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 precision voltage reference process
Received: 22 November 2002 / Accepted: 20 February 2003 / Published online: 19 February 2004 Springer-Verlag London Limited 2004
Abstract A multiprocess performance analysis chart (MPPAC), based on the process capability index Cpm,
called Cpm MPPAC, is developed to analyse the
man-ufacturing quality of a group of processes in a multiple process environment. The Cpm MPPAC conveys critical
information about multiple processes regarding the departure of the process and process variability on one single chart. Existing research on MPPAC has been restricted to obtaining quality information from one single sample of each process, ignoring sampling errors. The information provided from the existing MPPAC chart, therefore, is unreliable and misleading, resulting in incorrect decisions. In this paper, the natural esti-mator of Cpm is considered based on multiple samples.
Based on the natural estimator of Cpm, sampling errors
are considered by providing an explicit formula with Matlab to obtain the estimation accuracy of the Cpm.
The sampling accuracy of Cpm is tablulated for sample
size determination so that engineers/practitioners can use it for in-plant applications. An example of multiple PVR processes is presented to illustrate the applicability of CpmMPPAC 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 a numerical measures of process potential and process performance. The two most commonly used process capability indices are Cp and Cpk, introduced by Kane
[1]. These two indices are defined in the following:
Cp¼ USL LSL 6r Cpk¼ min USL l 3r ; l LSL 3r ;
where USL and LSL are the upper and the lower spec-ification limits, respectively, l is the process mean, and r is the process standard deviation. The index Cpmeasures
the process variation relative to the manufacturing tol-erance, which reflects only the process potential. The Cpk
index measures process performance based on the pro-cess yield (percentage of conforming items) without considering the process loss (a new criteria for process quality championed by Hsiang and Taguchi [2]). Taking process departure into consideration (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 Cpmindex is 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).
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 proces yield would
be at least 99.982%. Some commonly used values of
W. L. Pearn (&)
Department of Industrial Engineering and Management, National Chiao Tung Univ., Ta Hsuch Rd.,
Hsinchu 1001, Taiwan, R.O.C. E-mail: roller@cc.nctu.edu.tw M. H. Shu
Department of Commerce Automation & Management, National Pingtung Institute of Commerce, Taiwan, R.O.C. B. M. Hsu
Department of Industrial Engineering and Management, Cheng Shiu University, Taiwan, R.O.C.
Cpm are 1/3 (process is incapable), 1/2 (process is
incapable), 1.00 (process is normally called capable), 1.33 (process is normally called satisfactory), 1.67 (process is normally call good), and 2.00 (process is normally called super).
Statistical process control charts have been widely used to monitor individual factory manufacture pro-cesses on a routine basis. Those charts are essential tools for the control and improvement of these processes. In the multiprocess environment, where a group of pro-cesses need to be monitored and controlled, it could be difficult and time-consuming for factory engineers or supervisors to analyse the individual chart in order to evaluate 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
imple-mented to illustrate and analyse the performance of a group of processes in a multiple process environment by including the departure of the process mean from the target value, process variability, capability zones, and expected fallout outside specification limits on a single chart. Pearn and Chen [7] proposed a modification to MPPAC that combined the more advanced process capability index, Cpm, to identify the problems causing
the processes to fail to centre around the target. Pearn et al. [8] introduced 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 calculating one single 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 as to 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, a new control chart is introduced for Cpm
MPPAC. The natural estimator of Cpm, based on multiple
samples, is investigated. The sampling errors are also considered and a Matlab program is developed to deter-mine the overall number of observations and sub-samples required for a Cpmestimating accuracy. An example of
PVRP 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 the focus, as well as to provide an easy way to qualify the process improvement by comparing the locations on the chart of the processes before and after the improvement effort. Since Cpm
simultaneously measures process variability and centre-ing, a CpmMPPAC would provide a convenient way to
identify problems in process capability after statistical control is established. Based on the definition, first Cpm=h is set for various h values, then a set of (l,r)
values satisfying the equation: r d=3 2 þ l T d=3 2 ¼ 1 h 2
can be plotted on the contour (indifference curve) of Cpm=h. These contours are semicircles centered at (T, 0)
with radius 1/h. The more capable the process, the smaller the semicircle. The six contours are plotted on the Cpm MPPAC for the six values, Cpm=1/3, 1/2, 1,
1.33, 1.67, and 2, as shown in Fig. 1. On the Cpm
MPPAC, it is noted that:
(a) The parallel line and perpendicular line through the plotted point intersecting the vertical axis and hor-izontal axis at points represented (r/(d/3))2 and ((l)T)/(d/3))2, respectively.
(b) The distance between T and the point, which the perpendicular line through the plotted point inter-secting the horizontal axis, denotes the departure of process mean from target.
(c) The distance between 0 and the point, which the parallel line through the plotted point intersecting the vertical axis, denotes the process variance.
Table 1 Various values of Cpm=0.95(0.01)2.00 and the
corre-sponding 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
(d) For the points inside the semicircle of contour (indifference curve) Cpm=h, the corresponding Cpm
values are larger than h. For the points outside the semicircle of contour Cpm=h, the corresponding
Cpm values 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, the process vari-ability 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 vari-ability is contributed mainly by the process depar-ture.
In general, the process parameters l and r2are un-known. But, in practice l and r2 can be estimated by sample data obtained from stable processes. In the next section, estimating Cpm and 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 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 size n, are
cho-sen randomly from a stable process which follows a normal distribution N(l,r2). Let Xi¼Pnj¼1xij=n
and Si¼ nð Þ1Pnj¼1xij Xi 2 1=2
be the i-th sam-ple mean and the samsam-ple standard deviation, respec-tively. The following natural estimator of Cpm is
considered: ~ CpmM ¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2pþ X T 2 r ; where X¼Pms i¼1Xi=ms and Sp2¼ Pms i¼1Si2=ms.
If the process follows the normal distribution N(l,r2), then C~M pm¼ ffiffiffiffi N p USLLSL 6r hNS2 P r2 þ N Xm 2 r2 þ N mTð Þ2 r2 i1=2 ; where Pms i¼1n¼ N .
The NSP2=r2 and N X l =r2 are distributed as or-dinary central Chi-square distribution with N)ms and
one degree of freedom, v2Nms and v21, respectively. Therefore, ~ CpmM USL LSL 6r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N v2 Nmsþ1;k s ¼ CP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N v2 Nmsþ1;k s ;
where v2N ;k denotes the noncentral Chi-square distribu-tion with N degrees of freedom and noncentral param-eter k=N((l)T)/r)2. The r-th moment (about zero) can be obtained as the following:
E ~CMPm r¼ pffiffiffiffiNCP r E v2 Nmsþ1;k r=2 ¼ ffiffiffiN p CP ffiffi 2 p r exp k 2 P1 j¼0 k=2 ð Þj j! C 2jþN mðð sþ1rÞ=2Þ C 2jþN mðð sþ1Þ=2Þ n o
The probability density function (PDF) of natural estimator of Cpmcan be easily attained as the following,
where C0¼ 3pffiffiffiffiNCp, N)ms=N*, and x>0. f xð Þ ¼2 1N ð Þ=2C0 Nð þ1Þ 3ðNþ1Þ xðNþ2Þ exp k 2 C02 18x2 X1 j¼0 ( kC02 36x2 j j!C N þ 1 þ 2j 2 1)
Using the method similar to that presented in Va¨nnman [11], an exact form of the cumulative distri-bution function of ~CPmM may be obtained. The cumulative distribution function of ~CPmM can 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 h / t þ npffiffiffiffiN þ / t n pffiffiffiffiN idt;
Fig. 1 The contours of Cpm
where b=d/r, n=(l)T)/r, G(·) is the cumulative dis-tribution function of the ordinary central Chi-square distribution v2Nms, and /(·) is the probability density function of the standard normal distribution N (0, 1). 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 Estimation accuracy of Cpm
For processes with target value setting to the mid-point of the specification limits (T=(USL+LSL)/2), the index may be rewritten as the following. 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;
where n=(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. 1. It should be noted, particularly, that Eq. 1 is an even function of n. Thus, for both n=n0and n=)n0the
same total observations N may be obtained. Z Rpm ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð1þn2Þ p 0 G R 2pmN1þ n2 t2 / t þ nh pNffiffiffiffi þ / t n pNffiffiffiffi idt¼ 1 c: ð1Þ
4.1 Estimation accuracy Rpmand parameter n
Since the process parameters l and 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 with the sample mean X and the sample standard deviation Sp.
Such an approach introduces additional sampling errors from estimating n to determine the sample accuracy, and certainly would make this 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 estimation of the distribution charac-teristic parameter n=(l)T)/r, the behaviour of the sample accuracy Rpm must be examined against the
parameter n=(l–T)/r.
Extensive calculations are performed 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
level c=0.90, 0.95, 0.975, and 0.99. The results indicate that: 1. The sample precision is increasing in n and is decreasing in ms, 2. The sample precision Rpmobtains its
minimum at n=0 in all cases. Hence, for practical pur-poses Eq. 1 may be solved with n=0 to obtain the re-quired sample accuracy for a given N, msand c, without
having to further estimate the parameter n. Thus, the level of confidence c can be ensured, and the decisions made based on such approach are indeed more reliable. Fig. 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 Sample size determination for Cpm MPPAC
Using Eq. 1, the estimation accuracy of Rpm may be
computed. Three auxiliary functions for evaluating Rpm
are included here: (a) the cumulative distribution func-tion of the chi-square v2Nms, G(·), (b) the probability density function of the standard normal distribution / (·), and (c) the function of numerical evaluate integration using the recursive adaptive Simpson quardrature, or ‘‘quad’’. The program sets (l)T)/r=0, reads the num-ber of 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
Fig. 2 Plots of Rmpvs |n| for N=200, ms=1, 10, 20, 40, 50, 100
[N1 ms1 r1]=read (‘Enter the total obser-vations, the number of subsamples, and con-fidence level:’);
global b N epsilon ecpm ms 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;
Table 2 Total number of sample observations, nms=N, number of samples, ms, and precision of estimation with c=0.90, 0.95, 0.975, 0.99
n 4 5 6 ms c 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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
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 ecpm ms 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. Tables 2 and 3 display 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 N=102 may be determined with ms=17
(each sample with six observations). Similarly, if Rpm=0.85 is chosen, then N=190 may be determined
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.
Table 3 Total number of sample observations, nms=N, number of samples, ms, and precision of estimation with c=0.90, 0.95, 0.975, 0.99
n 8 10 12 ms c 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 0.800 0.762 0.730 0.693 0.827 0.792 0.763 0.729 0.845 0.814 0.787 0.756 6 0.811 0.776 0.746 0.712 0.837 0.805 0.778 0.747 0.855 0.826 0.801 0.772 7 0.820 0.787 0.759 0.728 0.845 0.815 0.790 0.761 0.862 0.835 0.812 0.785 8 0.827 0.796 0.770 0.740 0.851 0.823 0.800 0.773 0.868 0.843 0.821 0.796 9 0.833 0.804 0.779 0.751 0.856 0.830 0.808 0.782 0.873 0.849 0.828 0.805 10 0.838 0.810 0.787 0.760 0.861 0.836 0.815 0.790 0.877 0.854 0.835 0.812 11 0.842 0.816 0.793 0.767 0.865 0.841 0.821 0.797 0.880 0.859 0.840 0.819 12 0.846 0.821 0.799 0.774 0.868 0.846 0.826 0.804 0.884 0.863 0.845 0.824 13 0.849 0.825 0.804 0.778 0.871 0.849 0.831 0.809 0.886 0.866 0.849 0.829 14 0.852 0.829 0.809 0.785 0.874 0.853 0.835 0.814 0.889 0.870 0.853 0.834 15 0.855 0.832 0.813 0.790 0.876 0.856 0.838 0.818 0.891 0.873 0.856 0.838 16 0.857 0.835 0.817 0.795 0.879 0.859 0.842 0.822 0.893 0.875 0.860 0.842 17 0.860 0.838 0.820 0.799 0.881 0.861 0.845 0.826 0.895 0.878 0.862 0.845 18 0.862 0.841 0.823 0.802 0.882 0.864 0.848 0.829 0.897 0.880 0.865 0.848 19 0.864 0.843 0.826 0.806 0.884 0.866 0.850 0.832 0.898 0.882 0.867 0.851 20 0.865 0.846 0.829 0.809 0.886 0.868 0.853 0.835 0.890 0.884 0.870 0.853 21 0.867 0.848 0.831 0.812 0.887 0.870 0.855 0.838 0.901 0.885 0.872 0.856 22 0.868 0.850 0.833 0.815 0.889 0.872 0.857 0.840 0.902 0.887 0.874 0.858 23 0.870 0.851 0.836 0.817 0.890 0.873 0.859 0.842 0.904 0.888 0.875 0.860 24 0.871 0.853 0.838 0.820 0.891 0.875 0.861 0.845 0.905 0.890 0.877 0.862 25 0.872 0.855 0.840 0.822 0.892 0.876 0.863 0.847 0.906 0.891 0.879 0.864 26 0.874 0.856 0.841 0.824 0.893 0.878 0.864 0.849 0.907 0.892 0.880 0.866 27 0.875 0.858 0.843 0.826 0.894 0.879 0.866 0.850 0.908 0.894 0.881 0.867 28 0.876 0.859 0.845 0.828 0.895 0.880 0.867 0.852 0.908 0.895 0.883 0.869 29 0.877 0.860 0.846 0.830 0.896 0.881 0.869 0.854 0.909 0.896 0.884 0.871 30 0.878 0.862 0.848 0.831 0.897 0.882 0.870 0.855 0.910 0.897 0.885 0.872 31 0.879 0.863 0.849 0.833 0.898 0.883 0.871 0.857 0.911 0.898 0.886 0.873 32 0.880 0.864 0.850 0.835 0.899 0.884 0.872 0.858 0.912 0.899 0.887 0.875 33 0.880 0.865 0.852 0.836 0.899 0.885 0.873 0.860 0.912 0.900 0.889 0.876 34 0.881 0.866 0.853 0.838 0.900 0.886 0.875 0.861 0.913 0.900 0.890 0.877 35 0.882 0.867 0.854 0.839 0.901 0.887 0.876 0.862 0.913 0.901 0.890 0.878 36 0.883 0.868 0.855 0.840 0.901 0.888 0.877 0.863 0.914 0.902 0.891 0.879 37 0.883 0.869 0.856 0.841 0.902 0.889 0.877 0.864 0.915 0.903 0.892 0.880 38 0.884 0.870 0.857 0.843 0.903 0.890 0.878 0.865 0.915 0.903 0.893 0.881 39 0.885 0.870 0.858 0.844 0.903 0.890 0.879 0.866 0.916 0.904 0.894 0.882 40 0.885 0.871 0.859 0.845 0.904 0.891 0.880 0.867 0.916 0.905 0.895 0.883
6 Manufacturing quality control for multiple PVR processes
In the following, the production quality of a group of multiple processes is investigated for manufacturing the voltage reference devices. Voltage references are essen-tial 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 characteristics than some regulator chips, which can sometimes dissipate too much power. Another applica-tion of the voltage references is creating a precision, constant, current supply. In addition, voltage references are needed in the test equipment, which must be accu-rate, such as voltmeters, ohmmeters and ammeters.
Consider the following case taken from a microelec-tronics manufacturing factory that produces precision voltage reference devices. Twelve specific types of pre-cision voltage reference devices extensively used on the PC-based instrumentation and test equipment with dif-ferent precision voltage references specifications are selected in this study. Their precision voltage reference specifications are displayed in Table 4. A sample data collection plan is implemented in the factory on a daily basis to monitor/control production quality. The factory resource and sampling schedule allow the data collection plan be implemented with N=150 with ms=15 (each
sample with 10 observations). Looking at Tables 2 and 3, the estimation accuracy Rmp=0.856 is obtained, with
confidence c=0.95. The calculated overall sample mean, pooled sample standard deviation, the estimated ~CM
pm,
the minimum true value, and the maximum nonconfo-rmities are displayed in Table 5 and Table 6. Fig. 3 provides a plot of the Cpm MPPAC for the twelve
pro-cesses using the data summarised in Table 5. These process points were analysed in Fig. 3 and the following critical summary of the quality condition was obtained 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 point E is close to the target line, it signifies that the process mean is close to the target value, and the poor capability results mainly from the significant process variation. Thus, imme-diate quality improvement actions must be taken to reduce process variance.
(b) The plotted points H, D, and G lie outside of the contour of Cpm=1. This indicates that the capability
Cpmis less than 1. Since the point lies inside the two
45 lines envelop range, it indicates that the process variation measure, (r/(d/3)2, is more significant than the departure measure, ((l)T)/(d/3))2. Thus, reducing the process variance should be set to higher priority than reducing the process departure. (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 envelop range. It indicates that the departure measure, ((l)T)/(d/3))2 is higher than process variation measure, (r/(d/3)2. Thus, quality improvement effort for these processes should first focus on reducing their process departure from the target value T, then on the reduction of the process variance.
Table 5 The calculated statistics of the ten processes
Process X SP X T =d=3 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 4 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 6 The ~CM
pm, minimum 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
(d) The plotted points I and J are very close to the two 45 lines, and are outside the contour of Cpm=1.
This indicates the contributions of process mean departure and process variance are equally signifi-cant factors for the poor capability of both pro-cesses.
(e) The plotted points K and L lie inside the contours of Cpm=1.33 and Cpm=1, respectively. It shows that
both process capability values Cpm are greater than
1. Capabilities of both processes are consider satis-factory. They have lower priorities in allocating quality improvement efforts than other processes. (f) Process A is close to T and the amount of variation
is small. Therefore, process A is considered to be performing well. No immediate improvement actions 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 a
process capability index Cpm, is useful for manufacturing
quality control of a group of processes in a multiple process environment. In this paper, a new control chart, called Cpm MPPAC, was introduced that uses the
nat-ural estimator Cpm based on multiple samples. The
accuracy of the estimation was investigated as a function of the process characteristic parameter n=(l)m)/r, given a group of multiple control chart samples. Infor-mation regarding the true capability values and the maximum nonconformities (in PPM) is provided for production quality control. Appropriate sample sizes are then recommended to the proposed Cpm MPPAC for
multiple processes production quality control. This ap-proach ensures that the critical information conveyed from the CpmMPPAC, based on multiple control chart
samples, is more reliable than all other existing methods. A Matlab computer program was developed 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 process is given to illustrate the applicability and of the proposed Cpm
MPPAC.
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Fig. 3 The CpmMPPAC for the
