DOI 10.1007/s00170-004-2268-3
O R I G I N A L A R T I C L E Int J Adv Manuf Technol (2006) 27: 1038–1046
M.H. Shu · Chih-Hsiung Wang · B.M. Hsu
A review and extensions of sample size determination
for estimating process precision and loss with a designated accuracy ratio
Received: 11 February 2004 / Accepted: 21 May 2004 / Published online: 11 May 2005 ©Springer-Verlag London Limited 2005
Abstract Numerous process capability indices, including Cp, Cpk, Cpm, and Cpmk, have been proposed in the manufactur-ing industry providmanufactur-ing numerical measures of process capability based on various criteria. The index Cp provides a measure of process precision, which reflects the consistency of product qual-ity. The index Cpm, also called the Taguchi index, essentially measures process loss. Lower confidence bounds estimate the minimum process capability conveying critical information re-garding product quality, which is essential to quality assurance. Existing research works have focused on constructing lower con-fidence bounds, but investigation on the sample sizes required for a specified accuracy ratio of the estimation has been compara-tively neglected. The sample size determination is important, as it is directly related to the cost of the data collection plan. In this paper, we review some existing formulas for lower confidence bounds that can be used to determine the sample size required for a given estimating accuracy ratio. We then present a different ap-proach, with efficient MATLAB programs, to obtain the sample sizes using the UMVUE (uniformly minimum variance unbiased estimator) of Cp, and the MLE (maximum likelihood estimator) of Cpm. We also provide tables of the sample size information for the engineers/practitioners to use for their in-plant applications. Keywords Lower confidence bound· Maximum likelihood estimator· Process consistency · Process yield · Taguchi index· Uniformly minimum variance unbiased estimator M.H. Shu (u)
Department of Industrial Engineering & Management, National Kaohsiung University of Applied Sciences,
415 Chien Kung Road, Seng Min District, Kaohsiung 80778, Taiwan, R.O.C. E-mail: [email protected]
Tel.: +886-7-3814526 ext. 7103 Fax: +886-7-392-3375 C.-H. Wang
Department of Industrial Engineering & Management, I-Shou Uniuversity, Taiwan, R.O.C.
B.M. Hsu
Department of Industrial Engineering and Management, Cheng Shiu University,
Taiwan, R.O.C.
1 Introduction
Process capability indices, including Cp, Cpk, Cpm, and Cpmk, which establish the relationships between actual process per-formance and manufacturing specification, have been the focus of recent research in quality assurance and process capability an-alysis [1–4]. The first and long-honored process capability index that appeared in the literature is Cp, which is defined as [1] : Cp=
USL− LSL 6σ ,
where USL is the upper specification limit, L SL is the lower specification limit, andσ is the process standard deviation. The numerator of Cp gives the range over which the process meas-urements are allowable. The denominator gives the range over which the process is actually varying. The index Cp was de-signed to measure the magnitude of the overall process variation relative to the manufacturing tolerance, which is to be used for processes with data that are normal, independent, and in a sta-tistical control condition. Clearly, the index only measures the potential of a process to reproduce an acceptable product and does not take into account whether the process is centered.
The index Cpm, also called the Taguchi capability index, em-phasizes measuring process loss, the ability of the process to cluster around the target, which therefore reflects the degree of process targeting (centering). The index Cpm incorporates the variation of production items with respect to the target value and the specification limits preset in the factory [2, 5–7]. The index Cpm is defined as follows:
Cpm=
USL− LSL 6σ2+ (µ − T)2,
whereµ is the process mean and T is the target value setting to the mid-point of the specification limits T = (USL + LSL)2. The capability index Cpm is not primarily designed to provide an exact measure of the number of conforming items, i.e., the process yield. But Cpmconsiders the process departure(µ − T)2
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(rather than 6σ alone) in the denominator of the definition to re-flect the degrees of process targeting [2, 5]. We note thatσ2+ (µ − T)2= E(X − T)2which is the major part of the denomina-tor of Cpm. Since E(X − T)2is the expected loss, where the loss
of the characteristic X missing the target is often assumed to be well approximated by the symmetric squared error loss function, loss(X) = (X − T)2, the capability index Cpm has been referred
to as a loss-based index.
In general, the process meanµ and the process standard de-viationσ are unknown. Thus, in practice, Cp and Cpm must be
estimated using the sample data, and consequently, a great de-gree of uncertainty is introduced into capability assessments due to the sampling errors. A reliable approach for estimating the true value of the process capability Cp and Cpm is to construct the
lower confidence bounds. Lower confidence bounds are not only essential to yield assurance of production, but also can be used in capability testing for decision-making.
Several methods for constructing lower confidence bounds of Cpand Cpmhave been proposed in the literature. For Cp, Chou et
al. [8], and Franklin and Wasserman [9] provided the exact lower confidence bounds by using the nature estimator of Cp. For Cpm,
Marcucci and Beazley [10], Chan et al. [2], Boyles [11], and Kushler and Hurley [12], proposed various methods to obtain approximate lower confidence bounds by using two different es-timators of Cpm. A common approach taken in these works was
to use the ordinary central chi-square distribution to approximate the complicated non-central chi-square distribution, to obtain ap-proximate lower confidence bounds of Cpm. A straightforward
approach was taken by Zimmer et al. [13] to obtain the exact lower confidence bounds on Cpm by working with a non-central
chi-square distribution. Franklin [14] investigated the sample size required for a given estimating accuracy ratio for Cp and
Cpm. His approach provided an approximate formula for
calcu-lating the sample size required for a given estimating accuracy ratio on Cpand Cpm, which may not be reliable, particularly for
applications with low-cost sampling plans. In this paper, we re-view some existing formulas for lower confidence bounds that can be used to determine the sample size required for a given estimating accuracy ratio. We then present a different approach, with MATLAB programs, to obtain the sample sizes using the UMVUE of Cp, and the MLE of Cpm. We also provide tables of
the sample sizes for the engineers/practitioners to use for their in-plant applications.
2 Estimating C
pThe index Cp contains only one parameter,σ, to be estimated.
If a sample data is given, Chou and Owen [15] and Pearn et al. [16] proposed two different estimators of Cpas defined in the
following: ˆCp= USL− LSL 6S , ˜Cp= bn−1 USL− LSL 6S = bn−1ˆCp, (1) where S=ni=1(Xi− ¯X)
(n − 1)1/2 is the nature estimator of the process standard deviationσ, which can be obtained from a stably normal process, and the correction factor bn−1is defined
in the following: bn−1= Γ n− 1 2 Γ n− 2 2 n− 1 2 −1 .
In fact, the two estimators, ˆCp and ˜Cp, are asymptotical
equivalent. Pearn et al. [16] showed that the modified estima-tor ˜Cp is the uniformly minimum variance unbiased estimator
(UMVUE) of Cp. They also showed that the UMVUE, ˜Cp, is
consistent, asymptotically efficient, and that√n ˜Cp− Cp
con-verges to N 0, C2p/2
in distribution. Therefore, it is reasonable for reliability purposes that the estimator ˜Cpis used to evaluate
process performance in this paper. From Eq. 1, ˜Cpis distributed as
˜Cp∼ bn−1Cp n− 1 χ2 n−1 . (2)
From the results of Appendix I, the cumulative distribution function (CDF) and the probability density function (PDF) of ˜Cp
are F˜C p(x) = 1 − FK (n − 1)b2 n−1D2 9nx2 = 1 − FK (n − 1)b2 n−1C2p x2 , x > 0 , f˜C p(x) = 2(n − 1)b2n−1D2 9x3 fK (n − 1)b2 n−1D2 9nx2 =2(n − 1)b 2 n−1C2p x3 fK (n − 1)b2 n−1C2p x2 , x > 0 , where D=√nd/σ, FK(•) and fK(•) are the CDF and PDF of
the ordinary central chi-square distributionχn2−1.
3 Sample size determination for C
p3.1 A review of sample size determination for Cp
As United Technologies/Carrier Corporation [17] stated that “cheating on the amount of data during the capability study may represent short-term savings with long-term penalties.” Hence, using proper sample size in capability estimation and testing be-comes an essential element to capability analysis and quality assurance. Chou et al. [8] provided the formulas of theγ% lower confidence bounds, denoted as CLp(CH), by using the estimator
ˆCpas: CpL(CH)= ˆCp χ2 n−1(1 − γ) n− 1 ,
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whereχn2−1(1 − γ) is the lower (1 − γ)th percentile of the ordi-nary central chi-square with n− 1 degrees of freedom. Thus, the sample size needed for a given estimating accuracy ratio, R(CH)p ,
can be obtained as: n= 1 +χ 2 n−1(1 − γ) R(CH)p 2 , where R (CH) p = CLp(CH) ˆCp . (3)
Franklin [14] used Eq. 3 to provide an approximate formula for calculating the sample size required for a given estimating accuracy ratio on Cpas follows:
n ∼= 1 +2 9 1 Z(1−γ) 2 + 1+ Z(1−γ) 2 2 − RLp(CH) 2/32 ,
where Z(1 − γ) is the lower (1 − γ)th percentile for the standard normal distribution N(0, 1).
3.2 A different approach of the required sample size for Cp
Note that from Eq. 2, theγ% lower confidence bounds, denoted as CLp(SWH), can be obtained as CpL(SWH)= ˜Cp bn−1 χ2 n−1(1 − γ) n− 1 .
Thus, the exact sample sizes for a given estimating accuracy ratio, R(SWH)p , using the UMVUE of Cpis:
n= 1 + χ 2 n−1(1 − γ) b2 n−1 R(SWH)p 2, where R (SWH) p = CLp(SWH) ˜Cp .
Table 1. Sample size n required for Rγ≥ R(SWH)p , with R(SWH)p =
0.75(0.01)0.95, and γ = 0.90, 0.95, 0.975, 0.99 R(SWH)p nγ = 0.90R γ = 0.95 γ = 0.975 γ = 0.99 γ n Rγ n Rγ n Rγ 0.75 11 0.7559 19 0.7542 27 0.7516 39 0.7529 0.76 12 0.7657 20 0.7602 30 0.7638 42 0.7615 0.77 13 0.7744 22 0.7709 32 0.7710 46 0.7716 0.78 14 0.7821 24 0.7802 35 0.7806 50 0.7805 0.79 16 0.7954 27 0.7922 39 0.7917 55 0.7903 0.80 17 0.8012 30 0.8024 43 0.8012 61 0.8005 0.81 19 0.8113 33 0.8112 48 0.8114 68 0.8106 0.82 22 0.8239 37 0.8212 53 0.8201 76 0.8204 0.83 24 0.8309 42 0.8317 60 0.8305 86 0.8308 0.84 27 0.8400 47 0.8405 68 0.8404 97 0.8403 0.85 31 0.8501 54 0.8507 78 0.8506 111 0.8504 0.86 36 0.8603 62 0.8603 90 0.8605 128 0.8603 0.87 42 0.8701 73 0.8708 105 0.8705 149 0.8702 0.88 50 0.8804 86 0.8806 123 0.8800 176 0.8803 0.89 60 0.8903 103 0.8905 148 0.8903 210 0.8901 0.90 73 0.9001 125 0.9003 180 0.9002 256 0.9002 0.91 91 0.9100 155 0.9101 223 0.9101 317 0.9101 0.92 117 0.9202 198 0.9202 284 0.9201 403 0.9200 0.93 154 0.9301 260 0.9300 373 0.9300 530 0.9301 0.94 212 0.9401 357 0.9401 511 0.9400 725 0.9400
To compute the required sample sizes, n, a MATLAB com-puter program is developed. In developing the program, the in-verse of the chi-square cumulative distribution function χn2−1 is used as an auxiliary function for evaluating n. The program reads the desired estimating accuracy ratio R(SWH)p for the preset
confidence levelγ, and output the exact sample size n (always rounding up) and the corresponding exact estimating accuracy ratio Rγ. The MATLAB program is included in Appendix II. An illustrative example with actual input and output is included in Appendix III.
Table 1 displays the sample sizes, n, required for Rγ ≥ R(SWH)p with R(SWH)p = 0.75(0.01)0.95, and γ = 0.9 0.95, 0.975,
and 0.99. For example, if R(SWH)p = 0.84, then with confidence
levelγ = 0.975, we find that the roundup sample size needed is n= 68. We conclude that since a minimal sample size of n = 68 is required for 97.5% confidence, the true Cp is no less than
Rγ = 84.05% of the sample estimate ˜Cp. Thus, if the sample
es-timate ˜Cp= 1.3, then the true value of Cpis no less than 1.3 ×
84.05% = 1.092, with 97.5% confidence.
4 An application example: a PVC rigid pipe
designed for high impact applications
A factory owns the largest and finest production equipment for all types of PVC rigid pipes and fittings in Taiwan with a ca-pability meeting no rivalry and distribution channels spread all over the world. The PVC rigid pipe and fitting products fea-ture excellent pharmacopoeia and corrosion resistance, electrical insulation and fire retardance and due to their high strength, light weight, small fluid resistance causing no influence on wa-ter quality, and cheap and easy installation. They are supplied to electrical, fuel-gas, and chemical-engineering industries and are widely applied to engineering fields as conduits for agri-culture, fishery construction and sewage system construction, and they have become the most welcomed conduit material. One new type of PVC rigid pipes for high-impact products de-picted in Fig. 1 is used to study the process of measuring the capability of the manufacturing process. The quality character-istic of interest is the outside diameter of a PVC rigid pipe. A PVC rigid pipe is considered to be acceptable if the outside diameter falls within manufacturing specification limits. That is, the outside diameter must fall between (LSL, T, USL) = (47.6 mm, 48 mm, 48.4 mm). With active efficient quality con-trol efforts and an effective improvement plan, the PVC rigid pipe manufacturing process is assumed to be under statistical control, following a near-normal distribution. Quality engineers realize process consistency is an important criterion for a
1046 v=eval(str); L=length(v); if L>=n, v=v(1:n); else, v=[v,zeros(1,n-L)]; end for j=1:nargout eval([’a’, int2str(j),’=v(j);’]); end %---% cpm.m file. %---function Q1=cpm(t)
global n i b epsilon ecpm
Q1=chi2cdf(((b\^2*n(i)/(9*ecpm\^2))-t.\^2), n(i)-1).*... (normpdf((t+epsilon*sqrt(n(i)))) +normpdf((t-epsilon*sqrt(n(i))))); %---% cpm1.m file. %---function Q1=cpm1(t) global n i b epsilon e k Q1=chi2cdf(((b\^2*n(i)/(9*e(k)\^2))-t.\^2), n(i)-1).*... (normpdf((t+epsilon*sqrt(n(i)))) +normpdf((t-epsilon*sqrt(n(i))))); %---% The end.
%---Appendix V: An example of actual input and output
Input: size_cpm Enter values of desired estimating accuracy ratio and confidence level: 0.89,0.95Output: The sample size needed for Cpm is 114 (rounding up) for epsilon 0 with the estimating accuracy ratio 0.8903 at 0.95 level of confidence.
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