Qual. Reliab. Engng. Int. 2005; 21:29–42
Published online 15 November 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/qre.605
Research
Capability Testing Based on
C
pm
with Multiple Samples
Chien-Wei Wu and W. L. Pearn∗,†Department of Industrial Engineering & Management, National Chiao Tung University, Hsinchu, Taiwan, Republic of China
Numerous process capability indices have been proposed in the manufacturing industry to provide unitless measures on process performance, which are effective tools for quality improvement and assurance. Most existing methods for capability testing are based on the distribution frequency approaches. Recently, Bayesian approaches have been proposed for testing capability indices Cp and Cpm but
restricted to cases with one single sample. In this paper, we consider estimating and testing capability index Cpm based on multiple samples. We propose accordingly a
Bayesian procedure for testingCpm. Based on the Bayesian procedure, we develop a
simple but practical procedure for practitioners to use in determining whether their manufacturing processes are capable of reproducing products satisfying the preset capability requirement. A process is capable if all the points in the credible interval are greater than the pre-specified capability level. To make the proposed Bayesian approach practical for in-plant applications, we tabulate the minimum values of C∗(p) for which the posterior probability p reaches various desirable confidence
levels. Copyright c 2004 John Wiley & Sons, Ltd.
KEY WORDS: process capability indices; Bayesian approach; credible interval; unbiased estimator; posterior probability; multiple samples
1.
INTRODUCTION
P
rocess 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 capability analysis literature. Those capability indices, quantifying process precision, process accuracy, and process performance, are important for any production improvement activities and quality program implementation. The first process capability index appearing in the literature is the precision index Cp, which is defined by Kane1asCp=
USL− LSL 6σ
where USL is the upper specification limit, LSL is the lower specification limit, and σ is the process standard deviation. The numerator of Cpgives the range over which the process measurements are predefined.
∗Correspondence to: W. L. Pearn, Department of Industrial Engineering & Management, National Chiao Tung University, Hsinchu, Taiwan,
Republic of China.
The denominator gives the range over which the process is actually varying. The index Cp was designed to measure the magnitude of the overall process variation relative to the manufacturing tolerance, and is used for processes that are normally distributed, and are in statistical control. Clearly, the index only measures the precision of a process (product quality consistency), and does not take into account whether the process is centered.
In order to reflect the impact of the deviation of the process mean µ from the center point M of the specification limits on the process capability, several indices have been proposed, including
CPU= USL− µ 3σ , CPL= µ− LSL 3σ Cpk= min USL− µ 3σ , µ− LSL 3σ
However, both Cpand Cpkare independent of the target value T . Neither Cpnor Cpk takes the closeness of the process output to the target value T (on-target issues) into consideration. Taking the targeting as well as the process spread into consideration, a modification of Cpincorporating the Taguchi loss function, which has been referred to as Cpm, is introduced independently by Hsiang and Taguchi2and Chan et al.3. The process capability index Cpmis defined as Cpm= USL− LSL 6τ = d 3σ2+ (µ − T )2
where USL− LSL is the allowable tolerance range of the process, d is the half-interval length, and τ is the measure of the average product deviation from the target value T . The term τ2= σ2+ (µ − T )2=
E[(X − T )2] incorporates two variation components: (1) variation to the process mean and (2) deviation of the process mean from the target. The process capability index Cpm, sometimes called the Taguchi index, emphasizes the ability of clustering around the target, which therefore reflects the degrees of process targeting. The capability index Cpmis not primarily designed to provide an exact measure of the number of conforming items, i.e. the process yield. However, we note that E[(X − T )2] is the expected loss, where the process loss of a characteristic X missing the target is often assumed to be well approximated by the symmetric squared error loss function, loss (X)= (X − T )2. Hence, the capability index Cpmmay be termed as a loss-based index. The indices Cpand Cpk have been referred to as the first-generation process capability indices, and the index
Cpmis often called the second-generation process capability indices.
Pearn et al.4 proposed the process capability index Cpmk, which combines the merits of the three earlier indices Cp, Cpk and Cpm. The index Cpmkalerts the user if the process variance increases and/or the process mean deviates from its target value. The index Cpmk, referred to as the third-generation process capability index, has been defined as follows:
Cpmk= min USL− µ 3σ2+ (µ − T )2, µ− LSL 3σ2+ (µ − T )2
Note that the indices presented above are designed to monitor the performance for only normal and near-normal processes with symmetric tolerances. These indices have been shown to be inappropriate for cases with asymmetric tolerances. In practice, the process mean µ and the process variance σ2 are unknown. In order to calculate the index value, sample data must be collected and a great degree of uncertainty may be introduced into the capability assessments due to sampling error. The approach of simply looking at the calculated values of the estimated indices and then making a decision on whether the given process is capable is highly unreliable, since this ignores the sampling error. As the use of the capability indices grows more widespread, users are becoming educated and sensitive to the impact of the estimators and their sampling distributions, learning that capability measures must be reported in confidence intervals or via capability testing. Statistical properties of the estimators of those indices under various process conditions have been
investigated extensively, see Chan et al.3, Pearn et al.4, Bordignon and Scagliarini5, Borges and Ho6, Chang
et al.7, Hoffman8, Nahar et al.9, Noorossana10, Pearn et al.11, Pearn and Lin12, and Zimmer et al.13. Kotz
and Johnson14 presented a thorough review for the development of process capability indices in the past 10 years.
Existing research for capability testing has focused on the traditional frequency approaches. However, the sampling distributions are usually so complicated that this makes establishing the exact confidence interval very difficult. An alternative is to consider the Bayesian approach where we can specify a prior distribution for the parameter of interest, obtain the posterior distribution for the parameter, and then make inferences about the parameter using its posterior distribution given the observations. It is not difficult to obtain the posterior distribution when a prior distribution is given, even when the form of the posterior distribution is complicated, as one could always use numerical methods or Monte Carlo methods (Kalos and Whitlock15) to obtain an approximate point estimate or interval estimate. This is the advantage of the Bayesian approach over the traditional distribution frequency approach.
2.
ESTIMATION OF
C
pmThe process mean µ and the process variance σ2must be estimated form the sample. Thus, the estimated index ˆCpmis obtained by replacing µ and σ2by their estimators. Chan et al.3and Boyles16 proposed two different
estimators of Cpmrespectively defined as follows: ˆCpm(CCS)= d 3s2+ ( ¯x − T )2 and ˆCpm(B)= d 3s2 n+ (n/(n − 1))( ¯x − T )2 where ¯x = n i=1 xi/n, s2= n i=1 (xi− ¯x)2/(n− 1) and sn2= n i=1 (xi− ¯x)2/n
In fact, the two estimators, ˆCpm(CCS) and ˆCpm(B), are asymptotically equivalent. Note that ¯x and sn2 are the
maximum likelihood estimators (MLEs) of µ and σ2, respectively. Hence, the estimated ˆCpm(B)is the MLE of
Cpm. Further, the term sn2+ ( ¯x − T )2in the denominator of ˆCpm(B)is the uniformly minimum variance unbiased estimator (UMVUE) of the term σ2+ (µ − T )2in the denominator of Cpm, where
sn2+ ( ¯x − T )2= n
i=1
(xi− T )2/n and τ2= σ2+ (µ − T )2= E[(x − T )2]
Therefore, for reliability purposes, it is reasonable to use ˆCpm(B).
Under the assumption of normality, Kotz and Johnson17 obtained the rth moment, and calculated the first two moments, the mean, and the variance of ˆCpm. Zimmer and Hubele18provided tables of exact percentiles for the sampling distribution of the estimator ˆCpm. Zimmer et al.19 proposed a graphical procedure to obtain exact confidence intervals for Cpm, where the parameter ξ = (µ − T )/σ is assumed to be a known constant. Using a method similar to that presented in V¨annman20, Lin and Pearn21 obtained an exact form of the
cumulative distribution function of ˆCpm. Under the assumption of normality, the cumulative distribution function of ˆCpm can be expressed in terms of a mixture of the chi-square distribution and the normal distribution, for x > 0, where b= d/σ, ξ = (µ − T )/σ, G(·) is the cumulative distribution function of the chi-square distribution χn2−1, and φ(·) is the probability density function of the standard normal distribution
N (0, 1). FˆC pm(x)= 1 − b√n/(3x) 0 G b2n 9x2 − t 2 [φ(t + ξ√n)+ φ(t − ξ√n)] dt
2.1. Estimation of Cpmfor multiple samples
For a single sample, Boyles16 showed that ˆτ2= sn2+ ( ¯x − T )2 is the unbiased estimator of σ2+ (µ − T )2. Therefore, for cases where the data are collected as multiple samples, we consider m samples each of size ni
and suggest the following estimator of Cpm, where ¯xiis the ith sample mean, and si is the ith sample standard
deviation: ˆC∗ pm= d 3ˆτ2, ˆτ 2= m i=1 ni j=1(xij− T )2 m i=1ni (1) First, by taking the expectation of the numerator ofˆτ2, we obtain
E m i=1 ni j=1 (xij− T )2 = E m i=1 ni j=1 xij2 − 2T × E m i=1 ni j=1 xij + E m i=1 ni j=1 T2 = m i=1 ni j=1 E(xij2)− 2T × m i=1 ni j=1 E(xij)+ m i=1 niT2 = m i=1 ni(µ2+ σ2)− 2T × m i=1 niµ+ m i=1 niT2 = m i=1 ni[σ2+ (µ − T )2]
Thus, the estimator ˆτ2, such that E(ˆτ2)= σ2+ (µ − T )2, is the unbiased estimator of σ2+ (µ − T )2. However, for multiple control samples, we need to consider the variation between and within multiple samples. Thus, we define the ratio of total within sample variation (SSW) and total sum of square variation (SST) as
γ=SSW SST = m i=1 ni j=1(xij− ¯xi) 2 m i=1 ni j=1(xij− ¯¯x)2 = m i=1(ni − 1)s2p m i=1(ni− 1)sp2+ m i=1ni(¯xi− ¯¯x)2 (2) where s2p= m i=1 (ni − 1)s2i m i=1 (ni− 1)
is the pooled variance of these samples. The total sample variation about target value T can be decomposed as
m i=1 ni j=1 (xij− T )2= m i=1 ni j=1 (xij− ¯xi)2+ m i=1 ni j=1 (¯xi− ¯¯x)2+ m i=1 ni j=1 ( ¯¯x − T )2 = m i=1 (ni− 1)sp2+ 1− γ γ m i=1 (ni − 1)sp2+ m i=1 niδ2sp2 = 1 γ m i=1 (ni− 1) + m i=1 niδ2 sp2
Thus, the generation of the estimator of Cpmfor multiple samples defined in (1) can be rewritten as ˆC∗ pm= d 3spmi=1(ni− 1)/(γ m i=1ni)+ δ2 , δ=| ¯¯x − T | sp (3)
For the single sample, that is, m= 1, γ = 1, and sp= s, the estimator of Cpm, ˆCpm∗ = d/(3s(n− 1)/n + δ2), which can be reduced to the estimator ˆCpmdefined in Boyles16.
3.
BAYESIAN APPROACH FOR TESTING
C
pmA Bayesian procedure for assessing process capability was proposed in Cheng and Spiring22for the index Cp under the assumption that the process mean µ is equal to the target value T . However, the restriction of µ= T is not a practical assumption for many industrial applications. Shiau et al.23proposed a Bayesian procedure for the general situation without the restriction on the process mean. However, the research work focused on cases with one single sample. A common practice of process capability estimation in the manufacturing industry is to first implement a daily-based or weekly-based sample data collection plan for monitoring/controlling the process stability, then to analyze the past ‘in control’ data. Therefore, it is practical to develop a procedure for assessing process capability for cases with multiple samples. In addition, for practitioners’ convenience, we provide a simple but practical procedure for computing the posterior probability.
A 100p% credible interval is the Bayesian analogue of the classical 100p% confidence interval, where p is the confidence level for the interval. The credible interval covers 100p% of the posterior distribution of the parameter24. Assuming that the m samples are random samples taken from an independent and identically distributed (i.i.d.) normal distribution with mean µ and variance σ2(N (µ, σ2)). The measures of the ith sample
xi= {xi1, xi2, . . . , xini} with sample size ni. Then, the likelihood function for µ and σ is
L(µ, σ|x) = (2πσ2)−mi=1ni/2exp − m i=1 ni j=1(xij− µ)2 2σ2
The first step for the Bayesian approach is to find an appropriate prior. Usually, when there is little or no prior information, or there is only one parameter, one of the most widely used non-informative priors is the so-called reference prior, which is a non-informative prior that maximizes the difference between information (entropy) on the parameter provided by the prior and by the posterior. In other words, the reference prior allows the prior to provide as little information as possible about the parameter (see Bernardo and Smith25 for more details). Therefore, in this paper we adopt the following non-informative reference prior:
π(µ, σ )= 1/σ, 0 < σ < ∞
3.1. Posterior probability
The posterior probability density function (PDF) of (µ, σ ), f (µ, σ|x), may be expressed as follows:
f (µ, σ|x) ∝ L(µ, σ|x) × π(µ, σ) ∝ σ−(mi=1ni+1)exp − m i=1 ni j=1(xij− µ) 2 2σ2 since ∞ 0 ∞ −∞σ −(m i=1ni+1)exp − m i=1 ni j=1(xij− µ) 2 2σ2 dµ dσ = ∞ 0 σ−(mi=1ni+1)exp − 1 βσ2 × ∞ −∞exp − m i=1ni(µ− ¯¯x)2 2σ2 dµ dσ= π 2mi=1ni (α)βα
In order to satisfy the integration property, the probability over PDF is 1, so that f (µ, σ|x) = 2mi=1ni √ 2π(α)βασ −(m i=1ni+1)exp − m i=1 ni j=1(xij − µ)2 2σ2 α= m i=1 ni− 1 /2, β= m i=1 ni j=1 (xij− ¯¯x)2 2 −1 ¯¯x =m i=1 ni j=1 xij m i=1 ni= m i=1 ni¯xi m i=1 ni for −∞ < µ < ∞, 0 < σ < ∞ (4)
As we mentioned earlier, it is natural to consider the quantity p= Pr {process is capable|x} in the Bayesian approach. Since the index Cpmis the focus in this paper, we are interested in finding the posterior probability
p= Pr{Cpm> ω|x} for some fixed positive number ω. Therefore, given a pre-specified precision level ω > 0;
the posterior probability based on index Cpmthat a process is capable, is given as the following, where (·) is the cumulative distribution of the standard normal distribution, and γ and δ are defined as in (2) and (3):
p= Pr{Cpm> ω|x} = t 0 1 (α)yα+1 exp −1 y
[ (b1(y)+ b2(y))− (b1(y)− b2(y))] dy
t=m 2 i=1(ni− 1) ˆC∗ pm ω 2 m i=1(ni− 1) m i=1ni + γ δ 2 b1(y)= 2γmi=1ni m i=1(ni − 1)y δ b2(y)= m i=1 ni t y − 1 1/2 (5)
The derivations of (5) are given in AppendixA. Note that the posterior probability p depends on m, ni, γ , ω
and ˆCpm∗ only through m, ni, γ , δ and ˆCpm∗ /ω. C∗is denoted by C∗= ˆCpm∗ /ω. From expression (5) we can see that it is rather complicated to compute p without advanced computer programming skills. However, by noticing that there is a one-to-one correspondence between p and C∗ when m and ni are given, and the fact
that γ , δ and ˆCpm∗ can be calculated from the process data, we find that the minimum value of C∗required to ensure the posterior probability p reaching a certain desirable level can be useful in assessing process capability. This minimum value is denoted by C∗(p). Thus, the value C∗(p)satisfies
p= Pr{Cpm> ω|x} = Pr Cpm> ˆC∗ pm C∗(p) x
4.
THE TEST PROCEDURE
A 100p% credible interval for Cpmis[ ˆC∗pm/C∗(p),∞), where p is a number between 0 and 1, say 0.95, for a 95% confidence interval. This means that the posterior probability that the credible interval contains Cpmis p.
In our Bayesian approach we say that the process is capable in a Bayesian sense if all the points in this credible interval are greater than a pre-specified value of ω, say 1.00 or 1.33. When this happens, we have
p= Pr{Cpm> ω|x}. In other words, to see if a process is capable (with capability level ω and confidence
level p), we only need to check if ˆC∗pm> C∗(p)× ω.
Therefore, for users’ convenience in applying our Bayesian procedure, we tabulate the minimum values of
C∗(p)for various values of γ = 0.7(0.1)1.0 and δ = 0(0.5)2.0 with n = 5(5)20, m = 2(2)10 in TableI(a)–(d) and Table II(a)–(d) for p= 0.95, 0.99, respectively. For example, if ω = 1.33 is the minimum capability requirement, then for p= 0.95, with m = 10 of each sample size ni = n = 15 and γ = 0.9, δ = 0.5 we can
find C∗(p)= 1.1082 from TableI(c). Thus, the minimum value ˆCpm∗ required for the process to be capable is
C∗(p)× ω = 1.1082 × 1.33 = 1.4739. That is, if ˆCpm∗ is greater than 1.4739, we say that the process is capable in a Bayesian sense. The computer program for computing the required minimum values of C∗(p)is available from the authors.
From these tables we observe that for each fixed p, m, n and γ the value of C∗(p) decreases as δ increases. This phenomenon can be explained by the relationship of ˆCpm∗ in (3). For a fixed ˆCpm∗ , spbecomes smaller when δ becomes larger, and a smaller sp means that it is plausible that the underlying process is tighter (i.e. with smaller σ ). Since the estimation is usually more accurate with the data drawn from a tighter process, it is then plausible that the estimate ˆCpm∗ is more accurate with a smaller sp and the required minimum value C∗(p)is smaller, since we need only a smaller C∗(p) to account for the smaller uncertainty in the estimation. Intuitively, if the estimation error in our estimate is potentially large, then it is reasonable that we need a large ˆCpm∗ to be able to claim that the process is capable, and this means that the corresponding minimum value C∗(p) should be large as well. Thus, the value of C∗(p) decreases as
δ increases, and this pattern is consistent with Shiau et al.23. Alternatively, according to the definition of
γ, as (2) becomes larger, the variation between these multiple samples will become smaller when the other conditions are fixed. And the smaller the variation is between these multiple samples, the more stable the process. Thus, we only need a smaller C∗(p)to assess the process capability. Another observation from the tables is that the value of C∗(p) decreases as n and/or m increases for fixed δ, γ and p. This can also be explained by the same reasoning as above, since the estimation will be more accurate with a larger sample size.
As a result, to judge if a given process meets the capability requirement, we first determine the pre-specified value ω, the capability requirement, and the α-risk or the confidence level p for the interval. Checking the appropriate table (or run the program), we may obtain the critical value C∗(p)based on given values of p, m sub-samples of size niand γ calculated from samples. If the estimated value ˆCpm∗ is greater than the critical value
C∗(p)× ω, then we may conclude that the process meets the capability requirement (Cpm> ω). Otherwise, we do not have sufficient information to conclude that the process meets the present capability requirement. In this case, we would believe that Cpm≤ ω. In the following, we present a simple step-by-step procedure for testing the process precision. The practitioners can use the procedure on their in-plant applications to obtain reliable decisions.
Step 1. Decide the definition of ‘capable’ (ω, normally set to 1.00 or 1.33), and the confidence level p for the
interval (normally set to 0.99, 0.975 or 0.95). The chance of true Cpmlying in this interval is p.
Step 2. Calculate the value of the estimator ˆCpm∗ , γ and δ based on m multiple control samples of each sample size ni.
Step 3. Check the table and find the critical value C∗(p)based on given values of p, m subgroups of each sample size ni, and γ which is calculated in Step 2.
Step 4. Conclude that the process is capable (Cpm> ω) if the ˆCpm∗ value is greater than the critical value
T ab le I. C ritic al v alu es C ∗(p ) fo r m u ltip le sa mp le s w ith m = 2 (2 )10, δ = 0 .5 (0 .5 )2 .0, γ = 0 .7 (0 .1 )1 .0, p = 0 .95 0 0 .5 1 .0 1 .5 2 .0 δ 0. 7 0 .8 0. 9 1 0. 7 0 .8 0. 9 1 0. 7 0 .8 0. 9 1 0. 7 0 .8 0. 9 1 0. 7 0 .8 0. 9 1 (a ) n = 5 m = 2 1 .8318 1. 8318 1. 8318 1. 8318 1. 7410 1. 7302 1. 7180 1. 7097 1. 3495 1. 3293 1. 3120 1. 2971 1. 4430 1. 4207 1. 4012 1. 3841 1. 5571 1. 5559 1. 5368 1. 5194 m = 4 1 .4464 1. 4464 1. 4464 1. 4464 1. 4059 1. 4009 1. 3961 1. 3914 1. 2079 1. 1966 1. 1870 1. 1786 1. 2585 1. 2466 1. 2361 1. 2268 1. 3275 1. 3169 1. 3072 1. 2983 m = 6 1 .3264 1. 3264 1. 3264 1. 3264 1. 3006 1. 2973 1. 2941 1. 2910 1. 1600 1. 1516 1. 1444 1. 1381 1. 1975 1. 1887 1. 1810 1. 1741 1. 2471 1. 2396 1. 2327 1. 2264 m = 8 1 .2654 1. 2654 1. 2654 1. 2654 1. 2465 1. 2440 1. 2416 1. 2393 1. 1344 1. 1274 1. 1215 1. 1162 1. 1650 1. 1579 1. 1516 1. 1459 1. 2050 1. 1990 1. 1935 1. 1884 m = 10 1. 2276 1. 2276 1. 2276 1. 2276 1. 2127 1. 2107 1. 2008 1. 2068 1. 1178 1. 1118 1. 1066 1. 1021 1. 1443 1. 1381 1. 1327 1. 1278 1. 1783 1. 1733 1. 1686 1. 1643 (b) n = 10 m = 2 1 .4464 1. 4464 1. 4464 1. 4464 1. 4099 1. 4053 1. 4008 1. 3966 1. 3364 1. 3262 1. 3169 1. 3082 1. 2690 1. 2571 1. 2466 1. 2372 1. 2180 1. 2065 1. 1966 1. 1880 m = 4 1 .2654 1. 2654 1. 2654 1. 2654 1. 2485 1. 2462 1. 2440 1. 2419 1. 2100 1. 2043 1. 1990 1. 1941 1. 1712 1. 1642 1. 1579 1. 1522 1. 1406 1. 1335 1. 1274 1. 1221 m = 6 1 .2016 1. 2016 1. 2016 1. 2016 1. 1906 1. 1890 1. 1875 1. 1861 1. 1632 1. 1591 1. 1552 1. 1515 1. 1343 1. 1289 1. 1241 1. 1198 1. 1109 1. 1054 1. 1007 1. 0966 m = 8 1 .1674 1. 1674 1. 1674 1. 1674 1. 1592 1. 1580 1. 1569 1. 1557 1. 1375 1. 1341 1. 1309 1. 1279 1. 1137 1. 1093 1. 1053 1. 1017 1. 0942 1. 0896 1. 0857 1. 0822 m = 10 1. 1455 1. 1455 1. 1455 1. 1455 1. 1390 1. 1380 1. 1371 1. 1361 1. 1207 1. 1178 1. 1151 1. 1125 1. 1002 1. 0963 1. 0929 1. 0897 1. 0832 1. 0792 1. 0758 1. 0727 (c ) n = 15 m = 2 1 .3264 1. 3264 1. 3264 1. 3264 1. 3040 1. 3011 1. 2982 1. 2955 1. 2553 1. 2482 1. 2417 1. 2356 1. 2075 1. 1988 1. 1911 1. 1842 1. 1699 1. 1614 1. 1539 1. 1474 m = 4 1 .2016 1. 2016 1. 2016 1. 2016 1. 1909 1. 1895 1. 1880 1. 1866 1. 1643 1. 1602 1. 1564 1. 1528 1. 1357 1. 1304 1. 1256 1. 1213 1. 1123 1. 1069 1. 1022 1. 0980 m = 6 1 .1554 1. 1554 1. 1554 1. 1554 1. 1484 1. 1474 1. 1464 1. 1454 1. 1291 1. 1261 1. 1232 1. 1205 1. 1075 1. 1034 1. 0997 1. 0963 1. 0893 1. 0851 1. 0814 1. 0781 m = 8 1 .1302 1. 1302 1. 1302 1. 1302 1. 1249 1. 1241 1. 1233 1. 1225 1. 1094 1. 1069 1. 1045 1. 1023 1. 0914 1. 0880 1. 0849 1. 0821 1. 0762 1. 0727 1. 0695 1. 0668 m = 10 1. 1138 1. 1138 1. 1138 1. 1138 1. 1096 1. 1089 1. 1082 1. 1076 1. 0964 1. 0943 1. 0922 1. 0902 1. 0808 1. 0778 1. 0751 1. 0727 1. 0675 1. 0644 1. 0616 1. 0592 (d) n = 20 m = 2 1 .2654 1. 2654 1. 2654 1. 2654 1. 2493 1. 2472 1. 2451 1. 2430 1. 2122 1. 2066 1. 2015 1. 1967 1. 1741 1. 1671 1. 1608 1. 1551 1. 1434 1. 1364 1. 1302 1. 1248 m = 4 1 .1674 1. 1674 1. 1674 1. 1674 1. 1596 1. 1585 1. 1574 1. 1563 1. 1388 1. 1355 1. 1324 1. 1295 1. 1155 1. 1111 1. 1071 1. 1035 1. 0960 1. 0915 1. 0875 1. 0840 m = 6 1 .1302 1. 1302 1. 1302 1. 1302 1. 1250 1. 1242 1. 1234 1. 1226 1. 1097 1. 1072 1. 1049 1. 1026 1. 0919 1. 0885 1. 0854 1. 0826 1. 0767 1. 0731 1. 0700 1. 0672 m = 8 1 .1095 1. 1095 1. 1095 1. 1095 1. 1056 1. 1050 1. 1044 1. 1038 1. 0933 1. 0912 1. 0893 1. 0874 1. 0784 1. 0755 1. 0729 1. 0706 1. 0656 1. 0626 1. 0599 1. 0576 m = 10 1. 0961 1. 0961 1. 0961 1. 0961 1. 0929 1. 0924 1. 0919 1. 0914 1. 0824 1. 0806 1. 0789 1. 0773 1. 0695 1. 0669 1. 0646 1. 0626 1. 0582 1. 0556 1. 0532 1. 0511
T ab le II. Critic al v alu es C ∗(p ) fo r m u ltip le sa mp le s w ith m = 2 (2 )10, δ = 0 .5 (0 .5 )2 .0, γ = 0 .7 (0 .1 )1 .0, p = 0 .99 0 0 .5 1 .0 1 .5 2 .0 δ 0. 7 0 .8 0. 9 1 0. 7 0 .8 0. 9 1 0. 7 0 .8 0. 9 1 0. 7 0 .8 0. 9 1 0. 7 0 .8 0. 9 1 (a ) n = 5 m = 2 2 .3202 2. 3202 2. 3202 2. 3202 2. 1772 2. 1600 2. 1436 2. 1277 1. 1977 1. 8838 1. 8534 1. 8257 1. 7039 1. 6682 1. 6371 1. 6097 1. 5549 1. 5221 1. 4945 1. 4706 m = 4 1 .6688 1. 6688 1. 6688 1. 6688 1. 6085 1. 6011 1. 5938 1. 5869 1. 4907 1. 4747 1. 4602 1. 4468 1. 3866 1. 3687 1. 3529 1. 3889 1. 3104 1. 2935 1. 2790 1. 2663 m = 6 1 .4803 1. 4803 1. 4803 1. 4803 1. 4431 1. 4383 1. 4336 1. 4290 1. 3640 1. 3529 1. 3427 1. 3333 1. 2903 1. 2774 1. 2659 1. 2557 1. 2349 1. 2225 1. 2118 1. 2024 m = 8 1 .3870 1. 3870 1. 3870 1. 3870 1. 3603 1. 3567 1. 3532 1. 3498 1. 2995 1. 2907 1. 2726 1. 2751 1. 2406 1. 2301 1. 2209 1. 2126 1. 1956 1. 1854 1. 1767 1. 1670 m = 10 1. 3302 1. 3302 1. 3302 1. 3302 1. 3093 1. 3065 1. 3037 1. 3001 1. 2592 1. 2518 1. 245 1. 2386 1. 2093 1. 2003 1. 1924 1. 1853 1. 1707 1. 1619 1. 1543 1. 1477 (b) n = 10 m = 2 1 .6688 1. 6688 1. 6688 1. 6688 1. 6145 1. 6077 1. 6011 1. 5946 1. 5042 1. 4888 1. 4747 1. 4617 1. 4025 1. 3845 1. 3687 1. 3545 1. 3256 1. 3083 1. 2935 1. 2805 m = 4 1 .3870 1. 3870 1. 3870 1. 3870 1. 3631 1. 3599 1. 3567 1. 3536 1. 3069 1. 2985 1. 2907 1. 2835 1. 2498 1. 2394 1. 2301 1. 2218 1. 2047 1. 1944 1. 1854 1. 1776 m = 6 1 .2913 1. 2913 1. 2913 1. 2913 1. 2761 1. 2740 1. 2718 1. 2697 1. 2366 1. 2305 1. 2248 1. 2194 1. 1943 1. 1865 1. 1795 1. 1732 1. 1601 1. 1522 1. 1454 1. 1393 m = 8 1 .2408 1. 2408 1. 2408 1. 2408 1. 2297 1. 2280 1. 2264 1. 2247 1. 1984 1. 1934 1. 1888 1. 1844 1. 1638 1. 1573 1. 1515 1. 1463 1. 1355 1. 1289 1. 1232 1. 1181 m = 10 1. 2088 1. 2088 1. 2088 1. 2088 1. 2000 1. 1986 1. 1973 1. 1959 1. 1737 1. 1695 1. 1655 1. 1618 1. 1440 1. 1384 1. 1333 1. 1288 1. 1193 1. 1136 1. 1086 1. 1042 (c ) n = 15 m = 2 1 .4803 1. 4803 1. 4803 1. 4803 1. 4480 1. 4437 1. 4396 1. 4356 1. 3762 1. 3657 1. 3560 1. 3470 1. 3052 1. 2923 1. 2809 1. 2707 1. 2495 1. 2368 1. 2258 1. 2162 m = 4 1 .2913 1. 2913 1. 2913 1. 2913 1. 2767 1. 2746 1. 2725 1. 2705 1. 2382 1. 2322 1. 2265 1. 2213 1. 1964 1. 1886 1. 1816 1. 1753 1. 1623 1. 1544 1. 1475 1. 1414 m = 6 1 .2322 1. 2322 1. 2322 1. 2322 1. 2137 1. 2123 1. 2109 1. 2095 1. 1861 1. 1816 1. 1774 1. 1735 1. 1546 1. 1486 1. 1433 1. 1384 1. 1283 1. 1222 1. 1169 1. 1121 m = 8 1 .1864 1. 1864 1. 1864 1. 1864 1. 1793 1. 1782 1. 1771 1. 1760 1. 1571 1. 1535 1. 1500 1. 1468 1. 1311 1. 1262 1. 1217 1. 1177 1. 1092 1. 1040 1. 0995 1. 0956 m = 10 1. 1627 1. 1627 1. 1627 1. 1627 1. 1570 1. 1561 1. 1551 1. 1542 1. 1382 1. 1351 1. 1321 1. 1293 1. 1157 1. 1114 1. 1075 1. 1040 1. 0965 1. 0920 1. 0881 1. 0846 (d) n = 20 m = 2 1 .3870 1. 3870 1. 3870 1. 3870 1. 3643 1. 3613 1. 3582 1. 3553 1. 3101 1. 3019 1. 2943 1. 2872 1. 2540 1. 2436 1. 2344 1. 2261 1. 2089 1. 1985 1. 1895 1. 1816 m = 4 1 .2408 1. 2408 1. 2408 1. 2408 1. 2303 1. 2287 1. 2271 1. 2256 1. 2003 1. 1955 1. 1910 1. 1867 1. 1664 1. 1600 1. 1542 1. 1490 1. 1381 1. 1315 1. 1258 1. 1207 m = 6 1 .1864 1. 1864 1. 1864 1. 1864 1. 1794 1. 1784 1. 1773 1. 1762 1. 1576 1. 1540 1. 1506 1. 1473 1. 1318 1. 1268 1. 1224 1. 1183 1. 1099 1. 1047 1. 1002 1. 0962 m = 8 1 .1565 1. 1565 1. 1565 1. 1565 1. 1513 1. 1504 1. 1495 1. 1487 1. 1336 1. 1306 1. 1278 1. 1251 1. 1122 1. 1080 1. 1043 1. 1009 1. 0938 1. 0894 1. 0856 1. 0822 m = 10 1. 1371 1. 1371 1. 1371 1. 1371 1. 1329 1. 1321 1. 1314 1. 1307 1. 1178 1. 1153 1. 1128 1. 1105 1. 0992 1. 0956 1. 0923 1. 0893 1. 0831 1. 0792 1. 0759 1. 0729
Table III. Some recommended minimum capability requirements for special processes
Process type Capability requirement NCs (ppm)
Existing processes 1.33 66.07
New processes 1.50 6.80
Existing processes with safety, strength, or critical parameters 1.50 6.80 New processes with safety, strength, or critical parameters 1.67 0.54
5.
APPLICATION EXAMPLE
Peripheral devices such as drivers, printers, and CD-ROMs are connected to the host through a special bus called SCSI (Small Computer System Interface). The fast edge rated signals that are transmitted through the SCSI cable generate ringing on the bus. This will slow down communication between host and peripherals. The SCSI standard recommends proper resistor (Thevenin) termination at host and peripheral locations to eliminate transmission line effects. Dual Thevenin Termination Networks offer high integration and performance in a miniature QSOP or SOIC package, which saves spacious board space, provides manufacturing cost reduction and reliability efficiencies. A terminating resistor is used to reduce or eliminate unwanted reflections on a transmission line. It can perform this function only when its resistance value matches the characteristic impedance of the transmission line. The resistors used for terminating the transmission lines should be noiseless, stable and functional at high frequencies. Unlike thin film-based resistor networks, conventional thick film resistors used for terminating transmission lines are not stable over temperature and time and impose system performance limitations at very high frequencies.
5.1. Capability requirement
In the industry, some minimum capability requirements for special types of processes have been recommended. In particular, it is recommended that there be a minimum process capability of 1.33 for existing processes, and 1.50 for new processes; 1.50 also for existing processes on safety, strength, or critical parameters; and 1.67 for new processes on safety, strength, or critical parameters. The recommended guidelines for minimum quality requirements and the corresponding parts per million (ppm) of non-conformities (NCs) for those processes are summarized in TableIII.
The integrated passive networks are manufactured using advanced thin film technologies including ultra-stable and self-passivating tantalum nitride resistors, gold interconnect metallization and reliable MNOS capacitors to achieve excellent uniformity, performance and reliability. Thin film resistor technology is the preferred solution for all applications that require low noise, long-term stability and excellent performance at very high frequencies. To illustrate the application of assessing process capability for multiple control samples, we consider a real example taken from an electronic component manufacturer, located on the Science-Based Industrial Park in Taiwan, developing passive and active components for the personal computers, telecommunications, industrial controls, automotive parts, and avionics. The factory manufactures various types of resistors. For a particular model of the resistors investigated, the target value is set to T = 10.0 mil, and the tolerance of thickness is 2.0 mil, that is, the lower and upper specification limit are set to LSL= 8.0 mil and USL = 12.0 mil. If the characteristic data do not fall within the tolerance (LSL, USL), the lifetime or reliability of the resistors will be discounted. The collected sample data (10 samples each of size 15), which are under statistical control, are displayed in TableIV.
We now apply the Bayesian procedure in the following. A 100p% credible interval means the posterior probability that the true PCI lying in this interval is p. Let p be a high probability, say, 0.95. Suppose for this particular process under consideration to be capable, the process index must reach at least a certain
Table IV. The 10 samples each of 15 observations Samples 1 2 3 4 5 6 7 8 9 10 10.21 9.66 9.80 9.48 10.74 10.71 10.00 10.09 10.58 10.23 10.19 10.36 9.96 9.91 9.72 10.36 10.12 10.12 10.42 10.44 9.88 10.55 10.04 9.94 10.34 10.17 10.29 9.99 9.58 9.86 10.73 10.31 9.99 9.93 10.88 10.53 9.62 10.57 10.44 10.16 10.59 9.72 10.35 10.08 10.48 10.15 9.98 10.50 10.39 10.14 10.21 10.00 9.94 9.59 10.01 10.09 10.00 9.43 10.87 9.99 10.61 10.34 10.96 10.01 10.71 10.14 10.12 10.60 9.56 11.12 10.68 9.77 10.33 9.85 10.15 9.76 9.97 9.86 10.26 10.10 9.86 10.12 10.39 10.50 10.46 10.15 10.56 9.90 10.16 10.00 10.69 10.40 10.63 9.77 10.38 10.36 10.60 9.84 10.46 9.97 10.12 11.11 9.13 9.97 10.39 10.28 9.76 10.31 9.83 10.50 10.62 10.25 10.57 10.03 10.33 10.05 9.78 10.03 10.09 10.47 9.73 11.03 10.24 10.02 10.33 9.50 9.74 9.53 10.43 10.30 10.35 10.23 10.65 10.37 10.15 10.29 10.48 9.72 10.38 10.17 10.51 9.98 10.70 9.81 10.26 10.29 9.79 10.56 10.27 10.04
Table V. The calculated sample mean and the sample variance for the 10 samples
Sample i 1 2 3 4 5 6 7 8 9 10
¯xi 10.332 10.255 10.245 9.951 10.354 10.188 10.053 10.070 10.247 10.233
si2 0.110 0.178 0.207 0.066 0.085 0.083 0.096 0.141 0.129 0.097
level ω, say, 1.33. That is, the requirement for the process yield is no more than 2700 ppm. From the process data, we compute the lower bound of the credible interval for the index. The Bayesian testing procedure is simple. That is, if ˆCpm∗ > C∗(p)× ω, then we say that the process is capable in a Bayesian
sense.
The calculated sample mean ¯xi and the sample variance s2i for the ten sub-samples of size 15 are tabulated in TableV. Thus, ¯¯x = 10.1929 sp2= m i=1 si2 m= 0.1192 γ= m(n− 1)s 2 p m(n− 1)s2 p+ n m i=1(¯xi− ¯¯x)2 = 0.8816 δ=| ¯¯x − T | sp = 0.5587 ˆC∗ pm= d 3sp(n− 1)/(γ n) + δ2 = 1.6489
Next, we check the tables or run the computer software to obtain the critical value ˆC∗(p)× ω = 1.1069 ×
1.33= 1.4722 based on p = 0.95, m = 10, n = 15. Since the sample estimator ˆC∗pmfrom the samples, 1.6489, is greater than the critical value C∗= ˆC∗(p)× ω = 1.4722, we may conclude, with 95% confidence level, that
6.
CONCLUSIONS
Using process capability indices to quantify manufacturing process precision and performance is an essential part of implementing a quality improvement program. Most existing tests of the capability indices are obtained from the distributional frequency approaches. Statistical properties of the estimated Cpmbased on one single sample have been investigated extensively. But, the properties of the estimated Cpm based on multiple samples have been comparatively neglected. In this paper, we have considered the problem of estimating and testing process capability based on multiple samples. We accordingly proposed a Bayesian procedure for capability testing. Based on these multiple control samples, we also developed a simple step-by-step procedure. The practitioners can use the proposed procedure to determine whether their manufacturing processes are capable of reproducing products satisfying the preset precision requirements. A process is capable if all the points in the credible interval are greater than the pre-specified capability level ω. To make this Bayesian procedure practical for in-plant applications, we tabulated the minimum values of C∗(p)for which the posterior probability p reaches various desirable confidence levels.
Acknowledgements
The authors would like to thank the anonymous referees for their helpful comments and careful readings, which improved the paper.
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APPENDIX A.
DERIVATION OF EXPRESSION (
5
)
For the multiple control samples, given a pre-specified capability level ω > 0, the posterior probability based on index Cpmthat a process is capable is given as
p= Pr{Cpm> ω|x} = Pr USL− LSL 6τ > ω x= Pr τ < USL− LSL 6ω x = Pr σ2+ (µ − T )2< USL− LSL 6ω 2 x = ((USL−LSL)/6ω) 0 T+√a2−σ2 T−√a2−σ2 f (µ, σ|x) dµ dσ
Denote a= (USL − LSL)/6ω and g(σ) =√a2− σ2. Then
p= a 0 T+g(σ ) T−g(σ ) f (µ, σ|x) dµ dσ = a 0 T+g(σ ) T−g(σ ) 2mi=1ni √ 2π (α)βασ −(m i=1ni+1)exp − m i=1 ni j=1(xij− µ) 2 2σ2 dµ dσ = a 0 2mi=1ni √ 2π (α)βασ −(m i=1ni+1)exp − 1 βσ2 T+g(σ ) T−g(σ ) exp − m i=1ni(µ− ¯¯x)2 2σ2 dµ dσ = a 0 2σ−mi=1ni (α)βα exp − 1 βσ2 T − ¯¯x + g(σ) σ mi=1ni − T − ¯¯x − g(σ) σ mi=1ni dσ (A1) where α= m i=1 ni− 1 2, β= m i=1 ni j=1 (xij − ¯¯x)2 2 −1 , ¯¯x = m i=1 ni j=1 xij m i=1 ni
for−∞ < µ < ∞, 0 < σ < ∞, and (·) is the cumulative distribution of the standard normal distribution. By changing the variables, we let y= βσ2. Then, dy= 2βσ dσ, and
sp/σ= 2γ m i=1 (ni − 1)y
Thus, the posterior probabilityp for multiple control samples, which are given in (A1), can be simplified to p= Pr{Cpm> ω|x} = t 0 1 (α)yα+1 exp −1 y
× [ (b1(y)+ b2(y))− (b1(y)− b2(y))] dy
where b1(y)= T − ¯¯x σ mi=1ni = m i=1 ni T − ¯¯x sp %sp σ & = 2γ mi=1ni m i=1(ni− 1)y δ b2(y)= g(σ ) σ mi=1ni = m i=1 ni g(σ ) σ = m i=1 ni a2− σ2 σ2 1/2 = m i=1 ni a2 σ2− 1 1/2 = m i=1 ni βa2 y − 1 1/2 = m i=1 ni t y − 1 1/2 and t= βa2= 2γ a 2 m i=1(ni − 1)sp2 =m 2γ i=1(ni − 1)s2p USL− LSL 6ω 2 =m 2γ i=1(ni− 1)ω2 USL− LSL 6sp 2 =m 2γ i=1(ni− 1)ω2 USL− LSL 6ˆσ 2 ˆτ sp 2 =m 2γ i=1(ni− 1) ˆC∗ pm ω 2 ˆτ sp 2 =m 2 i=1(ni− 1) ˆC∗ pm ω 2 m i=1(ni − 1) m i=1ni + γ δ2
Therefore, the posterior probabilityp for multiple control samples, which is given in (5), can be derived.
Authors’ biographies
C. W. Wu received his MS degree from the Institute of Statistics at National Tsing Hua University, Taiwan,
ROC. Currently, he is a PhD student at the Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC.
W. L. Pearn is a Professor of Operations Research and Quality Assurance at the Department of Industrial
Engineering and Management, National Chiao Tung University, Taiwan, ROC. His research areas include process capability analysis, network optimization, queuing service management, applied statistics, and semiconductor manufacturing scheduling.
