Qual. Reliab. Engng. Int. 2007; 23:597–614
Published online 7 November 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/qre.836
Research
Measuring Process Capability
Based on
C
pmk
with Gauge
Measurement Errors
B. M. Hsu1, M. H. Shu2,∗,†and W. L. Pearn3
1Department of Industrial Engineering and Management, Cheng Shiu University, 840 Cheng Cing Road, Niaosong,
Kaohsiung 83347, Taiwan, Republic of China
2Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences,
415 Chien Kung Road, Seng-Min District, Kaohsiung 80778, Taiwan, Republic of China
3Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road,
Hsinchu 300, Taiwan, Republic of China
Due to their effectiveness and simplicity of use, the process capability indices Cp, Cpk, and Cpm have been popularly accepted in the manufacturing industry as management tools for evaluating and improving process quality. Combining the merits of those indices, a more advanced index, Cpmk, is proposed that takes into account process variation, process centering, and the proximity to the target value, and has been shown to be a very useful index for manufacturing processes with two-sided specification limits. Most research works related to Cpmk assume no gauge measurement errors. However, such an assumption inadequately reflects real situations even when highly advanced measurement instruments are employed. Conclusions drawn regarding process capability are therefore unreliable and misleading. In this paper, we conduct a sensitivity investigation for the process capability index Cpmk in the presence of gauge measurement errors. We consider the use of capability testing of Cpmk as a method for obtaining lower confidence bounds and critical values for true process capability when gauge measurement errors are unavoidable. The results show that using the estimator with sample data contaminated by measurement errors severely underestimates the true capability, resulting in an imperceptibly smaller test power. To measure the true process capability, three methods for the adjusted confidence bounds are presented and their performances are compared using computer simulation. Copyright c 2006 John Wiley & Sons, Ltd.
Received 4 January 2006; Revised 2 March 2006
KEY WORDS: gauge measurement error; lower confidence bound; critical value; process capability indices; generalized confidence interval
∗Correspondence to: 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, Republic of China.
†E-mail: workman@cc.kuas.edu.tw
Cpk= min USL− μ 3σ , μ− LSL 3σ (2) Cpm= USL− LSL 6σ2+ (μ − T )2 (3) Cpmk= min USL− μ 3σ2+ (μ − T )2, μ− LSL 3σ2+ (μ − T )2 (4) where USL is the upper specification limit, LSL is the lower specification limit,μ is the process mean, σ is the
process standard deviation, andT is the target value predetermined by the product designer or the manufacturing
engineer.
Process variation (product quality consistency), process departure (targeting), process yield, and process loss (relating to product reliability) are considered crucial benchmarks for measuring manufacturing quality. The indexCp measures only the distribution spread (process consistency/precision), which only reflects the consistency of the product quality characteristic. The yield-based indexCpkprovides lower bounds on process yield by taking the process location into consideration, which offsets some of the weaknesses inCp but can fail to distinguish between on-target and off-target processes (Hoffman7). The indexCpm takes the proximity of process mean from the target value into account, which is more sensitive to process departure than Cpk. Since the design ofCpmis based on the average process loss relative to the manufacturing tolerance, the index
Cpmprovides an upper bound on the average process loss.
The index Cpmk is constructed by combining the modifications to Cp that produced Cpk and Cpm, and therefore inherits the merits of both indices. We note that a manufacturing process satisfying the capability requirement ‘Cpk≥ c’ may not satisfy the capability condition ‘Cpm≥ c’. On the other hand, a process
satisfying the capability requirement ‘Cpm≥ c’ may not satisfy the capability requirement ‘Cpk≥ c’. However,
a manufacturing process does satisfy both capability requirements ‘Cpk≥ c’ and ‘Cpm≥ c’ if the process
satisfies the capability requirement ‘Cpmk≥ c’ since Cpmk≤ Cpk and Cpmk≤ Cpm. Thus, the index Cpmk
provides a greater level of quality assurance with respective to process yield and process loss to the customers than the other two indices. This is a desired property according to today’s modern quality theory, as a reduction of process loss (variation from the target) is just as important as increasing process yield (meeting the specifications). While Cpk remains the more popular and widely used index,Cpmk is considered a very useful index for processes with two-sided manufacturing specifications. For semiconductor or microelectronics manufacturing in particular,Cpmkis an appropriate index for capability measurement due to the high standard and stringent requirements on product quality and reliability.
2.
THE INDEX
C
pmkAND THE GAUGE MEASUREMENT ERROR
Most research works related toCpmkhave assumed no gauge measurement errors. For example, Chen and Hsu8 investigated the asymptotic sampling distribution of the estimatedCpmk. Wright9derived an explicit but rather
Table I. Guidelines for gauge capabilities
Gauge capability Result
λ < 0.1 Gauge system OK
0.1 < λ < 0.3 May be acceptable based on importance of application, cost of gauge, cost of repair, and so on 0.3 < λ Gauge system needs improvement; make every effort to identify the problems and have them corrected
studied the behavior ofCpmk for processes with asymmetric tolerances. Pearn et al.11 obtained an alternative but simpler form of the probability density function of the estimatedCpmkand considered the capability testing based onCpmk. Pearn and Lin12and Pearn and Shu13developed efficient Maple/Matlab computer programs to
calculate the critical values, thep-value, and the lower confidence bounds for estimating and testing process
capability based onCpmk.
However, capability analysis with no gauge measurement errors cannot reflect real situations closely, even with highly sophisticated and advanced measurement instruments (Bordignon and Scagliarini14,15). Any measurement error has some impact on the determination of true measurement systems and process capabilities. Montgomery and Runger16,17 and Burdick et al.18 noted that the quality of the collected data relies heavily on the gauge accuracy. Clearly, conclusions drawn regarding process capability, based on the empirical index values calculated from data contaminated with gauge measurement errors, are highly unreliable. To analyze the effect of gauge measurement errors on the true capability measure, Mittag19,20and Levinson21 quantified the percentage error on process capability evaluation in the presence of gauge measurement errors. Bordignon and Scagliarini14presented the statistical analysis on the estimation of confidence intervals forCp
with data contaminated with measurement errors.
Suppose that the measurement errors can be described as a random variableM∼ N(0, σ2
M). Montgomery
and Runger16,17expressed the gauge capability as
λ= 6σM
USL− LSL (5)
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. Montgomery22noted that the automotive industry action group recommended the guidelines for gauge acceptance given in Table I. In this paper, the gauge capability,λ, in Equation (5) provided by the gauge manufacturing factory is assumed to be
known.
The organization of this paper is as follows. In Section 3, we consider the sensitivity of theCpmkindex with gauge measurement errors. In Sections 4 and 5, the sampling distribution and bias and mean squared error (MSE) are studied when using ˆCpmkG as an estimator forCpmk. In Sections 6 and 7, we show that a large measurement error will cause underestimation of the true process capability. In Section 8 sampling distribution (standard distribution (SD) and mean squared distribution (MSD)) approaches and generalized confidence intervals (GCIs) approach are proposed to establish more reliable lower confidence bounds. In Section 9, a simulation study is conducted for the performance comparison of these methods. Section 10 concludes the paper.
3.
EMPIRICAL PROCESS CAPABILITY
C
pmkGSuppose that X∼ N(μ, σ2) represents the quality characteristic of the manufacturing process under
investigation. In practice, the observed variableG (with gauge measurement errors) is measured rather than the
true variableX. Assume that X and M are stochastically independent, then we have G∼ N(μ, σ2
G= σ2+ σM2)
(a) (b)
Figure 1. (a) Surface plot and (b) plots ofR1versusλ in [0,0.5] for Cp= 1.0(0.2)2.0 with ξ = 0.5
the true process capability,Cpmk, and the empirical process capability,CGpmk, can be expressed as
CpmkG Cpmk = 1+ ξ2 1+ λ2C2 p+ ξ2 (6)
whereξ= (μ − T )/σ. Since the variation of the observed data is larger than the variation of the original data,
the denominator of the indexCpmkbecomes larger and the true capability of the process is understated if the empirical dataG are used.
Figure 1(a) displays the surface plot of the ratioR1= CpmkG /Cpmkforλ in [0,0.5] with Cp∈ [1, 2]. Figure 1(b)
plots the ratioR1 versusλ for Cp= 1.0(0.2)2.0. These figures show that the measurement errors result in a
decrease in the estimated value. A small process variation has the same effect as the presence of a measurement error. SinceR1is small ifλ becomes large, the gauge becomes more important as the true capability improves.
For instance, if λ= 0.5, Cp= 2, and ξ = 0.5 (the ratio R1= 0.7454), CpmkG = 0.7454 with Cpmk= 1 and CpmkG = 1.8634 with Cpmk= 2.50. The empirical process capability diverges from the true process capability
as measurement errors increase.
4.
SAMPLING DISTRIBUTION OF ˆ
C
pmkGIn practice, sample data must be collected in order to estimate the empirical process capability CpmkG . For a stably normal process, the empirical data (observed data contaminated with errors)Gi, fori= 1, 2, . . . , n, is
collected. The maximum likelihood estimator (MLE) ofCpmkG is defined as ˆCG pmk= min ⎧ ⎨ ⎩ USL− ¯G 3 ˜S2 n+ ( ¯G − T )2 , ¯G − LSL 3 ˜S2 n+ ( ¯G − T )2 ⎫ ⎬ ⎭ (7)
where ¯G= ni=1Gi/n and ˜S2
n=
n
i=1(Gi− ¯G)2/n are the MLEs of μ and σG2. We note that the
statistic ˜S2
n+ ( ¯G − T )2=
n
i=1(Gi− T )2/n in the denominator of ˆCpmkG is the uniformly minimum variance
unbiased estimator ofσ2
G+ (μ − T )2= E[(G − T )2]. For processes with a symmetric manufacturing tolerance (T = m), the estimator ˆCpmkG can alternatively be expressed as follows
ˆCG pmk= d− | ¯G − m| 3 ˜S2 n+ ( ˜G − T )2 (8) whered= (USL − LSL)/2.
Obviously, if σM= 0, then the empirical process capability, CpmkG , reduces to the basic index, Cpmk. Pearn et al.6considered the MLE ofCpmk, expressed as
ˆCpmk= min ⎧ ⎨ ⎩ USL− ¯X 3 S2 n+ ( ¯X − T )2 , LSL− ¯X 3 S2 n+ ( ¯X − T )2 ⎫ ⎬ ⎭ = d− | ¯X − T | 3 S2 n+ ( ¯X − T )2 (9) where ¯X= ni=1Xi/n and Sn2= n
i=1(Xi− ¯X)2/n. Using the same technique as V¨annman22and Shu and
Chen23, the cumulative distribution function (CDF) of ˆCGpmk can be expressed in terms of a mixture of the Chi-square distribution and the normal distribution
FˆCG pmk(x)= 1 − bG√n/(1+3x) 0 FK (bG√n− t)2 9x2 − t 2 [φ(t + ξG√n)+ φ(t − ξG√n)] dt, (10)
forx > 0, where bG= d/σG= 3CpG,FK(·) is the CDF of the ordinary central Chi-square distribution χn2−1and
φ(·) is the probability density function (PDF) of the standard normal distribution N(0, 1) where ξG=μ− m σG = −CG p/3+ (CG p)2/9+ [(CpmkG )2− 1/9][(CpG)2− (CpmkG )2] (CpmkG )2− 1/9 CpG= Cp 1+ λ2C2 p , CpmkG = Cpmk 1+ ξ2 1+ λ2C2 p+ ξ2
Therth moment of the estimator ˆCG
pmkcan be obtained (V¨annman
23and Shu and Chen24) as
E[( ˆCpmkG )r] = 3−r r i=0 (−1)i r i D √ 2 r−i ∞ =0 βe−β/2 2! × (c− a) (b) (b− i/2) (c) (11) whereD=√nd/σG,β= n(μ − T )2/σG2,a= r/2, b = (1 + i + j)/2, and c = (n + i + j)/2.
Obviously, if theσM= 0, then the CDF of ˆCpmkreduces to FˆC pmk(x)= 1 − b√n/(1+3x) 0 FK (b√n− t)2 9x2 − t 2 [φ(t + ξ√n)+ φ(t − ξ√n)] dt (12) forx > 0, where b= d/σ = 3Cpand
ξ=μ− m σ = −Cp/3+ C2 p/9+ [Cpmk2 − 1/9][Cp2− Cpmk2 ] C2 pmk− 1/9
5.
BIAS AND THE MSE ANALYSIS
To investigate how measurement errors may affect the sampling distribution, we conduct bias and MSE analyses. Noting that from the expressionCpmk= (Cp− |ξ|/3)/
1+ ξ2, Pearn and Shu13 and Pearn and Lin12 show
that the lower confidence bounds and critical values for Cpmk can be obtained by setting ξ= 0.5 (for test
reliability purposes). We then setCpmk= (Cp− 1/6)/
√
1.25 and consider cases of (Cp, Cpmk)= (1.285, 1.00)
and (1.844, 1.50) as examples. Figures 2(a) and (b) plot the bias of ˆCpmkG versusn= 5(5)100 with λ = 0(0.1)0.5
forCp= 1.285 and Cpmk= 1.00 and Cp= 1.844 and Cpmk= 1.5, respectively. Note that when λ = 0, the bias
of ˆCpmkG is equal to the bias of ˆCpmk, and the bias of ˆCpmkG increases asλ increases or n decreases. Figures 3(a)
and (b) are the surface plots of the ratioR2= MSE( ˆCpmkG )/MSE( ˆCpmk) with n= 5(5)100 and λ in [0,0.5] for
Cp= 1.285 and Cpmk= 1.00 and Cp= 1.844 and Cpmk= 1.50, respectively. The maximum values of R2 in
(a) (b)
Figure 2. Plots of the bias of ˆCpmkG forn= 5(5)100, λ = 0(0.1)0.5 (bottom to top): (a) Cp= 1.285 and Cpmk= 1.00;
(b)Cp= 1.844 and Cpmk= 1.5
(a) (b)
Figure 3. Surface plot ofR2withn= 5(5)100 and λ in [0,0.5] for (a) Cp= 1.285 and Cpmk= 1.00; (b) Cp= 1.844 and
Cpmk= 1.50
6.
LOWER CONFIDENCE BOUND BASED ON ˆ
C
pmkGThe lower confidence bounds estimate the minimum process capability based on sample data. To find reliable 100γ % lower confidence bound L for Cpmk (whereγ is the confidence level), Pearn and Shu13 solved the
following equation: b√n/(1+3 ˆCpmk) 0 FK (b√n− t)2 9 ˆC2 pmk − t2 [φ(t + ξ√n)+ φ(t − ξ√n)] dt = 1 − γ (13)
Note that the parameterb can be expressed as b= 3Cp= 3L
1+ ξ2+ |ξ|. Since the process parameters μ
andσ are unknown, then the distribution parameter ξ= (μ − m)/σ is also unknown. To eliminate the need for
further estimation of the distribution characteristic parameterξ , Pearn and Shu13 investigated the behavior of
the lower confidence bound,L, against ξ . They performed extensive calculations to obtain the lower confidence
bound values forξ = 0(0.05)3.00, ˆCpmk= 0.7(0.1)3.0, and n = 10(5)200, and found that the lower confidence
bound obtains its minimum at ξ= 0.5 in all cases. Thus, for practical purposes they recommended solving
In practice, the observed sample data are contaminated with errors. Thus, ˆCpmkG is substituted into Equation (13) withξG= 0.5 to obtain the confidence bounds, which can be written as
bG√n/(1+3 ˆCpmkG ) 0 FK (bG√n− t)2 9( ˆCpmkG )2 − t 2 [φ(t + 0.5√n)+ φ(t − 0.5√n)] dt = 1 − γ (14)
We denote the bound originated from ˆCpmkG asLGwherebG= 3CGp = 3LG
√
1.25+ 0.5.
The confidence coefficient for the lower confidence boundLG(denoted asγG) is
γG= 1 − bG√n/(1+3 ˆCpmkG ) 0 FK (bG√n− t)2 9( ˆCpmkG )2 − t 2 [φ(t + ˆξG√n)+ φ(t − ˆξG√n)] dt (15) wherebG= 3LG
1+ ˆξG2 + |ˆξG| and γGis no less thanγ .
Figures 4(a)–(d) plotLGversusλ∈ [0, 0.5] with n = 30, 50, 70, 100, 150 for ˆCpmk= 1.00, 1.50 and ˆCp=
ˆCpmk+ R3,R3= 0.285 and 0.50 with 95% confidence level. It should be noted that for sufficiently large sample
size, we have ˆCG pmk= ˆCpmk √ 1.25 1.25+ λ2ˆC2 p (16) Therefore, we set ˆCpmkG = ˆCpmk √ 1.25 1.25+ λ2ˆC2
p to obtain ˆCpmkG . We see that in Figures 4(a)–(d),LG
decreases inλ, especially for large ˆCp values, and the reduction of LG is more significant for large ˆCpmk. A large measurement error results in significant underestimation of true process capability.
In current practice, a process is called ‘inadequate’ ifCpmk< 1.00, ‘marginally capable’ if 1.00≤ Cpmk<
1.33, ‘satisfactory’ if 1.33≤ Cpmk< 1.50, ‘excellent’ if 1.50≤ Cpmk< 2.00, and ‘super’ if 2.00≤ Cpmk.
If capability measures do not include the measurement errors, a significant underestimation of the true process capability may result in high production costs, reducing competitiveness. For instance, suppose that a process has a 95% lower confidence bound, 1.211 ( ˆCpmk= 1.50) with n = 50, which meets the threshold of an
‘excellent’ process. However, the bound may be calculated as 0.997 with measurement errorsλ= 0.42 and
the process is determined as ‘inadequate’.
7.
TESTING PROCESS CAPABILITY BASED ON ˆ
C
pmkGTo determine whether a given process meets the preset capability requirement, we could consider statistical testing with the null hypothesisH0: Cpmk≤ c (the process is not capable) versus the alternative H1: Cpmk> c
(the process is capable), where c is the required process capability. If the calculated process capability is
greater than the corresponding critical value, we reject the null hypothesis and conclude that the process is capable. The test rejects the null hypothesisH0: Cpmk≤ c if ˆCpmk≥ c0with type I error α (α-risk), which
is the chance of incorrectly concluding an incapable process (with Cpmk≤ c) as capable (with Cpmk> c).
That is,P ( ˆCpmk≥ c0| Cpmk= c) = α. Given values of the capability requirement c, sample size n, and risk α,
the critical valuec0can be obtained by solving Equation (17) using available numerical methods:
b√n/(1+3c0) 0 FK (b√n− t)2 9c2 0 − t2 [φ(t + ξ√n)+ φ(t − ξ√n)] dt = α (17)
(a) (b)
(c ) (d)
Figure 4. Plot ofLGversusλ with n= 30, 50, 70, 100, 150 with 95% confidence level: (a) ˆCp= 1.285, ˆCpmk= 1.00;
(b) ˆCp= 1.50, ˆCpmk= 1.00; (c) ˆCp= 1.785, ˆCpmk= 1.50; (d) ˆCp= 2.00; ˆCpmk= 1.50
whereb= 3c1+ ξ2+ |ξ|. The test power (where b = 3C
pmk 1+ ξ2+ |ξ|) is π(Cpmk)= P ( ˆCpmk≥ c0| Cpmk> c) = b√n/(1+3c0) 0 FK (b√n− t)2 9c20 − t 2 [φ(t + ξ√n)+ φ(t − ξ√n)] dt (18) To eliminate the need for estimating the characteristic parameterξ , we apply the technique used by Pearn
and Lin12 to examine the behavior of the critical values,c0, against the parameter ξ . We perform extensive
calculations to obtain the critical values forξ = 0(0.01)3, c = 1.00, 1.33, 1.50, 1.67, 2.00, 2.5, and 3.0, n =
10(50)300, and α= 0.05. The results show that the critical value obtains its maximum, uniformly, at ξ = 0.5 in
all cases. For practical purposes, we could greatly simplify the solution procedure by solving Equation (17) with
ξ = 0.5 to obtain the required critical values, without having to further estimate ξ. In practice, sample data are
collected, contaminated with measurement errors, to estimate the empirical process capability. Thus, theα-risk
corresponding to the test using the sample estimate ˆCpmkG (denoted byαG) becomes
P ( ˆCpmkG ≥ c0| Cpmk≤ c) = αG bG√n/(1+3c0) 0 FK (bG√n− t)2 9c02 − t 2 [φ(t + 0.5√n)+ φ(t − 0.5√n)] dt = αG (19) wherebG= 3.75c1.25+ λ2C2 p+ 0.5 and Cp= √ 1.25c+16.
(a) (b)
Figure 5. Plots ofαGwithn= 30, 50, 70, 100, 150 and λ ∈ [0, 0.5] for α = 0.05: (a) c = 1.00; (b) c = 1.50
The test power (denoted byπG) is
πG(Cpmk)= P ( ˆCpmkG ≥ c0| Cpmk> c) πG(Cpmk)= bG√n/(1+3c0) 0 FK (bG√n− t)2 9c2 0 − t2 [φ(t + 0.5√n)+ φ(t − 0.5√n)] dt (20) wherebG= 3.75Cpmk 1.25+ λ2C2 p+ 0.5 and Cp= √ 1.25Cpmk+16.
Earlier discussions indicate that the true process capability would be severely underestimated if ˆCpmkG is used. The probability that ˆCpmkG is greater thanc0would be less than that of using ˆCpmk. Thus, theα-risk using ˆCpmkG is
less than theα-risk using ˆCpmk(α) for Cpmk. The test power using ˆCpmkG is also smaller than the test power using ˆCpmk. That is,πG< π. Figures 5(a) and (b) are the plots of αGwithn= 30, 50, 70, 100, 150, λ ∈ [0, 0.5], and α= 0.05 for c = 1.00 and 1.50, respectively. Figures 6(a) and (b) plot πG versusλ with n= 100, α = 0.05,
andCpmk= c1(0.02)c2 (wherec1= c + 0.2 and c2= c + 1) for c = 1.00 and 1.50, respectively. Note that if
λ= 0, then αG= α and πG= π. In Figures 5(a) and (b), αGdecreases asλ or n increases, and the decreasing
rate is more significant with largec. In fact, for large λ, αGis smaller than 10−2. In Figures 6(a) and (b),πG
decreases asλ increases, but increases as n increases. The reduction of πGinλ is more significant for large c.
In the presence of measurement errors, the test powerπG decreases. For instance, in Figure 6(b) theπGvalue (c= 1.50, n = 100) for Cpmk= 2.30 is πG= 0.9957 if there is no measurement error (λ = 0). However, when λ= 0.5, πGsignificantly decreases to 0.0834, a reduction of 0.912.
In previous sections we pointed out the problems associated with using CpmkG as an estimator for Cpmk. In this situation the lower confidence bound is underestimated, and theα-risk and the test power decrease with
measurement errors. The probability of passing non-conforming product units decreases, but the probability of correctly judging a capable process as capable also decreases. Since the lower confidence bound is severely underestimated and the test power becomes low, the producers cannot firmly state that their processes meet the capability requirement even if their processes are sufficiently capable. Good product units would be incorrectly rejected in this case. These incorrect decisions may lead to unnecessary costs for the producers. Improving the gauge capability and providing sufficient training for the operators are both essential to reduce measurement error. Nevertheless, measurement errors are inevitable in most industry applications. In order to provide a better capability assessment, three methods for adjusted confidence bounds are proposed. A simulation study is also conducted for the performance comparison of three methods.
(a)
(b)Figure 6. Plots of πG versus λ with n= 100, α = 0.05: (a) c = 1.00, Cpmk= 1.2(0.20)2.00; (b) c = 1.50,
Cpmk= 1.70(0.20)2.50
8.
METHODS FOR ADJUSTED CONFIDENCE BOUNDS
8.1. Sampling distribution approaches
From Equation (10), suppose that the desired confidence coefficient isγ and the adjusted confidence interval of
ˆCG
pmkwith the adjusted lower confidence boundLAcan be established as P ( ˆCpmkG > LA)= γ then 1− bG√n/(1+3 ˆCGpmk) 0 FK (bG√n− t)2 9( ˆCpmkG )2 [φ(t + ξG√n)+ φ(t − ξG√n)] dx = γ (21) wherebG= 3LA(1+ ξ2) 1+ ξ2+ λ2C2 p+ |ξG|
Equation (21) involves the unknown parameterξ and Cp. As recommended by Bolyes25we directly replaced
these variables by their MLEs. Thus, an adjusted lower confidence bound based on sampling distribution of ˆCpmkG
(denoted as LSD
A ) can be obtained by solving Equation (21) with ∗bG= 3LSDA (1+ ˆξ2G)(1+ ˆξ2)/
1+ ˆξ2+ λ2ˆC2
p+ |ˆξG| for given sample data, LSL, USL, γ , and λ:
∗bG√n/(1+3 ˆCG pmk) 0 FK (∗bG√n− t)2 9( ˆCpmkG )2 [φ(t + ˆξG √ n)+ φ(t − ˆξG √ n)] dx = 1 − γ (22)
In finding the lower confidence bounds, Pearn and Shu13 recommended placing ˆξ = 0.5 for reasons
of quality assurance purposes to eliminate additional sampling errors from estimating ξ . Therefore, we
may solve Equation (23) to obtain the adjusted lower confidence bounds (denoted as LMSD
A ) with ∗∗bG=
3.75LMSD A
1.25+ λ2ˆC2
p+ 0.5 for given sample data, LSL, USL, γ , and λ:
∗∗bG√n/(1+3 ˆCG pmk) 0 FK (∗∗bG√n− t)2 9( ˆCpmkG )2 [φ(t + 0.5√n)+ φ(t − 0.5√n)] dx = 1 − γ (23)
8.2. GCI approach
Tsui and Weerahandi26introduced the concept of generalized inference for testing hypotheses and constructing GCIs in situations such as the SD approach (see Equations (22) and (23)) where the exact confidence intervals are not exact. GCIs have been used in recent articles for a variety of problems including the construction of tolerance intervals by Liao et al.27 and the development of tests for variance components by Mathew and
Webb28. Hamada and Weerahandi29 and Adamec and Burdick30 used GCIs to handle measurement error
problems. Burdick et al.31,32presented GCIs for misclassification rates in a gauge R&R study. Daniels et al.33 proposed using GCIs for comparing capability measures when there is no measurement error. To compute a GCI for monitoring process capability based onCpmkwith gauge measurement errors, one must define generalized pivotal quantities (GPQs) forμ and σ2. Using the previous notation ofG∼ N(μ, σ2
G= σ2+ σM2), using the
method of Iyer and Patterson34 as described in Appendix B2 of Burdick et al.32, the following GPQs can be defined forμ and σ2
G μ(GPQ)= ¯G − Z σ2 G(GPQ) n (24) σG2(GPQ)=n ˜S 2 n W (25)
where Z is a standard normal variable and W is a χ2 random variable with n− 1 degrees of freedom.
SinceσM2 = [(USL − LSL)λ/6]2is known, then the GPQ forσ2is
σ2(GPQ)= max(ε, σG2(GPQ)− σM2) (26) whereε is a small positive quantity to maintain non-negative variance components, for example 0.0001.
The following procedure can be used to construct a 100(1− α)% lower confidence bound on Cpmk:
(1) compute ¯G and ˜S2
nfor the collected data and denote the realized value as ¯g and ˜sn2;
(2) simulate N= 2000 values of μ(GPQ) and σ2(GPQ) using Equations (24)–(26) by simulating N
independent values ofZ and W ;
(3) for each simulated pair ofμ(GPQ) and σ2(GPQ), compute Cpmk(GPQ) where Cpmk(GPQ)= min USL− μ(GPQ) 3σ2(GPQ)+ (μ(GPQ) − T )2, μ(GPQ)− LSL 3σ2(GPQ)+ (μ(GPQ) − T )2 (27) (4) order theN = 2000 values of Cpmk(GPQ) from the least to greatest;
(5) the 100γ % lower bounds on ˆCpmkG , denoted asLGCIA , is the value in positionN× α of the ordered step
in (4).
9.
ADJUSTED LOWER CONFIDENCE BOUND COMPARISONS:
A SIMULATION STUDY
The important consideration in choosing methods for determining the adjusted lower confidence bound is the performance of each method. In order to ascertain the performance of the adjusted lower confidence bound methods (Equation (22) forLSD
A , Equation (23) for LMSDA , and Equations (24)–(27) for LGCIA ) a simulation
study is conducted (the Matlab program for performing this calculation is available upon request). Random samples ofn= 20(10)150 different sample sizes are drawn 2000 times from processes with different gauge
capabilitiesλ= 0, 0.1, 0.2, 0.25 and alternative values of μ and σ2for whichξ= 0(0.05)1, so as to detect any
dependence on the coverage rate for the three methods. Tables II–VI present the mean value (ME) of ˆCpmkG , the percentage coverage rate (CR) observed in the simulation for a nominal confidence level of 95%, and mean
λ= 0.2 λ= 0.25 LSDA LMSDA LGCIA LSDA LMSDA LGCIA ˆCG pmk ˆCpmkG n ME CR MLCB CR MLCB CR MLCB ME CR MLCB CR MLCB CR MLCB 20 0.9598 0.9575 0.7105 0.9780 0.6546 0.9900 0.6121 0.9578 0.9615 0.7091 0.9840 0.6528 0.9945 0.6106 50 0.9585 0.9625 0.8116 0.9885 0.7683 0.9930 0.7542 0.9576 0.9635 0.8117 0.9890 0.7675 0.9950 0.7530 70 0.9616 0.9560 0.8153 0.9830 0.8026 0.9888 0.7942 0.9572 0.9635 0.8370 0.9845 0.7986 0.9905 0.7904 100 0.9566 0.9690 0.8600 0.9910 0.8262 0.9930 0.8223 0.9625 0.9600 0.8655 0.9850 0.8317 0.9885 0.8280 150 0.9644 0.9610 0.8893 0.9890 0.8601 0.9905 0.8599 0.9645 0.9660 0.8897 0.9885 0.8601 0.9890 0.8598
Table III. The simulated results for 95% lower confidence bounds of SD, MSD, and GCI methods withCpmk= 1.6007
(LSL= −5.2, USL = 5.2, μ = 0.25, σ = 1.0) at N = 2000 λ= 0.0 λ= 0.1 LSDA LMSDA LGCIA LSDA LMSDA LGCIA ˆCG pmk ˆCpmkG n ME CR MLCB CR MLCB CR MLCB ME CR MLCB CR MLCB CR MLCB 20 1.6632 0.9400 1.2014 0.9530 1.1667 0.9745 1.1101 1.6275 0.9380 1. 1927 0.9575 1.1571 0.9780 1.0949 50 1.6288 0.9380 1.3359 0.9545 1.3188 0.9690 1.2991 1.5962 0.9380 1.3272 0.9575 1.3083 0.9730 1.2852 70 1.6194 0.9435 1.3705 0.9550 1.3587 0.9655 1.3457 1.5944 0.9375 1.3693 0.9515 1.3541 0.9665 1.3386 100 1.6043 0.9555 1.3961 0.9665 1.3880 0.9705 1.3799 1.5927 0.9370 1.4061 0.9515 1.3949 0.9590 1.3852 150 1.6108 0.9400 1.4390 0.9480 1.4335 0.9515 1.4291 1.5876 0.9405 1.4383 0.9575 1.4299 0.9625 1.4248 λ= 0.2 λ= 0.25 LSDA LMSDA LGCIA LSDA LMSDA LGCIA ˆCG pmk ˆCpmkG n ME CR MLCB CR MLCB CR MLCB ME CR MLCB CR MLCB CR MLCB 20 1.5785 0.9170 1.2155 0.9360 1.1720 0.9725 1.0833 1.5238 0.9045 1.2136 0.9315 1.1659 0.9750 1.0593 50 1.5346 0.9270 1.3306 0.9465 1.3035 0.9665 1.2683 1.4956 0.9020 1.3370 0.9415 1.3037 0.9675 1.2589 70 1.5273 0.9265 1.3654 0.9525 1.3430 0.9710 1.3188 1.4931 0.9095 1.3760 0.9425 1.3465 0.9665 1.3154 100 1.5277 0.9280 1.4042 0.9490 1.3851 0.9580 1.3694 1.4766 0.9190 1.3951 0.9515 1.3705 0.9710 1.3502 150 1.5255 0.9250 1.4386 0.9535 1.4218 0.9600 1.4136 1.4860 0.9060 1.4418 0.9520 1.4191 0.9630 1.4080
value of the lower confidence bounds (MLCB) for true process capabilityCpmk. The CR and MLCB entries are used as a basis for evaluating the performance of various methods. Bold numbers denote acceptable performance measures.
Obviously, the ratio of CR and MLCB is negative. The lower the value of CR, the closer it is to the actual value of the MLCB. This is true, because the lower CR means that the much lower confidence bounds do not cover the actual value, and the MLCB is much closer to the actual value ofCpmk.
An advantage of obtaining the adjusted lower confidence intervals of the SD approach of Equation (22) and MSE approach of Equation (23),LSDA andLMSDA , respectively, are that both can be written in closed form and, unlike the GCI method, do not require Monte Carlo simulation.
Table IV. The simulated results for 95% lower confidence bounds of SD, MSD, and GCI methods withCpmk= 1.3416 (LSL= −5, USL = 5, μ = 0.5, σ = 1) at N = 2000 λ= 0.0 λ= 0.1 LSD A LMSDA LGCIA LSDA LMSDA LGCIA ˆCG pmk ˆCGpmk n ME CR MLCB CR MLCB CR MLCB ME CR MLCB CR MLCB CR MLCB 20 1.4043 0.9435 0.9887 0.9560 0.9737 0.9660 0.9497 1.3817 0.9350 0.9856 0.9465 0.9701 0.9665 0.9389 50 1.3595 0.9480 1.0959 0.9525 1.0926 0.9595 1.0836 1.3487 0.9430 1.0999 0.9470 1.0965 0.9600 1.0832 70 1.3502 0.9585 1.1276 0.9620 1.1259 0.9645 1.1195 1.3455 0.9385 1.1368 0.9410 1.1351 0.9530 1.1248 100 1.3507 0.9515 1.1640 0.9525 1.1630 0.9570 1.1583 1.3395 0.9485 1.1672 0.9500 1.1663 0.9550 1.1582 150 1.3495 0.9505 1.1968 0.9525 1.1963 0.9555 1.1931 1.3336 0.9510 1.1957 0.9525 1.1953 0.9610 1.1894 λ= 0.2 λ= 0.25 LSDA LMSDA LGCIA LSDA LMSDA LGCIA ˆCG pmk ˆCGpmk n ME CR MLCB CR MLCB CR MLCB ME CR MLCB CR MLCB CR MLCB 20 1.3325 0.9195 0.9864 0.9305 0.9714 0.9655 0.9190 1.3028 0.9085 0.9917 0.9195 0.9772 0.9660 0.9077 50 1.3051 0.9275 1.0999 0.9345 1.0967 0.9450 1.0706 1.2761 0.9205 1.1012 0.9255 1.0983 0.9550 1.0622 70 1.3004 0.9335 1.1345 0.9370 1.1332 0.9580 1.1117 1.2716 0.9245 1.1353 0.9290 1.1344 0.9575 1.1044 100 1.2976 0.9415 1.1673 0.9445 1.1666 0.9610 1.1492 1.2688 0.9285 1.1676 0.9350 1.1671 0.9610 1.1428 150 1.2932 0.9375 1.2010 0.9398 1.2001 0.9585 1.1818 1.2622 0.9245 1.2014 0.9342 1.2011 0.9585 1.1763
Table V. The simulated results for 95% lower confidence bounds of SD, MSD, and GCI methods withCpmk= 1.800
(LSL= −7.5, USL = 7.5, μ = 0.75, σ = 1) at N = 2000 λ= 0.0 λ= 0.1 LSDA LMSDA LGCIA LSDA LMSDA LGCIA ˆCG pmk ˆCGpmk n ME CR MLCB CR MLCB CR MLCB ME CR MLCB CR MLCB CR MLCB 20 1.8636 0.9475 1.3445 0.9520 1.3158 0.9615 1.3196 1.8281 0.9420 1.3497 0.9440 1.3289 0.9630 1.3056 50 1.8263 0.9455 1.4996 0.9490 1.4846 0.9520 1.4921 1.7912 0.9370 1.5014 0.9375 1.4946 0.9520 1.4120 70 1.8235 0.9535 1.5460 0.9555 1.5352 0.9570 1.5412 1.7884 0.9420 1.5473 0.9415 1.5443 0.9565 1.5322 100 1.8160 0.9500 1.5842 0.9545 1.5756 0.9520 1.5810 1.7833 0.9480 1.5871 0.9460 1.5867 0.9580 1.5757 150 1.8076 0.9595 1.6187 0.9630 1.6122 0.9620 1.6168 1.7782 0.9395 1.6244 0.9380 1.6260 0.9470 1.6157 λ= 0.2 λ= 0.25 LSDA LMSDA LGCIA LSDA LMSDA LGCIA ˆCG pmk ˆCGpmk n ME CR MLCB CR MLCB CR MLCB ME CR MLCB CR MLCB CR MLCB 20 1.7295 0.9035 1.3674 0.8955 1.3714 0.9595 1.2652 1.6669 0.8820 1.3834 0.8625 1.4107 0.9615 1.2384 50 1.6905 0.9170 1.5044 0.9020 1.5198 0.9525 1.4507 1.6332 0.9000 1.5096 0.8715 1.5403 0.9530 1.4305 70 1.6885 0.9200 1.5461 0.9040 1.5649 0.9575 1.5019 1.6269 0.9035 1.5497 0.8685 1.5832 0.9580 1.4840 100 1.6857 0.9230 1.5860 0.8960 1.6069 0.9610 1.5499 1.6241 0.8985 1.5881 0.8580 1.6226 0.9600 1.5341 150 1.6791 0.9300 1.6204 0.8975 1.6432 0.9560 1.5914 1.6176 0.9130 1.6213 0.8640 1.6578 0.9575 1.5778
It can be noted that the lower bound of the SD method, LSD
A , is a better performance measure in the
absence of gauge measurement errors, i.e. λ= 0.0. When measurement errors are unavoidable, for small ξ
(say 0≤ ξ < 0.15), the SD method performs well providing the most accurate CRs of the methods studied here. The MSD and GCI methods,LMSD
A andLGCIA , keep type I error (α-risk) from exceeding a predetermined
value (such as 0.05 or 0.01) to provide necessary protection to the customers. However, the conservative lower confidence bounds for the true value of the same index can lead to a higher level of type II error. For 0.15≤ ξ < 0.40, the MSD and GCI methods have the same acceptable performance measures with accurate
CRs for all of the cases studied. However, the SD method ensures that type I error (α-risk) is greater than a
λ= 0.2 λ= 0.25 LSDA LMSDA LGCIA LSDA LMSDA LGCIA ˆCG pmk ˆCpmkG n ME CR MLCB CR MLCB CR MLCB ME CR MLCB CR MLCB CR MLCB 20 1.9804 0.8780 1.6658 0.8430 1.7228 0.9560 1.5139 1.8799 0.8580 1.7208 0.7640 1.8373 0.9580 1.5805 50 1.9441 0.8950 1.8132 0.8290 1.8775 0.9495 1.7216 1.8525 0.8675 1.8234 0.7575 1.9418 0.9515 1.6878 70 1.9366 0.8905 1.8568 0.8150 1.9277 0.9575 1.7789 1.8453 0.8640 1.8639 0.7215 1.9870 0.9580 1.7488 100 1.9341 0.8885 1.8992 0.7940 1.9736 0.9610 1.8337 1.8427 0.8590 1.9037 0.6875 2.0260 0.9615 1.8071 150 1.9270 0.8995 1.9354 0.7630 2.0147 0.9560 1.8885 1.8351 0.8740 1.9378 0.6460 2.0635 0.9505 1.8584
Table VII. Summary of the most effective methods for adjusted lower confidence bounds
λ= 0.0 λ= 0.1 λ= 0.2 λ= 0.25
0.0≤ ξ < 0.15 SD SD SD SD
0.15≤ ξ < 0.4 SD, MSD, GCI SD, MSD, GCI MSD, GCI MSD, GCI
ξ≥ 0.4 SD, MSD, GCI GCI GCI GCI
0.40≤ ξ, the GCI method performs very well and the CRs achieved by the lower confidence bounds are quite
robust and close to the nominal values for all of the methods studied. On the other hand, the CRs of the SD and MSD methods are significantly lower than the state level for all of the methods studied. Table VII shows the most effective methods for adjusted lower confidence bounds withλ= 0.0, 0.1, 0.20, 0.25 and 0.0 ≤ ξ < 0.15,
0.15≤ ξ < 0.4, and ξ ≥ 0.4.
As a result of this discussion we present the following generalizations for practitioners of real-world factory applications:
(a) if no measurement errors exist, the adjusted lower confidence bound is in support of the use of the SD approach,LSD
A ;
(b) when 0≤ ξ < 0.15, the adjusted lower confidence bound is in support of the use of the SD approach,
LSD A ;
(c) when 0.15≤ ξ < 0.40, the adjusted lower confidence bound is in support of the use of the MSD and GCI
approaches,LMSDA andLGCIA ;
(d) when 0.4≤ ξ, the adjusted lower confidence bound is in support of the use of the GCI approach LGCIA .
10.
APPLICATION EXAMPLE: PRECISION VOLTAGE REFERENCE
When a data conversion system is designed, the system accuracy greatly depends on the accuracy of the voltage established by the internal or external DC voltage reference. The voltage reference is used to produce a precise
(a) (b)
Figure 7. The eight-pin SOIC-30 and eight-pin TO-99 packages for PVR
value of the output voltage for setting the full-scale input of the data conversion system. In an analog-to-digital converter (ADC), the DC voltage reference together with the analog input signal are used to generate the digitized output signal. In a digital-to-analog converter (DAC), the DAC selects and produces an analog output from the DC reference voltage according to the digital input signal presented at the input of the DAC. Any errors in the reference voltage over the operating temperature range will adversely affect the linearity and spurious free dynamic range (SFDR) of the ADC/DAC. With the emergence of the portable battery-operated environment, low voltage and low power are key goals of the industry. The voltage reference can also be used in constructing a precision regulated supply that could have better characteristics than some regulator chips, which can occasionally dissipate too much power. In addition, voltage references are needed in the design of products which must be accurate, such as voltmeters, ohmmeters, and ammeters.
Consider the following case taken from a manufacturing factory in Taiwan making one type of low-power, fast-warm-up, and highly stable 15 V precision voltage reference (PVR). The output voltage is extremely insensitive to both line and load variations and can be externally adjusted with minimal effect on drift and stability. This PVR is offered in eight-pin SOIC-30 and eight-pin TO-99 packages, as depicted in Figure 7. They are ideal for communications equipment, data acquisition systems, instrumentation and process control, high-precision power supplies, and battery powered equipment. They may also be used in portable battery powered equipment (such as notebook computers, PDAs, DVMs, GPS, etc.). Initial accuracy is one critical quality characteristic of this PVR which has a significant impact on the PVR quality/reliability. This characteristic is usually only valid at room temperature, and it provides a starting point for most of the other specifications. The output voltage tolerance of a reference measured without a load applied after the device is turned on and warmed up. Manufacturers specify a reference with a small initial error so they do not have to perform room-temperature system calibration after assembly.
For this particular model of PVR product, the specification limits are T = 15 V, USL = 15.025, and LSL= 14.975. A total of 70 observations are collected and displayed in Table VIII. A histogram and a normal
probability plot show that the collected data follow the normal distribution. A Shapiro–Wilk test is applied to further justify the assumption. To determine whether the process is ‘excellent’ (Cpmk> 1.33) with unavoidable
measurement errorsλ= 0.24, we first determine that c = 1.33 and α = 0.05. Then, based on the sample data of
70 observations, we obtain the sample mean ¯g = 15.0014, the sample standard deviation ˜sn= 0.0049 ( ¯g and ˜sn
are the realized sample values for ¯G and ˜Sn), and the point estimator ˆCpmkG = 1.5526. Since 0.15 ≤ ˆξ < 0.4, the
GCI method is suggested for this PVR process capability assessment. The Matlab computer program (available upon request) readsT = 15 V, USL = 15.025, LSL = 14.975, 70 observations, and λ = 0.24 (provided by the
gauge manufacturing factory), and the desired confidence coefficientγ= 0.95, so the 95% lower confidence
bound of the true process capability can be obtained asLGCIA = 1.3812. We thus can assure that the production
Moreover, similar to the adjusted lower confidence bound, we obtain the adjusted critical value 1.498 for the MSD method based on α= 0.05, λ = 0.24, and n = 70. Since ˆCGpmk> 1.498, we therefore conclude that the
process is ‘excellent’. However, we also see that if we ignore the measurement errors and evaluate the critical value without any correction, the critical value may be calculated asc0= 1.585. In this case we would say that
the process is not ‘excellent’ since ˆCpmkG is no greater than the uncorrected critical value 1.585.
11.
CONCLUSIONS
Measurement errors are unavoidable in most industry applications. In this paper, we conducted a sensitivity study for process capability, Cpmk, in the presence of gauge measurement errors. The statistical properties and capability testing of estimating Cpmk were investigated to obtain lower confidence bounds and critical values for true process capability testing. In estimating and testing the capability, the estimator ˆCpmkG , using the sample data contaminated with the measurement error, severely underestimates the true capability and decreases testing power. The lower confidence bounds must be adjusted to improve the accuracy of capability assessment. The SD approaches and the GCI approach were presented to obtain the adjusted lower confidence bounds. An intensive simulation study was used to compare performance of the attained confidence levels and the average interval lengths of both approaches. The result recommends the appropriate method to practitioners for the real-world factory applications.
Acknowledgements
Work on this paper was partially funded by National Science Council of Taiwan, under grant NSC92-2213-E-251-005. The authors wish to thank the chief editor and the anonymous reviewers for encouraging and helpful comments that resulted in an improved presentation of our research.
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![Figure 1. (a) Surface plot and (b) plots of R 1 versus λ in [0,0.5] for C p = 1.0(0.2)2.0 with ξ = 0.5](https://thumb-ap.123doks.com/thumbv2/9libinfo/7671675.141429/4.850.187.688.85.313/figure-surface-plot-plots-r-versus-λ-c.webp)
![Figure 5. Plots of α G with n = 30, 50, 70, 100, 150 and λ ∈ [0, 0.5] for α = 0.05: (a) c = 1.00; (b) c = 1.50](https://thumb-ap.123doks.com/thumbv2/9libinfo/7671675.141429/9.850.131.705.81.338/figure-plots-α-g-n-λ-α-c.webp)
