Abstract. Process capability indices have been proposed in the manufacturing industry to provide numerical measures on process reproduction capability, which are effective tools for quality assurance and guidance for process improvement. In process capability analysis, the usual practice for testing capability indices from sample data are based on traditional distribution frequency approach. Bayesian statistical techniques are an alternative to the frequency approach. Shiau, Chiang and Hung (1999) applied Bayesian method to index Cpm and the index Cpk but under the restriction that the
process mean l equals to the midpoint of the two specification limits, m. We note that this restriction is a rather impractical assumption for most factory applications, since in this case Cpkwill reduce to Cp. In this paper, we consider
testing the most popular capability index Cpk for general situation – no
restriction on the process mean based on Bayesian approach. The results obtained are more general and practical for real applications. We derive the posterior probability, p, for which the process under investigation is capable and propose accordingly a Bayesian procedure for capability testing. To make this Bayesian procedure practical for in-plant applications, we tabulate the minimum values of ^Cpk for which the posterior probability p reaches
desirable confidence levels with various pre-specified capability levels. Key words: Bayesian approach, posterior distribution, process capability indices, posterior probability
1 Introduction
Process capability indices (PCI), Cp, Cpk, Cpmand Cpmkhave been proposed in
the manufacturing industry and the service industry providing numerical measures on whether a process is capable of reproducing items within
DOI 10.1007/s001840400333
Process capability assessment for index C
pkbased on
bayesian approach
W. L. Pearn and Chien-Wei Wu
Department of Industrial Engineering & Management, National Chiao Tung University, Taiwan Received August 2003
specification limits preset in the factory (see Kane (1986), Chan, Cheng and Spiring (1988), Pearn, Kotz and Johnson (1992), Kotz and Lovelace (1998)). These indices have been defined as:
Cp¼ USL LSL 6r ; Cpk¼ min USL l 3r ; l LSL 3r ; Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q ; Cpmk¼ min USL l 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q ; l LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q 8 > < > : 9 > = > ;: where USL is the upper specification limit, LSL is the lower specification limit, l is the process mean, r is the process standard deviation (overall process variation), and T is the target value. The index Cp considers the overall
process variability relative to the manufacturing tolerance, reflecting product quality consistency. The index Cpktakes the magnitude of process variance as
well as process departure from target value, and has been regarded as a yield-based index since it providing lower bounds on process yield. The index Cpm
emphasizes on measuring the ability of the process to cluster around the target, which therefore reflects the degrees of process targeting (centering). Since the design of Cpm is based on the average process loss relative to the
manufacturing tolerance, the index Cpm provides an upper bound on the
average process loss, which has been alternatively called the Taguchi index. The index Cpmk is constructed from combining the modifications to Cp that
produced Cpk and Cpm, which inherits the merits of both indices.
Process yield is currently defined as the percentage of the processed product units passing the inspections. Units are inspected according to specification limits placed on various key product characteristics and sorted into two categories: passed (conforming) and rejected (defectives). Thus, yield is one of the commonly understood basic criteria for interpretations of the process capability. Suppose a proportion conforming items is the primary concern, then most natural measure is the proportion itself called the yield, which we refer to as X defined as:
Y ¼ Z USL
LSL
dFðxÞ ¼ F ðUSLÞ F ðLSLÞ; ð1Þ
where FðxÞ is the cumulative distribution function of the measured charac-teristic X , USL and LSL are the upper and the lower specification limits respectively. As the index Cpk provides a lower bound on the process yield, a
widely used criterion for measuring process, it has become the most popular capability index used in the industry. Existing methods for testing the capa-bility indices have focused on using the traditional but long time been widely used distribution frequency approaches. The usual practice of judging process capability by evaluating the point estimates of process capability indices is highly unreliable, as there is no assessment on the error distributions of these estimates. A point estimate to the index is not very useful in making reliable decision. Interval estimation approach, in fact, is more appropriate and widely accepted. But the frequency distributions of these estimates are usually complicated that it is very difficult to obtain exact interval estimates. A process is usually defined as a capable process if its capability exceeds a
pre-specified value w. A reliable approach for testing process capability is to establish an interval estimate, for which we can assert with a reasonable degree of certainty that it contains the true PCI value. However, the con-struction of such an interval estimate is not trivial, since the distributions of the commonly used PCI estimators are usually quite complicated.
An alternative is to use the Bayesian approach, which essentially specifies a prior distribution for the parameter of interest, and to obtain the posterior distribution of the parameter and then infer about the parameter by only using its posterior distribution given the observations. It is not difficult to obtain the posterior distribution when a prior distribution is given, even in the case where the form of the posterior distribution is complicated, as one could always use numerical methods or Monte Carlo methods to obtain an approximate but quite accurate interval estimate. This is the advantage of the Bayesian approach over the traditional distribution frequency approach (Kalos and Whitlock (1986)).
In this paper, we consider testing the most popular capability index Cpk
using Bayesian approach. We obtain the posterior probability p for which the process under investigation is capable, and propose accordingly a Bayesian procedure for capability testing. To make this Bayesian procedure practical for in-plant applications, we tabulate the minimum values of ^Cpk for which
the posterior probability p reaches various desirable confidence levels. An application example to the oil-hydraulic cylinders manufacturing process is presented to illustrate the applicability of the proposed approach. Finally, some concluding remarks are made in Section 7.
2 Distribution frequency approach for Cpk
Utilizing the identity minfa; bg ¼ ða þ bÞ ja bj=2, the definition of the index Cpk can be alternatively written as:
Cpk¼
d jl mj
3r ; ð2Þ
where d ¼ ðUSL LSLÞ=2 is half of the length of the specification interval, m¼ ðUSL þ LSLÞ=2 is the mid-point between the lower and the upper speci-fication limits. The natural estimator ^Cpkis obtained by replacing the process
mean l and the process standard deviation r by their conventional estimators
xand s, which may be obtained from a process that is demonstrably stable (under statistical control).
^ Cpk¼ d jx mj 3s ¼ 1 jx mj d ^ Cp; ð3Þ
where x¼Pni¼1xi=n and s¼ ½Pni¼1ðxi xÞ2=ðn 1Þ1=2. Under the
assump-tion of normality, Kotz and Johnson (1993) obtained the r-th moment, and the first two moments as well as the mean and the variance of ^Cpk. In addition,
numerous methods for constructing approximate confidence intervals of Cpk
have been proposed in the literature. Examples include Chou, Owen and Borrego (1990), Zhang, Stenback and Wardrop (1990), Franklin and Wass-erman (1991), Kushler and Hurley (1992), Nagata and Nagahata (1994), Tang, Than and Ang (1997), Hoffman (2001), and many others.
Kotz and Johnson (2002) presented a thorough review for the develop-ment of process capability indices during the years 1992 to 2000. Further-more, from the estimated ^Cpk defined in (1), since ^Cp is distributed as
ðn 1Þ1=2Cpðv1n1Þ, and n1=2jx mj=r is distributed as the folded normal
distribution with parameter n1=2jl mj=r (see Leone, Nelson and
Notting-ham (1961) for details about this distribution). Thus, ^Cpk is a mixture of v1n1
and the folded normal distribution (Pearn, Kotz and Johnson (1992)). The probability density function of ^Cpkcan be obtained as (Pearn, Chen and Lin
(1999)), where D¼ ðn 1Þ1=2d=r, a¼ ½ðn 1Þ=n1=2. fC^pkðyÞ ¼ 4AnP 1 ‘¼0 P‘ðkÞB‘D nþ2‘ a2‘þ1 R1 0 ð1 yzÞ 2‘zn1 exp D2 18a2 a2z2þ 9ð1 yzÞ 2 n o dz; y 0; 4AnP 1 ‘¼0 P‘ðkÞB‘D nþ2‘ a2‘þ1 R1=y 0 ð1 yzÞ 2‘zn1 exp D2 18a2 a2z2þ 9ð1 yzÞ 2 n o dz; y >0; 8 > > > > > > > > < > > > > > > > > : ð4Þ P‘ðkÞ ¼ eðk=2Þðk=2Þ‘ ‘! ; An¼ 1 3n12n=2Cððn 1Þ=2Þ; B‘¼ 1 2‘Cðð2‘ þ 1Þ=2Þ:
Using the integration technique similar to that presented in Va¨nnman (1997), Pearn and Lin (2003) first obtain an exact and explicit form of the cumulative distribution function of the natural estimator ^Cpk, under the
assumption of normality. The cumulative distribution function of ^Cpk is
ex-pressed in terms of a mixture of the chi-square distribution and the normal distribution: FC^pkðyÞ ¼ 1 Z bpffiffin 0 G ðn 1Þðb ffiffiffi n p tÞ2 9ny2 ! /ðt þ npffiffiffinÞ þ /ðt npffiffiffinÞ dt; ð5Þ for y > 0, where b¼ d=r, n ¼ ðl mÞ=r, GðÞ is the cumulative distribution function of the chi-square distribution with degree of freedom n 1, v2
n1, and
/ðÞ is the probability density function of the standard normal distribution Nð0; 1Þ. Based on the cumulative distribution function of ^Cpk, Pearn and Lin
(2003) implemented the statistical theory of the hypotheses testing, and devel-oped a simple but practical procedure accompanied with convenient tabulated critical values, for engineers/practitioners to use for decisions making in their factory applications. Formulae for computing the power of the corresponding test are also obtained. Pearn and Shu (2003) further developed an efficient algorithm with Matlab computer program to find the exact (rather than just approximate) lower confidence bounds conveying critical information regard-ing the true process capability. An illustrative application of the lower confi-dence bound to the power distribution switch was given. Their investigations are all based on traditional distribution frequency approaches.
3 Bayesian approach for Cpk
Cheng and Spiring (1989) proposed a Bayesian procedure for assessing pro-cess capability index Cp. Shiau, Hung and Chiang (1999) derived the posterior
distributions for C2p, Cpm2 under the restriction that process mean l equals to
the target value T , and C2
pk under the restriction that the process mean l
equals to the midpoint of the two specification limits, m, with respect to the two priors (a non-informative and a Gamma prior). Shiau, Chiang and Hung (1999) applied Bayesian method to index Cpm relaxing the restriction on
l¼ T . Wu and Pearn (2003) generalized this Bayesian procedure for testing process capability index Cpm to cases where data are collected over time as
multiple samples. Shiau, Chiang and Hung (1999) also applied a similar Bayesian approach for testing the index Cpk but under the restriction l¼ m.
We note that this restriction is a rather impractical assumption for most factory applications, since in this case Cpkwill reduce to Cp. In the following,
we consider a Bayesian procedure for the capability index Cpk for general
situation – no restriction on the process mean. Thus, the results obtained are more general and practical for real applications. 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 parameter (Berger (1980)). Assuming that the measures x¼ fx1; x2; ; xng are random sample taken from independent and
identically distributed (i.i.d.) Nðl; r2Þ, a normal distribution with mean l and
variance r2. Then, the likelihood function for l and r is
Lðl; rjxÞ ¼ 2pr 2 n=2 exp Pn i¼1ðxi lÞ2 2r2 ( ) ð6Þ The most important problem in Bayesian inference is how to specify an appropriate prior distribution. If prior information about the parameters is available, it should be incorporated in the prior density. If we have no prior information, we want a prior with minimal influence on the inference. There are mainly two types of priors: informative and non-informative. Ideally, a Bayesian should subjectively elicit a prior on the basis of available informa-tion, expert opinion or past experience. Informative prior distributions summarize the evidence about the parameters concerned from many sources and often have a considerable impact on the results. For an example of informative priors, conjugate priors, although being widely used, can only be justified if enough information is available to believe that the true prior dis-tribution belongs to the specified family; otherwise, the main justification for using conjugate prior is their mathematical tractability.
On the other hand, non-informative prior, Bayesian analysis often leads to the procedures with approximate frequency validity while retaining the Bayesian flavor, thus allowing certain amount of reconciliation between the two conflicting paradigms of statistics and providing with mutual justifica-tion. Box and Tiao (1973) define a non-informative prior as prior, which provides little information relative to the experiment. Bernardo and Smith (1993) use a similar definition, they say that non-informative prior have minimal effect relative to the idea, on the final inference. And Kass and Wasserman (1996) stated two interpretations of non-informative priors.
Therefore, the first step for the Bayesian approach is to find an appro-priate prior. Usually, when there is only a little or no prior information is available, or only one parameter of interest, one of the most widely used non-informative priors is the so-called reference prior, which is a non-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 information about the parameter as little as possible (see Bernardo and Smith (1993) for more details). For this reason, in this paper we adopt the following non-informative reference prior,
pðl; rÞ ¼ 1=r; 0 < r <1: ð7Þ
We note that the parameter space of the prior is infinite, hence the reference prior is improper, which means that it does not integrate to one. However, it is not always a serious problem, since the prior incorporate with ordinary likelihood will lead to proper posterior. Furthermore, the credible interval obtained from a non-informative prior has a more precise coverage proba-bility than that obtained from any other priors. The posterior probaproba-bility density function (PDF), fðl; rjxÞ of ðl; rÞ may be expressed as the following:
fðl; rjxÞ / Lðl; rjxÞ pðl; rÞ / rðnþ1Þ exp Pn i¼1ðxi lÞ2 2r2 ! Since Z 1 0 Z 1 1 rðnþ1Þ exp Pn i¼1ðxi lÞ2 2r2 ! dldr ¼ Z 1 0 rðnþ1Þexp 1 br2 Z 1 1 exp nðl xÞ 2 2r2 ! dl " # dr¼ ffiffiffiffiffi p 2n r CðaÞba
And in order to satisfy the integration property, probability over PDF is 1, so that fðl; rjxÞ ¼ 2 ffiffiffi n p ffiffiffiffiffiffi 2p p CðaÞbar ðnþ1Þ exp P n i¼1ðxi lÞ2 2r2 ! ð8Þ where a¼ ðn 1Þ=2, b ¼ ½Pni¼1ðxi xÞ2=21¼ ½ðn 1Þs2=21.
Subsequently, we consider the quantity Pr{process is capable |x} in the Bayesian approach. Since the index Cpk is our focus in this paper, so we are
interested in finding the posterior probability p¼ PrfCpk> wjxg for some
fixed positive number w.
4 The posterior probability
Given a pre-specified capability level w > 0, the posterior probability based on index Cpk that a process is capable can be derived in the following way.
From equation (8), we have the posterior probability density function (PDF) fðl; rjxÞ of ðl; rÞ as the following, where
a¼ ðn 1Þ=2; b ¼ ½Xn i¼1ðxi xÞ 2=21 ¼ ½ðn 1Þs2=21; fðl; rjxÞ ¼ 2 ffiffiffi n p ffiffiffiffiffiffi 2p p CðaÞbar ðnþ1Þ exp P n i¼1ðxi lÞ2 2r2 ! :
Therefore, given a pre-specified capability level w > 0, the posterior proba-bility based on index Cpk that a process is capable is given as
p¼ PrfCpk> wjxg ¼ Pr d l mj j 3r > w x ¼ Pr l mfj j < d 3rwjxg ¼ Z 1 0 Z mþd3rx mdþ3rx fðl;rjxÞdl dr ¼ Z 1 0 Z mþd3rx mdþ3rx 2pffiffiffin ffiffiffiffiffiffi 2p p CðaÞbar ðnþ1Þ exp Pn i¼1ðxi lÞ2 2r2 ! dl dr¼ Z 1 0 2pffiffiffin ffiffiffiffiffiffi 2p p CðaÞbar ðnþ1Þ exp 1 br2 Z mþd3rw mdþ3rw exp nðl xÞ 2 2r2 ! dl dr ¼ Z 1 0 2rn CðaÞbaexp 1 br2 U mþ d 3rw x r=pffiffiffin U m d þ 3rw x r=pffiffiffin dr ¼ Z 1 0 2rn CðaÞbaexp 1 br2 U d ðx mÞ s=pffiffiffin s r 3 ffiffiffi n p w þ U d ðm xÞ s=pffiffiffin s r 3 ffiffiffi n p w 1 dr Next, we consider the two cases for derivation of posterior probability p as follows:
CASE I: x m
If x m, then ^Cpk¼dðxmÞ3s anddðmxÞ3s ¼dþðxmÞ3s ¼ ^Cpkþ23d where d¼jxms j.
Thus, p¼ PrfCpk> wjxg ¼ Z 1 0 2rn CðaÞbaexp 1 br2 U 3pffiffiffinC^pk s r 3 ffiffiffi n p w h þU 3pffiffiffin C^pkþ 2 3d s r 3 ffiffiffi n p w 1 dr CASE II: m > x
If m > x, thendðxmÞ3s ¼dþðmxÞ3s ¼ ^Cpkþ23d anddðmxÞ3s ¼ ^Cpk where d¼jxms j.
Thus, p¼ PrfCpk> wjxg ¼ Z 1 0 2rn CðaÞbaexp 1 br2 U 3pffiffiffin C^pkþ 2 3d s r 3 ffiffiffi n p w þU 3pffiffiffinC^pk s r 3 ffiffiffi n p w 1idr From both cases,
p¼ PrfCpk> wjxg ¼ Z 1 0 2rn CðaÞbaexp 1 br2 U 3pffiffiffinC^pk s r 3 ffiffiffi n p w h þ U 3pffiffiffin C^pkþ 2 3d s r 3 ffiffiffi n p w 1 dr
By changing the variable, let y¼ br2, then dy¼ 2br dr, and s r¼ ffiffiffiffiffiffiffiffiffiffiffi 2 ðn1Þy q . Therefore, the posterior probability p may be rewritten as:
p¼ Prfthe process is capablejxg ¼ PrfCpk> wjxg
¼ Z 1
0
1
CðaÞyaþ1exp
1 y U 3pffiffiffinC^pk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðn 1Þy s 3pffiffiffinw ! " þU 3pffiffiffin C^pkþ 2 3d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðn 1Þy s 3pffiffiffinw ! 1 # dy ¼ Z 1 0 1
CðaÞyaþ1exp
1 y U 3pffiffiffin C^pk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðn 1Þy s w ! " # ( þU 3pffiffiffin ð ^Cpkþ 2 3dÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðn 1Þy s w ! " # 1 ) dy ¼ Z 1 0 1
CðaÞyaþ1exp
1 y
U bf ½ 1ðyÞ þ U b½ 2ðyÞ 1g dy; ð9Þ
where a¼ ðn 1Þ=2, d ¼jxms j, b1ðyÞ ¼ 3 ffiffiffin p ^ Cpk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðn 1Þy s w ! ; b2ðyÞ ¼ 3pffiffiffin ð ^Cpkþ 2 3dÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðn 1Þy s w ! ;
and UðÞ is the cumulative distribution function of the standard normal dis-tribution. Note that the posterior probability p depends on n, w, d and ^Cpk.
5 Bayesian procedure for testing Cpk
As we can see it is rather complicated to compute posterior probability p in (9) without advanced computer programming skills. However, by noticing that there is a one-to-one correspondence between p and Cwhen n and w are given, and by the fact that ^Cpk can be calculated from the process data, we
find that the minimum value of C required to ensure the posterior proba-bility p reaching a certain desirable level, can be useful in assessing process capability. Denote this minimum value of Cfor probability p as CðpÞ. Thus, we can find the value of CðpÞ satisfies equation (9) for various p, where p is a
number between 0 and 1, say 0.95, for 95% confidence interval, which means that the posterior probability that the credible interval contains the true value of Cpk is p. Suppose for this particular process under consideration to be
1.33. From expression (9) we have the probability p¼ PrfCpk> wjxg based
on the observed process data. Moreover, to see if a process is capable (with capability level w and confidence level p), we only need to check if
^
Cpk> CðpÞ. Throughout this paper it is assumed that the process measure-ments are independent and identically distributed from a normal distribution, and the process is under statistical control. We remark that estimation of these capability indices is meaningful only when the process is under statis-tical control.
To make this Bayesian procedure practical for in-plant applications, we calculate the values of CðpÞ for various values of n= 10(5)160 and d=
0(0.5)2.0 with posterior probability p= 0.90, 0.95, and 0.99 and w= 1.00, 1.33, 1.50, 2.00. Tables 1(a)-1(c) summarize the values of CðpÞ with w= 1.00,
for p= 0.90, 0.95, and 0.99, respectively. Tables 2(a)-2(c) summarize the values of CðpÞ with w= 1.33, for p= 0.90, 0.95, and 0.99, respectively.
Tables 3(a)-3(c) summarize the values of CðpÞ with w= 1.50, for p= 0.90, 0.95, and 0.99, respectively. And the values of CðpÞ with w= 2.00, for p= 0.90, 0.95, and 0.99 are displayed in Tables 4(a)-4(c), respectively. Interested readers may visit the following website for details of those tables: http:// www.nctu.edu.tw/~qtqm/paper/cpktables/. For example, if w= 1.33 is the minimum capability requirement, then for p= 0.95, n= 100, d= 0.5, CðpÞ = 1.5173 by checking Table 2(b). Thus, the value ^Cpk calculated from sample
data must satisfy ^Cpk‡ 1.5173 to conclude that Cpk‡ 1.33 (process is capable).
From these tables we observe that for each fixed p and n the value of CðpÞ
decreases as d increases. Figures 1–4 display the value of CðpÞ versus d= jx mj=s for sample size n= 10(10)50 from top to bottom in plots, with w= 1.00 and p= 0.95. This phenomenon can be explained by the following argument. For a fixed ^Cpk, since
^ Cpk¼ d jx mj 3s ¼ d=s d 3 ; ð10Þ
then s becomes smaller when d becomes larger, and a smaller s means that it is plausible that the underlying process is tighter (i.e. with smaller r). Since the estimation is usually more accurate for data drawn from a tighter process, it is then plausible that the estimate ^Cpk is more accurate with a smaller s. In this
case the required minimum value is smaller, so we need only a smaller CðpÞ
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 CðpÞ to be able to claim that the process is capable, and this means that the corresponding minimum value CðpÞ should be large as well. Thus the value of CðpÞ decreases as d increases. Another observation from the tables is that the value of CðpÞ decreases as n increases for fixed d and p. This can also be seen from the same argument as above, a larger n implies that
^
Cpk is more accurate.
6 Capability testing with applications
In current practice, a process is called ‘‘Inadequate’’ if Cpk< 1.00; it indicates
that the process is not adequate with respect to the production tolerances (specifications), either process variation (r2) needs to be reduced or process
‘‘Capable’’ if 1.00 £Cpk<1.33; it indicates that caution needs to be taken
regarding to process distribution, some process control is required. A process is called ‘‘Satisfactory’’ if 1.33£Cpk<1.50; it indicates that process quality is
satisfactory, material substitution may be allowed, and no stringent quality control is required. A process is called ‘‘Excellent’’ if 1.50 £Cpk< 2.00; it
indicates that process quality exceeds satisfactory. Finally, a process is called ‘‘Super’’ if Cpk ‡ 2.00. Many companies have recently adopted criteria for
1.5 0.0 0.5 1.0 2.0 1.0 1.2 1.4 1.6 1.8 2.0
Fig. 1. Plots of CðpÞ versus d for w ¼ 1:00, p ¼ 0:95, and n ¼ 10; 20, 30, 40, and 50 (top to bottom in plot) 0.0 0.5 1.0 1.5 2.0 1.4 1 .6 1.8 2 .0 2.2 2 .4 2.6
Fig. 2. Plots of CðpÞ versus d for w ¼ 1:33, p ¼ 0:95, and n ¼ 10; 20; 30; 40, and 50 (top to bottom in plot)
evaluating their processes that include process capability objectives more stringent. For example, Motorola’s ‘‘Six Sigma’’ program essentially requires the process capability at least 2.0 to accommodate the possible 1.5r process shift (see Harry (1988)). Table 1 summarizes some commonly used capability requirements and fractions of nonconformities (in ppm) corresponding to process conditions.
We now describe a Bayesian procedure in the following. A 100p% credible interval means the posterior probability that the true PCI lies in this interval
0.0 0.5 1.0 1.5 2.0
2.5
3.0
3.5
4.0
Fig. 4. Plots of CðpÞ versus d for w ¼ 2:00, p ¼ 0:95, and n ¼ 10; 20; 30; 40, and 50 (top to
bottom in plot) 0.0 0.5 1.0 1.5 2.0 1.6 2.0 2.2 2.4 2.6 2.8 3.0 1.8
Fig. 3. Plots of CðpÞ versus d for w ¼ 1:50, p ¼ 0:95, and n ¼ 10; 20; 30; 40, and 50 (top to
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 level w, say, 1.33. Next, from the process data, we compute (or check the tables) the lower bound of the credible interval for the Cpkindex. Thus, if
^
Cpk > CðpÞ, then we say that the process is capable in a Bayesian sense. Otherwise, we do not have sufficient information to conclude that the process meets the preset capability requirement, and then we tend to believe that the process is incapable in this case.
To illustrate how we apply the proposed procedure to actual data col-lected from the factory. We consider the following example taken from a company engaged mainly in making oil-hydraulic cylinder components and oil-hydraulic cylinder (oil-hydraulic propeller) assembly. Oil-hydraulic equipment is required for automation and oil-hydraulic cylinders are the main component of such equipment. The pistons are one of the most critical parts of oil-hydraulic cylinders. A typical piston for the oil-hydraulic cylin-ders has a 20 mm inner diameter. When the oil goes through the oil-hydraulic cylinder, it can exert pressure and make the piston move. The two points C are the grooves on the piston that must be fitted with the U-shaped oil seal to prevent the oil from leaking when the piston move. If the oil leaks, it affects the efficiency and performance of the oil-hydraulic cylinder. There are two points called A and two points called B, which are the prominent parts of the piston holding two U-shaped oil seals to make them assuming the pressure from the oil-hydraulic cylinder. Since it is the U-shaped oil seals, and is not the main body of the piston in direct contact with the tube of the oil-hydraulic cylinder, then it is essential to make the piston grooves (called point C) complying with the required manufacturing specifications.
The manufacturing specifications for the grooves of the piston are set to the followings: USL = 13.25 mm, LSL = 13.15 mm, target value T = 13.20 mm. The capability requirement for this particular model of oil-hydraulic cylinder was defined as "Satisfactory" if Cpk > 1.33. The process has been
justified to be well in-controlled, and is near normally distributed. The col-lected sample data (a total of 150 observations) are displayed in Table 2. The sample mean x= 13.201 and sample standard deviation s = 0.00969 are first calculated. For n= 150, we calculate the value of the estimator ^
Cpk ¼ ðd jx mjÞ=ð3sÞ= 1.6925, and d ¼ x mj j=s= 0.103. By solving the
posterior probability (9), the critical value is found to be CðpÞ= 1.4869
based on w= 1.33, p = 0.95, and n= 150. Note that the computer program for calculating CðpÞ is available from authors. Since ^C
pk = 1.6925 is greater
than the critical value CðpÞ = 1.4869 in this case, it is therefore concluded with 95% confidence (a= 0.05) that the grooves of the piston manufacturing process satisfies the requirement ‘Cpk > 1.33’. Thus, at least 99.9934% of the Table 1. Some commonly used capability requirements and nonconformities corresponding to process conditions
Process conditions Cpkvalues Non-conformities
Incapable Cpk<1:00 > 2700 ppm
Capable 1:00 Cpk<1:33 < 2700 ppm
Satisfactory 1:33 Cpk<1:50 < 66 ppm
Excellent 1:50 Cpk<2:00 < 6.795 ppm
produced oil-hydraulic cylinders are conformed to the manufacturing speci-fications, which are considered satisfactory and reliable in terms of product quality (originally set by the product designers or the manufacturing engi-neers).
7 Conclusions
In the last decade, numerous process capability indices have been proposed to provide measure on whether a process is capable of reproducing items meeting the quality requirement preset by the product designer. Those indices are effective tools for process capability analysis and quality assurance. In process capability analysis, the usual practice for estimating the capability indices from sample data are based on the traditional distribution frequency approach. An alternative is to use the Bayesian approach. The Bayesian approach specifies a prior distribution for the parameter of interest, to obtain the posterior distribution for the parameter, then infer about the parameter using it posterior distribution given the observations. This paper considers estimating and testing capability index Cpk using Bayesian approach. The
posterior distribution of Cpk is derived and an accordingly Bayesian
proce-dure for capability testing is proposed. For users’ convenience in applying our Bayesian procedure, we tabulate the minimum values of ^Cpk required to
ensure the posterior probability p reaching various pre-specified capability levels.
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