具有多重機遇原因變異的奈米科技製程其整合製程監控與績效量測之研究

行政院國家科學委員會專題研究計畫 成果報告

具有多重機遇原因變異的奈米科技製程其整合製程監控與

績效量測之研究(第 3 年)

研究成果報告(完整版)

中 華 民 國 99 年 10 月 31 日

Metrika (2008) 68:65–82 DOI 10.1007/s00184-007-0143-6

Generalized confidence intervals for the process capability index Cpm

Bi-Min Hsu · Chien-Wei Wu · Ming-Hung Shu

Received: 29 October 2006 / Published online: 19 July 2007 © Springer-Verlag 2007

Abstract Process capability indices have been proposed to the manufacturing indus-try for measuring process reproduction capability. The Cpmindex takes into account the degree of process targeting (centering), which essentially measures process per-formance based on average process loss. To properly and accurately estimate the capa-bility index, numerous conventional approaches have been proposed to obtain lower limits of the classical confidence intervals (CLCLs) for providing process capability information. In particular, lower confidence limits (LCLs) not only provide critical information regarding process performance but are used to determine if an improve-ment was made in reducing the nonconforming percent and the process expected loss. However, the conventional approach lacks for exact confidence intervals for Cpminvolving unknown parameters which is a notable shortcoming. To remedy this, the method of generalized confidence intervals (GCIs) is proposed as an extension of classical confidence intervals (CCIs). For evaluating practical applications, two lower limits of generalized confidence intervals (GLCLs) for Cpmusing generalized

B. M. Hsu

Department of Industrial Engineering and Management, Cheng Shiu University, 840 Cheng Cing Road, Niaosong, Kaohsiung 83347, Taiwan C.-W. Wu

Department of Industrial Engineering and Systems Management, Feng Chia University, 100 Wenhwa Road,

Seatwen, Taichung 40724, Taiwan M.-H. Shu (B)

Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Seng-Min District, Kaohsiung 80778, Taiwan

e-mail: [email protected]

66 B. M. Hsu et al.

pivotal quantities (GPQs) are considered, (i) to assess the minimum performance of one manufacturing process/one supplier, and (ii) to assess the smallest performance of several manufacturing processes/several suppliers for equal as well as unequal process variances.

Keywords Generalized confidence intervals· Generalized pivotal quantities · Lower confidence limits· Process capability index.

1 Introduction

Process yield, process expected loss and process capability indices (PCIs) are three basic means that have been widely applied in measuring the manufacturing pro-cess/supplier potential and performance. Of the three, PCIs are easily understood and can be straightforwardly applied to the manufacturing industry. Especially, sup-pliers and manufacturers nowadays require their products be high quality with very low fraction of nonconformities (NC), often measured in parts per million (PPM). Tra-ditional methods for measuring fraction of N C become inapplicable for those high quality processes since any manufacturing sample of reasonable size likely contains no defective product items.

For this reason, recently developed PCIs, including Cp, Cpkand Cpm, have received substantial attention in the manufacturing industries, particularly, for companies mak-ing microelectronics devices and accessories demandmak-ing strict quality requirements. The Cpand Cpkindices are appropriate measures of progress for quality improvement paradigms in which reduction of variability is the guiding principle and process yield is the primary measure of success. Those two indices are defined byKane(1986) as

Cp= USL− LSL 6σ , Cpk = min
USL− µ 3σ , µ − LSL 3σ  ,

where USL and LSL refer to the upper and lower specification limits, respectively, which are fixed values specified by the user.µ stands for the process mean, and σ stands for the process standard deviation of the in-control process. However, they are not related to the cost of failing to meet customer’s requirement. Taguchi, on the other hand, emphasizes the loss in a product’s worth when one of its characteristics departs from the customer’s target valueτ = (USL + LSL)/2. To help account for this,Hsiang and Taguchi(1985) introduced the index Cpmdefined as (1), sometimes called the Taguchi index or loss-based capability index, which was also proposed independently byChan et al.(1988). The Cpm index emphasizes on measuring the ability of the process to cluster around the target, which therefore reflects the degrees of process targeting (centering).

Cpm=

USL− LSL 6σ2_{+ (µ − τ)}2 =

d

3σ2_{+ (µ − τ)}2, (1)

where d= (USL − LSL)/2 refers to the half length of the allowable tolerance of the process. The term expected squared error loss E(X − τ)2= σ2+(µ−τ)2incorporates two variation components: (i) variation of the process mean and (ii) deviation of the

Generalized confidence intervals for the process capability index Cpm 67

process mean from the target. By observing the definition of Cpm, it is easy to see that if the process variance increases (decreases) then the denominator will increase (decrease) and Cpmwill decrease (increase). Also, if the process’s mean moves away from (closer to) the target value, then the denominator will increase (decrease) and Cpm will decrease (increase). Obviously, Cpmis sensitive toτ with an additional penalty for being off-target.Ruczinski(1996) obtained a lower bound on the process yield as Y i eld ≥ 2(3Cpm) − 1 for Cpm>

√

3/3, or equivalently NC% ≤ 2(−3Cpm)

for Cpm> √

3/3, where (•) is the cumulative distribution function of the standard

normal distribution N(0, 1) and NC% is the percentage of nonconforming items. Since the knowledge of Cpm is estimated from data collected from samples of manufactured/supplied items with the sample size n, it is important to know the statis-tical properties of the estimator in order to prevent misleading interpretations.Boyles

(1991) proposed the following estimator, ˆCpm,

ˆCpm= d 3  S2₊ ¯_{X− τ}2 , (2)

where ¯X = _in₌₁Xi/n and S2 = _in₌₁(Xi − ¯X)2/n. Under normal conditions, the statistical properties of the conventional approach for the estimator of Cpmhave been investigated byChan et al.(1988),Boyles(1991),Kushler and Hurley(1992),

Pearn et al.(1992),Vännman and Kotz(1995),Vännman(1995),Zimmer and Hubele

(1997),Kotz and Lovelace(1998),Wright(2000),Zimmer et al.(2001),Pearn and Shu(2003), andPerakis and Xekalaki(2004). Unfortunately, in practice the conven-tional approaches involving unknown parameters lack exact confidence intervals for Cpm, which is a notable shortcoming.

To avoid this, the method of generalized confidence intervals (GCIs), a pioneer work developed byWeerahandi(1993,1995), is proposed as an extension of classi-cal confidence interval (CCIs). GCIs approach has been used in recent articles for a variety of problems including construction of tolerance intervals byLiao et al.(2005) and development of the tests for variance components byMathew ands Webb(2005).

Hamada and Weerahandi(2000) andAdamec and Burdick (2003) used GCIs to han-dle measurement error problems.Daniels et al.(2005) proposed GCIs for comparing two capability measures when there are no measurement errors.Chang and Huang

(2000) dealt with GCIs for the maximum value of functions of means, quantiles, and signal-to-noise ratios.

With rapid advancement of the manufacturing technology, current companies in the manufacturing sectors have implemented the mass production method to produce large amounts of standardized products on several production lines. Besides, most of large manufacturers increase their level of supplies from several suppliers (out-sourcing) and are relying more heavily on their supply chain as a source of their competitive advan-tage. In these practical situations, manufacturers are not only interested in assessing the minimum performance of one manufacturing process from one production line/one supplier, but also in considering the lowest performance of manufacturing processes from several production lines/several suppliers for quality assurance.

68 B. M. Hsu et al.

In this paper, we applied the idea of GCIs to obtain lower confidence limits (LCLs) that are exact probability expressions in a sense of generalization but are approxima-tions for the conventional lower confidence limits, denoted as GLCLs (lower limits of generalized confidence intervals) for Cpm. GLCLs not only provide critical informa-tion regarding process performance but are used to determine if an improvement was made in reducing non-conforming percent and process expected loss. For practical applications, two GLCLs for Cpmbased on the notations of generalized pivotal quan-tities (GPQs) are considered, (i) to assess the minimum performance of one manu-facturing process/one supplier performance, (ii) to assess the smallest performance of several manufacturing processes/several suppliers for equal as well as unequal process variances.

To clarify the presentation, the remainder of this paper is organized as follows. In Sect.2, a brief introduction of GCIs idea is given. Section3 assesses the minimum performance of one manufacturing process/one supplier based on the GLCL for Cpm. We tabulate the minimum value for making decision to illustrate the applicability of the proposed approach. In Sect.4, derivations of the GLCL for Cpm, the lowest performance of several manufacturing processes/several suppliers for equal as well as unequal process variances are investigated along with an application example. For evaluating the minimum performance of one manufacturing process/one supplier and the smallest performance of several manufacturing processes/several suppliers, the detailed derivations on 100(1 − α)% GLCLs for Cpm and min

1≤i≤kCpmi are shown in Appendices A and B, respectively. Finally, some conclusions are drawn in Sect.5.

2 Generalized confidence intervals (GCIs)

We now give a brief introduction to the concepts and definitions of GCIs. The method of GCIs is valuable whenever standard pivotal quantities are either nonexistent or difficult to obtain. To illustrate the problem and to formulate the GCI, consider an observable random variable Xi from populationπi, i =1, 2, . . . , k, with cumulative distribution function F_ψi, whereψi = (θi, ϑi) is a vector of unknown parameters, θiis the

param-eter of interest, and ϑi is a vector of nuisance parameters. For convenience, let xi be an observation of Xi, X = (X1, X2, . . . , Xk), x = (x1, x2, . . . , xk), ϑ =  ϑ1, ϑ2, . . . , ϑk  , andψ =ψ1, ψ2, . . . , ψk 

. Based on x, our goal is to derive a 100(1 − α)% GCI for the situations whenθ = θi when k= 1 and for θ = min

1≤i≤kθiwith k> 1. It is well known that CCIs in statistical problems involving nuisance parameters are available only in some special cases. To remedy this,Weerahandi(1993,1995) proposed GPQs and derived GCIs as an extension of CCIs. The following definitions are taken fromWeerahandi(1995):

Definition 1 Let R= R(X; x, ψ) be a function of X and possibly x and ψ as well,

Then R is said to be a generalized pivotal quantity (GPQ) if it satisfies the following conditions,

[1] R has a probability distribution free of unknown parameters;

[2] the observed pivotal, defined as robs = r(x; x, ψ) does not depend on the nuisance parameterϑ.

Generalized confidence intervals for the process capability index Cpm 69

Definition 2 Let be the real parameter space of θ. If a subset C1_−αof the real sample space of R satisfies P(R ∈ C1−α) = 1 − α, then the subset C of the real parame-ter space given by C(1 − α) = {θ ∈ | robs∈ C1−α} is said to be a 100(1 − α)% general confidence interval (GCI) forθ.

3 Measuring the minimum performance of one manufacturing process/one supplier based on the GLCL for Cpm

To compute the GLCL for Cpm when measuring the minimum performance of one manufacturing process/one supplier quality performance, one must define a GPQ for Cpm. Consider the identity

C_pmGPQ= d 3  ns2 W +  ¯x − Z_√s W − τ 2, (3)

where n is the sample size for X1, Z is a standard normal variable, Z = √

n ¯X− µ/ σ ∼ N(0, 1), W is a chi-square random variable with n − 1 degrees of freedom, W = nS2/σ2∼ χ_n2₋₁, and Z and W are independent.

Let Cpmgpq= d 3  ns2 w +  ¯x − z_√s w − τ 2. (4)

Clearly, the observed value of RX1; x1, ψ1  = CGPQ pm is r  x1; x1, ψ1  = Cgpq pm and the distribution of CpmGPQis free from the unknown parameters. Therefore, CpmGPQis a GPQ from Definition1. Based on Definition2, we consider P



CpmGPQ≥ C

= 1 − α, then C is denoted as the 100(1 − α)% GLCL for Cpmmeasuring the minimum per-formance of one manufacturing process/one supplier quality. C can be obtained by solving the following equation:

1− α = ∞ nC2_/ˆC2 pm(1+δ2)  ⎧ ⎨ ⎩ ⎛ ⎝√wδ +  w ˆC2 pm(1 + δ2) C2 − n ⎞ ⎠ − ⎛ ⎝√wδ −  w ˆC2 pm(1 + δ2) C2 − n ⎞ ⎠ ⎫ ⎬ ⎭fχ2 n−1(w)dw, (5) whereδ = ( ¯x − τ)/s and f_χ2

n−1(·) is the probability density function of chi-square

ran-dom variable with n−1 degrees of freedom. The detailed derivation of the 100(1 − α)% GLCL for Cpmin (5) is presented in Appendix A.

70 B. M. Hsu et al.

3.1 Simulation results

Using a simulation study, we now examine the performance of the GLCLs. We use LSL =−3 and USL=3 and draw 2,000 random samples of size n ∈ {25, 50, 100, 150} from processes in four scenarios of a experiment design using(µ, σ) = (1, 1), (1, 0.5),

(0, 1) and (0, 0.5), respectively. These provide Cpm= 0.7071, 0.8944, 1.00 and 2.00,

respectively, so as to detect the influence of the percentage coverage rate (CR) and the mean value of lower confidence bounds (MLCB) for the GLCI methods. The esti-mated CR is then the proportion of times the true value of Cpm exceeds the LCB. The estimated MLCB is the average of the 2,000 values of LCBs. The mean value of ˆCpm(ME) is simply the average of the 2,000 values of ˆCpm. Table1presents the ME, CR and MLCB derived from the simulation for each pair ofn, Cpm



for a nominal confidence level of 95% for the Cpm. The CR and MLCB entries are used as a basis for evaluating the performance of various methods. The CRs indicate that the performance of the GLCLs for Cpmis quite satisfactory.

3.2 The minimum value MV(1 − α) for making decision

From (5), we find that the minimum value of C/ ˆCpm, denoted by MV(1 − α), is required to ensure the confidence level reaching a certain desirable value 1− α. The minimum value MV(1 − α) equivalently satisfies

1− α = P  C_pmGPQ≥ C |x  = PC_pmGPQ≥ ˆCpm× MV (1 − α) |x  , (6)

which is useful in practice to evaluate the minimum performance of the process. Based on (5), we tabulate M V(1 − α) for various n=10(5)150 (range from 10 to 150 by the increment of 5) and|δ| = 0(0.5)2.0 in Tables2,3and4for 1− α = 0.90, 0.95 and 0.975, respectively. For example, for 1− α = 0.95, n = 60 and δ = 0.5, we can find

M V(1 − α) = 0.841 from Table3. In this case, if ˆCpm = 1.5 is calculated from the

observed data, the GLCL on Cpm, C can be obtained as ˆCpm× MV(1 − α) = 1.2615. We therefore conclude that the true value of the process capability Cpm, is no less than 1.2615 with 95% level of confidence. We thus can assure that the process yield is 99.985% (Y i eld ≥ 2(3 × 1.2615) − 1), and the number of the NC is less than 155 PPM (NC%×106≤ 2(−3 × 1.2615) × 106). If you use ˆCpm= 1.5 instead of 1.2615, you get yield = 99.999% instead of 99.985% and the NC is less than 7 PPM instead of 155 PPM.

3.3 An application example-power distribution switch

Consider the following case taken from a manufacturing plant making various types of power-distribution switches (PDS). The PDS is made for applications where heavy capacitive loads and short circuits are likely to be encountered. These devices are around 33 and 80 m N-channel MOSFET high-side power switches. The functional block diagram of a single 33 m PDS is displayed in Fig.1. The switch is controlled

123

Generalized confidence intervals for the process capability index Cpm 71 Ta b le 1 Simulated results for 95% GLCLs o f Cpm for L SL =− 3, USL = 3, and four scenarios (µ ,σ ) of a experiment design at 2,000 times µ = 1,σ = 1 µ = 1,σ = 0. 5 µ = 0,σ = 1 µ = 0,σ = 0. 5 Cpm = 0. 7071 Cpm = 0. 8944 Cpm = 1. 0000 Cpm = 2. 0000 n ME CR MLCB ME CR MLCB ME CR MLCB ME CR MLCB 25 0. 7194 0. 9650 0. 5679 0. 9027 0. 9550 0. 7737 1. 0309 0. 9695 0. 7533 2. 0655 0. 9720 1. 5095 50 0. 7128 0. 9615 0. 6077 0. 8991 0. 9505 0. 8091 1. 0184 0. 9650 0. 8299 2. 0413 0. 9610 1. 6642 100 0. 7090 0. 9610 0. 6356 0. 8959 0. 9520 0. 8327 1. 0070 0. 9630 0. 8788 2. 0119 0. 9615 1. 7561 150 0. 7086 0. 9535 0. 6485 0. 8965 0. 9480 0. 8448 1. 0072 0. 9550 0. 9038 2. 0096 0. 9590 1. 8038

123

72 B. M. Hsu et al.

Table 2 MV(1 − α) for n = 5(5)150, |δ| = 0(0.5)2.0, and 1 − α = 0.90

n |δ| = 0.0 |δ| = 0.5 |δ| = 1.0 |δ| = 1.5 |δ| = 2.0 10 0.612 0.643 0.704 0.759 0.801 15 0.695 0.720 0.768 0.812 0.845 20 0.742 0.763 0.803 0.840 0.868 25 0.774 0.791 0.827 0.859 0.884 30 0.797 0.812 0.843 0.872 0.895 35 0.814 0.828 0.856 0.882 0.903 40 0.828 0.840 0.866 0.891 0.910 45 0.839 0.850 0.874 0.897 0.915 50 0.848 0.858 0.881 0.903 0.920 55 0.856 0.866 0.887 0.908 0.924 60 0.863 0.872 0.892 0.912 0.927 65 0.869 0.877 0.896 0.915 0.930 70 0.875 0.882 0.900 0.918 0.933 75 0.879 0.887 0.904 0.921 0.935 80 0.884 0.890 0.907 0.924 0.937 85 0.887 0.894 0.910 0.926 0.939 90 0.891 0.897 0.913 0.928 0.941 95 0.894 0.900 0.915 0.930 0.942 100 0.897 0.903 0.917 0.932 0.944 105 0.900 0.905 0.919 0.934 0.945 110 0.902 0.908 0.921 0.935 0.947 115 0.905 0.910 0.923 0.937 0.948 120 0.907 0.912 0.925 0.938 0.949 125 0.909 0.914 0.926 0.939 0.950 130 0.911 0.916 0.928 0.941 0.951 135 0.913 0.917 0.929 0.942 0.952 140 0.914 0.919 0.931 0.943 0.953 145 0.916 0.920 0.932 0.944 0.954 150 0.918 0.922 0.933 0.945 0.954

by a logic enable compatible with 5-V logic and 3-V logic. Gate drive is provided with an internal charge pump designed to control the power-switch rise times and fall times to minimize current surges during switching. The charge pump requires no external components and allows operation from supplies as low as 2.7 V. When the output load exceeds the current-limit threshold or a short is present, the switch limits the output current to a safe level by switching into a constant-current mode, pulling the over-current logic output low. When continuous heavy overloads and short cir-cuits increase the power dissipation in the switch, causing the junction temperature to rise, a thermal protection circuit shuts off the switch to prevent damage. Recovery from a thermal shutdown is automatic once the device has cooled sufficiently. Internal

Generalized confidence intervals for the process capability index Cpm 73

Table 3 MV(1 − α) for n = 5(5)150, |δ| = 0(0.5)2.0, and 1 − α = 0.95

n |δ| = 0.0 |δ| = 0.5 |δ| = 1.0 |δ| = 1.5 |δ| = 2.0 10 0.545 0.577 0.642 0.703 0.752 15 0.637 0.664 0.717 0.767 0.806 20 0.691 0.713 0.759 0.802 0.835 25 0.727 0.746 0.786 0.824 0.854 30 0.753 0.770 0.806 0.841 0.868 35 0.774 0.789 0.821 0.853 0.878 40 0.790 0.803 0.833 0.863 0.886 45 0.803 0.815 0.843 0.871 0.893 50 0.814 0.825 0.852 0.878 0.899 55 0.823 0.834 0.859 0.884 0.904 60 0.832 0.841 0.865 0.889 0.908 65 0.839 0.848 0.871 0.893 0.912 70 0.845 0.854 0.875 0.897 0.915 75 0.851 0.859 0.880 0.901 0.918 80 0.856 0.864 0.884 0.904 0.921 85 0.861 0.868 0.887 0.907 0.923 90 0.865 0.872 0.891 0.910 0.925 95 0.869 0.876 0.893 0.912 0.927 100 0.872 0.879 0.896 0.914 0.929 105 0.876 0.882 0.899 0.917 0.931 110 0.879 0.885 0.901 0.919 0.932 115 0.882 0.887 0.903 0.920 0.934 120 0.884 0.890 0.905 0.922 0.935 125 0.887 0.892 0.907 0.924 0.937 130 0.889 0.894 0.909 0.925 0.938 135 0.891 0.896 0.911 0.927 0.939 140 0.893 0.898 0.913 0.928 0.940 145 0.895 0.900 0.914 0.929 0.941 150 0.897 0.902 0.916 0.930 0.942

circuitry ensures the switch remains off until valid input voltage is present. The short-circuit current threshold characteristic of the PDS process is essential for the product reliability performance, which has a significant impact to the product quality.

With a focus on the critical characteristic, short-circuit current threshold, we exam-ine a 33 m type of PDS product. The manufacturing specifications of short-circuit current threshold are set to LSL= 0.7A, USL = 1.3A and τ = 1.0A, respectively. According to the loss concepts introduced by Taguchi, the measured value of short-circuit current threshold closer to the target value will result in higher yield and lower loss. Therefore, the factory engineers have been recommended to use index Cpmfor such application in judging whether products meet specifications and taking action to

74 B. M. Hsu et al.

Table 4 MV(1 − α) for n = 5(5)150, |δ| = 0(0.5)2.0, and 1 − α = 0.975

n |δ| = 0.0 |δ| = 0.5 |δ| = 1.0 |δ| = 1.5 |δ| = 2.0 10 0.490 0.523 0.590 0.655 0.708 15 0.588 0.616 0.674 0.728 0.772 20 0.647 0.671 0.721 0.768 0.806 25 0.687 0.708 0.752 0.795 0.828 30 0.717 0.735 0.775 0.813 0.844 35 0.739 0.755 0.792 0.828 0.857 40 0.757 0.772 0.806 0.839 0.866 45 0.772 0.786 0.817 0.849 0.874 50 0.785 0.797 0.827 0.857 0.881 55 0.795 0.807 0.835 0.864 0.887 60 0.805 0.815 0.842 0.870 0.891 65 0.813 0.823 0.848 0.875 0.896 70 0.820 0.830 0.854 0.879 0.900 75 0.827 0.836 0.859 0.884 0.903 80 0.832 0.841 0.864 0.887 0.906 85 0.838 0.846 0.868 0.891 0.909 90 0.843 0.850 0.871 0.894 0.912 95 0.847 0.854 0.875 0.897 0.914 100 0.851 0.858 0.878 0.899 0.916 105 0.855 0.862 0.881 0.902 0.918 110 0.858 0.865 0.884 0.904 0.920 115 0.862 0.868 0.886 0.906 0.922 120 0.865 0.871 0.888 0.908 0.924 125 0.868 0.874 0.891 0.910 0.925 130 0.870 0.876 0.893 0.912 0.927 135 0.873 0.879 0.895 0.913 0.928 140 0.875 0.881 0.897 0.915 0.929 145 0.878 0.883 0.899 0.916 0.930 150 0.880 0.885 0.901 0.918 0.932

improve the process if necessary (Chan et al. 1988;Pearn and Shu 2003). Historical data based on routine process monitoring has been justified to be in statistical control and run under the desired stable conditions. The process distribution is also justified and shown to be fairly close to the normal distribution using Shapiro–Wilks normality test. The collected sample of size n = 50 is taken from the stable process and the calculated sample mean ¯x = 0.9730A, sample variance s2= 0.002916A2, δ = −0.5 and ˆCpm = 1.6563 with n = 50, 1 − α = 0.95 and δ = −0.50. By ckecking Table3, the MV(1 − α) = 0.825, therefore, the GLCL for Cpmcan be obtained as

C= ˆCpm× MV(1 − α) = 1.6563 × 0.825 = 1.3664.

123

Generalized confidence intervals for the process capability index Cpm 75

Fig. 1 The functional block diagram of a single 33 m PDS

4 Measuring the smallest performance for several manufacturing processes/several suppliers based on the GLCL for Cpm

In many practical situations, manufacturers are not only interested in assessing one production line/supplier performance, but also in considering the lowest performance of manufacturing processes from several production lines/suppliers for quality assur-ance. Letπ1, π2, . . . , πk be k populations where observations Xi j fromπi are inde-pendently distributed as Nµi, σ_i2  , where i = 1, 2, . . . , k, j = 1, 2, . . . , n. Let n∗= k i=1(n − 1), ¯Xi = n j=1Xi j/n, Si2 = n j=1  Xi j− ¯Xi 2 /n, S2 p = k i=1 n j=1  Xi j − ¯Xi 2

(kn), and ¯xi, s_i2and s2pare the observed values of ¯Xi, Si2and S2p,

respec-tively. We want to construct a 100(1 − α)% GLCL for min 1≤i≤kCpmi.

4.1 Whenσ₁2= σ₂2= · · · = σ_k2= σ2are unknown

To compute a GLCL on Cpm for measuring the smallest performance of k manufac-turing processes/k suppliers whenσ₁2= σ₂2= · · · = σ_k2= σ2are unknown, one must define a GPQ for each Cpm_i, i = 1, 2, . . . , k. Consider the identity

C_pmGPQ i = d 3  kns2 p W +  ¯xi− Zi  ks2 p W − τ 2, for i = 1, 2, . . . , k. (7)

where Zi is a standard normal variable, Zi = √

n ¯Xi − µi 

/σ ∼ N(0, 1), and W is a chi-square random variable with n∗degrees of freedom, W = knS2p/σ2∼ χn2∗.

76 B. M. Hsu et al. Let Cpmgpqi = d 3  kns2 p w +  ¯xi − zi  k wsp− τ 2, for i = 1, 2, . . . , k. (8)

Similarly, CGPQpmi for i = 1, 2, . . . , k are GPQs. We consider

P 

min CpmGPQ_i ≥ C∗

= 1 − α, then C∗ _{is denoted as the 100(1 − α)% GLCL for}

min 1≤i≤kCpmi whenσ 2 1 = σ 2 2 = · · · = σ 2 k = σ

2_{are unknown. C∗can be obtained as:}

1− α = ∞ 9knC∗2s2 p/d2 k  i=1 ⎧ ⎨ ⎩ ⎛ ⎝w kδi+  w ˆC2 pm_i(1 + δ2i) kC∗2 − n ⎞ ⎠ − ⎛ ⎝w kδi−  w ˆC2 pm_i(1 + δi2) kC∗2 − n ⎞ ⎠ ⎫ ⎬ ⎭ fχ2 n∗(w)dw (9) whereδi = ( ¯xi − τ)/spand f_χ2

n∗(·) is the probability density function of chi-square

random variable with n∗ degrees of freedom. Detailed derivations of C∗, the

100(1 − α)% GLCL for min

1≤i≤kCpmi, in (9) are presented in Appendix B. Note that the confidence level 100(1 − α)% depends on n, n∗, C∗, δi and ˆCpmi. Therefore, based

on the above results (9), the GLCL for min

1≤i≤kCpmi, C

∗_{can be obtained with observed}

data from each manufacturing process by numerical integration computation using the bisection search technique including the recursive adaptive Simpson quadrature integration method (Burden and Faires 2004) (the program is available upon request).

4.2 Whenσ_i2are unequal and unknown

To compute a GLCL on Cpmfor measuring the smallest performance of k manufac-turing processes/k suppliers whenσ_i2are unequal and unknown, one must define a GPQ for each Cpmi, i = 1, 2, . . . , k. Consider the identity

CpmGPQ_i = d 3  ns_i2 Wi +  ¯xi− Zi√s_Wi i − τ 2, (10)

where Zi is a standard normal variable, Zi = √

n ¯Xi − µi 

/σ ∼ N(0, 1), and Wi is

a chi-square random variable with n− 1 degrees of freedom, Wi = nS_i2/σ_i2∼ χ_n2₋₁.

123

Generalized confidence intervals for the process capability index Cpm 77 Let Cgpqpm_i = d 3  ns2 i wi +  ¯xi− zi√wsi i − τ 2, for i = 1, 2, . . . , k. (11)

Similarly, CpmGPQ_i for i= 1, 2, . . . , k are GPQs. We consider P 

min CpmGPQ_i ≥ C∗∗

= 1− α, then C∗∗ is denoted as the 100(1 − α)% GLCL for min

1≤i≤kCpmi whenσ 2

i are unequal and unknown. C∗∗can be obtained as:

k  i=1 ∞ n(3C∗∗s∗/d)2 ⎧ ⎨ ⎩ ⎛ ⎝√wiδi+  wi ˆCpm2 _i(1 + δi2) C∗∗2 − n ⎞ ⎠ − ⎛ ⎝√wiδi −  wi ˆCpm2 i(1 + δ 2 i) C∗∗2 − n ⎞ ⎠ ⎫ ⎬ ⎭ fχ2 n−1(wi)dwi = 1 − α, (12) where s∗ = max

1≤i≤k(si) and δi = ( ¯xi − τ)/si. The detailed derivations of C

∗∗_{, the}

100(1 − α)% GLCL for min

1≤i≤kCpmi, in (12) are presented in Appendix B. Note that the confidence level 100(1 − α)% depends on n, C∗∗, δi, s∗ and ˆCpm_i. Therefore, based on the above results (12), C∗∗can be obtained with observed data from each manufacturing process by numerical integration computation (Burden and Faires 2004) (the program is available upon request).

4.3 PDS with four production lines

The PDS manufacturer has implemented the mass production method to produce large amounts of standardized PDS products on several production lines. In these practical situations, suppose that the PDS manufacturer faces four production lines of the PDS typically set up for long production runs and decides to assess the smallest perfor-mance of these four manufacturing processes based on the GLCL of Cpmin terms of the short-circuit current threshold quality. Table5summarizes the statistics of repeated determinations, n= 50, from each production line data of the PDS. To test homoge-neity of variance, apply Bartlett’s test and find a p value >0.1 which accepts the four samples with equal variances. Applying the proposed method at (9), the associated GLCLs for the smallest Cpmwith various confidence levels, 0.9, 0.95 and 0.975, are displayed in Table6. With 95% level of confidence in Table6, we therefore conclude that the true value of the smallest process capability Cpm, is no less than 1.092. We thus can assure that the process yield is 99.895%, and the number of the N C is less than 1053 PPM.

78 B. M. Hsu et al.

Table 5 The statistics result of four manufacturing processes

Number of production lines 1 2 3 4

¯xi 1.073 0.966 0.959 1.052

s2_i 6.25×10−4 0.0030 0.0022 0.0037

δi 1.4917 −0.6947 −0.8378 1.0626

ˆCpm_i 1.2960 1.5465 1.6033 1.2476

Table 6 GLCLs for the smallest value of Cpm

Confidence levels 0.9 0.95 0.975

GLCLs 1.120 1.092 1.069

5 Conclusion

Process yield, process expected loss and process capability indices (PCIs) are three basic means that have been widely applied in measuring the manufacturing pro-cess/supplier potential and performance. To properly and accurately estimate the capa-bility index, numerous conventional approaches have been proposed to obtain lower limits of the classical confidence intervals (CLCLs) for providing process capabil-ity information. The conventional approach lacks for exact confidence intervals for Cpminvolving unknown parameters which is a notable shortcoming. In this paper, we applied the idea of generalized confidence intervals (GCIs) to obtain lower confidence limits (LCLs) that are exact probability expressions in a sense of generalization but are approximations for the conventional lower confidence limits, denoted as GLCLs, for Cpm. Two GLCLs for Cpmbased on the notations of generalized pivotal quantities (GPQs) are developed, (i) to assess the minimum performance of one manufacturing process/one supplier performance, and (ii) to assess the smallest performance of sev-eral manufacturing processes/sevsev-eral suppliers for equal as well as unequal process variances. Rather than Monte Carlo simulations which are popular used in GCIs meth-odology, we presented detailed theoretical derivations that both GLCLs for Cpmcan be written in closed forms. We also developed the programs of the numerical integra-tion for evaluating the 100(1 − α)% GLCLs for Cpmwhen both practical situations happen in real application.

Acknowledgments The authors thank the anonymous referees for their constructive suggestions and comments that resulted in an improved present of our research. Work on this paper was partially funded by National Science Council of Taiwan under grant NSC92-2213-E-251-005.

Appendix A

Derivation of Eq. (5)

To compute GLCL on Cpm for measuring one manufacturing process/one supplier quality performance, we consider 1− α = P



CpmGPQ≥ C

, then

Generalized confidence intervals for the process capability index Cpm 79 1− α = P ⎛ ⎜ ⎜ ⎝ d 3  ns2 W +  ¯x − Z_√s W − τ 2 ≥ C ⎞ ⎟ ⎟ ⎠ = P ⎛ ⎝d ≥ 3C  ns2 W +  ¯x − Z√s W − τ 2⎞ ⎠ = P ⎧ ⎨ ⎩−  1 W  d2_W 9C2 − ns 2  ≤ ¯x − Z√s W − τ ≤  1 W  d2_W 9C2 − ns 2 ⎫_⎬ ⎭ = w≥n(3Cs/d)2
 √ w s ( ¯x − τ + )  − √ w s ( ¯x − τ − )  f_χ2 n−1(w)dw where =  1 w  d2_w 9C2 − ns2 . Let δ = ¯x−τ_s ,s_d = 1 3 ˆCpm √ 1+δ2, then √_w s = d2_w 9s2_C2 − n =  w ˆC2 pm(1+δ2) C2 − n.

Therefore, the 100(1 − α)% GLCL for Cpmcan be obtained by solving the follow-ing equation: ∞ nC2_/ˆC2 pm(1+δ2)  ⎧ ⎨ ⎩ ⎛ ⎝√wδ +  w ˆC2 pm(1 + δ2) C2 − n ⎞ ⎠ − ⎛ ⎝√wδ −  w ˆC2 pm(1 + δ2) C2 − n ⎞ ⎠ ⎫ ⎬ ⎭ fχ2 n−1(w)dw = 1 − α. Appendix B

Derivations of Eqs. (9) and (12)

Case I Whenσ₁2= σ₂2= · · · = σ_k2= σ2are unknown

To compute a 100(1 − α)% GLCL on Cpmfor measuring the smallest performance of k manufacturing processes whenσ₁2 = σ₂2 = · · · = σ_k2 = σ2are unknown. We consider 1− α = P  min CpmGPQ_i ≥ C∗ , then

123

80 B. M. Hsu et al. 1− α = P ⎛ ⎝ d 3 ! " " #kns2p W + $ ¯xi−Zi  ks2p W −τ %2 ≥ C ∗_{, for all i = 1, 2, . . . , k} ⎞ ⎠ = P ⎧ ⎨ ⎩−  1 W  d2_W 9C∗2 − kns 2 p  ≤ ¯xi− Zi  ks2 p W − τ ≤  1 W  d2_W 9C∗2− kns 2 p  , for all i = 1, 2, . . . , k ⎫ ⎬ ⎭ = w≥kn(3C∗sp/d)2 k  i=1 &  $ w ks2 p ( ¯xi− τ + ) % −  $ w ks2 p ( ¯xi − τ − ) %' f_χ2 n∗(w)dw where =  1 w  d2_w 9C∗2 − kns2p . Letδi=¯xi_s− τ_p , sp d = 1 3 ˆC_pmi  1+ δ2 i , then_ksw2 p =  d2_w 9ks2 pC∗2−n =  w ˆC2 pmi(1+ δi2) kC∗2 − n. Therefore, the 100(1 − α)% GLCL for min

1_≤i≤kCpmi whenσ 2

1 = σ22= · · · = σk2= σ2_{are unknown, is obtained as:}

∞ 9knC∗2s2 p  d2 k  i=1 ⎧ ⎨ ⎩ ⎛ ⎝w kδi +  w ˆC2 pmi(1 + δ 2 i) kC∗2 − n ⎞ ⎠ − ⎛ ⎝w kδi−  w ˆC2 pm_i(1 + δi2) kC∗2 − n ⎞ ⎠ ⎫ ⎬ ⎭ fχ2 n∗(w)dw = 1 − α.

Case II Whenσ_i2are unequal and unknown

To compute a GLCL on Cpmfor measuring the smallest performance of k man-ufacturing processes/k suppliers whenσ_i2 are unequal and unknown. We consider 1− α = P  min 1≤i≤kC GPQ pmi ≥ C∗∗  , then

123

Generalized confidence intervals for the process capability index Cpm 81 1− α = P $ _d 3  ns2_i Wi+  ¯xi−Zi√si Wi−τ 2 ≥ C ∗∗_{, for all i = 1, 2, . . . , k}% = P ⎧ ⎨ ⎩−  1 Wi  d2_Wi 9C∗∗2 − ns 2 i  ≤ ¯xi− Zi si √ Wi − τ ≤  1 Wi  d2_Wi 9C∗∗2 − ns 2 i  , for all i = 1, 2, . . . , k ⎫ ⎬ ⎭ = k i=1 wi≥n  3C∗∗max 1≤i≤k(si)/d 2
 √_w i si ( ¯xi− τ + i)  − √_w i si ( ¯xi− τ − i)  f_χ2 n−1(wi)dwi where i =  1 wi  d2_w i 9C∗∗2− ns 2 i . Letδi = ¯xi_s−τ i , s ∗ _{= max} 1≤i≤k(si), si d = 1 3 ˆC_pmi  1_+δ2 i , then √wi i si =  d2_w i 9s2 iC∗∗2− n =  wiˆC2_pmi(1+δ2i) C∗∗2 − n.

Therefore, the 100(1 − α)% GLCL for min

1≤i≤kCpmi whenσ 2

i, i = 1, 2, . . . , k are

unequal and unknown, is obtained as:

k  i₌₁ ∞ n(3C∗∗s∗/d)2 ⎧ ⎨ ⎩ ⎛ ⎝√wiδi +  wi ˆCpm2 i(1 + δ 2 i) C∗∗2 − n ⎞ ⎠ − ⎛ ⎝√wiδi−  wi ˆCpm2 _i(1 + δ2i) C∗∗2 − n ⎞ ⎠ ⎫ ⎬ ⎭fχ2 n−1(wi)dwi = 1 − α. References

Adamec E, Burdick RK (2003) Confidence intervals for a discrimination ratio in a gauge R&R study with three random factors. Qual Eng 15:383–389

Boyles RA (1991) The Taguchi capability index. J Qual Technol 23:17–26 Burden R, Faires J (2004) Numerical analysis. 8thedn. Brooks/Cole, Belmont

Chan LK, Cheng SW, Spiring FA (1988) A new measure of process capability: Cpm. J Qual Technol 20:162–173

Chang YP, Huang WT (2000) Generalized confidence intervals for the largest value of some functions of parameters under normality. Stat Sin 5:805–820

Daniels L, Edgar B, Burdick RK, Hubele N (2005) Using confidence intervals to compare process capability indices. Qual Eng 17:23–32

82 B. M. Hsu et al.

Hamada M, Weerahandi S (2000) Measurement system assessment via generalized inference. J Qual Tech-nol 32(3):241–253

Hsiang TC, Taguchi G (1985) A tutorial on quality control and assurance—the Taguchi methods. ASA Annual Meeting, Las Vegas, Nevada, USA

Kane VE (1986) Process capability indices. J Qual Technol 18(1):41–52

Kotz S, Lovelace CR (1998) Process capability indices in theory and practice. Arnold, London Kushler RH, Hurley P (1992) Confidence bounds for capability indices. J Qual Technol 24(4):188–195 Liao CT, Lin TY, lyer HK (2005) One and two-sided tolerance intervals for general balanced mixed models

and unbalanced one-way random model. Technometrics 47:323–335

Mathew T, Webb DW (2005) Generalized p values and confidence intervals for variance components: applications to army testing and evaluation. Technometrics 47:312–322

Pearn WL, Kotz S, Johnson NL (1992) Distributional and inferential properties of process capability indices. J Qual Technol 24(4):216–231

Pearn WL, Shu MH (2003) Lower confidence bounds with sample size information of Cpmapplied to production yield assurance. Int J Prod Res 41(15):3581–3599

Perakis M, Xekalaki E (2004) A new method for constructing confidence intervals for the index Cpm. Qual Reliab Eng Int 20:651–665

Ruczinski I (1996) The relation between Cpmand the degree of includence. Doctoral Dissertation, Univer-sity of Würzburg, Würzburg, Germany

Vännman K (1995) A unified approach to capability indices. Stat Sin 5:805–820

Vännman K, Kotz S (1995) A superstructure of capability indices-distributional properties and implications. Scand J Stat 22:477–491

Weerahandi S (1993) Generalized confidence intervals. J Am Stat Assoc 88:899–905, correct in (1994), 89:726

Weerahandi S (1995) Exact statistical methods for data analysis, Springer, New York

Wright PA (2000) The cumulative distribution of process capability index Cpm. Stat Probab Lett 47: 249– 251

Zimmer LS, Hubele NF (1997) Quantiles of the sampling distribution of Cpm. Qual Eng 10:309–329 Zimmer LS, Hubele NF, Zimmer WJ (2001) Confidence intervals and sample size determination for Cpm.

Qual Reliab Eng Int 17:51–68

JOHN WILEY & SONS, LTD., THE ATRIUM, SOUTHERN GATE, CHICHESTER P019 8SQ, UK

*** PROOF OF YOUR ARTICLE ATTACHED, PLEASE READ CAREFULLY ***

After receipt of your corrections your article will be published initially within the online version of the journal.

PLEASE NOTE THAT THE PROMPT RETURN OF YOUR PROOF CORRECTIONS WILL ENSURE THAT THERE ARE NO UNNECESSARY DELAYS IN THE PUBLICATION OF

YOUR ARTICLE

READ PROOFS CAREFULLY

ONCE PUBLISHED ONLINE OR IN PRINT IT IS NOT POSSIBLE TO MAKE ANY FURTHER CORRECTIONS TO YOUR ARTICLE

§ This will be your only chance to correct your proof

§ Please note that the volume and page numbers shown on the proofs are for position only

ANSWER ALL QUERIES ON PROOFS (Queries are attached as the last page of your proof.)

§ List all corrections and send back via e-mail to the production contact as detailed in the covering e-mail, or mark all corrections directly on the proofs and send the scanned copy via e-mail. Please do not send corrections by fax or post

CHECK FIGURES AND TABLES CAREFULLY

§ Check sizes, numbering, and orientation of figures

§ All images in the PDF are downsampled (reduced to lower resolution and file size) to facilitate Internet delivery. These images will appear at higher resolution and sharpness in the printed article

§ Review figure legends to ensure that they are complete § Check all tables. Review layout, titles, and footnotes

COMPLETE COPYRIGHT TRANSFER AGREEMENT (CTA) if you have not already signed one

§ Please send a scanned signed copy with your proofs by e-mail. Your article cannot be published

unless we have received the signed CTA

OFFPRINTS

§ 25 complimentary offprints of your article will be dispatched on publication. Please ensure that the correspondence address on your proofs is correct for dispatch of the offprints. If your delivery address has changed, please inform the production contact for the journal – details in the covering e-mail. Please allow six weeks for delivery.

Additional reprint and journal issue purchases

§ Should you wish to purchase a minimum of 100 copies of your article, please visit http://www3.interscience.wiley.com/aboutus/contact_reprint_sales.html

§ To acquire the PDF file of your article or to purchase reprints in smaller quantities, please visit http://www3.interscience.wiley.com/aboutus/ppv-articleselect.html. Restrictions apply to the use of reprints and PDF files – if you have a specific query, please contact [email protected]. Corresponding authors are invited to inform their co-authors of the reprint options available

§ To purchase a copy of the issue in which your article appears, please contact [email protected] upon publication, quoting the article and volume/issue details

§ Please note that regardless of the form in which they are acquired, reprints should not be resold, nor further disseminated in electronic or print form, nor deployed in part or in whole in any marketing, promotional or educational contexts without authorization from Wiley. Permissions requests should be directed to mailto: [email protected]

UNCORRECTED PROOF

QRE 996

pp: 1–16 (col.fig.: Nil) PROD. TYPE: COM

ED: CHENBAGAM E PAGN: G SHARMILA -- SCAN: QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL

Qual. Reliab. Engng. Int. (2008)

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/qre.996

Research

1

Generalized Confidence Intervals

for Assessing Process Capability of

Multiple Production Lines

Chien-Wei Wu1,∗,†, Ming-Hung Shu2and F. T. Cheng1

1_{Department of Industrial Engineering and Systems Management, Feng Chia University, 100 Wenhwa Road, Seatwen,} Taichung 40724, Taiwan

3

2_{Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, 415} Chien Kung Road, Seng-Min District, Kaohsiung 80778, Taiwan

5

Process capability indices (PCIs) have become popular as unit-less measures on whether a process is capable of reproducing items meeting the quality requirement.

7

A reliable approach for testing process capability is to establish an interval estimate, for which we can assert that it contains the true PCI value with a reasonable degree

9

of certainty. However, the construction of such an interval estimate is not trivial, since the distribution of the commonly used C_{pk index involves unknown} param-11

eters. In this paper, we adopt the concept of generalized confidence intervals and generalized pivotal quantities to derive the generalized lower confidence bounds for

13

providing critical information on process performance. Two practical applications in the area of process capability were considered, they include (i) assessing whether

15

a process under investigation is capable and (ii) providing the lowest performance of the manufacturing processes from several production lines or several suppliers

17

for quality assurance. The applicability of the derived results is also illustrated with examples. Copyright©2008 John Wiley & Sons, Ltd.

19

KEY WORDS: decision making; generalized confidence intervals; generalized pivotal quantities; production lines; quality assurance

21

1. INTRODUCTION

M

23

to evaluate the ability of a process satisfying the specified requirements. Several PCIs are available in the literature (see Kotz and Lovelace1and Kotz and Johnson2), which are designed to provide 25

a common and easily understood language for quantifying process performance, and are dimensionless functions of process parameters and manufacturing specifications. Let USL and LSL denote the upper and 27

lower specification limits, d=(USL−LSL)/2 and M =(USL+LSL)/2 are the half-length and the midpoint

∗_{Correspondence to: Chien-Wei Wu, Department of Industrial Engineering and Systems Management, Feng Chia University, 100} Wenhwa Road, Seatwen, Taichung 40724, Taiwan.

†E-mail: [email protected]

Contract/grant sponsor: National Science Council of Taiwan; contract/grant number: NSC 96-2221-E-035-014

Copyright_q 2008 John Wiley & Sons, Ltd.

UNCORRECTED PROOF

QRE 996

C.-W. WU, M.-H. SHU AND F. T. CHENG

of the specification tolerances, respectively. The two widely used indices are Cpand Cpk defined by

1 Cp= USL−LSL 6
= d 3
Cpk= min
USL− 3
, − LSL 3
 =d−|− M| 3

where and
are the mean and the standard deviation of quality characteristic X, respectively. Note that the Cpand Cpk indices are designed to monitor and measure the performance of only normal and near-normal

3

processes with symmetric tolerances. Generally, the Cpk index takes into account the process variation as

well as the location (mean) of the process relative to the specification limits, while the Cp index reflects

5

only the magnitude of the process variation. The Cpk index has been regarded as a yield-based index since

it provides bounds on the process yield, that is, 2
(3Cpk)−1≤Yield≤
(3Cpk) for a fixed value of Cpk.

7

As the knowledge of Cpk is estimated from the collected samples of manufactured/supplied items, it

is important to know the statistical properties of the following natural estimator for preventing misleading 9 interpretations: ˆCpk= d−| ¯X − M| 3S 11

where ¯X=n_i=1Xi/n is the sample mean and S =[ni=1(Xi− ¯X)2/(n −1)]1/2 is the sample standard

deviation. Discussions and analysis of Cpk index on point estimation and construction of confidence intervals

13

have been investigated by many statisticians and quality researchers including Kotz and Lovelace1, Kushler and Hurley3, V¨annman and Kotz4, Hoffman5, Pearn and Shu6, Pearn and Wu7,8, Mathew et al.9, Wu10 15

and many others. However, construction of the exact confidence intervals on Cpk is complicated since the

sampling distribution of ˆCpkinvolves the joint distribution of two non-central t -distributed random variables,

17

or alternatively, the joint distribution of the folded-normal and chi-square random variables, with unknown parameters even when the samples are given (Pearn et al.11, Pearn and Lin12).

19

Weerahandi13introduced the concept of generalized pivotal quantities (GPQs) and generalized confidence intervals (GCIs), and demonstrated how to use them to derive confidence interval procedures for situations 21

when exact frequentist intervals are unavailable or difficult to apply. During this decade, the idea of GCIs and generalized tests has been used by many authors to obtain useful inference procedures in non-standard 23

problems, see e.g. Weerahandi14, Hamada and Weerahandi15, Chang and Huang16, McNally et al.17, Burdick and Park18, Krishnamoorthy and Lu19, Mathew and Webb20 and Kurian et al.21. Chiang22 proposed a 25

systematic approach for constructing GPQs for functions of variance components in balanced mixed linear model problems. Iyer and Patterson23 have provided a general recipe for the construction of generalized 27

test variables and GPQs. Although no general theory appears to exist that can provide a mathematical justification for the use of generalized inference, they also illustrated several applications of this approach to 29

a variety of problems. Krishnamoorthy and Mathew24derived one-sided tolerance limits for the observable random variable and the unobservable random effect in a one-way random model with balanced as well 31

as unbalanced data. Liao et al.25 further developed procedures for one- and two-sided tolerance intervals for general linear models in which there exists a set of independent scaled chi-squared random variables. 33

Although GCIs do not always have exact frequentist coverage, a number of simulation studies reported in the literature appear to support the claim that coverage probabilities of GCIs are sufficiently close to their 35

nominal value so that they are in fact useful procedures in practical applications. Hannig et al.26identified an important subclass of GPQs, which called fiducial generalized pivotal quantities (FGPQs), and showed 37

that under some mild conditions, GCIs constructed using FGPQs have correct frequentist coverage, at least asymptotically. The concept of fiducial inference was originally introduced by Fisher27and subsequently 39

refined by Fraser28. In addition, Liao et al.25 described three general approaches for constructing FGPQs and demonstrated their usefulness by deriving some previously unknown GPQs and GCIs. Connections 41

between GPQs and fiducial inference are also discussed.

Copyrightq 2008 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. (2008)

DOI: 10.1002/qre

UNCORRECTED PROOF

QRE 996

3

In this paper, we first apply the idea of GCIs to obtain the lower confidence bound (LCB) of the commonly 1

used index Cpk for assessing the process performance. The LCB not only gives a clue on the minimum level

of the actual performance of the process, which is tightly related to the fraction of non-conforming, but is also 3

useful in making decisions for capability testing. As rapid advancement of the manufacturing technology, current companies in the automotive and electronics sectors have implemented the mass production method 5

to produce large amounts of standardized products on several production lines. Besides, most of large manufacturers increase their level of supplies from several suppliers (out-sourcing) and are relying more 7

heavily on their supply chain as a source of their competitive advantage. In these practical situations, manufacturers are not only interested in measuring the capability of the process under investigation, but also 9

in considering the lowest performance of manufacturing processes from several production lines/suppliers for quality assurance. Let1,2,...,kbe h populations where observations Xi j fromiare independently

11

distributed as N(_i,
2_i), where i =1,2,...,h, j =1,2,...,n. The Cpk index is a function of the unknown

mean and variance. We consider thati is better thanj if Cpki>Cpkj. Therefore, our goal is to construct 13

an interval estimation of min_1≤i≤hCpki.

This paper is organized as follows. In Section 2, we first introduced some basic definitions and notations 15

of GCIs and GPQs. The formula of a 100(1−)% generalized lower confidence bound (GLCB) for Cpk is

derived. In Section 3, a testing procedure and the values of GLCB with various parameters are provided for 17

practitioners to assess the process performance in their factory. Further, we consider the lowest performance of manufacturing processes from several production lines/suppliers in Section 4. That is, a 100(1−)% 19

GLCB for min_1≤i≤hCpki is developed for quality assurance. This concept is illustrated with application examples in Section 5. In the final section some conclusions are given.

21

2. GENERALIZED LOWER CONFIDENCE BOUND FOR C_pk OF THE PROCESS

In this section, some basic concepts of GCIs and CPQs are introduced and reviewed from Iyer and Patterson23 23

and Burdick et al.29. The definitions and notations below will be used throughout the paper unless otherwise stated. Construction of a GCI requires a GPQ with a distribution that is free of the model parameters. 25

Suppose we wish to construct a GCI for. Let X represent a random variable with the distribution function

F(x,,), where  is the parameter of interest,  is a nuisance parameter and x is the observed value of X.

27

Definition 1. Let R = R(X, x,,) is a function of X, the observed value x, parameters  and . Then R is

said to be a GPQ if it satisfies the following two conditions: 29

(i) For fixed x , the distribution of R is free from any unknown parameters.

(ii) The observed value of R, Robs= R(x, x,,), does not depend on nuisance parameters.

31

Definition 2. Let  be the parameter space of . If a subset C_1− of the sample space of R

satis-fies P{R(X, x,,)∈C_1−}=1−, then the subset C of the parameter space given by C(1−)=

33

{∈|Robs∈C1−} is said to be a 100(1−)% GCI for .

Under the assumption of normality, suppose ¯X and S2 are independent random variables with ¯X∼

35

N(,
2/n) and (n −1)S2/
2∼_n2₋₁, where  and
2 are unknown constants. The pair ( ¯X, S2) may be

viewed as the sample mean and sample variance (complete and sufficient statistics) from an i.i.d. N(,
2) 37

sample of size n. Write1= and 2=
. Consider the functions f1and f2defined by

Z= f1( ¯X, S2,,
)= ¯X −
/√n V= f2( ¯X, S2,,
)=(n −1)S 2
2 39

Copyrightq 2008 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. (2008)

DOI: 10.1002/qre

UNCORRECTED PROOF

QRE 996

C.-W. WU, M.-H. SHU AND F. T. CHENG

Clearly, Z and V are independent, where Z is a standard normal distribution and V is a chi-square 1

distribution with n−1 degrees of freedom. Thus, the joint distribution of (Z, V ) is free from any model parameters. Inverting the functions f1and f2, we get

3 1= = g1( ¯X, S2, Z, V )= ¯X − Z  (n −1)S2 nV 2=
= g1( ¯X, S2, Z, V )=  (n −1)S2 V

In order to construct the GCI for and
, we obtain the GPQs, R_and R
are given by

R_= ¯x − Z  (n −1)s2 nV = ¯x − Z R
√ n R
=  (n −1)s2 V

where ¯x and s2are the observed values of ¯X and S2, respectively, and(R_, R
) is free from any unknown 5

parameters. Thus, a GPQ for Cpk is given by (the same one is also conducted by Mathew et al.9)

RCpk= d−|R_− M| 3R
= d−   ¯x− Z  (n −1)s2 nV − M    3  (n −1)s2 V (1) 7

Then, the 100(1−)% GLCB for Cpk can be derived as follows:

1− = P(RCpk≥C)= P ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ d−   ¯x− Z  (n −1)s2 nV − M    3  (n −1)s2 V ≥C ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = P ⎛ ⎝ ¯x − Z  (n −1)s2 nV − M   ≤d −3C  (n −1)s2 V ⎞ ⎠ = P ⎧ ⎨ ⎩− ⎛ ⎝d −3C  (n −1)s2 V ⎞ ⎠≤ ¯x − Z  (n −1)s2 nV − M ≤d −3C  (n −1)s2 V ⎫ ⎬ ⎭ = P  nV n−1  ¯x − M s − d s  +3√nC≤ Z ≤  nV n−1  ¯x − M s + d s  −3√nC, V ≥9(n −1)  Cs d 2

Copyrightq 2008 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. (2008)

DOI: 10.1002/qre

UNCORRECTED PROOF

QRE 996

where
(·) denotes the standard normal distribution function and f_2

n−1(·) is the probability density function

1

of chi-square distribution with n−1 degrees of freedom. Next, we consider the two situations for derivation of (2) as 3 Case I: ¯x ≥ M If ¯x ≥ M, then 5 ˆCpk= d−( ¯x − M) 3s , d+( ¯x − M) s =3 ˆCpk+2 and 7 s d= 1 3 ˆCpk+ , =| ¯x − M| s Thus, the expression (2) can be rewritten as

9 1− =  _∞ 9(n−1)C2_{/(3 ˆC} pk+)2

 nv n−1(3 ˆCpk+2)−3 √ nC  +
 3  nv n−1 ˆCpk−3 √ nC  −1  f_2 n−1(v)dv (3) Case II: ¯x<M 11 If ¯x<M, then ˆCpk= d+( ¯x − M) 3s and d−( ¯x − M) s =3 ˆCpk+2 13 where =| ¯x − M| s 15

Thus, the expression (2) can be rewritten as

1− =  _∞ 9(n−1)C2_{/(3 ˆC} pk+)2

 3  nv n−1 ˆCpk−3 √ nC  +
 nv n−1(3 ˆCpk+2)−3 √ nC  −1  f_2 n−1(v)dv (4) 17

Copyrightq 2008 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. (2008)

DOI: 10.1002/qre

UNCORRECTED PROOF

QRE 996

C.-W. WU, M.-H. SHU AND F. T. CHENG

Fortunately, Equations (3) and (4) obtained, respectively, from Case I and Case II are identical. Therefore, 1

without considering two cases, the 100(1−)% GLCB for Cpk, C, can be obtained by solving the following

equation: 3 1− =  _∞ 9(n−1)C2_{/(3 ˆC} pk+)2

 nv n−1(3 ˆCpk+2)−3 √ nC  +
 3  nv n−1 ˆCpk−3 √ nC  −1  f_2 n−1(v)dv (5)

3. ASSESSING PROCESS PERFORMANCE BASED ON THE GLCB

5

3.1. Simulation results

The generalized variable approach is not necessarily exact in the classical sense even if it can be computed 7

exactly using numerical procedures in some situations. Therefore, this paper derives the GCIs in an exact manner but it does not guarantee that the GCIs have exact coverage probabilities. For this reason, a series 9

of simulations were undertaken to investigate the performance of the GCIs method for Cpk index. Without

loss of generality, the values LSL=4, M =10 and USL=16 were used for our simulation study. For this

11

simulation, a class of normal processes with various process means=10,11,12,13 and process standard

deviations
=1.0,1.5,2.0 were considered. These combination values of (,
) and the corresponding values 13

of Cpk are summarized in Table I. The results include a total of 12 pairs of different(,
) combinations,

which are chosen to represent processes with different degrees of departure from process mean and to include 15

processes vary from ‘incapable’ (i.e. values of Cpk less than 1.0) to ‘capable’.

For each combination of(,
), a sample of size n =25,50,75,100,125 and 150 was drawn. The estimated 17

ˆCpkand GLCB for Cpk, denoted by LG_pk, can be calculated for each single simulation. This single simulation

was then replicated N=2000 times. Thus, we were able to calculate the expected values of ˆCpk, LG_pk and its

19

coverage probability (CP) for each sample size n based on 2000 trials. The expected value of ˆCpk (denoted

by E( ˆCpk)) and the expected value of LG_pk (denoted by E(LG_pk)) are simply the average of the 2000 values

21

of ˆCpk and LGpk, respectively. The estimated CP is the proportion of times that the calculated LGpk were

actually smaller than the corresponding true value of Cpk index. For each combination listed in Table I, the

23

values of E( ˆCpk), CP and E(LGpk) for nominal confidence level of 0.95 with various n are shown in Table II.

Table I. Different(,
) combinations used in the simulation study

Case 
Cpk 1 10 1.0 2.000 2 1.5 1.333 3 2.0 1.000 4 11 1.0 1.667 5 1.5 1.111 6 2.0 1.000 7 12 1.0 1.333 8 1.5 0.889 9 2.0 0.667 10 13 1.0 1.000 11 1.5 0.667 12 2.0 0.500

Copyrightq 2008 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. (2008)

DOI: 10.1002/qre