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Journal of Applied Statistics
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Testing process capability based
on C
pm
in the presence of random
measurement errors
W. L. Pearn a , M. H. Shu b & B. M. Hsu c aDepartment of Industrial Engineering & Management , National Chiao Tung University
b
Department of Industrial Engineering & Management , National Kaohsiung University of Applied Sciences
c
Department of Industrial Engineering & Management , Cheng Shiu University , Taiwan
Published online: 20 Aug 2006.
To cite this article: W. L. Pearn , M. H. Shu & B. M. Hsu (2005) Testing process capability based
on Cpm in the presence of random measurement errors, Journal of Applied Statistics, 32:10, 1003-1024, DOI: 10.1080/02664760500164951
To link to this article: http://dx.doi.org/10.1080/02664760500164951
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Testing Process Capability Based
on C
pm
in the Presence of Random
Measurement Errors
W. L. PEARN
, M. H. SHU** & B. M. HSU
†Department of Industrial Engineering & Management, National Chiao Tung University,
**Department of Industrial Engineering & Management, National Kaohsiung University of Applied Sciences,†Department of Industrial Engineering & Management, Cheng Shiu University, Taiwan
ABSTRACT Process capability indices have been widely used in the manufacturing industry
providing numerical measures on process performance. The index Cp provides measures on
process precision (or product consistency). The index Cpm, sometimes called the Taguchi index,
meditates on process centring ability and process loss. Most research work related to Cpand Cpm
assumes no gauge measurement errors. This assumption insufficiently reflects real situations even with highly advanced measuring instruments. Conclusions drawn from process capability analysis are therefore unreliable and misleading. In this paper, we conduct sensitivity investigation on process capability Cp and Cpm in the presence of gauge measurement errors. Due to the
randomness of variations in the data, we consider capability testing for Cp and Cpm to obtain
lower confidence bounds and critical values for true process capability when gauge measurement errors are unavoidable. The results show that the estimator with sample data contaminated by the measurement errors severely underestimates the true capability, resulting in imperceptible smaller test power. To obtain the true process capability, adjusted confidence bounds and critical values are presented to practitioners for their factory applications.
KEYWORDS: Gauge measurement error, lower confidence bound, critical value, process capability analysis.
Introduction
Process capability indices, which establish the relationships between the actual process performance and the manufacturing specifications, have been the focus of recent research in quality assurance and process capability analysis. The first process capability index Cp,
which was introduced outside of Japan by Juran et al. (1974) has been defined as
Cp¼
USL LSL
6s (1)
Vol. 32, No. 10, 1003 – 1024, December 2005
Correspondence Address: M. H. Shu, Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Seng-Min District, Kaohsiung 80778, Taiwan. Email: workman@cc.kuas.edu.tw
0266-4763 Print=1360-0532 Online=05=101003 – 22 # 2005 Taylor & Francis DOI: 10.1080=02664760500164951
where USL is the upper specification limit, LSL is the lower specification limit, andsis the process standard deviation. The numerator of Cpgives the size of the range over which the
process measurements can vary, and the denominator gives the size of the range over which the process is actually varying. Obviously, it is desirable to have a Cpas large as
possible. Small values of Cp would not be acceptable, since this indicates that the
natural range of variation of the process does not fit within the tolerance band. Under the assumption of that process data are normal, independent, and in control, Kocherlakota (1992) developed a general guideline for the percentage NC (non-conforming units) associated with Cp, assuming that the process is perfectly centred at the midpoint of the
specification range (see Table 1). Mizuno (1988) presented detailed criteria for Cp,
which had been widely used in US industries. Clearly, the index Cp only measures
process potential to reproduce acceptable product and does not take into account whether the process is centred.
The index Cpm, sometimes called the Taguchi index, adequately reveals the ability of
the process to cluster around the target, which reflects the degrees of process targeting (centring). The index Cpm incorporates with the variation of production items with
respect to the target value and the specification limits preset in the factory (see Hsiang & Taguchi, 1985; Chan et al., 1988; Kotz & Johnson, 1993; Kotz & Lovelace, 1998). The index Cpmis defined in the following:
Cpm¼
USL LSL
6pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2þ(mT)2 (2)
wheremdenotes the process mean and T refers to the target value often set to the midpoint of the specification limits (T ¼ m ¼ (USL þ LSL)/2). The capability index Cpmis not
pri-marily designed to provide an exact measure on the number of conforming items, but con-siders the process departure (m2 T )2 (rather than 6s alone) in the denominator of the definition to reflect process targeting (Hsiang & Taguchi, 1985; Chan et al., 1988). We note that s2þ(mT)2 ¼E(X T)2 which is the major part of the denominator of
Cpm. Since E(X T)2 is the expected loss of the characteristic, X (missing the target) is
assumed to be based on the well approximated symmetric squared error loss function, loss(X) ¼ k(X T)2, the capability index Cpmhas been referred to as a loss-based index.
Process Capability with Gauge Measurement Errors
Most research works related to Cpand Cpmhave assumed no gauge measurement errors.
For examples, Kane (1986), Kocherlakota (1992), Mizuno (1988), Marcucci & Beazley (1988), Boyles (1991), Pearn et al. (1992), Zimmer & Hubele (1997), Zimmer et al. (2001), and Pearn & Shu (2003). Such assumption, however, does not accommodate
Table 1.Minimum proportion NC associated with various values of Cp
Amount of process data
within specification range Cp Minimum % NC
6s 1.00 0.27 1022
8s 1.33 0.6334 1024
10s 1.67 0.5733 1026
12s 2.00 0.1973 1028
closely real situations even with highly advanced measuring instruments. Any measure-ment error has some impacts on the determination of process capability. Montgomery & Runger (1993, 1993b) noted that quality of the collected data relies very much on the gauge accuracy. Clearly, conclusions about process capability based on the empirical index values are not reliable. To analyse the effects of measurement errors on true capability measure, Mittag (1994, 1997) and Levinson (1995) quantified the percentage error on process capability indices evaluation with the presence of measurement errors.
Suppose that the measurement errors can be described as a random variable M N(0,s2
M), Montgomery & Runger (1993) expressed the gauge capability as l¼ 6sM
USL LSL100% (3) For the measurement system to be deemed acceptable, the measurement variability due to the measurement system must be less than a predetermined percentage of the engineering tolerance. The automotive industry action group recommended the following guidelines (Table 2) for gauge acceptance.
In this paper, we consider sensitivity of the indices Cpand Cpmwith gauge measurement
errors. Because of the random variations in the data, we present some statistical analysis to obtain reliable lower confidence bounds and critical values for capability estimation and testing purposes.
Testing Cpwith Gauge Measurement Errors
Considering the process capability in the measurement error system, we denote X N(m,
s2) the relevant quality characteristic of a manufacturing process. Because of measure-ment errors, the observed variable G N(mG¼m,s2
G¼s2þs2M) is measured with X
and M stochastically independent, instead of measuring the true variable X. The empirical process capability index CGP is obtained after substituting sG for s, and we have the
relationship between the true process capability CPand the empirical process capability
CPGstated below.
CpG¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCp 1 þl2C2 p
q (4)
Since the variation of data observed is larger than the variation of the original data, the true process capability will be under-estimated. Table 3 lists some process capabilities with l¼ 0.05(0.05)0.50 for various true process capability indices Cp¼ 0.50, 1.00,
1.33, 1.50, 1.67, 2.00, and 2.50. Obviously, the gauge becomes more important as the true capability improves (Levinson, 1995).
Table 2.Guidelines for gauge capabilities
Gauge capability Result
l , 10% Gauge system O.K.
10% , l , 30% May be acceptable based on importance of application, cost of gauge, cost of repair, and so on.
30% , l Gauge system needs improvement; make every effort to identify the problems and have them corrected.
Sampling Distribution of ^CGp
Suppose that {Xi, i ¼ 1, . . . , n} denotes the random sample of size n from the quality
characteristics X. To estimate the precision index Cp, we consider the natural estimator
^
CP defined below, where S ¼ ½
Pn
i¼1(Xi X)=(n 1)1=2 is the conventional estimator
ofs, which may be obtained from a stable process, ^
Cp¼
USL LSL
6S (5)
Chou & Owen (1989) have shown the probability density function (PDF) of ^CPcan be
expressed as: fC^p(x) ¼ 2 (pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(n 1)=2Cp)n1 G½(n 1)=2 (x) nexp½(n 1)C2 p(2x 2)1 (6)
By adding the well-known correction factor
Dn1 ¼G n 1 2 G n 2 2 1 ffiffiffiffiffiffiffiffiffiffiffi 2 n 1 r (7)
to ^CP, such as ~CP ¼Dn1C^P, Pearn et al. (1998) showed that ~CP is the uniformly
minimum variance unbiased estimator (UMVUE) of Cp. In real applications, the sample
observations are not {Xi, i ¼ 1, . . . , n} but {Gi, i ¼ 1, . . . , n}. The estimator of Cp
becomes ~ CGp ¼Dn1 USL LSL 6SG (8) where SG¼ ½ Pn
i¼1(Gi G)=(n 1)1=2. Based on the same arguments used in Chou &
Owen (1989) and Pearn et al. (1998), the PDF of ~CGP can be expressed as below
fC^G p(x) ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (n 1)=2 p Cp= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2C2 p q n1 G½(n 1)=2 (x) nexp (n 1)C 2 p(2x 2)1 1 þl2C2 p " # (9)
Note that it can be shown that Var( ~CGP) , Var( ~CP).
Table 3.Process capability with l ¼ 0.05(0.05)0.50 for various Cp
l Cp 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.49 0.49 0.49 0.49 0.49 1.00 1.00 1.00 0.99 0.98 0.97 0.96 0.94 0.93 0.91 0.89 1.33 1.33 1.32 1.30 1.29 1.26 1.24 1.21 1.17 1.14 1.11 1.50 1.50 1.48 1.46 1.44 1.40 1.37 1.33 1.29 1.24 1.20 1.67 1.66 1.65 1.62 1.58 1.54 1.49 1.44 1.39 1.34 1.28 2.00 1.99 1.96 1.92 1.86 1.79 1.71 1.64 1.56 1.49 1.41 2.50 2.48 2.43 2.34 2.24 2.12 2.00 1.88 1.77 1.66 1.56
Lower Confidence Bound Based on ^CGp
The 100g% lower confidence bounds of Cp, Lp, can be established as
P(LpCp) ¼ p K L2p Dn1 ffiffiffiffiffiffiffiffiffiffiffi n 1 p ~ Cp " #2 0 @ 1 A ¼g
where the statistic K ¼ (n 2 1)S2/s2 is distributed as xn212 , a chi-square with n 2 1
degrees of freedom. Thus, the lower confidence bounds Lpcan be obtained as:
Lp¼ ~ Cp Dn1 ffiffiffiffiffiffiffiffiffiffiffiffi x2 n1,g n 1 s (10)
where xn21,2 g is the upper 100gth percentile of thexn212 distribution. However, while
gauge measurement errors are unavoidable, ~CGPtaken as an estimator of Cp, the lower
con-fidence bounds with measurement errors, LpG, are
LGp ¼ ~ CGp Dn1 ffiffiffiffiffiffiffiffiffiffiffiffi x2 n1,g n 1 s (11)
and the confidence coefficientgG(the probability that the confidence interval contains the
actual Cpvalue with gauge measurement errors) is
gG¼p ~ CGp Dn1 ffiffiffiffiffiffiffiffiffiffiffiffi x2 n1,g n 1 s Cp 0 @ 1 A ¼ p K 1 1 þl2C2 p x2n1, g !
Because of the measurement errors, the confidence coefficients become small. For instance, when Cp¼ 2.00, n ¼ 100, and l¼ 0.50, the confidence coefficient is 0.26%,
which is much smaller than the stated confidence coefficient 95%.
Testing Process Capability Based on ^CG p
To determine whether a given process meets the present capability requirement and runs under the desired quality condition. We can consider the following statistical testing hypothesis, H0:Cpc versus H1:Cp .c. Process fails to meet the capability
requirement if Cpc, and meets the capability requirement if Cp. c. The critical
value c0can be determined by the following witha-riska(c0) ¼a(the chance of
incor-rectly judging an incapable process as capable), P( ~Cpc0jCp ¼c) ¼a, and c0 can be
obtained as: c0¼cDn1 ffiffiffiffiffiffiffiffiffiffiffiffi n 1 x2 n1,g s (12)
Meanwhile, the power of the test (the chance of correctly judging a capable process as capable) can be computed as
p(Cp) ¼ p( ~Cp. c0jCp) ¼ p K , C2 pD 2 n1(n 1) c2 0 !
In the presence of measurement errors, however, thea-risk (denoted byaG) and the power
of the test (denoted bypG) are as follows:
aG¼p( ~C G p c0jCp¼c) ¼ p KG x2 n1,g 1 þl2C2 p ! ; and pG(Cp) ¼ p( ~C G p . c0jCp) ¼ p KG, C2px2 n1,g (1 þl2C2 p)c2 ! where KG¼(n 1)S2G=s 2 G is distributed as x 2
n1. Since the process capability index
is estimated by using ~CGP instead of ~CP, the true capability of the process is
under-estimated. The probability of ~CGP being greater than c0 will be less than that of using
~
CP. Thus, thea-risk using ~C G
P to estimate Cpis less than that of using ~CPwhen estimating
Cp(aGa), and the power using ~C G
P in testing Cpis also less than the power using ~CP
(pGp).
Adjusted Confidence Bounds and Critical Values of Cp
We showed earlier that the confidence intervals do not meet the stated confidence coefficients. We also showed that both the a-risk and the test power decrease when the gauge measurement error increases. If the producers do not take account of the gauge measurement errors, capability estimation and testing results would be misleading, thus result in serious loss. In that case, the producers cannot anymore affirm that their processes meet the capability requirement even if their processes are sufficiently capable. The producers may incur a lot of cost because quantities of qualified product units are incorrectly rejected. Improving the gauge measurement accuracy and training the operators by proper education are essential for reducing the measurement errors. Nevertheless, measurement errors may be unavoidable in most manufacturing processes. In the following, we adjust the confidence intervals and critical values in order to ensure the intervals have the desired confidence coefficients and improve the power of the test with appropriate a-risk. Suppose that the desired confidence coefficient is
g, the adjusted confidence interval of Cp with lower confidence bounds LpA, can be
established as P(LAp Cp) ¼ p LAp ~ CGp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (n 1)D2n1K1 G (lC~ G p) 2 q ) 0 B @ 1 C A ¼p KG(LAp) 2 (n 1)D 2 n1 ( ~CGp)2(1 þ (lLA p) 2 2 4 3 5 0 @ 1 A ¼g
By some simplification, the 100g% adjusted lower confidence bound can be written as LAp ¼ ffiffiffiffiffiffiffiffiffiffiffiffi x2 n1,g q ~ CGp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (n 1)D2n1(lC~Gp)2x2 n1,g q (13)
With adjusted confidence bounds, we can ensure the interval would have the desired confidence coefficient. Moreover, in order to improve the power of the test, the adjusted critical values (denoted by c0A) are proposed to be satisfied c0A, c0. Since c0A, c0, the
probability of ~CGP being greater than c0A will be more than the probability of that ~C
G P
being greater than c0. In addition, both the a -risk and the power increase as c0A are
taken to be adjusted critical values for testing hypothesis. Suppose that the a-risk by adjusted critical values c0
A
isaA, the revised critical c0 A can be introduced by aA¼p( ~C G p c A 0jCp¼c) ¼ p KG c2D2 n1(n 1) (cA 0) 2(1 þ l2c2) :
To ensure that thea-risk is within the preset magnitude, we letaA¼a, thus c0Aand the
power (denoted bypA) can be obtained as
cA0 ¼cDn1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 (1 þl2c2)x2 n1,g s pA(CP) ¼ p( ~C G P . c A 0jCp) ¼ p KG, Cp c 2 1 þl2c2 1 þl2C2 p ! x2n1,g " # (14)
With adjusted critical values, the a-risk within the preset magnitude is ensured and a certain degree of power is improved. For the results to be practical and easily used, the tables of adjusted critical values for some commonly used capability requirements are tabulated in Tables 4 (a) – (d). Using those tables, the practitioner may skip the complex calculation and directly select the proper critical values for capability testing.
Extension to Multiple Samples
Many of the existing manufacturing factories have implemented a daily-based production control plan for monitoring/controlling process stability. A routine-basis data collection procedure is executed to run X and S control charts (for moderate sample sizes). The past ‘in control’ data consisting of multiple samples of msgroups, with variable sample
size ni¼(Xi1, Xi2, . . . , Xini), are then analysed to compute the manufacturing capability.
Thus, manufacturing information regarding the product quality characteristic is derived from multiple samples rather than one single sample. Under the assumption that these samples are taken from the normal distribution N(m,s2), we consider the following esti-mators of process mean and process standard deviation,
Xi¼ Xni j¼1 Xij=ni, Si¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xni j¼1 (Xij Xi)2=(ni1) v u u t
Table 4.Adjusted critical values of Cp
n
l
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
(a) Adjusted critical values c0 A
for Cp¼ 1.00, with n ¼ 10(10)100, l ¼ 0.05(0.05)0.50, and g ¼ 0.95
10 1.502 1.496 1.487 1.474 1.459 1.440 1.419 1.396 1.371 1.345 20 1.314 1.309 1.301 1.290 1.276 1.260 1.242 1.221 1.200 1.177 30 1.245 1.240 1.232 1.222 1.209 1.194 1.176 1.157 1.137 1.115 40 1.207 1.202 1.195 1.185 1.172 1.157 1.140 1.122 1.102 1.081 50 1.182 1.177 1.170 1.160 1.148 1.133 1.117 1.099 1.079 1.058 60 1.164 1.160 1.152 1.143 1.131 1.116 1.100 1.082 1.063 1.042 70 1.150 1.146 1.139 1.129 1.117 1.103 1.087 1.069 1.050 1.030 80 1.140 1.135 1.128 1.119 1.107 1.093 1.077 1.059 1.041 1.021 90 1.131 1.127 1.120 1.110 1.098 1.085 1.069 1.051 1.033 1.013 100 1.124 1.119 1.112 1.103 1.091 1.077 1.062 1.044 1.026 1.006
(b) Adjusted critical values c0Afor Cp¼ 1.33, with n ¼ 10(10)100, l ¼ 0.05(0.05)0.50, and g ¼ 0.95
10 1.995 1.982 1.961 1.932 1.898 1.857 1.813 1.765 1.716 1.665 20 1.746 1.734 1.716 1.691 1.660 1.625 1.586 1.545 1.501 1.457 30 1.654 1.643 1.626 1.602 1.573 1.540 1.503 1.463 1.422 1.380 40 1.603 1.593 1.576 1.553 1.525 1.492 1.457 1.419 1.379 1.338 50 1.570 1.560 1.543 1.521 1.493 1.462 1.427 1.389 1.350 1.310 60 1.547 1.536 1.520 1.498 1.471 1.440 1.405 1.368 1.330 1.291 70 1.529 1.519 1.502 1.480 1.454 1.423 1.389 1.352 1.314 1.276 80 1.514 1.504 1.488 1.467 1.440 1.410 1.376 1.340 1.302 1.264 90 1.503 1.493 1.477 1.455 1.429 1.399 1.365 1.330 1.292 1.254 100 1.493 1.483 1.467 1.446 1.420 1.390 1.356 1.321 1.284 1.246
(c) Adjusted critical values c0 A
for Cp¼ 1.50, with n ¼ 10(10)100, l ¼ 0.05(0.05)0.50, and g ¼ 0.95
10 2.249 2.230 2.200 2.160 2.112 2.057 1.997 1.934 1.869 1.804 20 1.968 1.951 1.925 1.890 1.848 1.799 1.747 1.692 1.635 1.579 30 1.864 1.849 1.824 1.791 1.750 1.705 1.655 1.603 1.549 1.496 40 1.807 1.792 1.768 1.736 1.697 1.653 1.604 1.554 1.502 1.450 50 1.770 1.755 1.732 1.700 1.662 1.619 1.571 1.522 1.471 1.420 60 1.743 1.729 1.705 1.674 1.637 1.594 1.548 1.499 1.449 1.398 70 1.723 1.709 1.686 1.655 1.618 1.576 1.530 1.482 1.432 1.382 80 1.707 1.693 1.670 1.639 1.603 1.561 1.515 1.468 1.419 1.369 90 1.694 1.680 1.657 1.627 1.590 1.549 1.504 1.456 1.408 1.359 100 1.683 1.669 1.646 1.616 1.580 1.539 1.494 1.447 1.399 1.350
(d) Adjusted critical values c0Afor Cp¼ 2.00, with n ¼ 10(10)100, l ¼ 0.05(0.05)0.50, and g ¼ 0.95
10 2.992 2.949 2.880 2.792 2.690 2.578 2.463 2.348 2.235 2.126 20 2.618 2.580 2.520 2.443 2.353 2.256 2.155 2.054 1.956 1.860 30 2.480 2.444 2.387 2.314 2.229 2.137 2.042 1.946 1.853 1.762 40 2.404 2.369 2.314 2.243 2.161 2.072 1.979 1.887 1.796 1.709 50 2.355 2.320 2.267 2.197 2.117 2.029 1.939 1.848 1.759 1.673 60 2.319 2.286 2.232 2.164 2.085 1.999 1.909 1.820 1.732 1.648 70 2.292 2.259 2.207 2.139 2.060 1.975 1.887 1.799 1.712 1.629 80 2.271 2.238 2.186 2.119 2.041 1.957 1.870 1.782 1.696 1.614 90 2.253 2.221 2.169 2.103 2.026 1.942 1.855 1.768 1.683 1.601 100 2.239 2.206 2.155 2.089 2.012 1.929 1.843 1.757 1.672 1.591
for the i – th sample mean and the sample standard deviation, respectively. Then, S2p¼ Pms
i¼1(ni1)S2i=
Pms
i¼1(ni1) are used for calculating the manufacturing capability Cp.
For cases with multiple samples the natural estimator of Cpcan be expressed below.
The sensitivity investigation, capability testing, and adjusted confidence bounds and criti-cal values for process capability Cpin the presence of gauge measurement errors based on
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.
~
CMp ¼DPms i¼1(ni1)
USL LSL 6Sp
Testing Cpmwith Gauge Measurement Errors
Similarly, in practice, the empirical process capability index CpmG is obtained after
substi-tutingsGfors. The relationship between the true process capability Cpmand the empirical
process capability CpmG can be expressed below, wherej¼(mT)=s.
CG pm Cpm ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þj2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2C2 pþj 2 q (15)
Since the variation of data observed is larger than the variation of the original data, the denominator of the index Cpmbecomes larger, and the true capability of the process is
understated if calculation of the process capability index is based on empirical data G. Figure 1(a) displays the surface plot of the ratio R1¼CpmG=Cpm for lin [0, 0.5] for
Cp[ ½1, 2 with j¼0:5. Figure 1(b) plots the ratio R1 versus l for Cp¼ 1.0(0.2)2.0
withj¼ 0.0. Those figures show that the measurement errors result in a downward distor-tion of the index Cpm. Small process variation has the same effect as the presence of
measurement error does. Since R1 would be small if l becomes large, the gauge
becomes more important as the true capability improves. For instance, if l¼ 0.5, Cp¼ 2, and j¼ 0.5 (the ratio R1¼ 0.7454), Cpm
G ¼ 0.7454 with C
pm¼ 1 (shrinks by
about 25.46%), and l¼ 0.5, Cp¼ 2, and j¼ 0.0 (the ratio R1¼ 0.7071),
Figure 1.(a) Surface Plot of R1versus l in [0,0.5] for Cp¼ 1.0(0.2)2.0 with j ¼ 0.5.; (b) Plots of R1
versus l in [0,0.5] for Cp¼ 1.0(0.2)2.0 (top to bottom) with j ¼ 0.0.
Cpm
G ¼ 1.7678 with C
pm¼ 2.50 (shrinks by about 29.29%). The empirical process
capa-bility diverges more from the true process capacapa-bility with large measurement errors.
Sampling Distribution of ^CG pm
In practice, sample data must be collected in order to estimate the empirical process capa-bility CpmG. The maximum likelihood estimator (MLE) of CpmG is defined as the following:
^ CGpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~S2nþ( G T)2 q ¼ d 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~S2nþ( G T)2 q (16) where G ¼ Sni¼1Gi, ~S 2 n ¼S n
i¼1(Gi G)2=n, and d ¼ (USL LSL)=2. We note that G and
~S2nare MLEs ofmands2
Grespectively. Hence the estimated index ^C G
pmis the MLE of C G
pm.
Furthermore, the term ~S2nþ( G T)2¼Si¼1n (GiT)2=n in the denominator of ^CGpmis the
UMVUE of s2
Gþ(mT)
2¼E(G T)2 in the denominator of CG
pm. Obviously, if the sM¼0, then the empirical process capability CpmG reduces to the basic index Cpm. As
with Boyles (1991), the MLE of Cpm can be expressed in equation (17), where X ¼
Sni¼1Xi=n and S2n ¼S n i¼1(Xi X)2=n: ^ Cpm¼ d 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 nþ( X T) 2 q (17)
From equation (16), it is easy to show that ^CG
pmis distributed as: ^ CGpm d 3sG ffiffiffiffiffiffiffiffiffiffi n x2 n, d2G s ¼ ^CGpm d 3sG ffiffiffiffiffiffiffiffiffiffi n x2 n, d2G s ¼CpG ffiffiffiffiffiffiffiffiffiffi n x2 n,d2G s ¼CGpm ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þd 2 G n s ffiffiffiffiffiffiffiffiffiffi n x2 n,d2G s wherex2 n,d2 G
denotes the non-central Chi-square distribution with n degrees of freedom and non-centrality parameterd2G¼nj
2
GwherejG¼(mT)=sG. We apply the method
simi-larly to that used in Pearn et al. (1992), Va¨nnman (1995), and Chen (1998), the cumulative distribution function (CDF) of ^CGpmk can be expressed in terms of a mixture of the Chi-square distribution and the normal distribution
FC^G pm(x) ¼ 1 ðbG ffiffin p =(3x) 0 FK (bG ffiffiffin p )2 9x2 t 2 ½f(t þjG ffiffiffi n p ) þf(t jG ffiffiffi n p )dt (18)
for x . 0 where bG¼ d/sG¼ 3CpG. FK(†) is the cumulative distribution function of the
ordinary central Chi-square distributionx2
n1andf(†) is the PDF of the standard normal
distribution N(0,1), where jG¼ mm sG ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CG p CG pm !2 v u u t 1, CG p ¼ Cp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2C2 p q , and CGpm¼ Cpm ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þj2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2C2 pþj 2 q :
Obviously, if thesM¼ 0, then the CDF of ^Cpmcan be easily obtained as: FC^pm(x) ¼ 1 ðbpffiffin=(3x) 0 FK (bpffiffiffin)2 9x2 t 2 ½f(t þjpffiffiffin) þf(t jpffiffiffin)dt (19)
for x . 0, where b ¼ d/s¼ 3Cpandj¼mms ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cp Cpm 2 1 r :
Lower Confidence Bound Based on ^CGpm
The lower confidence bounds estimate the minimum process capability based on sample data. To find reliable 100g% lower confidence bound Lpmfor Cpm, Pearn & Shu (2003)
solved equation (20). Note that the term b can be expressed as b ¼ 3Cp¼3L
ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þj2
p . Since the process parameters mandsare unknown, then the distribution characteristic parameterj¼(mm)=sis also unknown. To eliminate the need for further estimating the distribution characteristic parameterj, Pearn & Shu (2003) investigated the behaviour of the lower confidence bound Lpmagainst the parameterj. They performed extensive
cal-culations to obtain the lower confidence bound values L for j¼ 0(0.05)3.00, ^
Cpm¼ 0.7(0.1)3.0, n ¼ 10(5)200 with confidence coefficient g¼ 0.95. They found that
the lower confidence bound L obtains its minimum atj¼ 0.0 in all cases. Thus, for prac-tical purposes they recommended solving equation (20) withj¼ ^j ¼ 0.0 to obtain the required lower confidence bounds, without having to further estimate the parameterj.
ðbpffiffin=(3 ^Cpm 0 FK (bpffiffiffin)2 9 ^C2 pm t2 ! ½f(t þjpffiffiffin) þf(t jpffiffiffin)dt ¼ 1 g (20)
In practice, sample data are observed measurements contaminated with errors to esti-mate the empirical process capability. Thus, ^CG
pmis substituted into equation (18) andjG¼
0:0 to obtain the confidence bounds, which can be written as (we denote the bound origi-nated from ^CG pmas LGpm) follows, 2 ðbG ffiffin p =(3 ^CG pm) 0 FK (bG ffiffiffin p )2 9( ^CG pm) 2t 2 ! f(t)dt ¼ 1 g
where bG¼3CpG¼3LGpm. The confidence coefficient of the lower confidence bound LGpm
(denoted bygG) is the following.
gG¼1 2 ðbG ffiffin p =(3 ^Cpm) 0 FK (bG ffiffiffin p )2 9( ^Cpm)2 t2 ! f(t)dt (21)
ThegGis always not less thang
Figures 2 (a), (b) and Figures 3(a), (b) plot Lpm G
versusl [ [0, 0.5] with n ¼ 30, 50, 70, 100, 150 for ^Cpm¼ 1.00, 1.50 and ^Cp ¼ ^CpmþR3, R3¼ 0.33 and 0.67 with 95%
confi-dence level. It is noted that for sufficiently large sample size n, we have ^ CG^ pm¼ ^Cpm= p ð1 þl2C2 pÞ. Therefore, we set ^C G pm¼ ^Cpm= p ð1 þl2C2 pÞ to obtain ^C G pm in
Figures 2(a), (b) and Figures 3(a), (b). We see that in Figures 2(a), (b) and Figures 3(a), (b), LG
pm decreases inl, especially for large ^Cpvalues, and the decrement of LGpmis
more significant for large ^Cpm. A large measurement error results in significantly
under-estimating the true process capability.
In current practice, a process is called ‘inadequate’ if Cpm, 1:00, ‘marginally capable’
if 1:00 Cpm, 1:33, ‘satisfactory’ if 1:33 Cpm, 1:50, ‘excellent’ if
1:50 Cpm, 2:00, and ‘super’ if 2:00 Cpm. In fact, Ruczinski (1996) showed that
Yield 2F(3Cpm) 1, or the fraction of non-conformities 2F( 3Cpm). For
example, if a process has capability with Cpm1:25, then the production yield would
be at least 99.982%. If capability measures do not include the measurement errors, signifi-cant underestimation of the true process capability may result in high production cost, losing the power of competition. For instance, suppose that a process has a 95% lower confidence bound, 1.250 ( ^Cpm¼ 1.50) with n ¼ 50, which meets the threshold of
an ‘excellent’ process. But the bound may be calculated as 0.985 with measurement errorsl¼ 0.36 and the process is determined as ‘inadequate’.
Testing Process Capability Based on ^CG pm
To determine if a given process meets the preset capability requirement, we could consider the statistical testing with null hypothesis H0:Cpmc (process is not capable) and
Figure 2.(a) Plots of LpmG versus l with n ¼ 30, 50, 70, 100, 150 for ^Cp¼1:33 and ^Cpm¼1:00; (b)
Plots of LpmG versus l with n ¼ 30, 50, 70, 100, 150 for ^Cp¼1:67 and ^Cpm¼1:00.
Figure 3.(a) Plots of LpmG versus l with n ¼ 30, 50, 70, 100, 150 for ^Cp¼1:83 and ^Cpm¼1:50; (b)
Plots of Lpm G
versus l with n ¼ 30, 50, 70, 100, 150 for ^Cp¼2:17 and ^Cpm¼1:50.
alternative hypothesis H1:Cpm. c (process is capable), where c is the required process
capability. Given values of capability requirement c, sample size n, and riska, the critical value c0can be obtained by solving equation (2), using the available numerical methods.
ðbpffiffin=(3c0) 0 FK (bpffiffiffin)2 9c2 0 t2 ½f(t þjpffiffiffin) þf(t jpffiffiffin)dt ¼a (22)
where b ¼ 3cpffiffiffiffiffiffiffiffiffiffiffiffiffi1 þj2. Then, the test power can be expressed as the following, p(Cpm) ¼ P( ^Cpmc0jCpm. c) ¼ ðbpffiffin(3c0) 0 FK (bpffiffiffin)2 9c2 0 t2 ½f(t þjpffiffiffin) þf(t jpffiffiffin)dt where b ¼ 3Cpm ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þj2 p .
To eliminate the need for estimating the characteristic parameter j, we follow the method of Pearn & Lin (2004) to examine the behaviour of the critical values c0
against the parameter j. Extensive calculations to obtain the critical values c0 for
j¼ 0(0.01)3, c ¼ 1.00, 1.33, 1.50, 1.67, 2.00, 2.5, and 3.0, n ¼ 10(50)300, and
a¼ 0.05 are performed. The critical value c0 obtains its maximum at j¼ 0.0 in all
cases. For practice purposes, solving equation (22) with j¼ 0.0 to obtain the required critical values is recommended, without having to further estimate the parameterj. In practice, sample data are contaminated with measurement errors to estimate the empirical process capability. Thus, thea-risk corresponding to the test using the sample estimate
^ CGpmbecomes P( ^CpmG c0jCpmc) ¼aG, or 2 ðbG ffiffin p =(3c0) 0 FK (bG ffiffiffi n p )2 9c2 0 t2 f(t)dt ¼aG (23) where bG¼ 3c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2C2 p q and Cp¼c:
The test power (denoted bypG) is:pG(Cpm) ¼ P( ^CGpmc0jCpm. c). Thus, pG(Cpm) ¼ 2 ðbG ffiffin p =(3c0) 0 FK (bG ffiffiffin p )2 9c2 0 t2 f(t)dt (24) where bG¼ 3Cpm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2C2 p q and Cp ¼Cpm
Earlier discussions indicate that the true process capability would be severely underes-timated if ^CpmG is used. The probability of ^CGpmbeing greater than c0would be less than that
of using ^Cpm. Thus, thea-risk using ^CGpmis,aG, less than thea-risk if using ^Cpm,a, when
hypothesis testing Cpm. The test power if using ^CGpmis also less than the test power of using
^
Cpm. That ispG,p. Figures 4(a), (b) are the plots ofaGwith n ¼ 30, 50, 70, 100, 150,l
[ [0, 0.5] for c ¼ 1.00, 1.50, and a¼ 0.05. Figures 5(a), 5(b) plot pG versus l with
n ¼ 50, a¼ 0.05, for c ¼ 1.00, 1.50, and Cpm¼ (c þ 0.2)(0.20)(c þ 1). Note that for
l¼ 0,aG¼aandpG¼p. In Figures 4(a), (b),aGdecreases aslor n increases, and the
decreasing rate is more significant with large c. In fact, for largel,aGis smaller than 1022.
In Figures 5(a), (b),pGdecreases aslincreases, but increases as n increases. The
decre-ment ofpGinlis more significant for large c. In the presence of measurement errors,pG
decreases. For instance, in Figure 9(b), later, the pG values (c ¼ 1.50, n ¼ 50) for
Cpm¼ 2.1 is pG¼ 0.9556 if there is no measurement error (l¼ 0). But, when
l¼ 0.5,pGdecreases to 0.0257, the decrement of the power is 0.9299.
Adjusted Confidence Bounds and Critical Values of Cpm
In this section, we consider the adjustment of confidence bounds and critical values of Cpm
to provide better capability assessment. Suppose that the desired confidence coefficient is 100g% and the adjusted confidence interval of ^CGpmwith the adjusted lower confidence
Figure 4.(a) Plots of aGwith n ¼ 30, 50, 70, 100, 150 and l in [0,0.5] for c ¼ 1.00 and a ¼ 0.05;
(b) Plots of aGwith n ¼ 30, 50, 70, 100, 150 and l in [0,0.5] for c ¼ 1.50 and a ¼ 0.05.
Figure 5.(a) Plots of pGversus l with n ¼ 50, a ¼ 0.05 for c ¼ 1.00, Cpm¼ 1.2(0.20)2.00; (b)
Plots of pGversus l with n ¼ 50, a ¼ 0.05 for c ¼ 1.50, Cpm¼ 1.70(0.20)2.50.
bound LApmcan be established as: P( ^CGpm. LApm) ¼g 1 2 ðbA ffiffin p =(3 ^CG pm) 0 FK (bA ffiffiffin p )2 9( ^CG pm) 2 ! f(t)dx ¼g (25) where bA ¼3LApm= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2C2 p q
and Cp can be obtained by solving CGp ¼C G pm, thus,
Cp¼Lpm. Figures 6 and 7, are the comparisons among Lpm, LpmG , and LpmA for
^
Cpm¼1:00, 1.50 with n ¼ 50, where Lpmis the 95% lower confidence bound using ^Cpm,
LpmG is the 95% lower confidence bound using ^CGpm, and LpmA is the adjusted 95% lower
con-fidence bound using ^CGpm. It can be noted that ^CpmG ¼ ^Cpm=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2C^2 p
q
is used to obtain ^CpmG in equation (4). In this case, the probability that the lower confidence interval with bound Lpm
G
contains the actual Cpmvalue is greater than that of the interval with the bound Lpmor
Lpm A
, while the probability that the lower confidence interval with bound Lpmor Lpm A
contains the actual Cpmvalue is 0.95. From Figures 6 and 7, we see that the magnitude of lower
confidence bounds remained underestimated even if it is adjusted. But the magnitude of underestimation using the adjusted confidence bound is significantly reduced.
In order to improve the test power, we revise the critical values c0Ato satisfy c0A, c0.
Thus, the probability P( ^CG
pm. cA0) is greater than P( ^CGpm. c0). Both thea-risk and the
test power increase when we use c0Aas a new critical value in the testing. Suppose that
thea-risk using the revised critical value c0AisaA, the revised critical values c0Acan be
determined by P( ^CG pmcA0jCpmc) ¼aA, 2 ðbG ffiffin p =(3cA 0) 0 FK (bG ffiffiffin p )2 9(cA 0) 2 t 2 f(t)dt ¼aA (26) where bG¼3c= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2C2 p q
and Cpcan be obtained by solving equation CpG¼ CpmG, thus,
Cp¼ c. To ensure that thea-risk is within the preset magnitude, we letaA¼aand solve
Figure 6.Plots of Lpm, Lpm A
, and Lpm G
versus l with n ¼ 50 and for ^Cpm¼1:00 and ^Cp¼1:33.
the equation to obtain c0A. The power (denoted bypA) can be calculated as the following: pA(Cpm) ¼ P( ^CGpmc A 0jCpm. c) pA(Cpm) ¼ 2 ðbG ffiffin p =(3cA 0) 0 FK (bG ffiffiffin p )2 9(cA 0) 2 t 2 f(t)dt (27) where bG¼ 3Cpm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þl2c2 p :
Figures 8(a), (b) are plots ofpAversuslwith n ¼ 50,a¼ 0.05, for c ¼ 1.00, 1.50 and
Cpm¼ (c þ 0.2)(0.20)(c þ 1). From those figures, we see that the powers corresponding to
Figure 8.(a) Plots of pAversus l with n ¼ 50, a ¼ 0.05 for c ¼ 1.00, Cpm¼ 1.2(0.2)2.00 (bottom
to top); (b) Plots of pAversus l with n ¼ 50, a ¼ 0.05 for c ¼ 1.50, Cpm¼ 1.70(0.20)2.50 (bottom
to top).
Figure 7.Plots of Lpm, LpmA , and LpmG versus l with n ¼ 100 and for ^Cpm¼1:50 and ^Cp¼1:83.
Table 5.Adjusted critical values of Cpm
l n 30 50 70 100 120 150
(a) Adjusted critical values c0Afor n ¼ 30, 50, 70, 100, 120, 150, l ¼ 0.05(0.05)0.50, Cpm¼ 1.00,
and g ¼ 0.95 0.00 1.273 1.199 1.163 1.132 1.119 1.105 0.02 1.273 1.199 1.162 1.132 1.119 1.105 0.04 1.272 1.198 1.162 1.131 1.118 1.104 0.06 1.271 1.197 1.161 1.130 1.117 1.103 0.08 1.269 1.195 1.159 1.129 1.116 1.102 0.10 1.267 1.193 1.157 1.127 1.114 1.100 0.12 1.264 1.190 1.154 1.124 1.111 1.097 0.14 1.261 1.187 1.151 1.121 1.108 1.095 0.16 1.257 1.184 1.148 1.118 1.105 1.091 0.18 1.253 1.180 1.144 1.114 1.102 1.088 0.20 1.248 1.175 1.140 1.110 1.098 1.084 0.22 1.243 1.171 1.135 1.106 1.093 1.079 0.24 1.238 1.166 1.131 1.101 1.088 1.075 0.26 1.232 1.160 1.125 1.096 1.083 1.070 0.28 1.226 1.154 1.120 1.090 1.078 1.064 0.30 1.219 1.148 1.114 1.085 1.072 1.059 0.32 1.213 1.142 1.107 1.078 1.066 1.053 0.34 1.205 1.135 1.101 1.072 1.060 1.046 0.36 1.198 1.128 1.094 1.065 1.053 1.040 0.38 1.190 1.121 1.087 1.058 1.046 1.033 0.40 1.182 1.113 1.079 1.051 1.039 1.026 0.42 1.174 1.105 1.072 1.044 1.032 1.019 0.44 1.165 1.097 1.064 1.036 1.024 1.012 0.46 1.157 1.089 1.056 1.029 1.017 1.004 0.48 1.148 1.081 1.048 1.021 1.009 0.996 0.50 1.139 1.072 1.040 1.013 1.001 0.988
(b) Adjusted critical values c0Afor n ¼ 30, 50, 70,100,120,150, l ¼ 0.05(0.05)0.50, Cpm¼ 1.33,
and g ¼ 0.95 0.00 1.693 1.595 1.547 1.506 1.489 1.470 0.02 1.693 1.594 1.546 1.506 1.488 1.470 0.04 1.691 1.592 1.544 1.504 1.487 1.468 0.06 1.688 1.589 1.542 1.501 1.484 1.465 0.08 1.684 1.586 1.538 1.498 1.480 1.462 0.10 1.679 1.581 1.533 1.493 1.476 1.457 0.12 1.672 1.575 1.527 1.487 1.470 1.452 0.14 1.665 1.568 1.520 1.481 1.464 1.445 0.16 1.656 1.560 1.513 1.473 1.456 1.438 0.18 1.647 1.551 1.504 1.465 1.448 1.430 0.20 1.637 1.541 1.495 1.455 1.439 1.421 0.22 1.625 1.53 1.484 1.445 1.429 1.411 0.24 1.613 1.519 1.473 1.435 1.418 1.400 0.26 1.600 1.507 1.462 1.423 1.407 1.389 0.28 1.587 1.494 1.449 1.411 1.395 1.378 0.30 1.573 1.481 1.436 1.399 1.383 1.365 0.32 1.558 1.467 1.423 1.386 1.370 1.353 (Table continued )
Table 5. Continued l n 30 50 70 100 120 150 0.34 1.543 1.453 1.409 1.372 1.356 1.339 0.36 1.527 1.438 1.395 1.358 1.343 1.326 0.38 1.511 1.423 1.380 1.344 1.329 1.312 0.40 1.495 1.408 1.365 1.33 1.314 1.298 0.42 1.478 1.392 1.35 1.315 1.300 1.283 0.44 1.462 1.376 1.335 1.300 1.285 1.269 0.46 1.445 1.360 1.319 1.285 1.270 1.254 0.48 1.427 1.344 1.303 1.269 1.255 1.239 0.50 1.410 1.328 1.288 1.254 1.240 1.224
(c) Adjusted critical values c0Afor n ¼ 30, 50, 70,100,120,150, l ¼ 0.05(0.05)0.50, Cpm¼ 1.50,
and g ¼ 0.95 0.00 1.910 1.798 1.744 1.699 1.679 1.658 0.02 1.909 1.798 1.743 1.698 1.678 1.657 0.04 1.907 1.795 1.741 1.696 1.676 1.655 0.06 1.902 1.791 1.737 1.692 1.672 1.651 0.08 1.896 1.786 1.732 1.687 1.667 1.646 0.10 1.889 1.779 1.725 1.680 1.661 1.640 0.12 1.880 1.770 1.717 1.672 1.653 1.632 0.14 1.869 1.760 1.707 1.662 1.643 1.623 0.16 1.857 1.749 1.696 1.652 1.633 1.612 0.18 1.844 1.736 1.684 1.640 1.621 1.601 0.20 1.829 1.723 1.671 1.627 1.608 1.588 0.22 1.814 1.708 1.656 1.613 1.595 1.575 0.24 1.797 1.692 1.641 1.598 1.580 1.560 0.26 1.779 1.675 1.625 1.583 1.564 1.545 0.28 1.761 1.658 1.608 1.566 1.548 1.529 0.30 1.742 1.640 1.591 1.549 1.531 1.512 0.32 1.722 1.621 1.572 1.531 1.514 1.495 0.34 1.701 1.602 1.554 1.513 1.496 1.477 0.36 1.681 1.582 1.535 1.495 1.477 1.459 0.38 1.659 1.562 1.515 1.476 1.459 1.440 0.40 1.638 1.542 1.496 1.457 1.440 1.422 0.42 1.616 1.522 1.476 1.437 1.421 1.403 0.44 1.594 1.501 1.456 1.418 1.401 1.384 0.46 1.572 1.480 1.436 1.398 1.382 1.365 0.48 1.550 1.459 1.415 1.378 1.363 1.345 0.50 1.528 1.439 1.395 1.359 1.343 1.326
(d) Adjusted critical values c0 A for n ¼ 30, 50, 70,100,120,150, l ¼ 0.05(0.05)0.50, Cpm¼ 2.00, and g ¼ 0.95 0.00 2.547 2.398 2.326 2.265 2.239 2.211 0.02 2.545 2.396 2.324 2.263 2.237 2.209 0.04 2.539 2.390 2.318 2.258 2.232 2.204 0.06 2.529 2.381 2.309 2.249 2.223 2.195 0.08 2.515 2.368 2.297 2.237 2.211 2.183 0.10 2.497 2.351 2.281 2.221 2.196 2.168 0.12 2.477 2.332 2.262 2.203 2.177 2.150 (Table continued )
the adjusted critical values c0 A
remain stable in measurement error. We improve the test power to a certain degree. For instance, when we compare thepGvalues in Figure 5(b)
(c ¼ 1.50, n ¼ 50, Cpm¼ 2.1) to the pA values in Figure 8(b) (c ¼ 1.50, n ¼ 50,
Cpm¼ 2.1), we obtain that pG¼ 0.0257 and pA¼ 0.9557 with l¼ 0.5. In this case,
using the adjusted critical values c0A, we improve the test power by 0.930 (which is
rather significant). For our results to be practical, we tabulate the adjusted critical values for some commonly used capability requirements in Tables 5(a) – (d). Using those tables, the practitioner may omit the complex calculation and simply select the proper critical values for capability testing.
Application Example on a pH Sensor
The product investigated is the pH sensor combining process-hardened pH electrodes, a double junction reference electrode, temperature compensation element, and a solution ground in a durable, reliable, high performance design. It is ideal for process control applications in the most aggressive streams found in traditional processing industries, including chemicals, paper, metals and mining, utilities, food, pharmaceutical, and others. For a wide range of measurements where high accuracy is required at either both extreme ends of the pH scales, or a spherical glass with minimal sodium error. For applications involving abrasive processes, a rugged glass with a thicker membrane is recommended. The rugged glass is most accurate in the range of 1 to 12 pH. High temperature construction allows the sensor to be used for process pH measurements at temperatures up to 1208C (2508F). An integral 100 platinum RTD (resistance temperature detector), compatible with many common pH transmitters and monitors, is a standard feature. The reference electrode utilizes a double junction design to inhibit silver ions from contacting the process solution, thereby preventing junction fouling from silver
Table 5. Continued l n 30 50 70 100 120 150 0.14 2.453 2.309 2.240 2.181 2.156 2.129 0.16 2.426 2.284 2.215 2.157 2.132 2.106 0.18 2.396 2.256 2.188 2.131 2.107 2.080 0.20 2.365 2.226 2.159 2.103 2.079 2.053 0.22 2.331 2.195 2.129 2.073 2.049 2.024 0.24 2.296 2.162 2.097 2.042 2.018 1.993 0.26 2.260 2.128 2.063 2.010 1.986 1.961 0.28 2.222 2.092 2.029 1.976 1.953 1.929 0.30 2.184 2.056 1.994 1.942 1.920 1.896 0.32 2.145 2.020 1.959 1.908 1.886 1.862 0.34 2.106 1.983 1.923 1.873 1.851 1.828 0.36 2.067 1.946 1.887 1.838 1.817 1.794 0.38 2.028 1.909 1.852 1.803 1.783 1.760 0.40 1.989 1.872 1.816 1.769 1.748 1.726 0.42 1.950 1.836 1.781 1.734 1.714 1.693 0.44 1.912 1.800 1.746 1.700 1.681 1.660 0.46 1.874 1.765 1.712 1.667 1.648 1.627 0.48 1.837 1.730 1.678 1.634 1.615 1.595 0.50 1.801 1.696 1.644 1.602 1.583 1.563
compound precipitates. To help facilitate a noise-free signal, the sensor incorporates a solution ground post of titanium metal. 3/4 NPT threads are provided at both sensor ends to allow connection to insertion or submersion type mountings. The pH sensor is depicted in Figure 9.
The measuring accuracy of the pH sensor is an important factor that has significant effect on the pH sensor quality. A type of the pH sensor has the specification limits, T ¼ 0.0 pH, USL ¼ 0.05 pH, and LSL ¼ 20.05 pH. A total of 70 observations are collected and displayed in Table 6. Histogram and normal probability plots show that the collected data follow the normal distribution. The Shapiro – Wilk test is applied to further justify the assumption. To determine whether the process is ‘excellent’ (Cpm. 1.33) with unavoidable measurement errors l¼ 0.30, we first determine
that c ¼ 1.33 and a¼ 0.05. Then, based on the sample data of 70 observations, we obtain the sample mean G ¼ 0:0200, the sample standard deviation ~Sn¼0:0109, and
the point estimator ^CG
pm¼1:4629. From Table 5(b), we obtain the critical value
c0A¼ 1.436 based on a, l and n. Since ^CpmG . cA0, we therefore conclude that
the process is ‘excellent’. We also see that if we ignore the measurement errors and evalu-ate the critical value without any correction, the critical value may be calculevalu-ated as c0¼ 1.547. In this case we would reject that the process is ‘excellent’ since ^CGpm is no
Table 6.70 observations for the measuring accuracy (unit: pH)
0.0236 0.0433 0.032 0.0157 0.0129 0.0085 0.0406 0.0047 0.0047 0.0254 0.0257 0.0281 0.0246 0.0198 0.0335 0.0253 0.0115 0.0306 0.0097 0.0194 0.0304 0.0198 0.0172 0.0234 0.0311 0.0278 0.0035 0.0289 0.0044 0.0188 0.0289 0.0227 0.0186 0.0066 0.0265 0.0399 0.0168 0.0240 0.0040 0.0235 0.0057 20.0018 0.0385 20.0086 0.0178 0.0130 0.0240 0.0317 0.0200 0.0137 0.0188 20.0046 0.0322 0.0136 0.0187 0.0249 0.0157 0.0250 0.0167 0.0083 0.0204 0.0153 0.0154 0.0413 0.0089 0.0344 0.0273 0.0236 0.0158 0.0185
Figure 9.The pH sensor.
greater than the uncorrected critical value 1.547. Moreover, input T ¼ 0.0pH, USL ¼ 0.05pH, and LSL ¼ 20.05pH, 70 observations, l¼ 0.30 (provided by the gauge manufacturing factory), and the desired confidence coefficient g¼ 0.95 into the Matlab computer program (available upon request), the 95% lower confidence bound of the true process capability can be obtained as 1.415. We thus can ensure that the production yield is 99.9978%, and the number of the non-conformities is less than 21.78 PPM (Parts Per Million).
Conclusions
Gauge measurement errors have a significant impact on estimating and testing manufac-turing reproduction capability. In this paper, we conducted the sensitivity study for process capability Cpand Cpmin the presence of gauge measurement errors. We investigated the
statistical properties and capability testing of estimating Cpand Cpmto obtain lower
con-fidence bounds and critical values for true process capability testing when gauge measure-ment errors are unavoidable. In estimating the capability, the estimator ^CpGand ^CpmG using the sample data contaminated with the measurement error severely underestimates the true capability in the presence of measurement errors. The statistical testing is performed to determine whether the process meets the capability requirement, the test power decreases in the presence of gauge measurement errors. Since the measurement errors are unavoid-able in most industry applications, lower confidence bounds and critical values must be adjusted to improve the accuracy of capability assessment. For practical purposes, some adjusted critical values for Cpand Cpm are tabulated to the engineers for their factory
applications.
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![Figure 1. (a) Surface Plot of R 1 versus l in [0,0.5] for C p ¼ 1.0(0.2)2.0 with j ¼ 0.5.; (b) Plots of R 1](https://thumb-ap.123doks.com/thumbv2/9libinfo/7951451.157855/11.739.121.620.697.923/figure-surface-plot-r-versus-c-plots-r.webp)
![Figure 4. (a) Plots of a G with n ¼ 30, 50, 70, 100, 150 and l in [0,0.5] for c ¼ 1.00 and a ¼ 0.05;](https://thumb-ap.123doks.com/thumbv2/9libinfo/7951451.157855/16.739.110.639.75.307/figure-plots-g-n-l-c.webp)
