DOI 10.1007/s00170-003-1586-1 O R I G I N A L A R T I C L E
W.L. Pearn · P.C. Lin
Measuring process yield based on the capability index C
pm
Received: 29 June 2002 / Accepted: 6 November 2002 / Published online: 5 May 2004 Springer-Verlag London Limited 2004
Abstract Process capability indices Cp, Ca, Cpkand Cpmhave been proposed to the manufacturing industry as capability mea-sures based on various criteria including variation, departure, yield, and loss. It has been noted in recent quality research and capability analysis literature that both the Cpkand Cpm in-dices provide the same lower bounds on process yield, that is,
Yield2Φ(3Cpk) − 1 = 2Φ(3Cpm) − 1. In this paper, we in-vestigate the behaviour of the actual process yield in terms of the number of nonconformities (in ppm) for processes with a fixed index value of Cpk= Cpm, but with different degrees of process centring, which can be expressed as a function of the capability index Ca. The results illustrate that it is advanta-geous to use the index Cpm over the index Cpk when meas-uring process capability, since Cpm provides better customer protection.
Keywords Nonconformities· Process capability index · Process yield
1 Introduction
Process capability indices, including the precision index Cp, the accuracy index Ca, and the yield-based index Cpk have been proposed in the manufacturing industry, as well as the service industry, providing numerical measures on whether a process is capable of reproducing items within specification limits preset in the factory. These indices have been defined as the
follow-W.L. Pearn
Department of Industrial Engineering & Management, National Chiao Tung University,
Taiwan, R.O.C. P.C. Lin (u)
Center of General Education,
National Chin-Yi Institute of Technology, Taiping, Taichung 411, Taiwan, R.O.C. E-mail: linpc@ncit.edu.tw ing [1–4]: Cp= USL− LSL 6σ , Ca= 1 −|µ − m| d , Cpk= min USL− µ 3σ , µ − LSL 3σ , Cpm= USL− LSL 6σ2+ (µ − T )2,
where USL is the upper specification limit, LSL is the lower spe-cification limit,µ is the process mean, σ is the process standard deviation, m= (USL + LSL)/2 is the midpoint of the specifica-tion interval, and d= (USL − LSL)/2 represents the half-length of the specification tolerance.
The index Cp, which is a function of the process standard deviationσ and the specification limits, has been referred to as the precision index. It is defined to measure the consistency of the process quality characteristic relative to the manufacturing tolerance. The index Ca, a function of the process mean and the specification limits, has been referred to as the accuracy in-dex, which is defined to measure the degree of process centring relative to the manufacturing tolerance. The index Cpm, often re-garded as a loss-based index, may be rewritten as Cpm= Cp/[1+ 3Cp(1 − Ca)]1/2, and is a function of the two basic indices Cp and Ca.
In recent quality research and capability analysis, it has been often noted that the Cpm index provides both a lower bound on the process yield, that is, Yield= Φ[(USL − µ)/σ] −
Φ[(LSL − µ)/σ]2Φ(3Cpm) − 1, and an upper bound on the fraction of the nonconformities, P(NC) = 1 − Φ[(USL −
µ)/σ] + Φ[(LSL − µ)/σ]2Φ(−3Cpm). In this paper, we in-vestigate the behaviour of the actual process yield in terms of the number of nonconformities (in ppm) for processes with a fixed index value of Cpk= Cpm, but with different degrees of process centring, which can be expressed as a function of the capability index Ca. The results illustrate the advantage of using the index
Cpmover the index Cpkwhen measuring process capability, since
Cpmprovides better customer protection.
2 Process yield measuring based on C
pmIn general, if the process characteristic, X, follows the normal distribution, N(µ, σ2), then the fraction of the nonconformities (NC), may be expressed as:
P(NC) = 1 − Pr (LSLXUSL) = Pr (X < LSL) + Pr (X > USL) = Pr X− µ σ < L SL− µ σ + Pr X− µ σ > USL− µ σ = Φ L SL− µ σ + 1 − Φ USL− µ σ ,
whereΦ(·) is the cumulated distribution function of the stan-dard normal distribution N(0, 1). Since USL = m + d and
L SL= m − d, then P(NC) = Φ m− d − µ σ + 1 − Φ m+ d − µ σ = Φ m− d − µ σ + Φ −m+ d − µσ = Φ −d+ µ − md ·dσ + Φ −d− µ + md ·σd . Therefore, P(NC) = Φ −(1 + δ)γ + Φ −(1 − δ)γ ,
Table 1. The corresponding nonconformities (in ppm) for Cpm= 1.0, 1.25, 1.5, 1.75and2.0 with various Ca
Ca 0.6667 0.6944 0.7222 0.7500 0.7778 0.8056 0.8333 0.8611 0.8889 0.9167 0.9444 0.9722 1.0000 Cpm 1.0 0 0.09 44.34 334.87 872.99 1468.46 1972.77 2328.70 2542.34 2648.91 2689.62 2699.16 2699.80 Ca 0.7333 0.7556 0.7778 0.8000 0.8222 0.8444 0.8667 0.8889 0.9111 0.9333 0.9556 0.9778 1.0000 Cpm 1.25 0 0 0.07 2.8 17.26 48.40 87.90 125.09 152.51 168.34 175.04 176.72 176.83 Ca 0.7778 0.7963 0.8184 0.8333 0.8519 0.8704 0.8889 0.9074 0.9259 .09444 0.9630 0.9815 1.0000 Cpm 1.5 0 0 0 0.01 0.14 0.71 1.93 3.56 5.10 6.15 6.65 6.79 6.80 Ca 0.8095 0.8254 0.8413 0.8571 0.8730 0.8889 0.9048 0.9206 0.9365 0.9524 0.9683 0.9841 1.0000 Cpm 1.75 0 0 0 0 0.0004 0.0046 0.0207 0.0530 0.0935 0.1276 0.1462 0.1517 0.1521 Ca 0.8333 0.8472 0.8611 0.8750 0.8889 0.9028 0.9167 0.9306 0.9444 0.9583 0.9722 0.9861 1.0000 Cpm 2.0 0 0 0 0 0 0 0.0001 0.0004 0.0009 0.0015 0.0018 0.0020 0.0020
whereδ = (µ − m)/d and γ = σ/d. Further, since P(NC) is an even function ofδ, then P(NC) may be rewritten as:
P(NC) = Φ −1+ |δ|γ + Φ −1− |δ|γ .
The yield as a function of Cpmis determined as follows. Since
Ca= 1 −|µ − m| d = 1 − |δ|, then, P(NC) = Φ −2− Ca γ + Φ −Ca γ . Therefore, Cpm= d 3σ2+ (µ − m)2 = 1 3γ2+ δ2, and so γ2= 1 (3Cpm)2− δ 2= 1 3Cpm+ δ 1 3Cpm− δ , γ = 1 3Cpm+ δ 1 3Cpm− δ = 1 3Cpm+ |δ| 1 3Cpm− |δ|
holds for 0|δ|1/(3Cpm), i.e., for 1 − 1/(3Cpm)Ca1. Consequently, we have the following relationship between the
Fig. 1. Plot of the actual NC vs Cafor Cpm= 1.0, 1.1, 1.2, 1.3 and 1.4 (top to bottom)
process yield and the index Cpmfor 1− 1/(3Cpm)Ca1:
P(NC) = Φ − 2− Ca 1 (3 Cpm)2− (1 − Ca) 2 + Φ − Ca 1 (3 Cpm)2− (1 − Ca) 2 .
Table 1 displays the number of nonconformities (in ppm) for
Cpm= 1.0, 1.25, 1.50, 1.75 and 2.0 with various values of Ca satisfying 1−1/(3Cpm)Ca1. Figure 1 plots the actual num-ber of nonconformities (in ppm) for Cpm= 1.0, 1.1, 1.2, 1.3 and 1.4 (from top to bottom in the plot) with 1−1/(3Cpm)Ca1. Note that for Cpm> 1.4, the curves are almost indistinguishable. Ruczinski [5] obtained a lower bound for the yield: Yield 2Φ(3Cpm) − 1, or P(NC)2Φ(−3Cpm) for Cpm>
√ 3/3. Table 2 displays the bound in ppm for Cpm= 0.99(0.01)2.00. For example, for Cpm= 1.24, the number of nonconformities is no greater than 200 ppm.
3 Comparisons between C
pmand C
pkUsing a similar technique as used for deriving the yield formula based on Cpm, we can obtain a yield measure formula based on
Cpk. We first rewrite the definition of the Cpkindex as:
Cpk= d − |µ − m| 3σ = 1− |(µ − m)/d| 3(σ/d) = 1− |δ| 3γ = Ca 3γ. Then, forδ0 and Cpk> 0, we have δ =1−Ca,γ =Ca/(3Cpk), and P(NC) = Φ −3Cpk(2 − Ca) Ca + Φ −3CpkCa Ca .
Table 2. Various values of Cpm= 0.99(0.01)2.00 and the corresponding nonconformities (in ppm) Cpm ppm Cpm ppm Cpm ppm 0.99 2977.997 1.33 66.073 1.67 0.544 1.00 2699.796 1.34 58.198 1.68 0.466 1.01 2445.537 1.35 51.218 1.69 0.398 1.02 2213.370 1.36 45.036 1.70 0.340 1.03 2001.565 1.37 39.566 1.71 0.290 1.04 1808.510 1.38 34.731 1.72 0.247 1.05 1632.705 1.39 30.460 1.73 0.210 1.06 1472.751 1.40 26.691 1.74 0.179 1.07 1327.350 1.41 23.369 1.75 0.152 1.08 1195.297 1.42 20.443 1.76 0.129 1.09 1075.475 1.43 17.867 1.77 0.110 0.10 966.848 1.44 15.603 1.78 0.093 1.11 868.460 1.45 13.614 1.79 0.079 1.12 779.425 1.46 11.868 1.80 0.067 1.13 698.926 1.47 10.337 1.81 0.056 1.14 626.211 1.48 8.996 1.82 0.048 1.15 560.587 1.49 7.822 1.83 0.040 1.16 501.414 1.50 6.795 1.84 0.034 1.17 448.107 1.51 5.898 1.85 0.029 1.18 400.127 1.52 5.115 1.86 0.024 1.19 356.981 1.53 4.432 1.87 0.020 1.20 318.217 1.54 3.837 1.88 0.017 0.21 283.421 1.55 3.319 1.89 0.014 1.22 252.215 1.56 2.869 1.90 0.012 1.23 224.254 1.57 2.477 1.91 0.010 1.24 199.223 1.58 2.137 1.92 0.008 1.25 176.835 1.59 1.842 1.93 0.007 1.26 156.828 1.60 1.587 1.94 0.006 1.27 138.967 1.61 1.365 1.95 0.005 1.28 123.034 1.62 1.174 1.96 0.004 1.29 108.835 1.63 1.008 1.97 0.003 1.30 96.193 1.64 0.865 1.98 0.003 1.31 84.946 1.65 0.742 1.99 0.002 1.32 74.950 1.66 0.636 2.00 0.002
On the other hand, forδ < 0 and Cpk> 0, we have δ = Ca− 1,
γ = Ca/(3Cpk), and P(NC) = Φ −3CCpkCa a + Φ −3Cpk(2 − CC a) a . Consequently, for Cpk> 0, P(NC) = Φ−3Cpk + Φ −3Cpk(2 − Ca) Ca .
We have P(NC)2Φ(−3C) and 0Ca1, i.e., L SL
µUSL, for Cpk= C. On the other hand, we have P(NC) 2Φ(−3C) and 1−1/(3C)Ca1, i.e., m−d/(3C)µm+
d/(3C), for Cpm= C. For example, if Cpk= 1.00 we only have the information of process yield through the upper bound
P(NC)2699.796 ppm. But, if Cpm= 1.00 we have the in-formation of process yield through the upper bound P(NC) 2699.796 ppm and the process centring measure 0.667Ca1. According to today’s modern quality improvement theories based on Taguchi’s quality philosophy, reduction of variation from the target value is the guiding principle. Therefore, atten-tion should focus on meeting the target instead of meeting the tolerances. Following this principle, ifµ is far away from the
Table 3. Bounds of P(NC) and Cafor Cpk= Cpm= C, respectively C Bound of P(NC) Bound of CCpk Cpm a Bound of P(NC) Bound of Ca 1.00 2699.796 ppm 0 Ca 1 2699.796 ppm 0.667 Ca 1 1.25 176.835 ppm 0 Ca 1 176.835 ppm 0.733 Ca 1 1.50 6.795 ppm 0 Ca 1 6.795 ppm 0.778 Ca 1 2.00 0.002 ppm 0 Ca 1 0.002 ppm 0.833 Ca 1
Fig. 2. The actual nonconformities curves for Cpk= 1.0 (line) and
Cpm= 1.0 (point), with various allowed Ca
Fig. 3. The actual nonconformities curves for Cpk= 1.25 (line) and
Cpm= 1.25 (point), with various allowed Ca
target T such that the corresponding Ca is small, then the pro-cess should not be considered capable even ifσ is so small that the P(NC) is also small. Table 3 displays the bounds of P(NC) and Ca for Cpk= Cpm= C, respectively. Figures 2–5 plot the actual number of nonconformities (in ppm) for Cpm= Cpk= 1.0, 1.25, 1.5 and 2.0, with the restrictions 1−1/(3Cpm)Ca
Fig. 4. The actual nonconformities curves for Cpk= 1.5 (line) and
Cpm= 1.5 (point), with various allowed Ca
Fig. 5. The actual nonconformities curves for Cpk= 2.0 (line) and
Cpm= 2.0 (point), with various allowed Ca
1 for Cpm, and 0Ca 1 for Cpk. The results illustrate the advantage of using the index Cpm over the index Cpk when measuring process capability, as Cpm provides better customer protection in terms of product quality loss.
4 Estimating and testing C
pmThe index Cpmcan be rewritten as:
Cpm=
d
3σ2+ (µ − T)2,
where d= (USL − LSL)/2 is half the length of the specifica-tion interval. In general, both the process meanµ and the process varianceσ2are unknown. The estimated index ˆC
by replacingµ and σ2 by their estimators. Chan et al. [2] and Boyles [6] proposed two different estimators for Cpm, respec-tively defined as:
ˆCpm(CCS)= d 3 S2+ ( ¯X − T )2 , ˆCpm(B)= d 3 S2 n+ ( ¯X − T )2 , where ¯X=ni=1Xi/n, S2=ni=1(Xi− ¯X)2/(n − 1) and S2n= n
i=1(Xi− ¯X)2/n. In fact, the two estimators, ˆCpm(CCS) and ˆCpm(B), are asymptotically equivalent. We note that ¯X and S2n are the MLEs of µ and σ2, respectively. Hence, the es-timated index ˆCpm(B) is the MLE of Cpm. Further, the term
S2n+ ( ¯X − T )2 in the denominator of ˆCpm(B) is the uniformly minimum variance unbiased estimator (UMVUE) of the term
σ2+ (µ − T )2 in the denominator of C
pm, where S2n+ ( ¯X −
T)2=n
i=1(Xi− T )2/n and σ2+ (µ − T )2= E[(X − T )2]. Therefore, it is reasonable for reliability purposes, that we let the estimator ˆCpm(B)evaluate the performances of normally dis-tributed processes in this paper and define the estimated index
ˆCpm= ˆCpm(B).
Using a method similar to that presented by Vännman [7], we obtain an exact form of the cumulative distribution function of ˆCpm. Under the assumption of normality, the cumulative distribu-tion funcdistribu-tion of ˆCpmcan be expressed in terms of a mixture of the chi-square distribution and the normal distribution (see Lin and Pearn [8]): FˆC pm(x) = 1− b√n/(3x) 0 G b2n 9x2− t 2 φ(t + ξ√n) + φ(t − ξ√n)dt, (1)
for x> 0, where b = d/σ, ξ = (µ − T)/σ, G(·) is the cumula-tive distribution function of the chi-square distributionχn2−1, and
φ(·) is the probability density function of the standard normal
distribution N(0, 1).
To test whether a given process is capable, we consider the following statistical testing hypotheses:
H0: CpmC (process is not capable),
H1: Cpm> C (process is capable).
Based on a givenα(c0) = α, the chance of incorrectly con-cluding an incapable process (CpmC) as capable (Cpm> C), the decision rule is to reject H0(CpmC) if ˆCpm> c0 and fails to reject H0otherwise. For processes with target value set-tings in the middle of the specification limits (T= m = (USL +
L SL)/2), which are fairly common situations, the index may
be rewritten as: Cpm= b/[3(1 + ξ2)1/2]. Given that Cpm= C,
b= d/σ can be expressed as b = 3C(1 + ξ2)1/2. Given a value of C (the capability requirement), Pr( ˆCpmc∗|Cpm= C), the
p-value corresponding to c∗, a specific value of ˆCpmcalculated
from the sample data, is:
p− value = b√n/(3c∗) 0 G b2n 9(c∗)2− t 2 φ(t + ξ√n) + φ(t − ξ√n)dt (2)
Given values ofα and C, the critical value c0can be obtained by solving the equation Pr( ˆCpmc0|Cpm= C) = α. Hence, given values of capability requirement C, parameter ξ, sample size n, and riskα, the critical value c0can be obtained by solving the following equation:
b√n/(3c0) 0 G b2n 9c20− t 2 φ(t + ξ√n) + φ(t − ξ√n)dt= α (3)
Lin and Pearn [8] then implemented the testing hypothesis the-ory using Eqs. 2 and 3, and provided efficient Maple programs to calculate the p-values as well as the critical values.
The Maple program reads the sample data and the preset ca-pability requirement C, and outputs the corresponding p-value and/or a critical value c0. Also, the decision is made to reject the null hypothesis H0: CpmC, or to not reject the null hypothesis. Based on the test, Lin and Pearn [8] developed a simple step-by-step procedure, which can be used for in-plant applications. The practitioners can use the proposed procedure to determine whether their process meets the preset capability requirement, and make reliable decisions.
5 Application example
The example presented below concerns the capability of a pro-cess that produces electrically erasable programmable read-only memory (EEPROM), which is user-modifiable read-only mem-ory that can be erased and reprogrammed (written to) repeatedly through the application of higher electrical voltage. The product investigated here is a 128-bit EEPROM organised as 16× 8 with a 2-wire serial interface. This EEPROM supports a bi-directional 2-wire bus and data transmission protocol. The output leakage current is an essential product characteristic, which has a signifi-cant impact on product quality. For the output leakage current of a particular model of EEPROM, the upper specification limit, USL, is set to 8 mA, the lower specification limit, LSL, is set to −8 mA, and the target, T, is set to 0 mA. Sample data are col-lected from 100 EEPROM chips, which are displayed in Table 4. Figure 6 displays the histogram of the 100 observations. Fig-ure 7 displays the normal probability plot of the sample data. The sample data appears to be normal. The Shapiro-Wilk test for normality is also performed, obtaining W= 0.9917. Thus, the sample data can be regarded as taken from a normal process. The sample mean ¯X= 0.54 and sample standard deviation Sn= 1.64 are calculated first. Hence, we can calculate the value of the estimator ˆCpm= d/(3(S2n+ ( ¯X − T )2)1/2) = 8/(3((1.64)2+
Table 4. The sample data of 100 observations 0.14 −0.45 0.10 3.13 −1.60 −3.32 1.65 0.71 3.95 1.28 −1.34 −1.97 0.04 −1.85 −0.69 2.80 1.09 1.86 2.79 1.54 0.97 −2.21 2.64 1.42 −1.71 −1.35 −0.83 1.91 2.58 0.92 2.54 −0.89 0.47 1.97 0.05 −0.39 0.23 −0.37 0.77 −0.96 2.42 −0.58 0.32 3.52 0.55 1.75 0.80 −0.80 0.60 2.48 3.23 −2.62 −0.18 0.47 0.64 1.31 1.45 2.29 1.29 0.13 −2.72 −0.26 3.60 −0.20 0.11 1.93 0.81 −1.23 0.56 −0.68 0.24 −0.01 1.92 1.63 3.94 1.51 −0.78 −1.77 −1.00 −0.65 2.26 0.80 4.21 0.02 −2.05 −1.49 3.46 1.68 −2.10 −0.05 −1.05 0.04 −0.40 1.75 −0.52 −1.10 1.34 1.57 1.86 −0.09
Fig. 6. Histogram of the sample data
(0.54 − 0)2)1/2) = 1.54. Using Maple computer software to cal-culate Eq. 2, we obtain the corresponding p− value = 0.026 for the preset capability requirement C= 1.33 and sample size
n= 100. We conclude that the EEPROM manufacturing process
meets the capability requirement “Cpm> 1.33” if the α-risk is set to be larger than 0.026. In this case, we believe that the process is capable, the number of the nonconformities is less than 67 ppm (from Table 2), and Ca1− 1/(3C) = 0.75.
6 Conclusions
The process capability indices Cp, Ca, Cpkand Cpm, have been proposed to the manufacturing industry as capability measures based on various criteria including variation, departure, yield, and loss. Both the Cpk and Cpm indices provided the same
Fig. 7. The normal probability plot
lower bounds on process yield, that is, Yield2Φ(3Cpk) − 1= 2Φ(3Cpm) − 1. In this paper, we investigated the behaviour of the actual process yield, in terms of the number of non-conformities (in ppm), for processes with fixed index value of
Cpk= Cpm, but with different degrees of process centring, which can be expressed as a function of the capability index Ca. We also compared the two indices Cpkand Cpmin terms of process yield and process loss. The results illustrated the advantage of using the index Cpmover the index Cpkwhen measuring process capability, as Cpmprovides better customer protection. We inves-tigated a real example taken from a factory to illustrate how to measure the process yield based on the index Cpm.
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