1.784 1.875 1.966 2.056 2.147 2.237 2.328 2.419 2.509 2.600 2.690 150 1.697 1.788 1.879 1.970 2.060 2.151 2.242 2.333 2.423 2.514 2.605 2.696 155 1.700 1.792 1.882 1.973 2.064 2.155 2.246 2.337 2.428 2.519 2.610 2.700 160 1.704 1.795 1.886 1.977 2.068 2.159 2.250 2.341 2.432 2.523 2.614 2.705 165 1.707 1.798 1.889 1.980 2.072 2.163 2.254 2.345 2.436 2.527 2.619 2.710 170 1.710 1.800 1.892 1.984 2.075 2.166 2.258 2.349 2.440 2.532 2.623 2.714 175 1.712 1.804 1.895 1.987 2.078 2.170 2.261 2.353 2.444 2.535 2.627 2.718 180 1.715 1.807 1.898 1.990 2.081 2.173 2.264 2.356 2.448 2.539 2.631 2.722 185 1.717 1.809 1.901 1.993 2.084 2.176 2.268 2.359 2.451 2.543 2.634 2.726 190 1.720 1.812 1.904 1.995 2.087 2.179 2.271 2.363 2.454 2.546 2.638 2.730 195 1.722 1.814 1.906 1.998 2.090 2.182 2.274 2.366 2.457 2.549 2.641 2.733 200 1.724 1.816 1.909 2.000 2.093 2.185 2.277 2.369 2.461 2.553 2.644 2.736
Table 4 The collected 150
sample observations (lA) 0.10 0.84 )0.28 )0.14 )0.46 )0.54 0.76 0.08 )0.90 0.58
0.16 0.01 0.64 )1.02 )2.33 0.24 0.22 )1.17 0.50 0.78 0.76 )2.03 1.03 0.00 )1.12 )0.63 )0.07 )1.60 )1.15 1.64 )0.43 0.38 2.55 1.54 1.39 0.88 1.63 )0.54 )0.15 )0.37 0.07 1.98 )1.26 )1.00 0.11 )0.05 2.28 0.54 )0.81 0.52 )0.25 1.35 )0.89 0.93 0.65 0.76 )0.34 )0.37 )1.06 0.22 0.14 )1.51 1.37 )0.43 1.27 0.97 0.34 )1.24 )0.89 )0.41 1.92 0.14 )0.20 0.84 )2.10 0.14 )0.66 1.41 )0.21 2.58 )0.44 )0.52 )1.29 )0.98 )0.48 1.21 0.98 )0.55 0.42 )0.05 )1.25 )0.90 0.58 0.32 )0.54 2.77 )2.37 0.22 0.10 )1.32 0.75 )1.13 1.94 )1.98 )0.89 0.81 1.32 0.23 1.40 2.18 )0.76 0.55 1.01 )0.31 0.03 0.22 0.47 )0.04 )0.04 0.59 0.27 )0.24 2.38 0.74 1.90 1.23 0.52 0.67 )1.44 )1.00 )0.46 0.29 0.79 )0.12 0.19 0.29 1.56 )0.06 0.24 0.91 0.82 )0.17 2.28 1.59 1.58 )0.99 3.07 )1.60 0.31 1.63
6.1 Matlab execution input and output
——————————————————————————————
Input:
Enter values of sample size, lower
specification limit, upper specification limit, target value, confidence level: 150,-5,5,0,0.95
Output:
The Sample Mean is 0.186589.
The Sample Standard Deviation is
1.08109.
The Epsilon 0.5.
The Estimate of Cpmk is 1.4625.
The true value of the manufacturing
capability Cpmk is no less than 1.299 with 0.95 level of confidence.
————————————————————————————
6.2 Multiple control chart samples application
Many of the existing 2WCT IC manufacturing facto-ries have implemented a daily-based production control plan for monitoring/controlling process stability. A routine data collection procedure is executed to run
X and S2 control charts (for moderate sample sizes).
The past ‘‘in control’’ data, consisting of multiple samples of m groups with variable sample size ni¼ xð i1; xi2; :::; xiniÞ), is then analysed to compute the
manufacturing capability. Thus, manufacturing infor-mation regarding product quality characteristics is de-rived from multiple samples rather than one single sample. Under the assumption that these samples are
taken from the normal distribution N(l, r2), we
consider the following estimators of process mean and process standard deviation:
Xi¼ Xni j¼1 xij=ni Si¼ ðniÞ1 Xni j¼1 xij Xi 2 " #1=2
for the ith sample mean and the sample standard devi-ation, respectively. Then, X ¼Pms
i¼1Xi=ms and Sp2¼
Pms
i¼1niS2i=
Pms
i¼1ni are used for calculating the
manu-facturing capability Cpmk. For cases with multiple
samples the natural estimator of Cpmkcan be expressed
as below. The derivations of the sampling distribution, lower confidence bounds, and the manufacturing capa-bility calculations for cases with multiple samples can be performed using the same techniques for cases with one single sample, although the derivations and calculations may be more tedious and complicated.
^ Cpmk¼ min USL X 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 pþ ð X T Þ 2 q ; X LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 pþ ð X T Þ 2 q 8 > < > : 9 > = > ;:
6.3 MPPAC control chart application
For factories having a group of processes that need to be monitored and controlled, it would be effective to use the MPPAC (multi-process performance analysis chart). The MPPAC can be used to illustrate and analyse the manufacturing capability for multiple processes, which conveys critical information regarding the departure of the process mean from the target value, process variability, and capability levels, and provides a guide-line of directions for capability improvement. Singhal
[13] introduced the Cpk MPPAC for monitoring
multiple processes. Pearn and Chen [6] proposed a
modification to the Cpk MPPAC, adding the more
Fig. 4 Histogram of the sample data
advanced capability index Cpmto identify the problems
causing the processes to fail to centre around the target. Pearn et al. [9] developed the MPPAC based on the
incapability index Cpp. Using the same technique, the
Cpmk MPPAC can be developed to monitor the
capa-bility for multiple 2WCT IC manufacturing processes.
Using the CpmkMPPAC, practitioners or engineers can
simultaneously analyse the performance of multiple
processes based on one single chart. The CpmkMPPAC
also prioritizes the order of the processes that the quality improvement effort should focus on, either to move the process mean closer to the target value or reduce the process variation. The developed confidence lower
bounds can then be applied to the Cpmk MPPAC to
ensure the accuracy of the MPPAC for given sample sizes.
7 Conclusions
Combining the merits of the two earlier indices, Cpkand
Cpm, the index Cpmk has been proposed to provide
numerical measures of process performance. The index
Cpmktakes into account the location of the process mean
between the two specification limits, the proximity to the target value, and the process variation. It has been shown to be a useful capability index for processes with two-sided specification limits. Based on the complicated probability density function of the natural estimator of
Cpmk, we developed an efficient algorithm to compute
the lower confidence bounds on Cpmk. The lower
confi-dence bound presents a measure of the minimum capa-bility of the process based on the sample data. We investigated the behaviour of the lower confidence bound values versus the process characteristic
parame-ter, n=(l)T)/r, and concluded that the lower
confi-dence bound obtains its minimal value at n=0.5. The proposed decision making procedure ensures that the risk of making wrong decision will be no greater than
the preset Type I error 1)c. We also provided a Matlab
computer program for engineers or practitioners to use
in measuring their processes. A real-world example on two-wire current transmitter integrated circuit (2WCT IC) manufacturing process, taken from a microelec-tronics device manufacturing factory, was investigated.
References
1. Chan LK, Cheng SW, Spiring FA (1988) A new measure of process capability: Cpm. J Qual Technol 20(3):162–175
2. Chen SM, Hsu NF (1995) The asymptotic distribution of the process capability index: Cpmk. Commun Stat Theory Methods
24:1279–1291
3. Jessenberger J, Weihs C (2000) A note on the behavior of Cpmk
with asymmetric specification limits. J Qual Technol 32(4):440– 443
4. Kane VE (1986) Process capability indices. J Qual Technol 18(1):41–52
5. Pearn WL, Kotz S, Johnson NL (1992) Distributional and inferential properties of process capability indices. J Qual Technol 24:216–231
6. Pearn WL, Chen KS (1997) Multiprocess performance analysis: a case study. Qual Eng 10(1):1–8
7. Pearn WL, Chen KS, Lin PC (1999) On the generalizations of the capability index Cpmkfor asymmetric tolerances. Far East
J Theor Stat 3(1):47–66
8. Pearn WL, Yang SL, Chen KS, Lin PC (2001) Testing process capability using the index Cpmk with an application. Int
J Reliability Qual Saf Eng 8(1):15–34
9. Pearn WL, Lin PC, Chen KS (2002) Estimating process capa-bility index Cpmk for asymmetric tolerances: distributional
properties. Metrika 54(3):261–279
10. Pearn WL, Lin GH (2003) A Bayesian like estimator of the process capability index Cpmk. Metrika 57:303–312
11. Pearn WL, Lin PC (2002) Computer program for calculating the p-value in testing process capability Index Cpmk. Qual
Reliability Eng Int 18(4):333–342
12. Pearn WL, Ko CH, Wang KH (2002) A multiprocess perfor-mance analysis chart based on the incapability index Cpp: an
application to the chip resistors. Microelectron Reliability 42(7):1121–1125
13. Singhal SC (1991) Multiprocess performance analysis chart (MPPAC) with capability zones. Qual Eng 4(1):75–81 14. Wright PA (1998) The probability density function of process
capability index Cpmk. Commun Stat Theory Methods
27(7):1781–1789
15. Montgomery DC (2001) Introduction to statistical quality control, 4th ed. John Wiley and Sons, New York.
