Manufacturing capability control for multiple
power-distribution switch processes based on modified
C
pk
MPPAC
W.L. Pearn
a,*, Ming-Hung Shu
ba
Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsin Chu 30050, Taiwan, ROC
b
Department of Management Science, Chinese Military Academy, 1 Wei-Wu Road, Fengshan, Kaohsiung 830, Taiwan, ROC Received 19 December 2002; received in revised form 19 March 2003
Abstract
A modification of the multiple process performance analysis chart based on process capability index Cpk, called
modified Cpk MPPAC, has been developed for controlling product quality/reliability of a group of multiple
manu-facturing processes. The modified CpkMPPAC conveys critical information of each individual process regarding process
accuracy and process precision from one single chart, which is an effective tool for controlling product quality/reliability for multiple processes. Existing MPPAC charts never considered sampling errors hence the capability information provided from those charts is often unreliable and misleading. In this paper, we develop an efficient algorithm to compute the lower confidence bounds of Cpk. The lower confidence bound presents the minimum true capability of the
process, which is essential to product reliability assurance. We apply the lower confidence bounds to the modified Cpk
MPPAC to provide reliable simultaneous capability control for multiple processes. A case involving multiple processes manufacturing power-distribution switch (PDS) is investigated. The modified CpkMPPAC incorporating with the lower
confidence bound is applied to the capability control of multiple PDS processes. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction
Process capability indices, including Cp, Cpk, Cpm, and
Cpmk [2,7,12], have been proposed in the manufacturing
industry to provide numerical measures on whether a process is capable of reproducing items meeting the quality/reliability requirement preset in the factory. These indices have been defined as:
Cp¼ USL LSL 6r ; Cpk¼ min USL l 3r ; l LSL 3r ; Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q ; Cpmk ¼ min USL l 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q ; l LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q 8 > < > : 9 > = > ;;
where USL is the upper specification limit, LSL is the lower specification limit, l is the process mean, r is the process standard deviation, and T is the target value. Statistical process control charts have been widely used for monitoring and controlling individual factory man-ufacture processes on a routine basis. Those charts are essential tools for product reliability control and
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*
Corresponding author. Tel.: 714261; fax: +886-35-722392.
E-mail address:[email protected](W.L. Pearn).
0026-2714/03/$ - see front matterÓ 2003 Elsevier Science Ltd. All rights reserved.
improvement. In the multiple manufacturing lines en-vironment where a group of processes need to be con-trolled, it could be difficult and time consuming for factory engineers or supervisors to analyze each indi-vidual chart to evaluate overall factory performance of process control activities. Singhal [24,25] introduced an MPPAC using process capability index Cpk, for
con-trolling product reliability of a group of multiple pro-cesses regarding process accuracy and process precision on a single chart. Process accuracy reflects the departure of process mean from the target value, and process precision reflects overall process variability. Pearn and Chen [15] proposed a modification to the Cpk MPPAC
combining the more-advanced process capability indi-ces, Cpm or Cpmk, to identify the problems causing the
processes failing to center around the target. Pearn et al. [19] introduced the MPPAC based on the incapability index. Chen et al. [3] extended the MPPAC for con-trolling product reliability with multiple characteristics where the manufacturing tolerances could be symmetric or asymmetric.
Existing research works in developing and applying those MPPAC control charts, however, never consid-ered sampling errors. Therefore, product reliability in-formation provided from those charts is often unreliable and misleading (more unreliable products than what is expected). In current practice of implementing those charts, practitioners simply plot the estimated index values on the chart then make conclusions on whether processes meet the capability requirement and directions need to be taken for further capability improvement. Their approach is highly unreliable since the estimated index values are random variables and sampling errors are ignored. A reliable approach is to first convert the estimated index values to the lower confidence bounds then plot the corresponding lower confidence bounds on the CpkMPPAC. The lower confidence bound not only
gives us a clue on the minimal actual performance of the process which is tightly related to the fractions of non-conforming units (unreliable products), but is also useful in making decisions for capability testing.
Construction of the exact lower confidence bounds on Cpk is complicated since the distribution of bCCpkinvolves
the joint distribution of two non-central t-distributed random variables, or alternatively, the joint distribution of the folded normal and the chi-square random vari-ables, with an unknown process parameter even when the samples are given [12]. Numerous methods for obtaining approximate confidence bounds of Cpk have been
pro-posed, including Bissell [1], Zhang et al. [28], Porter and Oakland [21,22], Nagata and Nagahata [11], Tang et al. [26] and many others. A different approach was taken by Chou et al. [4] and Levinson [9] who derived the exact lower confidence bounds by working with a bivariate non-central t-distribution. However, an impractical as-sumption made in their work was that the two sample
estimates, bCCpu¼ ðUSL X Þ=3S, and bCCpl¼ ðX LSLÞ=
3S must be the same. Several authors including Franklin and Wasserman [5], Kushler and Hurley [8], and Ro-dridguez [23] have commented that such lower confidence bounds on Cpk are rather conservative when bCCpu¼ bCCpl
is not satisfied, noting that the probability pð bCCpu¼
b C
CplÞ ¼ 0. Other investigations on the estimation of Cpk
include Pearn and Chen [13,14,16,17], and Pearn et al. [18]. In this paper, we overcome the difficulty by first obtaining an explicit form of the cumulative distribution function of the sampling distribution of Cpk. We then
apply direct integration techniques over the cumulative distribution function to obtain the lower confidence bounds of Cpk. A Matlab computer program is developed
for accurate computation of the lower confidence bounds. The behavior of the lower confidence bound against the distribution characteristic parameter, n¼ ðl mÞ=r, is investigated. Exact lower confidence bound ensuring type I error of estimating true Cpk no greater
than the preset value, 1 c, are obtained and used to construct the Cpk MPPAC for multiple
power-distribu-tion switch (PDS) processes capability control.
2. Power-distribution switch
Consider the following case taken from a manufac-turing factory making various types of PDS. The family of PDS is made for applications where heavy capacitive loads and short circuits are likely to be encountered. These devices are around 33 and 80 mX N-channel MOSFET high-side power switches. The functional block diagram of a single 33 mX PDS is displayed in Fig. 1. The switch is controlled 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 circuits increase the power dissipation in the switch, causing the junction temperature to rise, a ther-mal protection circuit shuts off the switch to prevent damage. Recovery from a thermal shutdown is auto-matic once the device has cooled sufficiently. Internal circuitry ensures the switch remains off until valid input voltage is present. Short-circuit current threshold char-acteristic of the PDS process is essential for product reliability performance, which has significant impact to product quality/reliability. Eight manufacturing lines need to be controlled and monitored at the same time in the factory making different types of PDS. There are eight manufacturing lines in the factory, which need to be simultaneously investigated and the short-circuit current threshold characteristic is of two-sided specifi-cation, using Cpk MPPAC for this typical multiple
pro-cesses environment is appropriate for product reliability control and improvement.
2.1. Manufacturing capability and PDS product reliability The index Cpk is yield-based which provides a lower
bound on the process yield; that is, 2Uð3CpkÞ 1 6
Yield 6 Uð3CpkÞ. A manufacturing process is said to be
inadequate if Cpk<1:00; it indicates that the process is
not adequate with respective to the manufacturing tol-erances, the process variation r2 needs to be reduced
(often using design of experiments). The fraction of unreliable PDS products for such process exceeds 2700 parts per million (ppm). A manufacturing process is said to be capable if 1:00 6 Cpk<1:33; it indicates that
cau-tion needs to be taken regarding the process consistency and some process control is required (usually using R or S control charts). The fraction of unreliable PDS products for such process is within 66–2700 ppm. A manufacturing process is said to be satisfactory if 1:33 6 Cpk<1:67; it indicates that process consistency is
satisfactory, material substitution may be allowed, and no stringent precision control is required. The fraction of unreliable PDS products for such process is within 0.54–66 ppm. A manufacturing process is said to be excellent if 1:67 6 Cpk<2:00; it indicates that process
precision exceeds satisfactory. The fraction of unreliable PDS products for such process is within 0.002–0.54 ppm. Finally, a manufacturing process is said to be super if CpkP2:00. The fraction of unreliable PDS
products for such process is less than 0.002 ppm. Table 1 summarizes the above five capability requirements for the PDS processes, the corresponding Cpk values, and
fractions of non-conformities (NC in ppm). Some min-imum capability requirements have been recommended in the manufacturing industry [10], for specific process types, which must run under some more designated
stringent quality conditions. For existing manufacturing processes, the capability must be no less than 1.33, and for new manufacturing processes, the capability must be no less than 1.50. For existing manufacturing processes on safety, strength, or critical parameters (such as manufacturing soft drinks or chemical solution bottled with glass containers), the capability must be no less than 1.50, and for new manufacturing processes on safety, strength, or critical parameters, the capability must be no less than 1.67.
3. The modified CpkMPPAC
Singhal [24] developed the Cpk MPPAC for
control-ling and monitoring multiple processes, which sets the priorities among multiple processes for capability im-provement and indicate if reducing the variability, or the departure of the process mean should be the focus of improvement. The CpkMPPAC provides an easy way to
process improvement by comparing the locations on the chart of the processes before and after the improvement effort. The modified CpkMPPAC is introduced by Pearn
and Chen [15], which incorporates the capability zones setting from using the more-advanced capability index Cpm. The modified CpkMPPAC is shown in Fig. 2. Four
contours for Cpk¼ 1:00, 1.33, 1.67, and 2.00 represent
different categories of process conditions as summarized in Table 1. The narrow lines form capability zones using Cpm measures. On the modified Cpk MPPAC, we note
that:
(a) The parallel line and perpendicular line through the plotted point intersecting the vertical axis (y-axis) and horizontal axis (x-axis) at points represented Cpu¼ ðUSL lÞ=ð3rÞ and Cpl¼ ðl LSLÞ=ð3rÞ,
respectively.
(b) The 45° target line represents the points where the process mean equal to the target (l¼ T ¼ m) and the values of Cpuand Cpl are equal.
(c) For the points fall below the target line, Cpk¼ Cpl.
On the other hand, for the points fall above the tar-get line, Cpk¼ Cpu< Cpl.
(d) For the points inside the area to the right of the 45° target line, represents processes where the process
Table 1
Some commonly used capability requirements and the process conditions Condition Cpk values ppm Inadequate Cpk<1:00 NC > 2700 Capable 1:00 6 Cpk<1:33 NC < 2700 Satisfactory 1:33 6 Cpk<1:67 NC < 66 Excellent 1:67 6 Cpk<2:00 NC < 0:54 Super 2:00 6 Cpk NC < 0:002
mean is towards the lower specification limit (pro-cess mean is lower than target value). On the other hand, for the points inside the area to the left of the 45° target line represents processes where the process mean is towards the upper specification limit (process mean is higher than target value).
(e) The origin point represents a process with Cpu¼
Cpl¼ 0 which means that the standard deviation of
the process is infinite. As the distance from origin of the projection of the plotted point on the target line increases, the variability of the corresponding process decreases.
In general, we never know the true values of the process parameters l and r2as well as C
pk. Hence, these
parameters need to be estimated and sampling error of the index Cpk needs to be considered for product
reli-ability purpose. In Section 4, sampling distribution of Cpk is obtained to compute the lower confidence bound
on Cpk.
4. Samplingdistribution of Cpk
Utilizing the identity minfx; yg ¼ ðx þ yÞ=2 jx yj= 2, the index Cpk can be alternatively written as:
Cpk¼
d jl mj 3r ;
where d¼ ðUSL LSLÞ=2 is half of the length of the specification interval, m¼ ðUSL þ LSLÞ=2 is the mid-point between the lower and the upper specification limits. The natural estimator bCCpk is obtained by
replac-ing the process mean l and the process standard devi-ation r by their conventional estimators X and S, which
may be obtained from a process that is demonstrably stable (under statistical control)
b C Cpk¼ d jX mj 3S ¼ 1 X m d b C Cp; where CCbp is distributed as ðn 1Þ 1=2 Cpv1n1, and
n1=2jX mj=r is distributed as the folded normal
distri-bution with parameter n1=2jl mj=r. Thus, bCC
pk is a
convolution of v1
n1and the folded normal distribution
[12]. The probability density function of bCCpk can be
obtained as [18], where D¼ ðn 1Þ1=2d=r, a¼ ½ðn 1Þ= n1=2. A brief derivation of the probability density function of bCCpk is included in Appendix A
fCC^pkðxÞ ¼ 4AnP1‘¼0P‘ðkÞB‘ D nþ2‘ a2‘þ1 R1 0 ð1 xzÞ 2‘ zn1 exp D2 18a2 a2z2þ 9ð1 xzÞ 2 n o dz; x 60; 4AnP1‘¼0P‘ðkÞB‘ D nþ2‘ a2‘þ‘ R1 x 0ð1 xzÞ 2‘ zn1 exp D2 18a2 a 2z2þ 9ð1 xzÞ2 n o dz; x >0; 8 > > > > > > > < > > > > > > > : P‘ðkÞ ¼ eðk=2Þðk=2Þ‘ ‘! ; An¼ 1 3n12n=2Cððn 1Þ=2Þ; B‘¼ 1 2‘Cðð2‘ þ 1Þ=2Þ:
Using the integration technique similar to that pre-sented in [27], we may obtain the following exact form of the cumulative distribution function of bCCpk, under the
assumption of normality.
4.1. Cumulative distribution function
The cumulative distribution function of bCCpk is
ex-pressed in terms of a mixture of the chi-square distri-bution and the normal distridistri-bution, for x > 0, where b¼ d=r, n ¼ ðl mÞ=r, Gð Þ is the cumulative distri-bution function of the chi-square distridistri-bution v2
n1, and
/ð Þ is the probability density function of the standard normal distribution Nð0; 1Þ FCC^ pkðxÞ ¼ 1 Z bpffiffin 0 G ðn 1Þðb ffiffiffi n p tÞ2 9nx2 ! ½/ðt þ npffiffiffinÞ þ /ðt npffiffiffinÞ dt: ð1Þ
5. Lower confidence bounds on Cpk
For processes with target value setting to the mid-point of the specification limits (T¼ m), the index may be rewritten as the following. We also note that when Cpk¼ C, b ¼ d=r can be expressed as b ¼ 3C þ jnj.
Thus, the index Cpk may be expressed as a function of
the characteristic parameter n Cpk¼ d jl mj 3r ¼ d=r jnj 3 ; where n¼ ðl mÞ=r.
Hence, given the sample of size n, the confidence level c, the estimated value bCCpkand the parameter n, the lower
confidence bounds C can be obtained using numerical integration technique with iterations, to solve the fol-lowing Eq. (2). In practice, the parameter n¼ ðl mÞ=r is unknown, but it can be calculated from the sample data as ^nn¼ ðX mÞ=S. It should be noted, particularly, that Eq. (2) is an even function of n. Thus, for both ^
n
n¼ ^nn0 and ^nn¼ ^nn0 we may obtain the same lower
confidence bound C Z bpffiffin 0 G ðn 1Þðb ffiffiffi n p tÞ2 9n bCC2 pk ! ½/ðt þ ^nnpffiffiffinþ /ðt ^nnpffiffiffinÞ dt ¼ 1 c: ð2Þ
5.1. Algorithm for the LCB
Using Eq. (2), we may compute the lower confidence bounds C. A Matlab program called the LCB is devel-oped. Three auxiliary functions for evaluating C are included here, (a) the cumulative distribution function of the chi-square v2
n1, Gð Þ, (b) the probability density
function of the standard normal distribution Nð0; 1Þ, /ð Þ, and (c) the function of numerical integration computation using the recursive adaptive Simpson quardrature––‘‘quad’’. The algorithm used is commonly known as the direct search method. We implement the algorithm, and develop the Matlab computer program (see Appendix A) to compute the minimal manufactur-ing capability
Step 1. Read the sample data (X1; X2; . . . ; Xn), LSL,
USL, T , and c.
Step 2. Calculate X , S, ^nn, and bCCpk.
Step 3. Compute an initial guess for C.
Step 4. Find the lower confidence bound C on Cpk.
Step 5. Output the conclusive message, ‘‘The true value of the manufacturing capability Cpk is no less
than the C with 100 c % level of confidence’’. 5.2. Lower confidence bounds C and parameter n
Since the process parameters l and r are unknown, then the distribution characteristic parameter, n¼ ðl mÞ=r is also unknown, which has to be estimated in real applications, naturally by substituting l and r by the sample mean X and the sample standard deviation S. Such approach introduces additional sampling errors
from estimating n in finding the lower confidence bounds, and certainly would make our approach (and of course including all the existing methods) less reliable. Consequently, any decisions made would provide less quality assurance to the customers. To eliminate the need for further estimating the distribution characteris-tic parameter n¼ ðl mÞ=r, we examine the behavior of the lower confidence bound values C against the pa-rameter n¼ ðl mÞ=r.
We perform extensive calculations to obtain the lower confidence bound values C for n¼ 0ð0:05Þ3:00, n ¼ 10ð5Þ200, bCCpk¼ 0:7ð0:1Þ3:0, and confidence level c ¼
0:95. Note that the parameter values we investigated, n¼ 0ð0:05Þ3:00, cover a wide range of applications with process capability CpkP0. It should be noted that in
common practice negative values of Cpk are normally
set to zero, indicating that process mean falls outside the manufacturing specification limits. The results indi-cate that (i) the lower confidence bound C is decreasing in n, and is increasing in n, (ii) the lower confidence bound C obtains its minimum at n¼ 1:00 in all cases, and stays at the same value for n P 1:00 for all C (with accuracy up to 104). Furthermore, we observe that for
n >30, the lower confidence bound C reaches its mini-mum at n¼ 0:50 and stays at the same value for n P 0:50, and for n P 100, reaches its minimum at n¼ 0:25 (with accuracy up to 104). Hence, for practical purpose
we may solve Eq. (2) with n¼ n ¼ 1:00 to obtain the required lower confidence bounds for given bCCpk, n, and
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. We note the above result is impossible to prove mathematically.
Fig. 3(a)–(f) plot the curves of the lower confidence bound, C, versus the parameter n for bCCpk¼ 0:7, 0.9, 1.2,
2.0, 2.5, 3.0, respectively, with confidence level c¼ 0:95. For bottom curve 1, sample size n¼ 30. For bottom curve 2, sample size n¼ 50, for bottom curve 3, sample size n¼ 70, for top curve 3, sample size n ¼ 100; for top curve 2, sample size n¼ 150; for top curve 1, sample size n¼ 200. Table 2 (see Appendix A) tabulates the lower confidence bound, C, for bCCpk¼ 0:7ð0:1Þ3:0, n ¼ 5ð5Þ200,
and c¼ 0:95 with the process parameter n set to n ¼ 1:0. For example, if bCCpk¼ 1:5, then with n ¼ 100 we find the
lower confidence bound C¼ 1:315, and so the minimal manufacturing capability is no less than 1.315, i.e., Cpk>1:315. Consequently, the manufacturing yield
(fraction of reliable products) is no less than 99.992% and the fraction of non-conformities (unreliable prod-ucts) is no greater than 79.80 ppm. We note that for other existing methods, either the confidence level c cannot be assured (PDS product reliability assurance is uncertain), or the lower confidence bounds C are too conservative (C is too small in this case). Our approach provides best reliability assurance to the PDS products.
We note that the lower confidence bound C calculated using the above proposed approach, is maximal (exact) which cannot be improved further.
6. Manufacturingcapability computation and data anal-ysis
We collected sample data of short-circuit current threshold from eight manufacturing processes making different kinds of PDS devices. One hundred observa-tions from each PDS process are taken and calculated for the sample mean, sample standard deviation, and the estimate bCCpk. The product codes, manufacturing
speci-fications, the estimated index values, the lower
confi-dence bounds, and the corresponding maximum non-conformities for each of the eight processes are tabu-lated in Tables 3 and 4. Fig. 4 plots the modified Cpk
MPPAC for the eight processes based on the minimum true values tabulated in Table 4. We analyze these pro-cess points in Fig. 4 and obtain the following critical summary information of the capability condition for all processes.
(a) The plotted points E and Hare not located within the contour of Cpk¼ 1:00. It indicates that the
pro-cess has a very low capability. Since the points E and Hare close to the 45° target line, both processes present that the process means are close to the target value, and the poor capabilities are mainly
contrib-0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0 1 2 3 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 0 1 2 3 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5 0 1 2 3 2.2 2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85 2.9 2.95 3 0 1 2 3 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 1 2 3 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0 1 2 3 (a) (b) (c) (d) (e) (f)
Fig. 3. Plots of C vsjnj for (a) bCCpk¼ 0:7, n ¼ 30, 50, 70, 100, 150, 200 (bottom to top); (b) bCCpk¼ 0:9, n ¼ 30, 50, 70, 100, 150, 200
(bottom to top); (c) bCCpk¼ 1:2, n ¼ 30, 50, 70, 100, 150, 200 (bottom to top); (d) bCCpk¼ 2:0, n ¼ 30, 50, 70, 100, 150, 200 (bottom to
uted by the significant process variation. Thus, im-mediate quality improvement actions must be taken for reducing the process variance for both processes. (b) The plotted points G and F lie within the contour of 1:00 6 Cpk<1:33. The point G lies inside the area
which is to the right of the 45° target line represents processes where the process mean is towards the lower specification limit (process mean is lower than target value). On the other hand, the point F lies inside the area, which is to the left of the 45° target line represents processes where the process mean is towards the upper specification limit (process mean
is higher than target value). Thus, quality improve-ment effort for these processes should be first focused on reducing their process departure from the target value T , then the reduction of the process variance. (c) Process B and D lie inside the contours of Cpk¼
1:33 6 Cpk<1:67. Both processes are considered
performing well and no immediate improvement activities needed to be taken, but both processes obviously departure from the target value T . There-fore, both processes may be improved by simply reducing their process departure from the target value T .
Table 3
Manufacturing specifications of the eight PDS products
Code A B C D Products T 1 A 1.2 A 500 mA 500 mA USL 1.3 A 1.5 A 650 mA 600 mA LSL 0.7 A 0.9 A 350 mA 400 mA E F G H Products T 250 mA 550 mA 250 mA 250 mA USL 320 mA 620 mA 310 mA 300 mA LSL 180 mA 480 mA 190 mA 200 mA Table 4
Calculated statistics, estimated Cpk, lower confidence bound, and the fractions of non-conformities (in ppm) of the eight PDS products
Code A B C D X 1.007153 A 1.25403 A 508.30 mA 483.76 mA S 0.047687 A 0.04502 A 27.65 mA 17.18 mA ðUSL X Þ=ð3SÞ 2.047 1.821 1.708 1.625 ðX LSLÞ=ð3SÞ 2.147 2.621 1.908 1.625 b C Cpk 2.047 1.821 1.708 1.625 LCB 1.799 1.599 1.499 1.425 ppm 0.0678 1.61 6.89 19.11 E F G H X 252.09 mA 570.89 mA 231.21 mA 245.61 mA S 27.91 mA 13.01 mA 10.02 mA 13.95 mA ðUSL X Þ=ð3SÞ 0.811 1.258 2.621 1.30 ðX LSLÞ=ð3SÞ 0.861 2.328 1.371 1.090 b C Cpk 0.811 1.258 1.371 1.090 LCB 0.7 1.099 1.2 0.949 ppm 35729 977.23 318.22 4413.3
(d) The plotted points C and A lie inside the contours of Cpk¼ 1:33 and Cpk¼ 1:67, respectively. Capabilities
of both processes are considered satisfactory and ex-cellent. They have lower priorities in allocating qual-ity improvement efforts than other processes. Table 5 displays the manufacturing capabilities and capability groupings for the eight PDS processes using the estimated Cpk values (uncorrected) and the lower
confidence bounds LCB (corrected) (with asterisks indicating incorrect groupings). The modified Cpk
MPPAC for the eight processes based on the estimated Cpk index values (an approach widely used in current
industrial applications) rather than using the lower confidence bounds, is displayed in Fig. 5. We note that such MPPAC obviously conveys unreliable information and is misleading, which should be avoided in real ap-plications.
7. Conclusions
Conventional investigations on manufacturing capa-bility control ignore sampling errors. In this paper, we considered the sample errors by finding the exact lower confidence bound for the Cpk. The lower confidence
bounds present a measure on the minimum capability of the process based on the sample data. Existing methods for computing the lower confidence bounds pro-vided only approximate or rather conservative bounds. We investigated the behavior of the lower confidence bound versus the process characteristic parameter n¼ ðl mÞ=r, which resulted that the lower confidence bound attains its minimal value at n¼ 1:0. The pro-posed decision making procedure ensures that the risk of making a wrong decision will be no greater than the preset Type I error 1 c. The proposed modified Cpk
MPPAC is useful for manufacturing capability control of a group of processes in a multiple process environ-ment. The modified CpkMPPAC prioritizes the order of
the processes for further capability improvement effort should focus on, either to move the process mean closer to the target value or reduce the process variation. The developed lower confidence bounds can be used to construct accurate modified Cpk MPPAC providing
in-formation regarding the true capability, and fractions of non-conforming products. The modified Cpk MPPAC is
applied to the PDS manufacturing process for control-ling PDS product reliability.
Acknowledgements
The authors would like to thank the anonymous referees for their careful readings and constructive crit-icisms, which improved the paper.
Fig. 4. The modified Cpk MPPAC groups for the eight PDS
processes.
Table 5
Estimated and corrected (LCB) capabilities, and their group-ings for the eight PDS processes
Code Estimated Cpk Grouping LCB Grouping A 2.047 Super 1.799 Excellent B 1.821 Excellent 1.599 Satisfactory C 1.708 Excellent 1.499 Satisfactory D 1.625 Satisfactory 1.425 Satisfactory E 0.811 Incapable 0.700 Incapable F 1.258 Capable 1.099 Capable G 1.371 Satisfactory 1.200 Capable H1.090 Capable 0.949 Incapable
Appendix A. Derivation of the probability density func-tion of bCCpk
Let X1; . . . ; Xn be a random sample of size n from a
normally distributed process Nðl; r2Þ. From a process
demonstrably stable (under statistical control), the nat-ural estimator bCCpk is obtained by replacing the process
mean l and the process standard deviation r by their conventional estimators X and S, which can be written as follows: b C Cpk¼ d jX mj 3S :
To derive the cumulative distribution function and the probability density function of bCCpk, we define
(1) D¼ ðn 1Þ1=2 d=r, (2) a¼ ½ðn 1Þ=ðnÞ1=2, (3) K¼ ðn 1ÞS2=r2, which is distributed as v2 n1, (4) Z¼ n1=2ðX mÞ=r, which is distributed as N ðd; 1Þ, where d¼ n1=2ðl mÞ=r,
(5) Y ¼ Z2, which is distributed as the ordinary
non-central chi-square distribution with one degree of freedom and non-centrality parameter d, v2
1ðdÞ.
Then, the probability density function of Y can be expressed as: fYðyÞ ¼ ek=2 2pffiffiffip X1 j¼0 hjðkÞ ð 1ÞjfYjðyÞ þ fYjðyÞ ¼e k=2 ffiffiffi p p X 1 k¼0 h2kðkÞfY2kðyÞ; for y > 0; where k¼ d2, h 2kðkÞ ¼ ð2kÞ k C½ð1 þ 2kÞ=2=ð2k!Þ and Y2kis distributed as v21þ2k.
We note that the estimator bCCpk can be rewritten as:
b C Cpk¼
D apffiffiffiffiY 3pffiffiffiffiK :
[Case I]: For x > 0, the cumulative distribution function of bCCpk is FCC^pkðxÞ ¼ P ð bCCpk6xÞ ¼ P D apffiffiffiffiY 3pffiffiffiffiK 6x ¼ 1 Z 1 0 P pffiffiffiffiK 6D a ffiffiffiffi Y p 3x jY ¼ y fYðyÞ dy: Since K is distributed as v2 n1, then P pffiffiffiffiK 6D a ffiffiffiy p 3x
¼ 0 for y >ðD=aÞ2 and x >0:
Hence, FCC^pkðxÞ ¼ 1 Z ðD=aÞ2 0 p pffiffiffiffiK 6D a ffiffiffiy p 3x fYðyÞ dy ¼ 1 Z ðD=aÞ2 0 FK ðD a ffiffiffiyp Þ2 9x2 ! fYðyÞ dy:
Substituting fYðyÞ, we obtain
FCC^pkðxÞ ¼ 1 ek=2 ffiffiffi p p X 1 k¼0 h2kðkÞ Z ðD=aÞ2 0 FK ðD apffiffitÞ2 9x2 ! fY2kðtÞ dt ! :
[Case II]: Similarly, for x < 0, the cumulative distri-bution function of bCCpk is FCC^pkðxÞ ¼ ek=2 ffiffiffi p p X 1 k¼0 h2kðkÞ Z 1 ðD=aÞ2 FK ðD apffiffitÞ2 9x2 ! fY2kðtÞ dt ! :
[Case III]: For x¼ 0,
FCC^ pkðxÞ ¼ 1 ek=2 ffiffiffi p p X 1 k¼0 h2kðkÞ Z ðD=aÞ2 0 fY2kðtÞ dt ! :
By LeibnitzÕs rule, taking the derivative of FCC^pkðxÞ
with respect to x we have the probability density func-tion of bCCpk: fCC^pkðxÞ ¼ 2ek=2 ffiffi p p P1 k¼0h2kðkÞ RðD=aÞ1 2fK ðDa ffi t p Þ2 9x2 ðDa ffit p Þ2 9x3 fY2kðtÞ dt ; x <0; 0; x¼ 0; 2ek=2 ffiffi p p P1k¼0h2kðkÞ RðD=aÞ2 0 fK ðDa ffi t p Þ2 9x2 ðDapffitÞ2 9x3 fY2kðtÞ dt ; x >0: 8 > > > > > > > > < > > > > > > > > :
Changing the variable z¼D a
ffiffi t p Dx
in the above integral, we can rewrite fCC^pkðxÞ as:
fCC^pkðxÞ ¼ 4ek=2 ffiffi p p P1 k¼0h2kðkÞ R1 0 fK D 2z2 9 fY2k D2ð1xzÞ2 a2 D4ð1xzÞz2 9a2 dz ; x <0; 0; x¼ 0; 4ek=2 ffiffi p p P1 k¼0h2kðkÞ R1=x 0 fK D 2z2 9 fY2k D2ð1xzÞ2 a2 D4ð1xzÞz2 9a2 dz ; x >0: 8 > > > > > > > > < > > > > > > > > : Let I2kðzÞ ¼ fK D2z2 9 fY2k D2ð1 xzÞ2 a2 ! D 4ð1 xzÞz2 9a2 :
Then fCC^pkðxÞ ¼ 4 ek=2 ffiffiffi p p X 1 k¼0 h2kðkÞ Z 1 0 I2kðzÞ dz for x 60:
Further, for positive odd integer n. Hence,
h2kðkÞ ¼ ð2kÞk ð2kÞ!C 1þ 2k 2 ¼ðk=2Þ k ffiffiffi p p k! : Thus, fCC^pkðxÞ ¼ 4 X1 k¼0 PkðkÞ Z 1 0 I2kðzÞ dz for x 6 0, where PkðkÞ ¼ eðk=2Þðk=2Þk k! :
On the other hand, since K is distributed as v2
n1and Y2kis distributed as v21þ2k, then fK D2z2 9 ¼ 2 ðn1Þ=2 Cððn 1Þ=2Þ D2z2 9 ðn3Þ=2 eD2z2=18 ; fY2k D2ð1 xzÞ2 a2 ! ¼ 2 ð2kþ1Þ=2 Cðð2k þ 1Þ=2Þ D2ð1 xzÞ2 a2 !ð2k1Þ=2 eD2ð1xzÞ2=ð2a2Þ: Hence, I2kðzÞ ¼ AnBk Dnþ2k a2kþ1ð1 xzÞ 2k zn1 exp D 2 18a2ða 2z2þ 9ð1 xzÞ2 Þ ; where An¼ 1 3n12n=2Cððn 1Þ=2Þ and Bk¼ 1 2kðð2k þ 1Þ=2Þ: Consequently, we have fCC^pkðxÞ ¼ 4An X1 k¼0 PkðkÞBk Dnþ2k a2kþ1 Z 1 0 ð1 xzÞ2kzn1 exp D 2 18a2ða 2 z2þ 9ð1 xzÞ2Þ dz for x 6 0:
Based on the similar derivation, we can obtain
fCC^pkðxÞ ¼ 4An X1 k¼0 PkðkÞBk Dnþ2k a2kþ1 Z 1=x 0 ð1 xzÞ2kzn1 exp D 2 18a2ða 2z2þ 9ð1 xzÞ2 Þ dz for x > 0:
Matlab Program for LCB
% -% Input the sample data (X1, X2, . . ., Xn), LSL, USL,
T, and c.
% -clear global
[n1 usl lsl r1]¼ read(ÔEnter values of sample size, upper specification limit, . . .
lower specification limit, confidence level:Õ); global b n epsilon ecpk
n¼ n1; r¼ r1; [data(1:n,1)]¼ textread(ÔPDS.datÕ,Ô%fÕ,n); % -% Compute X , S, ^nn, and bCCpk. % -mdata¼ mean(data); stddata¼ std(data); epsilon¼ (mdata-T)/stddata; ecpk¼ (min(usl-mdata,mdata-lsl))/(3*stddata); fprintf(ÔThe Sample Mean is %g.nnÕ,mdata); fprintf(ÔThe Sample Standard Deviation is %g.nnÕ,stddata)
fprintf(ÔThe Epsilon %g.nnÕ,epsilon)
fprintf(ÔThe Estimate of Cpk is %g.nnÕ,ecpk) % -% Compute a good initial value of C.
% -b¼ 0;d ¼ 0; c¼ 0.2:0.025:3; for i¼ 1:1:113 b¼ 0;d ¼ 0;y ¼ 0;b ¼ 3*c(i)+abs(epsilon); d¼ b*sqrt(n); y¼ quad(ÔcpkÕ,0,d); if (y-(1-r)) > 0 break end; end % -%Evaluate the lower confidence bound C on Cpk.
% -c¼ 0.2+0.025*(i-1):-0.001:0.2; for k¼ 1:(0.025*(i-1)*1000)+1 b¼ 0;d ¼ 0;y ¼ 0;b ¼ 3*c(k)+abs(epsilon); d¼ b*sqrt(n); y¼ quad(ÔcpkÕ,0,d); if ((1-r)-y) > 0.0001 break end; end % -% Output the conclusive message, ‘‘The true value of the process
% capability Cpkis no less than C with 100 c% level of
confidence’’
% -fprintf(ÔThe true value of the process capability Cpk is no less than %gÕ,c(k))
fprintf(Ôlevel of confidence.Õ) ppm¼ 1000000*2*normcdf(-3*c(k))
% -%Two function files included––read.m and cpk.m % -function Q1¼ cpk(t)
global n b epsilon ecpk
Q1¼ chi2cdf((((b*sqrt(n)-t).^22)./(9*ecpk^22))*((n-1)/ n), . . .
n-1).*(normpdf((t+epsilon*sqrt(n)))+normpdf((t-epsilon*sqrt(n))));
function [a1, a2, a3, a4, a5]¼ read(labl) if nargin¼ ¼ 0, labl ¼ Ô?Õ; end
n¼ nargout;str ¼ input(labl,ÔsÕ); str¼ [Ô[Ô,str,Õ]Õ]; v¼ eval(str);L ¼ length(v);
if L>¼ n, v ¼ v(1:n);
else, v¼ [v,zeros(1,n-L)]; end for j¼ 1:nargout
eval([ÔaÕ, int2str(j),Õ¼ v(j);Õ]); end
%
-Table 2
Lower confidence bounds C of Cpkfor bCCpk¼ 0:7ð0:1Þ1:8 (Panel A) and bCCpk¼ 1:9ð0:1Þ3:0 (Panel B), n ¼ 10ð5Þ200, c ¼ 0:95
n 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Panel A 10 0.371 0.438 0.503 0.568 0.632 0.696 0.759 0.822 0.885 0.948 1.010 1.072 15 0.435 0.508 0.581 0.653 0.724 0.795 0.865 0.936 1.006 1.075 1.146 1.215 20 0.472 0.549 0.626 0.702 0.777 0.852 0.927 1.001 1.076 1.150 1.224 1.298 25 0.497 0.577 0.656 0.735 0.813 0.890 0.968 1.045 1.123 1.200 1.277 1.353 30 0.516 0.597 0.678 0.759 0.839 0.918 0.998 1.077 1.157 1.236 1.315 1.394 35 0.530 0.613 0.695 0.777 0.859 0.940 1.021 1.102 1.183 1.264 1.344 1.425 40 0.541 0.625 0.709 0.792 0.875 0.957 1.040 1.122 1.204 1.286 1.368 1.450 45 0.550 0.635 0.720 0.804 0.888 0.972 1.055 1.138 1.222 1.305 1.388 1.471 50 0.558 0.644 0.729 0.814 0.899 0.984 1.068 1.152 1.236 1.320 1.404 1.488 55 0.565 0.651 0.737 0.823 0.909 0.994 1.079 1.164 1.249 1.334 1.418 1.503 60 0.571 0.658 0.745 0.831 0.917 1.003 1.089 1.174 1.260 1.345 1.430 1.516 65 0.576 0.664 0.751 0.838 0.924 1.011 1.097 1.183 1.269 1.355 1.441 1.527 70 0.581 0.669 0.756 0.844 0.931 1.018 1.105 1.191 1.278 1.364 1.451 1.537 75 0.585 0.673 0.761 0.849 0.937 1.024 1.111 1.198 1.286 1.373 1.459 1.546 80 0.588 0.677 0.766 0.854 0.942 1.030 1.117 1.202 1.292 1.380 1.467 1.555 85 0.592 0.681 0.770 0.858 0.947 1.035 1.123 1.211 1.299 1.387 1.474 1.562 90 0.595 0.684 0.774 0.862 0.951 1.040 1.128 1.216 1.305 1.393 1.481 1.569 95 0.598 0.687 0.777 0.866 0.955 1.044 1.133 1.221 1.310 1.398 1.487 1.575 100 0.600 0.690 0.780 0.870 0.959 1.048 1.137 1.226 1.315 1.403 1.492 1.581 105 0.603 0.693 0.783 0.873 0.962 1.052 1.141 1.230 1.319 1.408 1.497 1.586 110 0.605 0.696 0.786 0.876 0.965 1.055 1.145 1.234 1.323 1.413 1.502 1.591 115 0.607 0.698 0.788 0.878 0.968 1.058 1.148 1.238 1.327 1.417 1.506 1.596 120 0.609 0.700 0.791 0.881 0.971 1.061 1.151 1.241 1.331 1.421 1.511 1.600 125 0.611 0.702 0.793 0.883 0.974 1.064 1.154 1.244 1.335 1.424 1.514 1.604 130 0.613 0.704 0.795 0.886 0.976 1.067 1.157 1.248 1.338 1.426 1.518 1.608 135 0.614 0.706 0.797 0.888 0.979 1.069 1.160 1.250 1.341 1.431 1.521 1.612 140 0.616 0.707 0.799 0.890 0.981 1.072 1.162 1.253 1.344 1.434 1.525 1.615 145 0.617 0.709 0.801 0.892 0.983 1.074 1.165 1.256 1.346 1.437 1.528 1.618 150 0.619 0.711 0.802 0.894 0.985 1.076 1.167 1.258 1.349 1.440 1.531 1.621 155 0.620 0.712 0.804 0.895 0.987 1.078 1.169 1.260 1.352 1.443 1.534 1.624 160 0.621 0.713 0.805 0.897 0.989 1.080 1.171 1.263 1.354 1.445 1.536 1.627 165 0.623 0.715 0.807 0.899 0.990 1.082 1.173 1.265 1.356 1.447 1.539 1.630 170 0.624 0.716 0.808 0.900 0.992 1.084 1.175 1.267 1.358 1.450 1.541 1.632 175 0.625 0.717 0.810 0.902 0.994 1.085 1.177 1.269 1.360 1.452 1.543 1.635 180 0.626 0.718 0.811 0.903 0.995 1.087 1.179 1.271 1.362 1.454 1.546 1.637 185 0.627 0.720 0.812 0.904 0.997 1.089 1.181 1.272 1.364 1.456 1.548 1.639 190 0.628 0.721 0.813 0.906 0.998 1.090 1.182 1.274 1.366 1.458 1.550 1.642 195 0.629 0.722 0.814 0.907 0.999 1.091 1.184 1.276 1.368 1.460 1.552 1.644 200 0.630 0.723 0.815 0.908 1.000 1.093 1.185 1.277 1.369 1.462 1.554 1.646
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