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C-pm MPPAC for manufacturing quality control applied to the precision voltage reference process

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O R I G I N A L A R T I C L E

W. L. Pearn Æ M. H. Shu Æ B. M. Hsu

C

pm

MPPAC for manufacturing quality control applied

to the precision voltage reference process

Received: 22 November 2002 / Accepted: 24 April 2003 / Published online: 12 March 2004 Ó Springer-Verlag London Limited 2004

Abstract The multiprocess performance analysis chart (MPPAC) based on the process capability index Cpm,

called Cpm MPPAC, is developed for analysing the

manufacturing quality of a group of processes in a multiple process environment. The Cpm MPPAC

con-veys critical information of multiple processes regard-ing the departure of the process and process variability from one single chart. Existing research works on MPPAC are restricted to obtaining quality information from one single sample of each process ignoring sam-pling errors. The information provided from existing MPPAC charts, therefore, is unreliable and misleading and results in incorrect decisions. In this paper, we consider the natural estimator of Cpm based on multiple

samples. Based on the natural estimator of Cpm, we

consider the sampling errors by providing an explicit formula with the Matlab program to obtain the esti-mation accuracy of the Cpm. We tabulate the sampling

accuracy of Cpm for sample size determination so that

the engineers/practitioners can use it for their in-plant applications. An example of multiple precision voltage reference (PVR) processes is presented to illustrate the applicability of Cpm MPPAC for manufacturing

quality control.

Keywords Multiprocess performance analysis chart Æ Maximum likelihood estimator Æ Process capability index Æ Sample size determination

1 Introduction

Process capability indices (PCIs) have been widely used in various manufacturing industries, to provide numer-ical measures on process potential and process perfor-mance. The two most commonly used process capability indices are Cp and Cpk introduced by Kane [1]. These

two indices are defined in the following equation: Cp¼USLLSL6r

Cpk¼ min USLl3r lLSL3r

 

;

where USLand LSL are the upper and the lower speci-fication limits, respectively, lis the process mean and s is the process standard deviation. The index Cpmeasures

the process variation relative to the manufacturing tol-erance, with reflects only the process potential. The in-dex Cpk measures process performance based on the

process yield (percentage of conforming items) without considering the process loss (a new criteria for process quality championed by Hsiang and Taguchi [2]). Taking into consideration the process departure (which reflects the process loss), Chan et al. [3] developed the index Cpm, which measures the ability of the process to cluster

around the target. The index Cpmis defined as:

Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ l  Tð Þ2 q ¼ d 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ l  Tð Þ2 q ;

where T is the target value, and d=(USL)LSL)/2 is half of the length of the specification interval (LSL, USL). Ruczinski [4] showed that Yield=2F(3Cpm))1, or the

fraction of nonconformities =2F()3Cpm), where F(.)

is the cumulative function of the standard normal distribution. Table 1 displays various values of Cpm=

0.95(0.05) 2.00, and the corresponding nonconformities (in PPM). For example, if a process has capability with Cpm=1.25, then the manufacturing process yield would

be at least 99.982%. Some common used values of Cpm

are: 1/3 (the process is incapable), 1/2 (the process is

W. L. Pearn (&)

Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan ROC

E-mail: roller@cc.nctu.edu.tw M. H. Shu

Department of Commerce Automation & Management, National Pingtung Institute of Commerce, Taiwan ROC B. M. Hsu

Department of Industrial Engineering and Management, ChengShiu University, Taiwan ROC

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incapable), 1.00 (the process is normally called capable), 1.33 (the process is normally called satisfactory), 1.67 (the process is normally called good), and 2.00 (the process is normally called super).

Statistical process control charts have been widely used for monitoring individual factory manufacturing processes on a routine basis. Those charts are essential tools for the control and improvement of these pro-cesses. In the multiprocess environment where a group of processes need to be monitored and controlled, it could be difficult and time-consuming for factory engi-neers or supervisors to analyse the individual chart in order to evaluate the overall performance of factory process control activities. Singhal [5, 6] introduced the multiprocess performance analysis chart (MPPAC) using the process capability indices Cp and Cpk which

can be implemented for illustrating and analysing the performance of a group of processes in a multiple pro-cess environment by including the departure of the process mean from the target value, process variability, capability zones and expected fallout outside specifica-tion limits on a single chart. Pearn and Chen [7] pro-posed a modification to MPPAC combining the more advanced process capability index, Cpm, to identify the

problems causing the processes failing to centre around the target. Pearn et al. [8] introduced the MPPAC based on the incapability index. Chen et al. [9] presented a modification to the MPPAC.

With respect to these studies, there are some limita-tions and shortcomings included. First, the existing MPPACs based on the process capability indices are restricted to obtaining quality information by calculat-ing one scalculat-ingle sample data for each process. In practice, however, manufacturing information is often derived from multiple samples rather than one single sample,

particularly when a daily-based process control plan is implemented for monitoring process stability. Second, most existing MPPACs using process capability indices simply use the estimates of the indices on the chart and then make a conclusion on whether processes meet the capability requirement and directions need to be taken for further quality improvement. Their approach is highly unreliable since sampling errors are ignored. Therefore, in this paper, we introduce a new control chart of Cpm MPPAC. The natural estimator of Cpm

based on multiple samples is investigated. We also consider the sampling errors and develop a Matlab program for determining the overall number of obser-vations and sub-samples required by an estimating accuracy of the Cpm. An example of precision voltage

reference (PVR) processes is presented to illustrate the applicability of Cpm MPPAC for production quality

control.

2 The CpmMPPAC

Singhal [5] indicated that the MPPAC can be used to evaluate the performance of a single process as well as multiple-processes; to set the priorities among multiple processes for quality improvement, and to indicate whether reducing the variability, or the departure of the process mean should be focused upon; to provide an easy way to qualify the process improvement by com-paring the locations on the chart of the processes before and after the improvement effort. Since Cpm

simulta-neous measures process variability and centreing, a Cpm

MPPAC would provide a convenient way to identify problems in process capability after statistical control is established. Based on the definition, we first set Cpm=h,

for various h values, then a set of Cpm=h values

satis-fying the equation: r d=3  2 þ l T d=3  2 ¼ 1 h  2

can be plotted to form a contour (indifference curve) of Cpm=h. These contours are semicircles centreed at (T, 0)

with radius 1/h. The more capable the process, the smaller the semicircle is. Figure 1 plots the six contours on the CpmMPPAC for the six values, Cpm=1/3, 1/2, 1,

1.33, 1.67 and 2. On the CpmMPPAC, we note that:

[a] The horizontal line and vertical line through the plotted point intersect the vertical axis and hori-zontal axis at points represented by (r/(d/3))2and ((l)T)/(d/3))2, respectively.

[b] The distance between 0 and the intersection be-tween point the vertical line through the plotted point and the horizontal axis, denoting the depar-ture of the process mean from the target.

[c] The distance between 0 and the point, which the horizontal line through the plotted point intersect-ing the vertical axis, denotes the process variance.

Table 1 Various values of Cpm=0.95(0.01)2.00 and the

corres-ponding nonconformities (in PPM)

Cpm PPM 0.95 4371.923 1.00 2699.796 1.05 1632.705 1.10 966.848 1.15 560.587 1.20 318.217 1.25 176.835 1.30 96.193 1.35 51.218 1.40 26.691 1.45 13.614 1.50 6.795 1.55 3.319 1.60 1.587 1.65 0.742 1.70 0.340 1.75 0.152 1.80 0.067 1.85 0.029 1.90 0.012 1.95 0.005 2.00 0.002

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[d] For the points inside the semicircular contour (indifference curve) Cpm=h, the corresponding

Cpmvalues are larger than h. For the points outside

the semicircular contour Cpm=h, the corresponding

Cpmvalues are smaller than h.

[e] As the point gets closer to the target, the value of the Cpm becomes larger, and the process

perfor-mance is better.

[f] For processes with fixed values of Cpm, the points

within the two 45° lines envelop, and the process variability is contributed mainly by the process variance.

[g] For processes with fixed values of Cpm, the points

outside the two 45° lines envelop, the process varia-bility is contributed mainly by the process departure. In general, the process parameters land r2 are un-known. But, in practice land r2 can be estimated by sample data obtained from stable processes. In the next section, estimating Cpm and the estimation accuracy

based on multiple samples is investigated.

3 Estimating Cpm based on multiple samples

Kirmani et al. [10] indicated that a common practice of the process capability estimation in the manufacturing industry is to first implement a routine-basis data col-lection program for monitoring/controlling the process stability, then to analyse the past ‘‘in control’’ data. For multiple samples of ms groups each of with sizes n are

chosen randomly from a stable process which follows a normal distribution N(l,r2). Let Xi¼Pnj¼1xij

 n and Si¼ ð Þn1Pnj¼1 xij Xi

 2

h i1=2

be the ith sample mean and the sample standard deviation, respectively. We consider the following natural estimator of Cpm:

~ CpmM ¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 pþ X T 2 r ; where X ¼Pms i¼1Xi . ms and Sp2¼ Pms i¼1Si2  m2. If the

process follows the normal distribution N(l,r2), then

~ CMpm¼p USL  LSLffiffiffiffiN 6r NS2 p r2 þ N  X l2 r2 þ Nðl TÞ2 r2 2 6 4 3 7 5 1=2 ; whereXms i¼1n¼ N :

The NSp2.r2 and N  X l.r2 are distributed as ordinary central Chi-square distribution with N-msand

one degree of freedom, v2

Nmsand v 2 1, respectively. Therefore, ~ CpmM USL LSL 6r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N v2 Nmsþ1;k s ¼ CP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N v2 Nmsþ1;k s ; where v2

N ;k denotes the noncentral Chi-square

distribu-tion with N degrees of freedom and noncentral param-eter k=N((l)T)/r)2. The rth moment (about zero) can be obtained as the following equation:

E ~ CMPm r¼ pffiffiffiffiNCP r E v2Nmsþ1;kr=2 ¼ ffiffiffiffi N p CP ffiffiffi 2 p  r exp k 2   X 1 j¼0 k=2 ð Þj j!  C 2jðð þ N  msþ 1  rÞ=2Þ C 2jðð þ N  msþ 1Þ=2Þ ( ) :

The probability density function (PDF) of natural estimator of Cpm can be easily attained as the following

equation: f xð Þ ¼2 1N ð Þ=2C0 Nð þ1Þ 3ðNþ1Þ xðNþ2Þ exp  k 2 C02 18x2  X 1 j¼0 kC02 36x2 j j!C N þ 1 þ 2j 2   1 ( ) ; where C0¼ 3pffiffiffiffiNCP; N ms¼ N;and x > 0

Fig. 1 The contours of Cpm

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Using the method similar to that presented in Va¨nnman [11], we may obtain an exact form of the cumulative distribution function of ~CM

Pm. The cumulative

distribution function of ~CM

Pmcan be expressed in terms of

a mixture of the ordinary central Chi-square distribution and the normal distribution:

FC~M Pmð Þ ¼ 1 x Z bpffiffiffiN=ð Þ3x 0 G b 2N 9x2  t 2    / t þ nh pffiffiffiffiNþ / t  n pffiffiffiffiNidt;

where b=d/r, n=(l-T)/r, G(.) is the cumulative distri-bution function of the ordinary central Chi-square dis-tribution v2Nms, and F(.) is the probability density function of the standard normal distribution N(0, 1). We note that for cases with one single sample, ms=1, the

special case of multiple samples, the statistical properties of the estimator of Cpmare proposed by Chan et al. [3],

Boyles [12], Pearn et al. [13], Kotz and Johnson [14], Va¨nnman and Kotz [15] and Va¨nnman [11].

4 The estimation accuracy ofCpm

For processes with a target value setting to the mid-point of the specification limits (T=(USL+LSL)/2), the index may be rewritten as follows. We note that when Cpm=C, b=d/r can be expressed as b¼ 3C

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ n2 p

. Thus, the index Cpmmay be expressed as a function of

the characteristic parameter n.

Cpm¼

d 3rpffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2¼

d=r 3pffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2;

wheren=(l)T)/r. Hence, given the total number of observations N, the number of sub-samples mswith the

confidence level c, the parameter n, and the estimating accuracy Rpm can be obtained using numerical

integra-tion technique with iteraintegra-tions, to solve the following Eq. 11. It should be noted, particularly, that Eq. 1 is an even function of n. Thus, for both n=n0and n=)n0we

may obtain the same total observations N. Z Rpm ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð1þn2Þ p 0 G R 2pmN1þ n2 t2  / t þ nh pNffiffiffiffiþ / t  n pNffiffiffiffiidt¼ 1  c: ð1Þ

4.1 The estimation accuracy Rpmand parameter n

Since the process parameters land r are unknown, then the distribution characteristic parameter, n=(l)T)/r is also unknown, which has to be estimated in real appli-cations, naturally by substituting l and r by the sample mean Xand the sample standard deviation Sp. Such

approach introduces additional sampling errors from

estimating n in finding the sample accuracy, and cer-tainly would make our approach (and of course including all the existing methods) less reliable. Conse-quently, any decisions made would result in less pro-duction yield assurance to the factories, and provide less quality protection to the customers. To eliminate the need for further estimating the distribution characteris-tic parameter n=(l–T)/r, we examine the behaviour of the sample accuracy Rpm against the parameter n=(l–

T)/r.

We perform extensive calculations to obtain the Rpm

for n=0(0.1)3.00, the total number of observations N=200, ms=1, 10, 20, 40, 50 and 100 with confidence

levelc=0.90, 0.95, 0.975 and 0.99. The results indicate that (i) the sample precision is increasing as n increases, and is decreasing as ms increases, (ii) the sample

preci-sion Rpm obtains its minimum at n=0 in all cases.

Hence, for practical purpose we may solve Eq. 1 with n=0 to obtain the required sample accuracy for given N, ms and c, without having to further estimate the

parameter n. Thus, the level of confidence c can be en-sured, and the decisions made based on such approach are indeed more reliable. Figure 2 plots the curves of the sample accuracy Rpmversus the parameter n for N=200

and ms=1, 10, 20, 40, 50 and 100 with confidence level

c=0.95. For bottom curve 1, ms=100; for bottom curve

2, ms=50; for bottom curve 3, ms=40; for top curve 3,

ms=20; for top curve 2, ms=10; for top curve 1, ms=1.

5 The sample size determination forCpmMPPAC

Using Eq. 1, we may compute the estimation accuracy Rpm. Three auxiliary functions for evaluating Rpm are

included here: (a) the cumulative distribution function of the chi-square v2Nms, G(Æ), (b) the probability density function of the standard normal distribution F(Æ), and (c) the function of the numerical evaluate integration using the recursive adaptive Simpson quadrature—’’quad’’. The program sets (l)T)/r=0, reads the number of

Fig. 2 Plots of Rmp vs|n| for N=200, ms=1, 10, 20, 40, 50, 100

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sub-samples ms, the total number of observations N, and

the confidence level c, then outputs with the estimating precision Rpm. The program, actual executed inputs and

outputs are listed below.

Matlab Program for Computing the Accuracy

%---% This program calculates the sample % accuracy ofCpmfor given sample

% observations, the number of samples % and confidence level.

%---clear global

[N1 ms1 r1]=read(’Enter the total

observations, the number of subsamples, and confidence level:’);

global b N epsilon eCpmms N=N1; r=r1; ms=ms1; epsilon=0; eCpm=1.0; b=0;d=0; c=0.2:0.025:3; for i=1:1:113 b=0;d=0;y=0; b=3*c(i)*sqrt(1+epsilon^2); d=b*sqrt(N)/(3*eCpm); y=quad(’Cpm’,0,d); if (y-(1-r))>0 break end end c=0.2+0.025*(i-1):-0.001:0.2; for k=1:(0.025*(i-1)*1000)+1 b=0;d=0;y=0; b=3*c(k)*sqrt(1+epsilon^2); d=b*sqrt(N)/(3*eCpm); y=quad(’Cpm’,0,d); if ((1-r)-y)>0.0001 break end end

fprintf(’The Estimating Accuracy is

%g\n’,c(k)/eCpm)

%---% read.m file.

%---function [a1, a2, a3]=read(labl)

if nargin==0, labl=’?’; end n=nargout; str=input(labl,’s’); str=[’[’,str,’]’]; 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 b epsilon eCpmms Q1=chi2cdf(((b^2*N/(9*eCpm ^2))-t.^2), N-ms).*(normpdf((t+epsilon*sqrt(N)))+ normpdf((t-epsilon*sqrt(N)))); %---% The end.

%---Input Enter the total observations, the number of subsamples, and confidence level: 100,20,0.95

Output The estimating accuracy is 0.782

The sample size determination is essential to most factory applications, particularly for those implementing a routine-basis data collection plan for monitoring and controlling process quality. It directly relates to the sampling cost of a data collection plan. Table 2a and 2b displays the sample size N and number of samples ms

required and the corresponding precision of the esti-mation Rpm. For example,c=0.95, N=150, ms=30 gives

Rpm=0.802. Thus, the true value of Cpm, is no less

than ~CM

pm 0:802. On the other hand, if Rpm=0.802 is

chosen, then we may determine N=102 with ms=17

(each sample with six observations). Similarly, if Rpm=0.85 is chosen, then we may determine N=190

with ms=10 and c=0.975, N=198 with ms=6 and

c=0.90, or N=256 with ms=32 and c=0.975,

depend-ing on which sampldepend-ing plan is more appropriate to the application.

6 Manufacturing quality control for multiple PVR processes

In the following discussion, we investigate the production quality of a group of multiple processes for manufacturing the precision voltage reference devices. Voltage references are essential to the accuracy and performance of analog systems. They are used in many types of analog circuitry for signal processing, such as analog to digital (AD) or digital to analog (DA) converters and smart sensors. Precision voltage references can be used in constructing an accuracy regulated supply that could have better charac-teristics than some regulator chips, which sometimes can dissipate too much power. Another application of the voltage references is creating a precision constant current supply. In addition, voltage references are needed in the test equipment, which must be accurate, such as voltme-ters, ohmmeters and ammeters.

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Table 2 The total number of sample observations, nm s = N , number of samples, ms and the precision of estimation with c=0.90, 0.95, 0.975, 0.99 n 45 681 0 1 2 msc 0.90 0.95 0.975 0.99 0.9 0.95 0.975 0.99 0.90 0.95 0.975 0.99 0.90 0.95 0.975 0.99 0.9 0.95 0.975 0.99 0.90 0.95 0.975 0.99 5 0.682 0.630 0.587 0.538 0.727 0.680 0.641 0.596 0.759 0.715 0.679 0.637 0.800 0.762 0.730 0.693 0.827 0.792 0.763 0.729 0.845 0.814 0.787 0.756 6 0.692 0.649 0.608 0.563 0.740 0.697 0.661 0.619 0.771 0.731 0.697 0.659 0.811 0.776 0.746 0.712 0.837 0.805 0.778 0.747 0.855 0.826 0.801 0.772 7 0.708 0.663 0.626 0.583 0.751 0.711 0.676 0.637 0.781 0.744 0.712 0.676 0.820 0.787 0.759 0.728 0.845 0.815 0.790 0.761 0.862 0.835 0.812 0.785 8 0.717 0.675 0.640 0.599 0.760 0.722 0.689 0.652 0.789 0.754 0.724 0.690 0.827 0.796 0.770 0.740 0.851 0.823 0.800 0.773 0.868 0.843 0.821 0.796 9 0.725 0.685 0.651 0.613 0.767 0.731 0.700 0.665 0.795 0.762 0.734 0.702 0.833 0.804 0.779 0.751 0.856 0.830 0.808 0.782 0.873 0.849 0.828 0.805 10 0.731 0.694 0.661 0.625 0.773 0.739 0.709 0.676 0.801 0.770 0.743 0.712 0.838 0.810 0.787 0.760 0.861 0.836 0.815 0.790 0.877 0.854 0.835 0.812 11 0.737 0.701 0.670 0.635 0.778 0.745 0.718 0.685 0.806 0.776 0.750 0.721 0.842 0.816 0.793 0.767 0.865 0.841 0.821 0.797 0.880 0.859 0.840 0.819 12 0.742 0.708 0.678 0.644 0.783 0.751 0.725 0.694 0.810 0.781 0.757 0.728 0.846 0.821 0.799 0.774 0.868 0.846 0.826 0.804 0.884 0.863 0.845 0.824 13 0.747 0.713 0.685 0.652 0.787 0.757 0.731 0.701 0.814 0.786 0.763 0.735 0.849 0.825 0.804 0.778 0.871 0.849 0.831 0.809 0.886 0.866 0.849 0.829 14 0.751 0.719 0.691 0.659 0.791 0.761 0.736 0.708 0.818 0.791 0.768 0.741 0.852 0.829 0.809 0.785 0.874 0.853 0.835 0.814 0.889 0.870 0.853 0.834 15 0.755 0.723 0.696 0.666 0.794 0.766 0.741 0.714 0.821 0.795 0.772 0.747 0.855 0.832 0.813 0.790 0.876 0.856 0.838 0.818 0.891 0.873 0.856 0.838 16 0.758 0.728 0.702 0.672 0.797 0.770 0.746 0.719 0.823 0.798 0.777 0.752 0.857 0.835 0.817 0.795 0.879 0.859 0.842 0.822 0.893 0.875 0.860 0.842 17 0.761 0.731 0.706 0.677 0.800 0.773 0.750 0.724 0.826 0.802 0.781 0.756 0.860 0.838 0.820 0.799 0.881 0.861 0.845 0.826 0.895 0.878 0.862 0.845 18 0.764 0.735 0.710 0.682 0.802 0.776 0.754 0.728 0.828 0.805 0.784 0.760 0.862 0.841 0.823 0.802 0.882 0.864 0.848 0.829 0.897 0.880 0.865 0.848 19 0.766 0.738 0.714 0.687 0.805 0.779 0.758 0.733 0.830 0.807 0.787 0.764 0.864 0.843 0.826 0.806 0.884 0.866 0.850 0.832 0.898 0.882 0.867 0.851 20 0.769 0.741 0.718 0.691 0.807 0.782 0.761 0.737 0.832 0.810 0.790 0.768 0.865 0.846 0.829 0.809 0.886 0.868 0.853 0.835 0.890 0.884 0.870 0.853 21 0.771 0.744 0.721 0.695 0.809 0.785 0.764 0.740 0.834 0.812 0.793 0.771 0.867 0.848 0.831 0.812 0.887 0.870 0.855 0.838 0.901 0.885 0.872 0.856 22 0.773 0.747 0.724 0.699 0.811 0.787 0.767 0.744 0.836 0.814 0.796 0.774 0.868 0.850 0.833 0.815 0.889 0.872 0.857 0.840 0.902 0.887 0.874 0.858 23 0.775 0.749 0.727 0.702 0.812 0.789 0.770 0.747 0.838 0.817 0.798 0.777 0.870 0.851 0.836 0.817 0.890 0.873 0.859 0.842 0.904 0.888 0.875 0.860 24 0.777 0.752 0.730 0.705 0.814 0.791 0.772 0.750 0.839 0.818 0.801 0.778 0.871 0.853 0.838 0.820 0.891 0.875 0.861 0.845 0.905 0.890 0.877 0.862 25 0.778 0.754 0.733 0.708 0.816 0.793 0.774 0.752 0.841 0.820 0.803 0.783 0.872 0.855 0.840 0.822 0.892 0.876 0.863 0.847 0.906 0.891 0.879 0.864 26 0.780 0.756 0.735 0.711 0.817 0.795 0.777 0.755 0.842 0.822 0.805 0.785 0.874 0.856 0.841 0.824 0.893 0.878 0.864 0.849 0.907 0.892 0.880 0.866 27 0.782 0.758 0.738 0.714 0.818 0.797 0.779 0.758 0.843 0.824 0.807 0.787 0.875 0.858 0.843 0.826 0.894 0.879 0.866 0.850 0.908 0.894 0.881 0.867 28 0.783 0.760 0.740 0.717 0.820 0.799 0.781 0.760 0.844 0.825 0.809 0.789 0.876 0.859 0.845 0.828 0.895 0.880 0.867 0.852 0.908 0.895 0.883 0.869 29 0.784 0.762 0.742 0.719 0.821 0.800 0.783 0.762 0.846 0.827 0.810 0.792 0.877 0.860 0.846 0.830 0.896 0.881 0.869 0.854 0.909 0.896 0.884 0.871 30 0.786 0.763 0.744 0.721 0.822 0.802 0.784 0.764 0.847 0.828 0.812 0.793 0.878 0.862 0.848 0.831 0.897 0.882 0.870 0.855 0.910 0.897 0.885 0.872 31 0.787 0.765 0.746 0.724 0.823 0.803 0.786 0.766 0.848 0.829 0.814 0.795 0.879 0.863 0.849 0.833 0.898 0.883 0.871 0.857 0.911 0.898 0.886 0.873 32 0.788 0.766 0.748 0.726 0.824 0.805 0.788 0.768 0.849 0.831 0.815 0.797 0.880 0.864 0.850 0.835 0.899 0.884 0.872 0.858 0.912 0.899 0.887 0.875 33 0.789 0.768 0.749 0.728 0.825 0.806 0.789 0.770 0.850 0.832 0.817 0.799 0.880 0.865 0.852 0.836 0.899 0.885 0.873 0.860 0.912 0.900 0.889 0.876 34 0.790 0.769 0.751 0.730 0.826 0.807 0.791 0.772 0.851 0.833 0.818 0.800 0.881 0.866 0.853 0.838 0.900 0.886 0.875 0.861 0.913 0.900 0.890 0.877 35 0.791 0.771 0.752 0.732 0.827 0.809 0.792 0.774 0.851 0.834 0.819 0.802 0.882 0.867 0.854 0.839 0.901 0.887 0.876 0.862 0.913 0.901 0.890 0.878 36 0.792 0.772 0.754 0.733 0.828 0.810 0.794 0.775 0.852 0.835 0.821 0.804 0.883 0.868 0.855 0.840 0.901 0.888 0.877 0.863 0.914 0.902 0.891 0.879 37 0.793 0.773 0.755 0.735 0.829 0.811 0.795 0.777 0.853 0.836 0.822 0.805 0.883 0.869 0.856 0.841 0.902 0.889 0.877 0.864 0.915 0.903 0.892 0.880 38 0.794 0.774 0.757 0.737 0.830 0.812 0.796 0.778 0.854 0.837 0.823 0.806 0.884 0.870 0.857 0.843 0.903 0.890 0.878 0.865 0.915 0.903 0.893 0.881 39 0.795 0.775 0.758 0.738 0.831 0.813 0.797 0.780 0.855 0.838 0.824 0.808 0.885 0.870 0.858 0.844 0.903 0.890 0.879 0.866 0.916 0.904 0.894 0.882 40 0.796 0.776 0.760 0.740 0.832 0.814 0.799 0.781 0.855 0.839 0.825 0.809 0.885 0.871 0.859 0.845 0.904 0.891 0.880 0.867 0.916 0.905 0.895 0.883

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Consider the following case taken from a microelec-tronics manufacturing factory making precision voltage reference devices. Twelve specific types of precision voltage reference devices extensively used on PC-based instrumentation and test equipment with different precision voltage references specifications are selected in this study. Their precision voltage reference specifica-tions are displayed in Table 3. A sample data collection plan is implemented in the factory on a daily basis to monitor/control production quality. The factory re-source and sampling schedule allow the data collection plan be implemented with N=150 with ms=15 (each

sample with 10 observations). Checking Table 2a and

2b, we obtain the estimation accuracy Rpm=0.856, with

confidence c=0.95.

The calculated overall sample mean, the pooled sample standard deviation, the estimate ~CpmM, the mini-mum true value and the maximini-mum nonconformities are displayed in Table 4 and Table 5. Figure 3plots the Cpm

MPPAC for the 12 processes using the data summarised in Table 4. We analyse these process points in Fig. 3 and obtain the following critical summary information of the quality condition for all processes.

[a] The plotted point E is outside the contour of Cpm=1/2. It indicates that the process has a very

low capability. Since the point E is close to the target line, the process mean is close to the target value, and the poor capability is mainly contributed by the significant process variation. Thus, immedi-ate quality improvement actions must be taken for reducing the process variance.

[b] The plotted points H, D, and G lies outside of the contour of Cpm=1. It indicates that the capability

Cpmis less than 1. Since the point lies inside the two

45° lines envelope range, it indicates that the pro-cess variation measure, (r/(d/3))2, is more signifi-cant than the departure measure, ((l)T)/(d/3))2. Thus, reducing the process variance should be set to higher priority than that of reducing the process departure.

Table 4 The calculated statistics of the ten processes Process X SP X T . d=3 ð Þ h i2 SP= d=3ð Þ ½ 2 A 4.999529 0.001491 0.02 0.2 B 10.00111 0.000667 1.78 0.64 C 14.99325 0.004796 1.82 0.92 D 19.99795 0.002728 0.38 0.67 E 1.00003 0.00015 0.13 3.24 F 0.499996 1.49E)06 1.44 0.2 G 2.999946 7.87E)05 0.29 0.62 H 11.99864 0.002272 0.46 1.29 I 9.004948 0.005333 0.68 0.79 J 6.00337 0.0032 0.71 0.64 K 3.000087 0.000296 0.03 0.35 L 17.99944 0.002057 0.035 0.47

Table 3 The precision voltage reference specifications

Code V Precision LSL USL

A 5 ±0.2% 4.99 5.01 B 10 ±0.025% 9.9975 10.0025 C 15 ±0.1% 14.985 15.015 D 20 ±0.05% 19.99 20.01 E 1 ±0.025% 0.99975 1.00025 F 0.5 ±0.002% 0.49999 0.50001 G 3 ±0.01% 2.9997 3.0003 H 12 ±0.05% 8.994 9.006 I 9 ±0.2% 8.982 9.018 J 6 ±0.2% 5.988 6.012 K 3 ±0.05% 2.9985 3.0015 L 18 ±0.05% 17.991 18.009 Table 5 The ~CM

pmminimum true capability Cpm, and the maximum

nonconformities (in PPM) Process C~M pm Cpm PPM A 2.132 1.825 0.0438 B 0.643 0.550 98943 C 0.604 0.517 120900 D 0.976 0.835 8439 E 0.545 0.467 161210 F 0.781 0.669 44750 G 1.048 0.897 4331 H 0.756 0.647 52258 I 0.825 0.706 34175 J 0.861 0.737 27036 K 1.622 1.389 30.86 L 1.407 1.205 300.35

Fig. 3 The CpmMPPAC for the

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[c] The plotted points C, F and B lie outside the con-tour of Cpm=1. Since these points also lie outside

the two 45° lines envelope range, it indicates that the departure measure, ((l)T)/(d/3))2is higher than process variation measure, (r/(d/3))2. Thus, quality improvement effort for these processes should be first focused on reducing their process departure from the target value T, then only the reduction of the process variance can be considered.

[d] The plotted points I and J are very close to the two 45° lines, and are outside the contour of Cpm=1. It

indicates that the poor capability of both processes is contributed to equally significantly by the process mean departure and process variance.

[e] The plotted points K and L lie inside the contours of Cpm=1.33 and Cpm=1, respectively. It means

that both process capability values Cpmare greater

than 1. Capabilities of both processes are consider satisfactory. They have lower priorities in allocating quality improvement efforts than other processes. [f] Process A is near to 0 and its variation measure is

small. Therefore, process A is considered perform-ing well. No immediate improvement activities need to be taken.

7 Conclusions

Conventional investigations on manufacturing quality control are restricted to obtaining quality information based on one single sample for each process ignoring sampling errors. The proposed Cpm MPPAC using

process capability index Cpmis useful for manufacturing

quality control of a group of processes in a multiple process environment. In this paper, we introduced a new control chart called Cpm MPPAC using the natural

estimator of Cpm based on multiple samples. We

inves-tigated the accuracy of the estimation as a function of the process characteristic parameter n=(l)T)/r, given a group of multiple control chart samples. Information regarding the true capability values and the maximum nonconformities (in PPM) is provided for production quality control. Appropriate sample sizes are then rec-ommended to the proposed Cpm MPPAC for multiple

processes production quality control. This approach ensures that the critical information conveyed from the CpmMPPAC based on multiple control chart samples, is

more reliable than all other existing methods. We developed a Matlab computer program to calculate the estimating accuracy and provided convenient tables for practitioners to use in determining appropriate sample sizes needed for their factory applications. An example of PVR manufacturing processes is given to illustrate the applicability of the proposed CpmMPPAC.

References

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

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

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

4. Ruczinski I (1996) The relation between Cpmand the degree of

includence. Dissertation, University of Werg

5. Singhal SC (1990) A new chart for analyzing multiprocess performance. Qual Engin 2(4):379–390

6. Singhal SC (1991) Multiprocess performance analysis chart (MPPAC) with capability zones. Qual Engin 4(1):75–81 7. Pearn WL, Chen KS (1997) Multiprocess performance analysis:

a case study. Qual Engin 10(1):1–8

8. Pearn WL, Ko CH, Wang KH (2003) A multiprocess perfor-mance analysis chart based on the incapability index Cpp: an

application to the chip resistors. Microelect Reliab (in press) 9. Chen KS, Huang ML, Li RK (2001) Process capability analysis

for an entire product. Int J Prod Res 39(17):4077–4087 10. Kirmani SNUA, Kocherlakota K, Kocherlakota S (1991)

Estimation of s and the process capability index based on sub-samples. Comm Stat Theor Meth 20(1):275–291

11. Va¨nnman K (1997) Distribution and moments in simplified form for a general class of capability indices. Comm Stat Theor Meth 26:159–179

12. Boyles RA (1991) The Taguchi capability index. J Qualit Technol 23:17–26

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

14. Kotz S, Johnson NL (1993) Process capability indices. Chap-man & Hall, London, UK

15. Va¨nnman K, Kotz S (1995) A superstructure of capability indices—distributional properties and implications. Scand J Stat 22:477–491

數據

Table 1 Various values of C pm =0.95(0.01)2.00 and the corres-
Fig. 1 The contours of C pm
Fig. 2 Plots of R mp vs|n| for N=200, m s =1, 10, 20, 40, 50, 100
Table 3 The precision voltage reference specifications

參考文獻

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