Measuring PPM Non-conformities for Processes
with Asymmetric Tolerances
W. L. Pearn and C. H. Wu*
†Process yield has been the most basic and common criterion used in the manufacturing industry for evaluating process capability. TheCpkindex has been used widely in the manufacturing industry. In this note, we considered a generalization ofCpkindex which handles processes involving a targetT with asymmetric tolerances. Particularly, we established a formula for measuring the PPM non-conformities for given ratios of the two-side tolerances. We proved the validity of the established formula and tabulated the upper bounds on PPM non-conformities for various given Cpkindex values and ratios of the two-side tolerances. Copyright © 2012 John Wiley & Sons, Ltd.
Keywords: Asymmetric tolerances; non-conformity; process yield; target
1. Introduction
I
n manufacturing industry, the process yield is one of the major criteria for interpreting the process capability. Cpkindex is ayield-based index, which provides an upper bound on the non-conforming units in parts per million (NCPPM) for a normally distributed process. The Cpkindex was defined as
Cpk¼ min USL m 3s ; m LSL 3s ¼d m Mj j 3s : (1)
where m is the process mean, s is the standard deviation, M = (USL + LSL)/2 is the midpoint of the specification interval, and d = (USL LSL)/2. For normal processes distributed as N(m, s2), the bounds on the NCPPM for processes with a specific value of Cpk
can be represented as 2 2Φ 3Cpk 106⩾NCPPM⩾ 1 Φ 3C pk 106: (2)
where functionΦ is the cumulative probability function of the standard normal distribution. Table I displays some Cpkindex values
with the upper bounds of NCPPM for a normally distributed process.
2. The generalization
C
00pkindex
The formula presented in equation (2) only applies to processes with symmetric tolerances. For processes with asymmetric tolerances, Pearn and Chen1proposed a generalization of C
pkindex which was referred to as C
00
pk. The generalization C
00
pkindex is superior to other
existing generalizations of Cpkindex for processes with asymmetric tolerances. The C
00
pkindex was defined as (see Pearn and Chen1):
C00pk¼
d max df ðm TÞ=DU; dðT mÞ=DLg
3s (3)
where d* = min{DL, DU}, DU= USL T, DL= T LSL and T is the target value. Obviously, for processes with symmetric tolerances
(DU= DL), C
00
pkreduces to the Cpkindex which mentioned earlier in equation (1).
Pearn and Lin2investigated the statistical estimation of C00pkindex. Comparisons among several process capability indices (PCI) for
processes with asymmetric tolerances were proposed by Chen and Pearn.3Lin and Pearn4analyzed the large sample properties of the natural estimator of C00pk under general condition and provided an approximate confidence interval using the limiting distribution.
Department of Industrial Engineering & Management, National Chiao Tung University, Taiwan *Correspondence to: C. H. Wu, 1001 University Road, Hsinchu, Taiwan 300, ROC.
†E-mail: [email protected]
(wileyonlinelibrary.com) DOI: 10.1002/qre.1401 Published online 16 May 2012 in Wiley Online Library
The explicit forms of the cumulative distribution function and the probability density function for the natural estimator ^C
00
pkunder the
assumption of normality were derived by Pearn et al.5Various applications of PCI in manufacturing industry were considered by Pearn et al.6and Chen et al.7More recent researches on PCI include Pearn et al.,8Awad and Kovach,9Wu et al.,10and Yum and Kim.11
The measurement of process PPM non-conformities for normally distributed processes with asymmetric tolerance using C00pkindex
have never been investigated. In this note, we obtain an upper bound formula on the NCPPM for normally distributed processes with asymmetric tolerances and given ratios of the two-side tolerances.
3.
Non-conformity measures based on
C
00pkTo prove the formula, wefirst rewrite C00pkas follows:
C00pk¼
d dmax Rf U; RLg
3s ¼
dð1 max Rf U; RLgÞ
3s (4)
where the symbols RU= (m T)/DU and RL= (T m)/DL reflect the corresponding departure ratios on right and left tolerances,
respectively. For a process with characteristic X distributed as N(m, s2), the process yield can be represented as yield¼ P LSL < X < USLð Þ ¼ P LSL m½ð Þ=s < Z < USL mð Þ=s
¼ Φ ðUSLs mÞ Φ ðLSLs mÞ ¼ Φ ðUSLs mÞ þ Φ ðm LSLs Þ 1 ¼ Φ ðUSL TÞ s m T ð Þ s þ Φ ðT LSLÞ s T m ð Þ s 1 ¼ Φ DU s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 (5)
Letk = max{DU/DL, DL/DU} represents the larger one of the ratios of two-side tolerances. Four cases are discussed in the following:
Case 1 d* = DU, RU> RL,k = DL/DUand C 00 pk¼DUð1R3s UÞ yield ¼Φ DU s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 ¼ Φ 3C00 pk þ Φ DL DU DU s ð1 RLÞ 1 ⩾Φ 3C00 pk þ Φ DL DU DU s ð1 RUÞ 1 ¼ Φ 3C00 pk þ Φ 3kC00 pk 1; and yield¼ Φ DU s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 ⩽Φ DL s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 ¼ Φ DL s ð1 RUÞ þ Φ 3kC00 pk 1 < Φ 3kC00 pk : Case 2 d* = DU, RL> RU,k = DL/DUand C 00 pk¼DUð1R3s LÞ
Table I. Some specific values of Cpkand the upper bounds on the NCPPM
Cpk NCPPM 1.00 2699.796 1.25 176.835 1.33 66.073 1.45 13.614 1.50 6.795 1.60 1.587 1.67 0.544 2.00 0.002
432
yield¼ Φ DU s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 > Φ DU s ð1 RLÞ þ Φ DL DU 3C00pk 1 ¼ Φ 3C00 pk þ Φ 3kC00 pk 1; and yield¼ Φ DU s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 < Φ DL s ð1 RLÞ ¼Φ 3kC00 pk h i : Case 3 d* = DL, RU> RL,k = DU/DLand C 00 pk¼DLð1R3s UÞ yield¼ Φ DU s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 > Φ DU DL DL s ð1 RUÞ þ Φ DL s ð1 RUÞ 1 ¼ Φ 3kC00 pk þ Φ 3C00 pk 1; and yield¼ Φ DU s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 < Φ DU DL DL s ð1 RUÞ ¼ Φ 3kC00 pk h i : Case 4 d* = DL, RL> RU,k = DU/DLand C 00 pk¼DLð1R3s LÞ yield¼ Φ DU s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 > Φ DU s ð1 RLÞ þ Φ DL s ð1 RLÞ 1 ¼ Φ 3kC00 pk þ Φ 3C00 pk 1; and yield¼ Φ DU s ð1 RUÞ þ Φ DL s ð1 RLÞ 1 < Φ DL s ð1 RLÞ < Φ DU s ð1 RLÞ < Φ DU DL DL sð1 RLÞ ¼ Φ 3kC00 pk :
Figure 1. (a) Upper bounds on NCPPM and true NCPPM forfixed target T = 4 with 0 < m < 6. (b) Upper bounds on NCPPM and true NCPPM for fixed target T = 2 with 0< m < 6.
From cases 1–4, we establish the bounds on process yield based on the C00pkindex. Consequently, a two-sided bound on NCPPM for
normally distributed processes with asymmetric tolerances can be represented as follows: 2 Φ 3kC00pk Φ 3C00pk h i 106⩾NCPPM⩾ 1 Φ 3kC00 pk h i 106: (6)
For various values of process meanm, Figs. 1(a)–1(b) display the upper bound on NCPPM and true NCPPM for a normally distributed process with specifications LSL = 0, USL = 6 and variance s2= 1. In Figs. 1(a)–1(b), the dotted red line and the black line represent the NCPPM upper bounds and the true NCPPM of the process, respectively. We note that the true NCPPM is minimized bym = M = 3 for a given ratios of two-side specifications. The NCPPM upper bounds and the true NCPPM are plotted in Figure 2(a) 2(b) as a function of the target value T.
4.
Non-conformity bounds calculation
Table II displays the upper bounds on the NCPPM for various values of C00pk¼ 1:00 0:05ð Þ2:00 and the larger one of the ratios of
two-side tolerances k = max{DU/DL,DL/DU}, k = 1.00(0.05)1.50. For instance, for a normally distributed process with asymmetric
Figure 2. (a) Upper bounds on NCPPM and true NCPPM forfixed process mean m = 4 with 0 < T < 6. (b) Upper bounds on NCPPM and true NCPPM for fixed process mean m = 1 with 0< T < 6.
Table II. The upper bounds on NCPPM for various values of C00pkandk
C00pk k 1.000 1.050 1.100 1.150 1.200 1.250 1.300 1.350 1.400 1.450 1.500 1.00 2699.8 2166.3 1833.3 1630.2 1509.0 1438.3 1398.0 1375.5 1363.2 1356.7 1353.3 1.05 1632.7 1287.0 1081.5 962.24 894.77 857.52 837.46 826.92 821.52 818.82 817.50 1.10 966.85 748.54 625.14 557.25 520.90 501.96 492.36 487.62 485.34 484.28 483.80 1.15 560.59 426.18 354.12 316.61 297.67 288.36 283.94 281.89 280.98 280.58 280.41 1.20 318.22 237.52 196.58 176.47 166.91 162.51 160.54 159.70 159.34 159.20 159.14 1.25 176.84 129.59 106.95 96.488 91.815 89.800 88.961 88.624 88.493 88.444 88.427 1.30 96.193 69.205 57.030 51.742 49.531 48.640 48.295 48.166 48.120 48.104 48.099 1.35 51.218 36.179 29.803 27.209 26.196 25.816 25.679 25.632 25.616 25.611 25.609 1.40 26.691 18.514 15.264 14.028 13.579 13.422 13.370 13.353 13.348 13.346 13.346 1.45 13.614 9.275 7.662 7.090 6.896 6.834 6.815 6.809 6.807 6.807 6.807 1.50 6.795 4.548 3.769 3.512 3.431 3.407 3.400 3.398 3.398 3.398 3.398 1.55 3.319 2.183 1.817 1.704 1.672 1.663 1.660 1.660 1.660 1.660 1.660 1.60 1.587 1.026 0.858 0.810 0.798 0.794 0.794 0.793 0.793 0.793 0.793 1.65 0.742 0.472 0.397 0.377 0.372 0.371 0.371 0.371 0.371 0.371 0.371 1.70 0.340 0.213 0.180 0.172 0.170 0.170 0.170 0.170 0.170 0.170 0.170 1.75 0.152 0.094 0.080 0.077 0.076 0.076 0.076 0.076 0.076 0.076 0.076 1.80 0.067 0.040 0.035 0.034 0.033 0.033 0.033 0.033 0.033 0.033 0.033 1.85 0.029 0.017 0.015 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 1.90 0.012 0.007 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 1.95 0.005 0.003 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 2.00 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001
434
tolerances which satisfies C00pk¼ 1:40 and k = 1.30, the product’s fractions of defectives is at most 13.37 ppm. From Table II, for case of
k = max{DU/DL, DL/DU} = 1, the upper bounds on NCPPM are the same with the results which mentioned earlier in equation (2).
Obviously, if DU= DL(symmetric tolerance), then C
00
pk defined in equation (3) reduces to Cpkdefined in equation (1), and formula
we established in equation (6) reduced to equation (2).
Forfixed C00pkvalue, whenk increases, the upper bounds on NCPPM decrease and the bounds are closer to the true NCPPM. It is
evident since the larger the valuek, the smaller the value d*(C00pk). That is, when the tolerances become more asymmetric, it requires
the process to have a lower variance for keeping the C00pkvalue remains the same. For example, two on-target processes A and B with
same value of C00pk index and identical specifications (LSL, USL) = (0, 50) are considered. Since processes A and B have identical C
00
pk
index value, the expected proportions non-conforming are the same for both processes. On the cases thatmA= TA= 25 (kA= 1.00)
andmB= TB= 30 (kB= 1.50), because C
00
pk¼ 25= 3sð AÞ ¼ 20= 3sð BÞ ¼ C
00
pkBimplies that process B has smaller variance (sB< sA), process
B is better than process A.
5. Conclusions
In this note, the generalization C00pk index purposed by Pearn and Chen1was considered. Based on C
00
pk index, we established a
formula for measuring the PPM non-conformities for given ratios of the two-side tolerances. The validity of the established formula was also proved. The upper bounds on NCPPM and true NCPPM for various values of process mean and target were presented graphically. For practice and convenience, we tabulated the upper bounds on NCPPM for various C00pkindex values and given ratios
of the two-side tolerances.
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Authors' biographies
Wen-Lea Pearn received the Ph.D. degree in operations research from the University of Maryland, College Park. He is a Professor of Operations Research and Quality Assurance at the National Chiao-Tung University (NCTU), Hsinchu, Taiwan. He was with Bell Labora-tories, Murray Hill, NJ, as a Quality Research Scientist before joining the NCTU, and others. His current research interests include pro-cess capability, network optimization, and production management. Dr. Pearn’s publications have appeared in the Journal of the Royal Statistical Society, Series C, Journal of Quality Technology, European Journal of Operational Research, Journal of the Operational Research Society, Operations Research Letters, Omega, Networks, and the International Journal Productions Research.
Chia-Huang Wu received his MS degree in Applied Mathematics from National Chung-Hsing University. Currently, he is a PhD can-didate at the Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC.
