Qual. Reliab. Engng. Int. 15: 369–378 (1999)
A BAYESIAN PROCEDURE FOR PROCESS CAPABILITY
ASSESSMENT
JYH-JEN HORNG SHIAU1∗, CHUN-TA CHIANG2AND HUI-NIEN HUNG1 1Institute of Statistics, National Chiao Tung University, Hsinchu, Taiwan 300
2Department of Quality Management Technology, Center for Aviation and Space Technology, Industrial Technology Research Institute, Hsinchu, Taiwan 300.
SUMMARY
The usual practice of judging process capability by evaluating point estimates of some process capability indices has a flaw that there is no assessment on the error distributions of these estimates. However, the distributions of these estimates are usually so complicated that it is very difficult to obtain good interval estimates. In this paper we adopt a Bayesian approach to obtain an interval estimation, particularly for the index Cpm. The posterior probability p that the process under investigation is capable is derived; then the credible interval, a Bayesian analogue of the classical confidence interval, can be obtained. We claim that the process is capable if all the points in the credible interval are greater than the pre-specified capability level ω, say 1.33. To make this Bayesian procedure very easy for practitioners to implement on manufacturing floors, we tabulate the minimum values of ˆCpm/ω, for which the posterior probability p reaches the desirable level, say 95%. For the special cases where the process mean equals the target value for Cpmand equals the midpoint of the two specification limits for Cpk, the procedure is even simpler; only chi-square tables are needed. Copyright1999 John Wiley & Sons, Ltd. KEY WORDS: process capability indices; quality; Bayesian approach; confidence interval; credible interval; prior; posterior
1. INTRODUCTION
Process capability indices (PCIs) are unitless measures for the capability of a process in meeting specification limits. These indices have been widely used in assessing the capability of manufacturing processes by many companies during the last decade. More and more efforts have been devoted to studies and applications of PCIs. For example, Rado [1] presented how Imprimis Technology, Inc. used the PCIs to enhance product development, and the Cp and Cpk indices have been used in Japan and in the US automotive industry such as Ford Motor Company [2,3]. To incorporate the departure of the process mean µ from the target value T , the index Cpm was proposed [4]. This index has been getting more and more recognition in industries in recent years.
A capable process is usually defined as a process with a certain process capability index greater than a ∗Correspondence to: J.-J. H. Shiau, Institute of Statistics, National Chiao Tung University, Hsinchu, Taiwan 300.
E-mail:jyhjen@stat.nctu.edu.tw.
Contract/grant sponsor: National Science Council of the Republic of China; Contract/grant number: 2118-M-009-003; NSC87-2118-M-009-004.
pre-specified value ω. The usual practice is to estimate the PCI from process data. If the estimate is greater than the pre-specified value ω, say 1 or 1.33, then it is claimed that the process is capable. Of course, the estimate is not the index itself, so when the estimate is greater than ω, it does not guarantee that the index is greater than ω, and vice versa. Thus it is usually preferable to obtain an interval estimate, for which we can assert with a reasonable degree of certainty that it contains the true PCI value. However, the construction of such an interval estimate is not an easy task, since the distributions of the commonly used PCI estimators are usually quite complicated [4–8].
Therefore it is very natural to consider a Bayesian approach. By a Bayesian approach, it means that we first specify a prior distribution for the parameter of interest, obtain the posterior distribution of the parameter and then infer about the parameter by only using its posterior distribution given the observations. The reason why it is natural to consider a Bayesian approach is that for Bayesian estimation it is always very easy to obtain the posterior distribution when a prior distribution is given; and even when the form of the posterior distribution is complicated, it is still easy
CCC 0748–8017/99/050369–10$17.50 Received 10 October 1998
to use numerical methods or Monte Carlo methods [9] to obtain an approximate point estimate or interval estimate. This is a great advantage of the Bayesian approach over the classical frequentist approach.
More specifically, to assess the process capability, it is natural to consider the posterior probability Pr{process is capable|x}. Compared with the usual practice of just obtaining point estimates of PCIs, this Bayesian approach has the advantage of providing a statement on the posterior probability that the process is capable given the observed process data.
A nice Bayesian procedure for assessing process capability was proposed in Reference [5] for the index
Cp, also in Reference [4] for the index Cpmunder the assumption that the process mean µ is equal to the target value T . In general, Cpmis a better PCI than Cp [4]. However, the restriction that µ = T is a notable shortcoming, since the process mean may be quite deviated from the target value T in many industrial applications.
The main objective of this paper is to provide a Bayesian procedure for the general situation—no restriction on the process mean µ. In addition, for the restricted case in which µ = T , we provide a simple procedure for computing the posterior probability of the process being capable. Instead of using approximation or numerical integration as in Reference [4], this posterior probability can be obtained by simply looking up the commonly available chi-square tables. A similar Bayesian procedure was given in Reference [10] for the restricted case.
Throughout this paper it is assumed that the process measurements are independent and identically distributed from a normal distribution. In other words, the process is under statistical control. We remark that estimation of PCIs is meaningful only when the process is under statistical control.
This paper is organized as follows. We give a brief review on four popular PCIs—Cp, Cpk, Cpm, and Cpmk—in Section 2. In Section 3 we present a Bayesian procedure for assessing the process capability based on Cpm. All the derivations are given in the Appendix. In Section4we describe a Bayesian procedure based on Cpk, but only for the special case in which the process mean is equal to the midpoint of the two specification limits. In Section 5 we present some examples to illustrate the Bayesian procedure, and compare the results with those obtained from the procedure given in Reference [4]. Finally we conclude the paper in Section6.
2. A REVIEW ON SOME POPULAR PCIs
The index Cpis defined as
Cp=
USL− LSL 6σ
where USL and LSL denote the upper and lower specification limits respectively and σ is the process standard deviation of the quality characteristic of interest. The process standard deviation is usually unknown and can be estimated from a sample of
n measurements x1, x2, . . . , xn. The most common
estimate of σ is the sample standard deviation
s= 1 n− 1 n X i=1 (xi− ¯x)2 1/2 where ¯x = 1 n n X i=1 xi
is the sample mean. This gives an estimate of Cp,
ˆCp=
USL− LSL 6s
We remark that other estimates of σ can be used. For example, it is very common to use subgroup ranges to obtain an estimate of σ to guard against shifting of the mean in practice, since many processes in the industry may be just semi-stable.
In order to reflect the impact of the deviation of the process mean µ from the midpoint m of the specification limits on the process capability, several indices have been proposed, including
CPU=USL− µ 3σ CPL=µ− LSL 3σ and Cpk= min(CPL, CPU) (1) Cpkis sometimes defined as Cpk = (1 − k)Cp (2) where k = 2|m − µ|/(USL−LSL). The above two definitions of Cpk,(1)and(2), are algebraically equivalent [2].
These indices are usually estimated respectively by d CPU=USL− ¯x 3s d CPL= ¯x − LSL 3s
and
ˆCpk= min( dCPL, dCPU)
For the Cpkdefined in(2), it can be estimated by
ˆCpk= (1 − ˆk) ˆCp where
ˆk = 2|m − ¯x| USL− LSL
Both Cpand Cpkare independent of the target value
T . To account for the impact of the deviation of the
process mean from the target value, another PCI called
Cpmis defined [4] as Cpm= USL− LSL 6σ0 (3) where σ0 = [E(X − T )2]1/2 = [σ2+ (µ − T )2]1/2 (4) Chan et al. [4] estimated σ0 by
1 n− 1 n X i=1 (xi− T )2 1/2
In this paper, instead of using their estimator, we use ˆσ0 = 1 n n X i=1 (xi− T )2 1/2
to estimate σ0. The reason we use this estimator is that 1 n n X i=1 (xi− T )2
is both an unbiased estimator and the maximum likelihood estimator for σ02. The resulting estimator of Cpmis
ˆCpm=
USL− LSL 6ˆσ0
From(3)and(4), it is easy to see that Cpmand Cp have the relationship
Cpm=
Cp
r
1+ T−µσ 2
(5)
and the relationship between ˆCpmand ˆCpis
ˆCpm= ˆCp r n−1 n + T−x s 2
Thus by(5)it is clear that Cpm= Cpwhen µ= T . Combining the ideas of Cpk and Cpm, Pearn et al. [11] proposed another index called Cpmkdefined as
Cpmk= min ( USL− µ 3pσ2+ (µ − T )2 , µ− LSL 3pσ2+ (µ − T )2 )
The estimator bCpmk can be obtained by plugging in x for µ and s for σ . The study of this index is beyond the scope of this study.
There have been some studies on the distributions of these PCIs. When the process measurements follow a normal distribution, both dCPL and dCPU have a probability density function proportional to a non-central t distribution [4,6]. Chou and Owen [7] gave the exact distribution of ˆCpk, distribution mean, variance, and mean-squared error. Another interpretation for the distribution of ˆCpk was given in Reference [8], where it was shown that the distribution of ˆCpk is related to the folded normal distribution. Many properties of Cpm and ˆCpm were given in Reference [4]. More distributional and estimation properties for the above PCIs were given in Reference [11]. These studies indicated that the statistical distributions associated with these PCI estimators are quite complicated.
In the next section we derive a Bayesian interval estimate for Cpmand propose accordingly a Bayesian procedure for process capability assessment. Other approaches to obtaining interval estimates for PCIs have been suggested in the literature. For example, Bittanti et al. [12] suggested a curve-fitting approach based on the Pearson system of curves for PCI estimation, followed by application of the bootstrap to obtain an interval estimate. Their method is applicable to non-normal processes, but with fairly high computational cost.
The following two sections are more mathemat-ically/statistically involved. The proposed Bayesian procedure is illustrated by examples in Section 5. Readers who are not interested in the derivation of the procedure may skip to Section5.
3. A BAYESIAN PROCEDURE BASED ON Cpm
Cheng and Spiring [5] proposed a Bayesian approach for assessing process capability by finding a credible interval for the index Cp. A 100p% credible interval is the Bayesian analogue of the classical 100p% confidence interval, where p is a number between
0 and 1, say 0.95 for 95% confidence interval. It covers 100p% of the posterior distribution of the parameter [13]. Chan et al. [4] used the same approach to find an exact and an approximate credible interval for the index Cpm when µ = T . Without assuming
µ = T , we present a Bayesian procedure based on Cpmin this section.
Assume that the measurements{Xi, i = 1, . . . , n}
of the quality characteristic obtained from the process are independent and identically distributed (i.i.d.) from
N (µ, σ2). Denote x = (x1, x2, . . . , xn)T, where xi is
the observed value of Xi, i = 1, . . . , n. Then the
likelihood function for µ and σ is
L(µ, σ|x) = (2πσ2)−n/2 × exp − Pn i=1(xi− µ)2 2σ2 ! (6) For the Bayesian approach the first step is to find an appropriate prior. Usually, when there is little or no prior information, we use non-informative priors. When 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 information about the parameter as little as possible. See Reference [14] for more details. Also, with the reference prior the 100p% credible interval has the coverage probability close to
p up to the second order—in contrast to the first order
for any other priors—in the frequentist sense [15]. More specifically, the credible interval obtained from a non-informative prior has a more precise coverage probability than that obtained from any other priors.
However, when there is more than one parameter, it is not always possible to find the reference prior by maximizing the information difference. For this reason, Berger and Bernardo [16] suggested a step-by-step procedure for finding a multiparameter prior. In this paper we adopt this step-by-step procedure and the resulting prior is
π(µ, σ )= 1/σ 0 < σ < ∞, −∞ < µ < ∞
(7) As derived in the Appendix, the posterior probabil-ity densprobabil-ity function (PDF) of (µ, σ ) is
f (µ, σ|x) = √ 2n √ π 0(α)βα ! σ−(n+1) × exp − Pn i=1(xi− µ)2 2σ2 ! (8) where α= (n−1)/2 and β = [Pni=1(xi− ¯x)2/2]−1=
[(n − 1)s2/2]−1. A reparametrized version of(7)and (8)with σ replaced by σ2can be found in Problem 16 of Chap. 4 in Reference [13].
As mentioned before, it is natural to consider the quantity Pr{process is capable|x} in the Bayesian approach. Since the index Cpm is our major concern in this paper, we are interested in finding the posterior probability p = Pr{Cpm > ω|x} for some fixed positive number ω. Denote δ= |T − ¯x|/s. It is derived in the Appendix that
p= Z t 0 1 0(α)γαyα+1 exp − 1 γ y
× [8(b1(y)+ b2(y))− 8(b1(y)− b2(y))] dy (9) where t = 2 n ˆCpm ω !2 γ = 1 + n n− 1δ 2 b1(y)= s 2 y δ2 δ2+n− 1 n 1/2 b2(y)= √ n t y − 1 1/2
Note that the posterior probability p depends on n,
δ, ω and ˆCpmonly through n, δ and ˆCpm/ω. Denote
C∗= ˆCpm/ω.
From expression (9) we can see that it is very difficult to compute p for any process either on-line or off-on-line in practice without serious computer programming. However, by noticing that there is a one-to-one correspondence between p and C∗when n and δ are given, and by the fact that ˆCpmcan be easily 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 practice to assess the process capability. Denote this minimum value by C∗(p).
For users’ convenience in applying our Bayesian procedure in practice, we tabulate C∗(p) (for various
values of n and δ = |T − ¯x| /s) in Tables 1(a)– 1(c)for p = 0.90, 0.95, and 0.99 respectively. More specifically, the entries in these tables are values of
C∗(p) such that P Cpm> ˆCpm C∗(p) x ! = p (10)
Table 1(a). Values of C∗(p) for p= 0.90 |T − ¯x|/s n 0 0.5 1 1.5 2 5 2.3863 2.1643 1.8222 1.5857 1.4394 10 1.6326 1.5466 1.4068 1.3036 1.2360 15 1.4386 1.3856 1.2954 1.2255 1.1777 20 1.3464 1.3082 1.2406 1.1859 1.1477 25 1.2915 1.2617 1.2070 1.1613 1.1287 30 1.2546 1.2302 1.1838 1.1440 1.1154 35 1.2279 1.2072 1.1667 1.1312 1.1055 40 1.2075 1.1895 1.1534 1.1212 1.0976 45 1.1913 1.1754 1.1427 1.1131 1.0913 50 1.1781 1.1639 1.1339 1.1064 1.0860 55 1.1672 1.1542 1.1265 1.1008 1.0815 60 1.1578 1.1460 1.1201 1.0958 1.0776 65 1.1498 1.1389 1.1145 1.0915 1.0743 70 1.1428 1.1326 1.1097 1.0878 1.0713 75 1.1366 1.1271 1.1054 1.0845 1.0686 80 1.1312 1.1222 1.1015 1.0815 1.0663 85 1.1262 1.1178 1.0981 1.0788 1.0641 90 1.1217 1.1138 1.0949 1.0763 1.0621 95 1.1178 1.1102 1.0920 1.0741 1.0603 100 1.1141 1.1068 1.0894 1.0720 1.0587 110 1.1075 1.1010 1.0846 1.0684 1.0557 120 1.1020 1.0960 1.0806 1.0651 1.0532 130 1.0972 1.0916 1.0771 1.0624 1.0509 140 1.0929 1.0877 1.0739 1.0599 1.0490 150 1.0892 1.0843 1.0712 1.0577 1.0472 160 1.0859 1.0812 1.0687 1.0557 1.0456 170 1.0828 1.0785 1.0664 1.0539 1.0442 180 1.0801 1.0759 1.0644 1.0523 1.0429 190 1.0776 1.0736 1.0625 1.0509 1.0416 200 1.0753 1.0715 1.0608 1.0494 1.0405 210 1.0733 1.0695 1.0592 1.0482 1.0395 220 1.0712 1.0678 1.0577 1.0470 1.0386 230 1.0694 1.0661 1.0563 1.0459 1.0377 240 1.0678 1.0645 1.0550 1.0449 1.0369 250 1.0662 1.0631 1.0538 1.0439 1.0361 260 1.0647 1.0617 1.0527 1.0430 1.0353 270 1.0634 1.0605 1.0516 1.0421 1.0346 280 1.0621 1.0592 1.0507 1.0414 1.0339 290 1.0608 1.0581 1.0497 1.0406 1.0334 300 1.0597 1.0571 1.0488 1.0398 1.0328
Table 1(b). Values of C∗(p) for p= 0.95
|T − ¯x|/s n 0 0.5 1 1.5 2 5 2.9272 2.6268 2.1584 1.8293 1.6234 10 1.8319 1.7209 1.5389 1.4033 1.3139 15 1.5687 1.5017 1.3862 1.2952 1.2330 20 1.4465 1.3989 1.3127 1.2420 1.1925 25 1.3746 1.3377 1.2682 1.2092 1.1672 30 1.3265 1.2965 1.2377 1.1865 1.1496 35 1.2919 1.2665 1.2153 1.1697 1.1365 40 1.2655 1.2435 1.1979 1.1565 1.1262 45 1.2447 1.2254 1.1840 1.1460 1.1179 50 1.2277 1.2105 1.1726 1.1372 1.1110 55 1.2136 1.1979 1.1629 1.1298 1.1051 60 1.2017 1.1873 1.1547 1.1235 1.1001 65 1.1914 1.1782 1.1475 1.1179 1.0958 70 1.1824 1.1702 1.1412 1.1131 1.0918 75 1.1745 1.1631 1.1356 1.1088 1.0884 80 1.1674 1.1568 1.1306 1.1049 1.0853 85 1.1612 1.1511 1.1261 1.1014 1.0825 90 1.1554 1.1460 1.1221 1.0982 1.0800 95 1.1503 1.1413 1.1183 1.0953 1.0777 100 1.1456 1.1370 1.1149 1.0926 1.0755 110 1.1373 1.1294 1.1089 1.0879 1.0717 120 1.1302 1.1230 1.1036 1.0838 1.0685 130 1.1240 1.1174 1.0990 1.0802 1.0655 140 1.1187 1.1125 1.0950 1.0771 1.0630 150 1.1139 1.1080 1.0914 1.0742 1.0607 160 1.1097 1.1041 1.0882 1.0717 1.0586 170 1.1058 1.1005 1.0853 1.0693 1.0568 180 1.1023 1.0974 1.0827 1.0673 1.0550 190 1.0991 1.0944 1.0802 1.0653 1.0536 200 1.0961 1.0916 1.0781 1.0636 1.0521 210 1.0935 1.0892 1.0760 1.0619 1.0508 220 1.0910 1.0869 1.0741 1.0604 1.0496 230 1.0887 1.0848 1.0723 1.0590 1.0484 240 1.0866 1.0828 1.0706 1.0576 1.0473 250 1.0846 1.0809 1.0691 1.0564 1.0463 260 1.0827 1.0791 1.0677 1.0552 1.0454 270 1.0809 1.0775 1.0663 1.0542 1.0445 280 1.0793 1.0759 1.0650 1.0531 1.0437 290 1.0778 1.0745 1.0638 1.0521 1.0429 300 1.0763 1.0731 1.0627 1.0513 1.0421
Table 1(c). Values of C∗(p) for p= 0.99 |T − ¯x|/s n 0 0.5 1 1.5 2 5 4.5430 4.0165 3.1800 2.5761 2.1891 10 2.3203 2.1454 1.8567 1.6404 1.4974 15 1.8678 1.7660 1.5891 1.4496 1.3541 20 1.6689 1.5981 1.4685 1.3618 1.2872 25 1.5550 1.5011 1.3976 1.3096 1.2469 30 1.4804 1.4371 1.3501 1.2741 1.2194 35 1.4272 1.3910 1.3155 1.2482 1.1993 40 1.3871 1.3561 1.2890 1.2281 1.1836 45 1.3557 1.3284 1.2680 1.2122 1.1710 50 1.3303 1.3060 1.2508 1.1991 1.1607 55 1.3092 1.2874 1.2363 1.1879 1.1519 60 1.2914 1.2715 1.2239 1.1785 1.1445 65 1.2762 1.2579 1.2133 1.1703 1.1380 70 1.2629 1.2460 1.2039 1.1630 1.1322 75 1.2513 1.2356 1.1957 1.1567 1.1271 80 1.2409 1.2262 1.1884 1.1510 1.1226 85 1.2317 1.2179 1.1817 1.1459 1.1185 90 1.2233 1.2103 1.1756 1.1412 1.1148 95 1.2158 1.2035 1.1702 1.1369 1.1114 100 1.2089 1.1972 1.1652 1.1330 1.1083 110 1.1969 1.1861 1.1564 1.1260 1.1027 120 1.1865 1.1767 1.1487 1.1201 1.0979 130 1.1775 1.1684 1.1421 1.1148 1.0937 140 1.1697 1.1613 1.1363 1.1102 1.0900 150 1.1628 1.1549 1.1310 1.1061 1.0867 160 1.1566 1.1491 1.1263 1.1025 1.0837 170 1.1511 1.1440 1.1221 1.0990 1.0810 180 1.1460 1.1393 1.1183 1.0960 1.0786 190 1.1414 1.1350 1.1148 1.0933 1.0763 200 1.1372 1.1310 1.1116 1.0907 1.0743 210 1.1334 1.1275 1.1086 1.0883 1.0724 220 1.1298 1.1242 1.1058 1.0862 1.0706 230 1.1265 1.1210 1.1033 1.0841 1.0689 240 1.1234 1.1181 1.1009 1.0822 1.0674 250 1.1205 1.1154 1.0987 1.0804 1.0659 260 1.1178 1.1129 1.0966 1.0788 1.0646 270 1.1153 1.1106 1.0946 1.0772 1.0634 280 1.1129 1.1083 1.0928 1.0757 1.0621 290 1.1106 1.1062 1.0910 1.0743 1.0610 300 1.1085 1.1042 1.0893 1.0730 1.0599
We comment that the computations in creating these tables are rather involved and quite time-consuming.
According to Definition 3 on p. 102 of Refer-ence [13], we can see from(10)that [ ˆCpm/C∗(p),∞) is a 100p% credible interval for Cpm, which 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 or 1.33. When this happens, we have Pr{process is capable|x}> p. In other words, to see if a process is capable (with capability level
ω and confidence level p), we only need to check if
ˆCpm> ωC∗(p).
From these tables we observe that for each fixed p and n the value of C∗(p) decreases as δ increases.
This phenomenon can be explained by the following argument. For a fixed ˆCpm, since
ˆCpm= (USL − LSL) 6s r n− 1 n + δ 2 !
s becomes smaller when δ becomes larger, and a
smaller s 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
s. In this case 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. Another
observation from the tables is that the value of C∗(p)
decreases as n increases for fixed δ and p. This can also be explained by the same reasoning as above, since a larger n implies that ˆCpmis more accurate.
4. A BAYESIAN PROCEDURE FOR CpkWHEN
µ= m AND CpmWHEN µ= T
Owing to the complication of the distribution of ˆCpk, we can only discuss the special case in which µ =
m, where m is the midpoint of the two specification
limits. In this case, in fact, Cpkis reduced to Cp, since
Cpk = (d − |µ − m|)/3σ = d/3σ = Cp, where
ˆCpk = (USL−LSL)/6 ˜σ, where ˜σ = 1 n n X i=1 (xi − m)2 1/2
Suppose that the measurements are i.i.d. from
N (m, σ2). Then the likelihood function for σ is L(σ|x) = (2πσ2)−n/2 × exp − Pn i=1(xi− m)2 2σ2 !
Consider the non-informative reference prior
π(σ )= 1/σ 0 < σ < ∞
Then the posterior distribution of σ2 is an inverse Gamma distribution with the probability density function f (σ2|x) = 1 0(n/2) n˜σ2 2 !n/2 (σ2)−((n/2)+1) × exp −n˜σ2 2σ2 ! 0 < σ2< ∞
We remark that this posterior PDF is exactly the same as that of Cpm2 when µ= T , the case considered in Reference [4]. This is quite obvious, since the indices in both cases are reduced to the index Cp. Thus many results in Reference [4] for Cpm when
µ = T are applicable to Cpk when µ = m. Chan
et al. [4] tabulated approximate C∗(p) values for Cpm when µ= T . These values can be used for Cpkwhen
µ= m with some minor modification. However, there
is a more straightforward Bayesian procedure to assess the process capability in these two cases.
Let Y = n ˜σ2/2σ2. It can be derived easily that 2Y has a chi-square distribution with n degrees of freedom. Then the posterior probability of Cpk being greater than a value ω is
p= Pr{Cpk> ω|x} = Z a 0 f (σ|x) dσ = Z ∞ b 1 0(n/2)y (n/2)−1e−ydy= Pr{2Y > 2b}
where a= (USL−LSL)/6ω and b = (n/2)(ω/ ˆCpk)2. Thus, to compute p, we can use the commonly available chi-square tables. If p is greater than a desirable level, say 90% or 95%, then we may claim that the process is capable in a Bayesian sense with 90% or 95% confidence.
By the same nature, the Bayesian procedure based on Cpmunder the assumption µ = T is similar. Thus
we can summarize our Bayesian procedure for these two special cases as follows. Let C∗ = dPCI/ω, where d
PCI can be either ˆCpm or ˆCpk. Then the process is capable in a Bayesian sense with 100p% confidence if Pr{χ2
n > n(1/C∗)2} > p, where χn2 is a random
variable following the chi-square distribution with n degrees of freedom.
Note that ‘the degrees of freedom’ of the posterior distribution for Cpk when µ = m (or for Cpmwhen
µ = T ) are one more than those of the posterior
distribution for Cpgiven in Reference [5]. The reason is that σ is estimated by s in ˆCp, which uses an extra degree of freedom to estimate µ by x.
5. EXAMPLES AND DISCUSSION
In Section 3, we have derived a Bayesian process capability assessment procedure based on the index
Cpm. We have also provided tables (for various values of sample size n and off-target quantity δ = |T −
¯x|/s) of the minimum values C∗(p) of C∗ = ˆC
pm/ω required to ensure that the posterior probability p of the process being capable (i.e. p = P (Cpm >
ω| ˆCpm)) reaches the desirable confidence levels, such as 0.90, 0.95 and 0.99. With these tables the procedure is as simple as comparing ˆCpm with ω times the tabulated value C∗(p). If ˆCpm > ωC∗(p), then we claim that the process is capable in a Bayesian sense.
For example, when p= 0.9, n = 100 and δ = 0.5, we can find C∗(p) = 1.1068 from Table1(a). Thus, when ω is given, say ω = 4/3, the minimum ˆCpm required for the process to be capable is 1.1068× 4/3= 1.4757. That is, if ˆCpmis greater than 1.4757, we say that the process is capable in a Bayesian sense. For the special case in which µ= T , as described in Section4, we do not even need to use the tables given in Section 3. We need only look up the commonly available chi-square tables for the posterior probability
p of the process being capable (i.e. p = Pr{χn2 >
n(1/C∗)2}, with C∗ = ˆCpm/ω) and then judge the process capability by comparing this posterior probability with the desirable confidence level, say 0.95. In this case, if p > 0.95, then we may claim that the process is capable in a Bayesian sense with 95% confidence.
We first illustrate our procedure via an example given in Reference [4], which was first given in Reference [2]. In this example the measurements were taken on the radial length of machined holes with upper and lower specification limits of 20 and −20 units respectively and target value T = 0.
Table 2. Results of machined holes example Radial Length (×103inches)
Stage n ¯x s |T − ¯x|/s ˆCpm PT P 1 201 4.7 8.7 0.5402 0.67 0.0000 0.0000 2 96 10.4 21.1 0.4929 0.28 0.0000 0.0000 3 316 5.0 5.4 0.9259 0.91 0.0067 0.0032
to illustrate the two Bayesian procedures—the one proposed in Reference [4] and the Bayesian procedure proposed in this paper. Take ω= 1, which means that the process is capable if Cpm> 1. We summarize the results of this example in Table2, where P denotes Pr{Cpm > ω| ˆCpm} given in (9) and PT denotes the approximate posterior probability obtained in Reference [4].
From Table 2, we see that both PT and P are very small for all three stages, indicating that the process is incapable. Both Bayesian procedures have the same conclusion as the traditional procedure, since the values of ˆCpmare smaller than 1 in all three stages. At first glance, it is a little bit surprising to see that at stage 3, PT and P are so small for ω = 1 even when ˆCpm= 0.91. This can be explained by the high precision of the estimate ˆCpmresulting from the large sample size n = 316. Both posterior probabilities are too small in this example to show the effect of
δ= |T − ¯x| /s on PTand P .
Next we describe some scenarios to illustrate the effect of δ. Table 3gives two cases that are capable from the traditional point of view, i.e. ˆCpm > 1. First we notice that as δ increases, P increases while PT remains the same. This is because PT neglects the deviation of the process mean from the target value. From Table3, we observe that for both cases, if we require the posterior probability of the process being capable to be greater than 0.90, then the Bayesian procedure proposed in Reference [4] will claim that the process is incapable for all δ, while our procedure will claim that the process is capable when δ≥ 1.0.
On the other hand, let us compare the minimum values of ˆCpmin Tables1(a)–1(c)with the minimum values of Table 2(a) in Reference [4]. For δ = 0 the Bayesian procedure we proposed has the minimum values of ˆCpm larger than those values given in Reference [4], since we do not assume the known information µ= T . However, when δ gets larger, say
δ≥ 1.0, the minimum ˆCpmrequired for our procedure is less than that of Reference [4], which indicates that our procedure is more sensitive in claiming the process is capable.
Table 3. Results of examples in comparing PTand P .
Case n δ ˆCpm PT P 1 100 0.0 1.09 0.8858 0.8555 0.5 0.8858 0.8730 1.0 0.8858 0.9148 1.5 0.8858 0.9550 2.0 0.8858 0.9806 2 300 0.0 1.05 0.8826 0.8655 0.5 0.8826 0.8773 1.0 0.8826 0.9132 1.5 0.8826 0.9519 2.0 0.8826 0.9782
Finally we give a simple example to show how to use the tables in practice. Suppose a sample of size n = 50 is collected from a process and the data give that ˆCpm = 1.12 and |T − ¯x| /s = 1. Consider ω = 1 and p = 0.95. From Table 1(b), we find that C∗(p) = 1.1726, which implies that
the minimum value of ˆCpm—equal to ωC∗(p)— is 1.1726. Since 1.12 < 1.1726, we claim that this process is incapable in a Bayesian sense with 95% confidence. This shows that our procedure can differentiate processes with different δ values, which is definitely a desirable property for a process capability assessment procedure.
6. CONCLUSION
The index Cpm was proposed to take into account the departure of the process mean from the target value as well as the magnitude of the process variation [4]. However, the statistical distribution associated with its estimator ˆCpm is so complicated that it is very difficult to obtain an interval estimation of Cpm. Under a non-informative prior we obtain a simple Bayesian procedure for process capability assessment that provides a Bayesian credible interval estimation for Cpm. Thus this Bayesian procedure can serve as an alternative to the classical procedures in process capability assessment.
ACKNOWLEDGEMENTS
The authors would like to thank a chief editor, two referees and Professor Sheng-Tsaing Tseng of National Tsing Hua University for encouraging and helpful comments and suggestions. Jyh-Jen Shiau would also like to express her gratitude to the Department of Statistics, Stanford University for the hospitality. The revision of the paper was done during her visit there. This research was supported in part by the National Science Council of the Republic of China under grants 2118-M-009-003 and NSC87-2118-M-009-004.
APPENDIX
In this appendix we derive (8) and (9) given in Section3.
Derivation of (8)
From (6) and (7), we have the posterior PDF of
(µ, σ ) as f (µ, σ|x) ∝ L(µ, σ|x) × π(µ, σ) ∝ σ−(n+1)exp − Pn i=1(xi− µ)2 2σ2 ! . (11) Also Z ∞ 0 Z ∞ −∞σ −(n+1)exp −Pni=1(xi− µ)2 2σ2 ! dµ dσ = Z ∞ 0 σ−(n+1)exp − 1 βσ2 × Z ∞ −∞exp − n(µ− ¯x)2 2σ2 ! dµ dσ = r π 2n0 (α) β α (12) where α= (n−1)/2 and β = [Pni=1(xi− ¯x)2/2]−1= [(n − 1)s2/2]−1.
Then from(11)and (12)the posterior PDF (8)is obtained.
Derivation of (9)
Recall that σ02= σ2+ (µ − T )2and observe that ˆσ02= 1 n n X i=1 (xi− T )2 = 1 n Xn i=1 (xi− ¯x)2+ n( ¯x − T )2 .
Denote a= (USL−LSL)/6ω and g(σ) =√a2− σ2. Then p= Pr{Cpm> ω|x} = Pr USL− LSL 6σ0 > ω x = Pr{σ2+ (µ − T )2< a2|x} = Z a 0 Z T+√a2−σ2 T−√a2−σ2 f (µ, σ|x) dµ dσ = Z a 0 Z T+g(σ ) T−g(σ ) 2√n √ 2π 0(α)βασ −(n+1) × exp − Pn i=1(xi− µ)2 2σ2 ! dµ dσ = Z a 0 2√n √ 2π0(α)βασ −(n+1)exp− 1 βσ2 × Z T+g(σ ) T−g(σ ) exp −n(µ− ¯x) 2 2σ2 ! dµ dσ = Z a 0 2σ−n 0(α)βα exp − 1 βσ2 × 8 T − ¯x + g(σ) σ/√n −8 T − ¯x − g(σ) σ/√n dσ (13)
where 8 is the cumulative distribution function of the standard normal distribution.
Let β0= (Pni=1(xi−T )2/2)−1and y= β0σ2. Then
T − ¯x σ/√n = 1 √y(n(T − ¯x)2β0)1/2 =√1y 2n(T − ¯x) 2 Pn i=1(xi− T )2 !1/2 = s 2 y n(T − ¯x)2 Pn i=1(xi− ¯x)2+ n(T − ¯x)2 !1/2 = s 2 y δ2 δ2+n− 1 n 1/2 and g(σ ) σ/√n = n(a2− σ2) σ2 !1/2 =√n β 0a2 y − 1 !1/2 =√n t y − 1 1/2 where t = β0a2.
Let b1(y)= s 2 y δ2 δ2+n− 1 n 1/2 b2(y)= √ n t y − 1 1/2 γ =β β 0 = 1 + n n− 1δ 2 Observe that t= β0a2= 2 nˆσ02 USL− LSL 6ω 2 = 2 USL− LSL 6ˆσ0 2 nω2= 2 ˆC 2 pm nω2 Then(13)can be simplified to
p= Z t 0 1 0(α)γαyα+1 exp −1 γ y
× [8(b1(y)+ b2(y))− 8(b1(y)− b2(y))] dy as given in(9).
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Authors’ biographies:
Jyh-Jen Horng Shiau received her BS in mathematics from
National Taiwan University, MS in applied mathematics from the University of Maryland Baltimore County, MS in computer science and PhD. in statistics from the University of Wisconsin-Madison. She taught at Southern Methodist University, the University of Missouri at Columbia and National Tsing Hua University and worked for the Engineering Research Center of AT&T Bell Labs before she moved to Hsinchu, Taiwan where she is an associate professor in the Institute of Statistics at National Chiao Tung University. Her interests include industrial statistics and non-parametric function estimation. She is a member of the Institute of Mathematical Statistics.
Chun-Ta Chiang received his BS in mathematics from
National Tsing Hua University and MS in statistics from National Chiao Tung University. He is a quality engineer in the Department of Quality Management Technology, Center for Aviation and Space Technology, Industrial Technology Research Institute. His interests include industrial statistics, quality engineering and quality management.
Hui-Nien Hung received his BS in mathematics from
National Taiwan University, MS in mathematics from National Tsing-Hua University and PhD. in statistics from the University of Chicago. He is an associate professor in the Institute of Statistics at National Chiao Tung University. His interests include statistical inference, statistical computing and industrial statistics.
