Qual. Reliab. Engng. Int. 2005; 21:655–668
Published online 10 March 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/qre.681
Research
A Bayesian Approach to Obtain
a Lower Bound for the
C
pm
Capability Index
G. H. Lin1,∗,†, W. L. Pearn2and Y. S. Yang3
1Department of Transportation and Logistics Management, National Penghu Institute of Technology, Taiwan, Republic of
China
2Department of Industrial Engineering and Management, National Chiao Tung, University, Taiwan, Republic of China 3Department of Industrial Engineering, Dayeh University, Taiwan, Republic of China
The Taguchi capability indexCpm, which incorporates the departure of the process mean from the target value, has been proposed to the manufacturing industry for measuring manufacturing capability. A Bayesian procedure has been considered for testing process performance assuming µ = T , which was generalized without assuming µ = T . Statistical properties of the natural estimator of the index Cpm for normal processes have been investigated extensively. However, the investigation was restricted to processes with symmetric tolerances. Recently, a generalizedCpm, referred to as Cpm, was proposed to cover processes with asymmetric tolerances. Under the normality assumption, the statistical properties of the estimated Cpm including the exact sampling distribution, therth moment, expected value, variance, and the mean-squared error were obtained. In this paper, we use a Bayesian approach to obtain the interval estimation for the generalized Taguchi capability indexCpm. Consequently, the manufacturing capability testing can be performed for quality assurance. Copyright c 2005 John Wiley & Sons, Ltd.
KEY WORDS: asymmetric tolerances; process capability index
1.
INTRODUCTION
P
rocess capability indices (PCIs), whose purpose is to provide numerical measures on whether a manufacturing process is capable of reproducing items satisfying the quality requirements preset by the engineer (or the product designer), have recently been the research focus in quality assurance and engineering statistics literature. Examples include Boyles1, Bordignon and Scagliarini2, Borges and Ho3, Changet al.4, Hoffman5, Nahar et al.6, Noorossana7, Pearn et al.8, Pearn and Lin9, Zimmer et al.10, Lee et al.11. Kotz and Johnson12 and Spiring et al.13 provided a rather complete list of the PCI research papers from the
∗Correspondence to: G. H. Lin, Department of Transportation and Logistics Management, National Penghu Institute of Technology,
300 Liu-Ho Road, Makung, Penghu, Taiwan 88042, Republic of China.
†E-mail: ghlin@npit.edu.tw
past 10 years. The two basic capability indices Cpand Cpk, are defined in the following (Kane14): Cp= USL− LSL 6σ (1) Cpk= min USL− µ 3σ , µ− LSL 3σ (2) where USL and LSL are the upper and the lower specification limits, respectively, µ is the process mean, and σ is the process standard deviation. The index Cp only reflects the magnitude of the process variation
relative to the specification tolerance, and is therefore used to measure process precision. The index Cpk takes
into account process variation as well as the location of the process mean. The natural estimators of Cp and
Cpk can be obtained by substituting the sample mean ¯X=ni=1Xi/nfor µ and the sample variance Sn2−1= n
i=1(Xi− ¯X)2/(n− 1) for σ2in expressions (1) and (2). Chou and Owen15, Pearn et al.16and Kotz et al.17 investigated the statistical properties and the sampling distributions of the natural estimators of Cpand Cpk.
Boyles1noted that Cpkis a yield-based index. In fact, the design of Cpkis independent of the target value T ,
which can fail to account for process targeting (the ability to cluster around the target). For this reason, Chan et al.18introduced the index C
pmto take the process targeting issue into consideration. The index Cpmis defined
as follows:
Cpm=
USL− LSL
6σ2+ (µ − T )2 (3)
We note that the index Cpm is not originally designed to provide an exact measure on the number of
non-conforming items. However, Cpm includes the process departure (µ− T )2 (rather than 6σ alone) in the
denominator of the definition to reflect the degree of process targeting. Chan et al.18 proposed a Bayesian procedure under the restriction that µ= T , and investigated the statistical properties of the sampling distribution of Cpm. Boyles1suggested a maximum likelihood estimator (MLE) of Cpm by substituting the sample mean
¯X =n
i=1Xi/nfor µ and the MLE of σ2, Sn2= n
i=1(Xi− ¯X)2/n, in expression (3). Based on the suggested MLE of Cpm, Boyles1proposed two approximate 100(1− α)% lower confidence bounds using the normal and
chi-square distributions for Cpmfrom the distribution frequency point of view. Pearn et al.16 investigated the
statistical properties of the MLE estimator of Cpm. Shiau et al.19proposed a Bayesian procedure based on the
MLE of Cpmwithout the restriction µ= T on the process mean µ. Their results generalized those discussed in
Chan et al.18.
Pearn et al.16 proposed the process capability index Cpmk, which combines the merits of the three earlier
indices Cp, Cpkand Cpm. The index Cpmkalerts the user when the process variance increases and/or the process
mean deviates from its target value, which is designed to monitor the normal and the near-normal processes. The index Cpmk, referred to as the third-generation capability index and is defined as follows:
Cpmk= min USL− µ 3σ2+ (µ − T )2, µ− LSL 3σ2+ (µ − T )2 (4) We remark that those indices presented above are designed to monitor the performance for normal and near-normal processes with symmetric tolerances, which are shown to be inappropriate for cases with asymmetric tolerances.
2.
A GENERALIZATION OF
C
pmFOR ASYMMETRIC TOLERANCES
A process is said to have asymmetric tolerances if the upper tolerance, dU= USL − T , is unequal to the
with asymmetric tolerances, Chen et al.20 considered a generalization of the Taguchi capability index Cpm.
The generalization, referred to as Cpm , is defined as follows:
Cpm = d
∗
3√σ2+ A2 (5)
where d∗= min{dU, dL}, A = max{d(µ − T )/dU, d(T − µ)/dL}. Clearly, if the preset target value T = m =
(USL+ LSL)/2 (symmetric case), then d∗= d = (USL − LSL)/2, A = |µ − T |, and the generalization Cpm reduces to the original index Cpm. The factor A in the definition ensures that the generalization Cpm obtains its
maximal value at µ= T (process is on-target) regardless of whether the tolerances are symmetric (T = m) or asymmetric (T = m). An estimator of Cpm considered by Chen et al.20is defined as
ˆC pm= d∗ 3 S2 n+ ˆA2 (6)
where ˆA= max{d( ¯X − T )/dU, d(T − ¯X)/dL} and Sn2= n
i=1(Xi − ¯X)2/n. We note that if the production tolerance is symmetric, then ˆAmay be simplified as| ¯X − T | and the estimator ˆCpmbecomes the MLE of Cpm
discussed by Boyles1. Chen et al.20investigated the statistical properties of the estimated Cpm . They obtained
the exact distribution and the formulae for the rth moment, expected value, variance, and the mean-squared error under the normality assumption.
We note that the natural estimator ˆCpm can be rewritten as ˆC
pm=
C
3√K+ Y (7)
where C= n1/2d∗/σ, K= nSn2/σ2, and Y= [max{(d/dU)Z,−(d/dL)Z}]2with Z= n1/2( ¯X− T )/σ. On the
assumption of normality, the statistic K is distributed as χn2−1, Z is distributed as N (δ0,1), δ0= n1/2(µ− T )/σ,
and the probability density function of Y is
fY(y)= 1 2√y 1 d1 fZ(−√y/d1)+ 1 d2 fZ(√y/d2) , y >0 (8)
where d1= d/dLand d2= d/dU. Therefore, the probability density function of ˆCpm can be expressed as
fˆC pm(x)= C3 27x4 1 0 1 √ tfK C2(1− t) 9x2 1 d1 fZ −C √ t 3xd1 + 1 d2 fZ C√t 3xd2 dt, x >0 (9)
Chen et al.20 showed that the statistic Z2follows a non-central chi-square distribution with one degree of freedom and non-centrality parameter δ02. The distribution of Y is a weighted non-central chi-square distribution with one degree of freedom and non-centrality parameter δ02, under the assumption of normality. The probability density function of Y , in an alternative form, may be expressed as
fY(y)= e−λ/2 2√π ∞ j=0 (√2δ0)j j! 1+ j 2 2 i=1 (−1)ij di2 fYj y/di2 , y >0 (10)
Table I. C∗(p∗)for p∗= 0.90 with d/dL= 5/6 and d/dU= 5/4 δ= ( ¯x − T )/sn−1 n 0.0 0.5 1.0 1.5 2.0 5 2.4149 2.1576 1.7691 1.5361 1.4014 10 1.6450 1.5605 1.3998 1.2933 1.2270 15 1.4464 1.4006 1.2968 1.2223 1.1740 20 1.3520 1.3227 1.2448 1.1854 1.1460 25 1.2959 1.2754 1.2123 1.1620 1.1280 30 1.2582 1.2431 1.1897 1.1455 1.1153 35 1.2309 1.2195 1.1728 1.1331 1.1057 40 1.2100 1.2012 1.1596 1.2330 1.0981 45 1.1935 1.1866 1.1489 1.1154 1.0919 50 1.1801 1.1747 1.1401 1.1087 1.0867 55 1.1689 1.1646 1.1326 1.1031 1.0823 60 1.1594 1.1560 1.1261 1.0983 1.0785 65 1.1513 1.1486 1.1205 1.0940 1.0752 70 1.1441 1.1420 1.1156 1.0903 1.0722 75 1.1379 1.1362 1.1117 1.0870 1.0696 80 1.1323 1.1311 1.1072 1.0840 1.0672 85 1.1273 1.1264 1.1036 1.0812 1.0651 90 1.1228 1.1222 1.1004 1.0788 1.0632 95 1.1187 1.1184 1.0974 1.0765 1.0614 100 1.1149 1.1149 1.0947 1.0744 1.0597 110 1.1083 1.1087 1.0899 1.0707 1.0568 120 1.1027 1.1033 1.0857 1.0675 1.0542 130 1.0978 1.0987 1.0820 1.0647 1.0520 140 1.0935 1.0946 1.0788 1.0622 1.0500 150 1.0897 1.0909 1.0759 1.0599 1.0482 160 1.0863 1.0877 1.0733 1.0579 1.0466 170 1.0833 1.0847 1.0709 1.0561 1.0452 180 1.0805 1.0820 1.0688 1.0544 1.0439 190 1.0780 1.0796 1.0668 1.0529 1.0426 200 1.0757 1.0773 1.0650 1.0515 1.0415
where λ= δ20and Yjis distributed as χ12+j. Therefore, the probability density function of ˆCpm , in an alternative form, can be expressed as
fˆC pm(x)= 21−n/2Cnx−(n+1) 3n ((n− 1)/2) e−λ/2 2√π ∞ j=0 1 j! δ0C 3x j × 2 i=1 (−1)ij di−(j+1) 1 0 (1− y)(n−3)/2y(j−1)/2exp −C2 18x2(1− y + d −2 i y) dy , x >0 (11)
3.
A BAYESIAN PROCEDURE BASED ON
C
pmMost existing PCI research works in testing the manufacturing capability are based on the traditional distribution frequency approach. Shiau et al.19proposed a Bayesian approach for assessing process capability by finding a 100p% credible interval, which covers 100p% of the posterior distribution for the index Cpm. Assuming that
Table II. C∗(p∗)for p∗= 0.95 with d/dL= 5/6 and d/dU= 5/4 δ= ( ¯x − T )/sn−1 n 0.0 0.5 1.0 1.5 2.0 5 2.9665 2.6081 2.0771 1.7539 1.5651 10 1.8483 1.7380 1.5293 1.3892 1.3014 15 1.5789 1.5215 1.3883 1.2912 1.2280 20 1.4537 1.4182 1.3186 1.2416 1.1902 25 1.3800 1.3561 1.2759 1.2104 1.1663 30 1.3309 1.3140 1.2458 1.1886 1.1494 35 1.2955 1.2832 1.2236 1.1722 1.1367 40 1.2686 1.2594 1.2063 1.1594 1.1268 45 1.2473 1.2405 1.1924 1.1490 1.1187 50 1.2301 1.2250 1.1808 1.1404 1.1119 55 1.2157 1.2119 1.1711 1.1331 1.1062 60 1.2035 1.2008 1.1627 1.1267 1.1013 65 1.1931 1.1912 1.1554 1.1212 1.0969 70 1.1839 1.1828 1.1490 1.1164 1.0931 75 1.1759 1.1753 1.1433 1.1121 1.0897 80 1.1688 1.1686 1.1381 1.1082 1.0866 85 1.1624 1.1626 1.1335 1.1046 1.0839 90 1.1566 1.1572 1.1293 1.1014 1.0813 95 1.1514 1.1523 1.1255 1.0985 1.0790 100 1.1466 1.1477 1.1220 1.0958 1.0769 110 1.1382 1.1397 1.1157 1.0910 1.0731 120 1.1310 1.1328 1.1103 1.0869 1.0698 130 1.1248 1.1268 1.1055 1.0832 1.0669 140 1.1193 1.1215 1.1014 1.0800 1.0643 150 1.1145 1.1168 1.0976 1.0771 1.0620 160 1.1102 1.1126 1.0943 1.0745 1.0600 170 1.1063 1.1088 1.0912 1.0722 1.0581 180 1.1027 1.1054 1.0884 1.0700 1.0564 190 1.0995 1.1022 1.0859 1.0680 1.0548 200 1.0966 1.0993 1.0836 1.0662 1.0534
{X1, X2, . . . , Xn} is a random sample taken from N(µ, σ2), a normal distribution with mean µ and variance σ2, Shiau et al.19 adopted the prior π(µ, σ )= 1/σ and derived the posterior probability density function f (µ, σ|x) of (µ, σ) as follows: f (µ, σ|x) = 2n π σ−(n+1) βα(α) exp − n i=1(xi− µ)2 2σ2 (12)
where x= (x1, x2, . . . , xn),−∞ < µ < ∞, 0 < σ < ∞, α = (n − 1)/2, β = 2[(n − 1)Sn2−1]−1. Given a pre-specified capability level ω > 0, the posterior probability based on index Cpm that a process with symmetric
tolerances is capable is given as (Shiau et al.19):
p= t
0
(b1(y)+ b2(y))− (b1(y)− b2(y))
γαyα+1(α) exp − 1 γ y dy (13)
where is the cumulative distribution function of the standard normal distribution, b1(y)=
√
(2/y)[1 − (1/γ )], b2(y)=√n[(t/y) − 1], δ = | ¯x − T |/sn−1, γ = 1 + [(nδ2)/(n− 1)], t = 2 ˆC2pm/(nω2),
Table III. C∗(p∗)for p∗= 0.975 with d/dL= 5/6 and d/dU= 5/4 δ= ( ¯x − T )/sn−1 n 0.0 0.5 1.0 1.5 2.0 5 3.6036 3.1265 2.4314 2.0040 1.7525 10 2.0572 1.9176 1.6584 1.4839 1.3745 15 1.7098 1.6390 1.4758 1.3566 1.2790 20 1.5521 1.5093 1.3878 1.2937 1.2311 25 1.4604 1.4321 1.3342 1.2548 1.2012 30 1.3997 1.3801 1.2973 1.2278 1.1804 35 1.3562 1.3423 1.2700 1.2077 1.1648 40 1.3232 1.3132 1.2488 1.1920 1.1526 45 1.2973 1.2901 1.2317 1.1793 1.1427 50 1.2763 1.2711 1.2177 1.1688 1.1345 55 1.2588 1.2552 1.2058 1.1599 1.1275 60 1.2440 1.2417 1.1956 1.1522 1.1215 65 1.2314 1.2300 1.1867 1.1455 1.1163 70 1.2203 1.2198 1.1789 1.1396 1.1116 75 1.2106 1.2107 1.1720 1.1344 1.1075 80 1.2020 1.2026 1.1658 1.1297 1.1038 85 1.1943 1.1954 1.1602 1.1254 1.1004 90 1.1873 1.1888 1.1551 1.1216 1.0974 95 1.1810 1.1828 1.1505 1.1180 1.0946 100 1.1753 1.1773 1.1462 1.1148 1.0920 110 1.1652 1.1677 1.1387 1.1090 1.0874 120 1.1565 1.1593 1.1321 1.1040 1.0835 130 1.1491 1.1521 1.1264 1.0996 1.0800 140 1.1425 1.1457 1.1214 1.0957 1.0769 150 1.1367 1.1400 1.1169 1.0922 1.0742 160 1.1315 1.1350 1.1128 1.0891 1.0717 170 1.1269 1.1304 1.1092 1.0863 1.0694 180 1.1226 1.1262 1.1058 1.0837 1.0674 190 1.1188 1.1224 1.1028 1.0813 1.0655 200 1.1153 1.1189 1.1000 1.0792 1.0638 Posterior probability
If the production tolerance is asymmetric (USL− T = T − LSL), then the posterior probability based on index Cpm that a process with asymmetric tolerances is capable is given as follows:
p∗= t∗ 0
(b1(y)+ b∗U(y))− (b1(y)− bL∗(y))
γαyα+1(α) exp − 1 γ y dy, for¯x < T t∗ 0
(b1(y)+ b∗L(y))− (b1(y)− bU∗(y))
γαyα+1(α) exp − 1 γ y dy, for¯x > T (14)
where bL∗(y)= (dL/d)√n[(t∗/y)− 1], b∗U(y)= (dU/d)√n[(t∗/y)− 1], t∗= 2[d∗/(3ω)]2[nsn2+ n( ¯x − T )2]−1, ˆCpm = (d∗/3)[sn2+ (d/dL)2(T− ¯x)2]−1/2for ¯x < T , ˆCpm = (d∗/3)[sn2+ (d/dU)2(¯x − T )2]−1/2 for
¯x > T , d∗= dU for USL− T < T − LSL and d∗= dL for USL− T > T − LSL. The derivation of (14) is
given in AppendixA. We note that if the production tolerance is symmetric (USL− T = T − LSL), then d∗= d, t∗= t and b∗L(y)= b∗U(y)= b2(y)implies that p∗= p. Consequently, the posterior probability based
on index Cpmderived by Shiau et al.19is a special case of T = m (symmetric tolerance).
It is rather complicated and computationally inefficient to calculate p∗from (14). However, for fixed sample size n and given parameter value δ, there is a one-to-one correspondence between p∗ and C∗= ˆCpm /ω.
Table IV. C∗(p∗)for p∗= 0.99 with d/dL= 5/6 and d/dU= 5/4 δ= ( ¯x − T )/sn−1 n 0.0 0.5 1.0 1.5 2.0 5 4.6131 3.9473 2.9953 2.4036 2.0521 10 2.3472 2.1639 1.8335 1.6115 1.4727 15 1.8841 1.7932 1.5890 1.4405 1.3442 20 1.6801 1.6262 1.4754 1.3293 1.2823 25 1.5635 1.5284 1.4073 1.3099 1.2444 30 1.4870 1.4632 1.3610 1.2761 1.2183 35 1.4326 1.4159 1.3270 1.2510 1.1990 40 1.3917 1.3799 1.3007 1.2316 1.1839 45 1.3596 1.3513 1.2797 1.2160 1.1717 50 1.3336 1.3279 1.2624 1.2030 1.1616 55 1.3121 1.3084 1.2478 1.1921 1.1531 60 1.2940 1.2918 1.2353 1.1828 1.1458 65 1.2785 1.2774 1.2245 1.1746 1.1394 70 1.2650 1.2649 1.2150 1.1675 1.1337 75 1.2531 1.2538 1.2066 1.1611 1.1287 80 1.2426 1.2440 1.1990 1.1554 1.1242 85 1.2332 1.2351 1.1922 1.1502 1.1202 90 1.2248 1.2271 1.1861 1.1455 1.1165 95 1.2171 1.2198 1.1804 1.1412 1.1131 100 1.2102 1.2131 1.1753 1.1373 1.1100 110 1.1979 1.2014 1.1661 1.1303 1.1045 120 1.1874 1.1913 1.1582 1.1243 1.0997 130 1.1784 1.1825 1.1513 1.1190 1.0955 140 1.1705 1.1748 1.1452 1.1143 1.0918 150 1.1635 1.1679 1.1398 1.1101 1.0885 160 1.1572 1.1618 1.1349 1.1064 1.0855 170 1.1516 1.1562 1.1305 1.1030 1.0828 180 1.1465 1.1512 1.1265 1.0999 1.0804 190 1.1419 1.1466 1.1228 1.0970 1.0781 200 1.1376 1.1424 1.1194 1.0944 1.0760
Since the estimator ˆCpm can be calculated from the collected sample data, we could then find the minimum value C∗(p∗)of C∗ required to ensure the posterior probability p∗ reaching a certain desirable level, which is useful for practical applications in assessing process capability. For engineers to conveniently apply our Bayesian procedure based on Cpm in their factory applications, we calculate some C∗(p∗)for various values of sample sizes n and δ= ( ¯x − T )/sn−1.
In Tables I–VIII, we tabulate C∗(p∗) for an example process with asymmetric tolerance under the specifications d/dL= 5/6, and d/dU= 5/4 for p∗= 0.90, 0.95, 0.975, 0.99, respectively. We observe that
the C∗(p∗)value decreases as n increases for each fixed p∗and δ. We also note that for each fixed n and p∗, the C∗(p∗)value decreases as positive δ value increases and increases as negative δ value increases. In fact, the entries in these tables are values of C∗(p∗)that satisfy P{Cpm >[ ˆCpm /C∗(p∗)]x} = p∗, and[ ˆCpm /C∗(p∗),∞) is a 100p∗% credible interval (similar to the 100p∗% confidence interval using the distribution frequency approach) for Cpm . Hence, the posterior probability that the credible interval contains the true Cpm value is p∗. In our Bayesian approach, we say that a process with asymmetric production tolerance is capable if all the points fall within this credible interval are greater than a pre-specified value of ω. When this occurs, we have P {process with asymmetric production tolerance is capable|x} > p∗. Therefore, to test whether a process is capable or not (with capability level ω and credible level p∗), we only need to check if ˆCpm > ωC∗(p∗).
For the special case in which µ= T , the formula for Cpm = (d∗/3)(σ2+ A2)−1/2 reduces to C pm=
Table V. C∗(p∗)for p∗= 0.90 with d/dL= 5/6 and d/dU= 5/4 δ= ( ¯x − T )/sn−1 n −0.25 −0.5 −1.0 −1.5 −2.0 5 2.3256 2.1840 1.8757 1.6407 1.4845 10 1.6002 1.5437 1.4195 1.3189 1.2490 15 1.4156 1.3807 1.3003 1.2325 1.1842 20 1.3285 1.3033 1.2424 1.1898 1.1516 25 1.2769 1.2569 1.2073 1.1635 1.1313 30 1.2422 1.2256 1.1833 1.1454 1.1172 35 1.2171 1.2029 1.1656 1.1319 1.1067 40 1.1979 1.1854 1.1520 1.1248 1.0985 45 1.1827 1.1715 1.1411 1.1131 1.0919 50 1.1704 1.1601 1.1321 1.1061 1.0864 55 1.1600 1.1506 1.1246 1.1003 1.0818 60 1.1513 1.1425 1.1181 1.0953 1.0778 65 1.1437 1.1355 1.1125 1.0909 1.0743 70 1.1371 1.1294 1.1076 1.0871 1.0712 75 1.1313 1.1240 1.1033 1.0837 1.0685 80 1.1261 1.1192 1.0994 1.0806 1.0661 85 1.1215 1.1148 1.0959 1.0779 1.0639 90 1.1173 1.1109 1.0928 1.0754 1.0619 95 1.1134 1.1074 1.0899 1.0731 1.0600 100 1.1099 1.1041 1.0873 1.0710 1.0584 110 1.1038 1.0983 1.0826 1.0673 1.0554 120 1.0985 1.0934 1.0786 1.0641 1.0528 130 1.0939 1.0891 1.0751 1.0613 1.0505 140 1.0899 1.0853 1.0720 1.0589 1.0485 150 1.0863 1.0820 1.0692 1.0567 1.0467 160 1.0831 1.0790 1.0667 1.0547 1.0451 170 1.0802 1.0763 1.0645 1.0529 1.0437 180 1.0776 1.0738 1.0625 1.0513 1.0424 190 1.0752 1.0716 1.0606 1.0498 1.0411 200 1.0731 1.0695 1.0589 1.0484 1.0400
n(Cpm / ˆCpm )2=in=1(Xi − T )2/σ2 is distributed as χ2(n), the ordinary chi-square distribution with n degrees of freedom. The posterior probability for a capable process that is on target with asymmetric tolerance, Cpm = d∗/(3σ ), can be expressed as p∗= P {Cpm > ω|x} = P {χ2(n) > n/C∗2}, where C∗= d∗/(3ωˆτ).
Thus, to compute p∗, we only need to check the commonly available chi-square tables for the posterior probability p∗. 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. We note that our computational results have been verified to be identical to those in Shiau et al.19for symmetric tolerance cases with USL− T = T − LSL, and identical to those in Chan et al.18for on-target cases with µ= T .
4.
APPLICATION OF THE PROCEDURE
To demonstrate how we may apply the proposed procedure to the actual data and judge whether the process is capable, we consider the following case taken from a microelectronic manufacturing factory making current transmitters. The process investigated is one that makes a monolithic 4–20 mA, two-wire current transmitter integrated circuit (2WCT IC) designed for bridge input signals. This device provides complete bridge excitation, instrumentation amplifier, linear circuitry, and the current output circuitry necessary for high-impedance strain
Table VI. C∗(p∗)for p∗= 0.95 with d/dL= 5/6 and d/dU= 5/4 δ= ( ¯x − T )/sn−1 n −0.25 −0.5 −1.0 −1.5 −2.0 5 2.8546 2.6647 2.2410 1.9132 1.6928 10 1.7931 1.7199 1.5572 1.4246 1.3320 15 1.5414 1.4967 1.3930 1.3049 1.2418 20 1.4253 1.3931 1.3152 1.2472 1.1977 25 1.3572 1.3320 1.2686 1.2122 1.1706 30 1.3118 1.2909 1.2369 1.1882 1.1519 35 1.2791 1.2612 1.2138 1.1706 1.1381 40 1.2543 1.2385 1.1960 1.1569 1.1273 45 1.2346 1.2204 1.1818 1.1459 1.1187 50 1.2186 1.2057 1.1701 1.1369 1.1115 55 1.2053 1.1934 1.1604 1.1293 1.1054 60 1.1940 1.1829 1.1520 1.1228 1.1003 65 1.1842 1.1739 1.1448 1.1171 1.0957 70 1.1757 1.1660 1.1384 1.1122 1.0918 75 1.1682 1.1590 1.1328 1.1077 1.0882 80 1.1616 1.1528 1.1278 1.1038 1.0851 85 1.1556 1.1473 1.1233 1.1002 1.0822 90 1.1502 1.1422 1.1192 1.0970 1.0796 95 1.1453 1.1377 1.1155 1.0941 1.0773 100 1.1408 1.1335 1.1121 1.0914 1.0751 110 1.1329 1.1261 1.1061 1.0866 1.0712 120 1.1261 1.1197 1.1009 1.0825 1.0679 130 1.1203 1.1142 1.0964 1.0788 1.0650 140 1.1151 1.1094 1.0924 1.0757 1.0624 150 1.1105 1.1050 1.0889 1.0728 1.0601 160 1.1064 1.1012 1.0857 1.0702 1.0580 170 1.1027 1.0977 1.0828 1.0680 1.0561 180 1.0994 1.0946 1.0802 1.0659 1.0544 190 1.0964 1.0917 1.0778 1.0640 1.0529 200 1.0935 1.0891 1.0756 1.0622 1.0514
gage sensors. The instrumentation amplifier can be used over a wide range of gain, accommodating a variety of input signals and sensors. Linear circuitry consists of a second, fully independent instrumentation amplifier that controls the bridge excitation voltage. It provides the second-order correction to the transfer function, typically achieving a 20 : 1 improvement in nonlinearity, even with low-cost transducers. Total unadjusted error of the complete current transmitter, including the linearized bridge is low enough to permit use without adjustment in many applications such as industrial process control, factory automation, SCADA remote data acquisition, weighting systems, and accelerometers.
The total unadjusted error of the 2WCT IC is an essential product characteristic, which has significant impact to product quality. The unadjusted error has an USL, of 14 µA, target value T is set to 6 µA, and the LSL is set to−6 µA. Therefore, the factory engineers have been recommended to use Cpmfor such applications with
asymmetric tolerance for determining whether products meet specifications and taking action to improve the process if necessary. The calculation shows that
d= 10, dU= 8, dL= 12, d∗= 8, n= 100, ¯x = 7.5599, sn−1= 1.5599,
Table VII. C∗(p∗)for p∗= 0.975 with d/dL= 5/6 and d/dU= 5/4 δ= ( ¯x − T )/sn−1 n −0.25 −0.5 −1.0 −1.5 −2.0 5 3.4684 3.2253 2.6707 2.2356 1.9399 10 1.9917 1.9009 1.6981 1.5323 1.4161 15 1.6655 1.6108 1.4835 1.3752 1.2974 20 1.5187 1.4797 1.3848 1.3018 1.2412 25 1.4337 1.4032 1.3263 1.2579 1.2072 30 1.3775 1.3523 1.2871 1.2280 1.1840 35 1.3372 1.3157 1.2585 1.2062 1.1669 40 1.3067 1.2878 1.2366 1.1894 1.1536 45 1.2826 1.2657 1.2192 1.1759 1.1430 50 1.2631 1.2477 1.2049 1.1649 1.1342 55 1.2469 1.2327 1.1930 1.1556 1.1269 60 1.2331 1.2200 1.1828 1.1477 1.1205 65 1.2213 1.2090 1.1740 1.1408 1.1150 70 1.2110 1.1994 1.1664 1.1347 1.1102 75 1.2019 1.1910 1.1595 1.1294 1.1059 80 1.1938 1.1834 1.1535 1.1246 1.1021 85 1.1866 1.1767 1.1480 1.1203 1.0986 90 1.1801 1.1706 1.1431 1.1164 1.0955 95 1.1742 1.1651 1.1386 1.1128 1.0926 100 1.1687 1.1600 1.1345 1.1096 1.0900 110 1.1592 1.1511 1.1272 1.1038 1.0853 120 1.1511 1.1434 1.1209 1.0988 1.0813 130 1.1440 1.1368 1.1155 1.0944 1.0778 140 1.1378 1.1309 1.1108 1.0906 1.0747 150 1.1323 1.1258 1.1064 1.0872 1.0719 160 1.1273 1.1211 1.1026 1.0841 1.0694 170 1.1229 1.1169 1.0991 1.0813 1.0672 180 1.1189 1.1132 1.0960 1.0788 1.0651 190 1.1152 1.1097 1.0931 1.0765 1.0632 200 1.1118 1.1065 1.0905 1.0744 1.0615
Consider ω= 1 and p∗= 0.95. From TableIIor running the Mathcad program as displayed in AppendixB, we find that C∗(p∗)= 1.1220, which implies that the minimum value of ˆCpm is equal to ωC∗(p∗)= 1.1220. Since 1.07 < 1.1220, we claim that this process is incapable in a Bayesian sense with 95% confidence.
5.
CONCLUSION
Existing developments and applications of the Taguchi capability index Cpm have focused on processes with
symmetric tolerances. The generalization Cpm , which incorporates the departure of the process mean from
the target value and the magnitude of the process variation, was proposed to the manufacturing industry for handling processes with asymmetric tolerances. However, the sampling distribution of the natural estimator
ˆC
pmconsidered by Chen et al.20 is rather complicated to deal with computationally for obtaining an interval
estimation of Cpm . Under the assumption of a non-informative prior we obtained a simple Bayesian procedure
for process capability assessment, which allows one to proceed with the Bayesian credible interval estimation for Cpm , which is similar to the classical confidence interval using the distribution frequency approach.
This Bayesian procedure can provide an efficient alternative to the classical distribution frequency approach in assessing process capability for asymmetric tolerances.
Table VIII. C∗(p∗)for p∗= 0.99 with d/dL= 5/6 and d/dU= 5/4 δ= ( ¯x − T )/sn−1 n −0.25 −0.5 −1.0 −1.5 −2.0 5 4.4445 4.1214 3.3653 2.7617 2.3462 10 2.2682 2.1531 1.8938 1.6815 1.5323 15 1.8308 1.7624 1.6033 1.4678 1.3703 20 1.6402 1.5919 1.4745 1.3718 1.2967 25 1.5317 1.4942 1.3997 1.3155 1.2532 30 1.4607 1.4300 1.3501 1.2778 1.2238 35 1.4102 1.3840 1.3143 1.2504 1.2024 40 1.3722 1.3492 1.2870 1.2295 1.1859 45 1.3424 1.3218 1.2654 1.2128 1.1727 50 1.3182 1.2996 1.2477 1.1991 1.1619 55 1.2982 1.2811 1.2330 1.1877 1.1528 60 1.2813 1.2654 1.2205 1.1779 1.1451 65 1.2668 1.2520 1.2097 1.1695 1.1384 70 1.2542 1.2402 1.2003 1.1621 1.1325 75 1.2431 1.2299 1.1920 1.1556 1.1272 80 1.2332 1.2207 1.1846 1.1497 1.1226 85 1.2244 1.2125 1.1779 1.1445 1.1184 90 1.2164 1.2051 1.1719 1.1397 1.1145 95 1.2092 1.1983 1.1664 1.1354 1.1111 100 1.2027 1.1922 1.1614 1.1314 1.1079 110 1.1911 1.1813 1.1526 1.1244 1.1022 120 1.1812 1.1720 1.1450 1.1183 1.0973 130 1.1726 1.1640 1.1384 1.1131 1.0931 140 1.1651 1.1569 1.1326 1.1084 1.0893 150 1.1584 1.1506 1.1274 1.1043 1.0860 160 1.1524 1.1450 1.1228 1.1006 1.0830 170 1.1471 1.1400 1.1186 1.0973 1.0803 180 1.1422 1.1354 1.1148 1.0942 1.0778 190 1.1378 1.1312 1.1114 1.0914 1.0755 200 1.1338 1.1274 1.1082 1.0889 1.0734 Acknowledgements
The authors would like to thank the anonymous referees for their helpful comments, which significantly improved the paper. This research was partially supported by the National Science Council of the Republic of China (NSC-93-2213-E-346-002).
REFERENCES
1. Boyles RA. The Taguchi capability index. Journal of Quality Technology 1991; 23:17–26.
2. Bordignon S, Scagliarini M. Statistical analysis of process capability indices with measurement errors. Quality and Reliability Engineering International 2002; 18(4):321–332.
3. Borges WS, Ho LL. A fraction defective based capability index. Quality and Reliability Engineering International 2001; 17(6):447–458.
4. Chang YS, Choi IS, Bai DS. Process capability indices for skewed populations. Quality and Reliability Engineering International 2002; 18(5):383–393.
5. Hoffman LL. Obtaining confidence intervals for Cpkusing percentiles of the distribution of ˆCp. Quality and Reliability Engineering International 2001; 17(2):113–118.
6. Nahar PC, Hubele NF, Zimmer LS. Assessment of a capability index sensitive to skewness. Quality and Reliability Engineering International 2001; 17(4):233–241.
7. Noorossana R. Process capability analysis in the presence of autocorrelation. Quality and Reliability Engineering International 2002; 18(1):75–77.
8. Pearn WL, Lin GH, Chen KS. Distributional and inferential properties of the process accuracy and process precision indices. Communications in Statistics—Theory and Methods 1998; 27(4):985–1000.
9. Pearn WL, Lin PC. Computer program for calculating the p-value in testing process capability index Cpmk. Quality and Reliability Engineering International 2002; 18(4):333–342.
10. Zimmer LS, Hubele NF, Zimmer WJ. Confidence intervals and sample size determination for Cpm. Quality and Reliability Engineering International 2001; 17(1):51–68.
11. Lee JC, Hung HN, Pearn WJ, Kueng TL. On the distribution of the estimated process yield index Spk. Quality and Reliability Engineering International 2002; 18(2):111–116.
12. Kotz S, Johnson NL. Process capability indices: A review, 1992–2000. Journal of Quality Technology 2002; 34(1):1– 19.
13. Spiring FA, Leung B, Cheng S, Yeung A. A bibliography of process capability papers. Quality and Reliability Engineering International 2003; 19:445–460.
14. Kane VE. Process capability indices. Journal of Quality Technology 1986; 18:41–52.
15. Chou YM, Owen DB. On the distributions of the estimated process capability indices. Communication in Statistics— Theory and Methods 1989; 18:4549–4560.
16. Pearn WL, Kotz S, Johnson NL. Distributional and inferential properties of process capability indices. Journal of Quality Technology 1992; 24:216–231.
17. Kotz S, Pearn WL, Johnson NL. Some process capability indices are more reliable than one might think. Applied Statistics 1993; 42:55–62.
18. Chan LK, Cheng SW, Spiring FA. A new measure of process capability: Cpm. Journal of Quality Technology 1988; 20:162–175.
19. Shiau JH, Chiang CT, Hung HN. A Bayesian procedure for process capability assessment. Quality and Reliability Engineering International 1999; 15:369–378.
20. Chen KS, Pearn WL, Lin PC. A new generalization of the capability index Cpm for asymmetric tolerances. International Journal of Reliability, Quality and Safety Engineering 1999; 6:383–398.
APPENDIX A.
DERIVATION OF EXPRESSION (
14
)
Given ω > 0, the posterior probabilityp∗= P {Cpm > ω|x} = P d∗ 3√σ2+ A2> x = P A2+ σ2< d∗ 3 2 x = a∗ 0 T+gU(σ ) T−gL(σ ) f (µ, σ|x) dµ dσ = a∗ 0 T+gU(σ ) T−gL(σ ) 2n π exp[−1/(βσ2)] σn+1βα(α) exp −n 2 ¯x − µ σ 2 dµ dσ = a∗ 0 2 exp[−1/(βσ2)] σnβα(α) √n √ 2π σ T+gU(σ ) T−gL(σ ) exp −n 2 µ− ¯x σ 2 dµ dσ Let z=√n(µ− ¯x)/σ, then dz = (√n/σ )dµ a∗ 0 2 exp[−1/(βσ2)] σnβα(α) √n √ 2πσ T+gU(σ ) T−gL(σ ) exp −n 2 µ− ¯x σ 2 dµ dσ = a∗ 0 2 exp[−1/(βσ2)] σnβα(α) 1 √ 2π √ n[T +gU(σ )− ¯x]/σ √ n[T −gL(σ )− ¯x]/σ exp −z 2 2 dz dσ
= a∗ 0 2 exp[−1/(βσ2)] σnβα(α) T − ¯x + gU(σ ) σ/√n − T − ¯x − gL(σ ) σ/√n dσ If ¯x < T , then P∗= a∗ 0 2 exp[−1/(βσ2)] σnβα(α) T − ¯x + gU(σ ) σ/√n − T − ¯x − gL(σ ) σ/√n dσ If ¯x > T , then P∗= a∗ 0 2 exp[−1/(βσ2)] σnβα(α) T − ¯x + gU(σ ) σ/√n − T − ¯x − gL(σ ) σ/√n dσ = a∗ 0 2 exp[−1/(βσ2)] σnβα(α) −¯x − T − gU(σ ) σ/√n − −¯x − T + gL(σ ) σ/√n dσ = a∗ 0 2 exp[−1/(βσ2)] σnβα(α) 1− ¯x − T − g U(σ ) σ/√n − 1− ¯x − T + g L(σ ) σ/√n dσ = a∗ 0 2 exp[−1/(βσ2)] σnβα(α) ¯x − T + gL(σ ) σ/√n − ¯x − T − gU(σ ) σ/√n dσ
where a∗= d∗/(3ω), gL(σ )= (dL/d)(a∗2− σ2)1/2, gU(σ )= (dU/d)(a∗2− σ2)1/2, α= (n − 1)/2, β =
2[(n − 1)s2
n−1]−1, is the cumulative distribution function of the standard normal distribution.
Let y= βσ2, β= 2/ni=1(xi − T )2, b1(y)=√2/y√1− (1/γ ), γ = 1 + [n/(n − 1)]δ2, δ= ( ¯x − T )/sn−1, bL∗(y)= (dL/d)√n[(t∗/y)− 1], b∗U(y)= (dU/d)√n[(t∗/y)− 1], t∗= 2[d∗/(3ω)]2[nsn2+ n( ¯x − T )2]−1.
The posterior probability based on index Cpm that a process with asymmetric or symmetric tolerances is
capable is given as p∗= t∗ 0 exp[−1/(γy)] yα+1γα(α)
[b1(y)+ bU∗(y)] − [b1(y)− b∗L(y)]
dy (A1) for¯x < T , and p∗= t∗ 0 exp[−1/(γy)] yα+1γα(α)
[b1(y)+ bL∗(y)] − [b1(y)− b∗U(y)]
dy (A2)
APPENDIX B.
MATHCAD PROGRAM FOR THE
C
∗(p
∗) VALUES
Input the values of parameters:
p:= 0.95, n := 100, w := 1, dU:= 8, dL:= 12, d := 10, δ := 1 α=n− 1 2 , γ= 1 + n n− 1δ 2, t (Cpm)= (n− 1) + (nδ2)(d/dU)2 (n− 1) + (nδ2) (2C2 pm) nw2 , b(y)= 2 y δ2 δ2+ (n − 1)/n 1/2 bL(y, Cpm)= dL d √ n t (Cpm) y − 1 1/2 , bU(y, Cpm)= dU d √ n t (Cpm) y − 1 1/2 , LNGA(n)= n−1 i−1 ln(i) if mod (2n, 2)= 0 ln((n− floor(n))) + n−1 i=n−floor(n) ln i otherwise
Set the initial value for solving:
Cpm= 1.07 Given p= Cpm 0 exp −1
γ y(α+ 1) ln(y) − LNGA(α) − α ln(y)
(pnorm(b1(y)+ bL(y, cpm,0, 1)
− pnorm(b1(y)− bU(yCpm,0, 1)) dy
The calculated value C∗(p∗)based on above setting:
Find(Cpm)= 1.121 953 93 Authors’ biographies
G. H. Lin is an Associate Professor at the Department of Transportation and Logistics Management, National
Penghu Institute of Technology, Penghu, Taiwan.
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. He has worked at AT&T Bell Laboratories at Switch Network Control and Process Quality Centers before joining National Chiao Tung University.
