A multiprocess performance analysis chart based on
the incapability index C
pp
: an application to the chip resistors
W.L. Pearn
a,*, C.H. Ko
a, K.H. Wang
ba
Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsin Chu 30050, Taiwan, ROC
b
Department of Applied Mathematics, National Chung Hsin University, Taiwan, ROC Received 21 January 2002; received in revised form 5 March 2002
Abstract
Statistical process control charts, such as the XX, R, S2, S, and MRcharts, have been widely used in the manufac-turing industry for controlling/monitoring process performance, which are essential tools for any quality improvement activities. Those charts are easy to understand, which effectively communicate critical process information without using words and formula. In this paper, we introduce a new control chart, called the Cppmultiple process performance analysis chart (MPPAC), using the incapability index Cpp. The CppMPPAC displays multiple processes with the de-parture, and process variability relative to the specification tolerances, on one single chart. We demonstrate the use of the Cpp MPPAC by presenting a case study on some resistor component manufacturing processes, to evaluate the factory performance. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
Process capability indices (PCIs) have been widely used in various manufacturing industries, to provide numerical measures on process potential and process performance. The two most commonly used process capability indices are Cpand Cpkintroduced by Kane [1]. These two indices are defined in the following:
Cp¼ USL LSL 6r ; Cpk¼ min USL l 3r ; LSL l 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 production toler-ance, which reflects only the process potential. The index Cpkmeasures process performance based on the process yield (percentage of conforming items) without
consid-ering the process loss (a new criteria for process quality championed by Hsiang and Taguchi [2]). Taking into the consideration of 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 ðl T Þ2 q ¼ d 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðl T Þ2 q ;
where T is the target value, and d¼ ðUSL LSLÞ=2 is half of the length of the specification interval (LSL, USL).
Based on the index Cpm, Greenwich and Jahr-Schaffrath [4] introduced an incapability index, called Cpp, which is a simple transformation of the Taguchi index Cpm. The index Cppis defined as:
Cpp¼ 1 Cpm 2 ¼ l T D 2 þ r D 2 ;
where D¼ d=3. Some commonly used values of Cpp, 9.00 (process is incapable), 4.00 (process is incapable), 1.00 (process is normally called capable), 0.57 (process is normally called satisfactory), 0.44 (process is normally called good), and 0.25 (process is normally called super), and the corresponding Cpm values are listed in Table 1.
www.elsevier.com/locate/microrel
*
Corresponding author. Tel.: 714261; fax: +886-35-722392.
E-mail address:roller@cc.nctu.edu.tw(W.L. Pearn).
0026-2714/02/$ - see front matterÓ 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 7 1 4 ( 0 2 ) 0 0 0 7 1 - 9
If we denote the first termðl T Þ2=D2as C
ia, and the second item r2=D2 as C
ip, then Cpp can be rewritten as Cpp¼ Cipþ Cia. The sub-index Cipmeasures the relative variability, which has been referred to as the imprecision index. Some commonly used values of Cip, 1.00, 0.56, 0.44, 0.36, and 0.25, and the corresponding quality conditions are listed in Table 2. Note that those values of Cipare equivalent to Cp¼ 1:00, 1.33, 1.50, 1.67, and 2.00 respectively, covering a wide range of the precision requirements used for most real-world applications.
On the other hand, the sub-index Cia measures the relative departure, which has been referred to as the inaccuracy index. The advantage of using the index Cpp, is that it provides an uncontaminated separation be-tween information concerning the process precision and process accuracy. The separation suggests a direction the practitioners may consider on the process parame-ters to improve the process quality.
Based on the sub-indices Cipand Cia, we introduce a control chart called the Cpp multiple process perfor-mance analysis chart (MPPAC), using the incapability index Cpp. The CppMPPAC displays multiple processes with the relative departure, and process variability rel-ative to their specification tolerances on one single chart. We demonstrate the use of the Cpp MPPAC by pre-senting a case study taken from a resistor component manufacturing company located on an Industrial Park in Taiwan, to evaluate the factory performance.
2. Estimation of Cip, Cia, Cpp 2.1. Estimation of Cip
To estimate the process imprecision, we consider the natural estimator ^CCipdefined in the following, where the
sample standard deviation Sn1 is calculated as Sn1¼ P
ðXi XXÞ 2
=ðn 1Þ
h i1=2
, which is the conventional es-timator of the process standard deviation r,
^ C Cip¼ 1 n 1 Xn i¼1 Xi XX 2 D2 ¼ S2 n1 D2 : The natural estimator ^CCip can be rewritten as:
^ C Cip¼ Cip n 1 ðn 1Þ ^CCip Cip ¼ Cip n 1 Xn i¼1 Xi XX 2 r2 : If the process characteristic is normally distributed, Pearn and Lin [5] showed the natural estimator ^CCip distributed as ½Cip=ðn 1Þv2n1, where v2n1 is a chi-square distribution with (n 1) degrees of freedom. Pearn and Lin [5] showed that the natural estimator ^CCip is the uniformly minimum-variance unbiased estimate (UMVUE) of Cip, which is consistent, and asymptoti-cally efficient. Pearn and Lin [5] also showed that the statisticpffiffiffinð ^CCip CipÞ converges to N ð0; 2Cip2Þ in distri-bution. Thus, in real-world applications, using ^CCipwhich has all desired statistical properties as an estimate of Cip, would be reasonable.
Note that by multiplying the constant cn¼ ðn 1Þ=n to the UMVUE ^CCip, we can obtain the maximum like-lihood estimate (MLE) of Cip. Pearn and Lin [5] showed that the MLE ^CC0ipis consistent, asymptotically unbiased and efficient. They also showed that the statistic
ffiffiffi n p
ð ^CC0ip CipÞ converges to N ð0; 2C2ipÞ in distribution. Since the constant cn<1, then the MLE ^CC0ip¼ cnCC^ip underestimates Cip but with smaller variance. In fact, we may calculate the mean square error MSEð ^CC0
ipÞ ¼ ½ð2n 1Þ=n2ðC
ipÞ 2
. Hence, MSE ð ^CCipÞ MSEð ^CCip0Þ ¼ ½ð3n 1Þ=n2ðn 1ÞðC
ipÞ2>0, for all sample size n. Therefore, the MLE ^CCip0 has a smaller mean square error than that of the UMVUE ^CCip, hence is more reliable, particularly, for short production run applications (such as accepting a supplier providing short production runs in QS-9000 certification). For short run applications (with n 6 35) we recommend using the MLE ^CC0ip rather than the UMVUE ^CCip. For other applications with sample sizes n > 35, the difference between the two es-timators is negligible (less than 0.52%).
2.2. Estimation of Cia
For the process inaccuracy index Cia, we consider the natural estimator ^CCiadefined as the following
^ C Cia¼
ð XX T Þ2 D2 ;
where the sample mean XX ¼Pn
i¼1Xi=n is the conven-tional estimator of the process mean l. We note that the estimator ^CCiacan be written as the following:
Table 2
Some commonly used precision requirements
Quality condition Precision requirement Capable 0:56 6 Cip61:00 Satisfactory 0:44 6 Cip60:56 Good 0:36 6 Cip60:44 Excellent 0:25 6 Cip60:36 Super Cip60:25 Table 1
Some commonly used Cppand equivalent Cpm
Cpp Cpm 9.00 0.33 4.00 0.50 1.00 1.00 0.57 1.33 0.44 1.50 0.25 2.00
^ C Cia¼ Cip n n ^CCia Cip ¼Cip n nð XX T Þ2 r2 :
If the process characteristic follows the normal distribution, then the estimator ^CCia is distributed as ½Cip=nv21ðdÞ, where v
2
1ðdÞ represents a non-central chi-square distribution, which has one degree of freedom with non-centrality parameter d¼ nðl T Þ2=r2¼ nC
ia= Cip. We note that since the estimator ^CCipis a function of Sn1only, and the estimator ^CCiais a function of XX only, then the two estimators ^CCip and ^CCia are statistically in-dependent.
Since XX is the MLE of l, then by the invariance property of the MLE, the natural estimator ^CCia is the MLE of Cia. Noting that Eð ^CCiaÞ ¼ Ciaþ ðCip=nÞ, and Eð ^CCipÞ ¼ Cip, the corrected estimator ~CCia¼ ^CCia ð ^CCip=nÞ must be unbiased for Cia. Pearn and Lin [5] showed that
~ C
Cia is the UMVUE of Cia, which is consistent and as-ymptotically efficient, and thatpffiffiffinð ~CCia CiaÞ converges to Nð0; 4CipCiaÞ in distribution. Thus, in real-world ap-plications using the UMVUE ~CCia, which has all desired statistical properties, as an estimate of Cia would be reasonable.
We note that the MLE ^CCiahas smaller variance than the UMVUE ~CCia. But, we can show that MSEð ~CCiaÞ ¼ 4CipCia=nþ ½2=nðn 1ÞðCipÞ
2
, and so MSEð ~CCiaÞ MSEð ^CCiaÞ ¼ ½ð3 nÞ=n2ðn 1ÞðCipÞ
2
, which is less than 0 for n P 4. Therefore, the UMVUE ~CCiahas a smaller mean square error than that of the MLE ^CCia, and is more reliable for applications with n P 4.
2.3. Estimation of Cpp
To estimate the process incapability Cpp, a combined measure of process imprecision and process inaccuracy, we consider the natural estimator ^CCpp defined as the following, which also can be rewritten as a function of Cip. ^ C Cpp¼ 1 n Xn i¼1 Xi XX 2 D2 þ X X T 2 D2 ¼ Cip n n ^CCpp Cip ¼Cip n Xn i¼1 Xi T ð Þ2 r2 :
If the process characteristic follows the normal dis-tribution Nðl; r2Þ, then the estimator ^CC
pp is distributed as½Cip=nv2nðdÞ, where v2nðdÞ is a non-central chi-square distribution with n degrees of freedom and non-cen-trality parameter d¼ nðl T Þ2=r2¼ nC
ia=Cip.
If the process characteristic follows the normal dis-tribution, Pearn and Lin [5] showed that ^CCppis the MLE, which is also the UMVUE of Cpp. We also can show that the estimator ^CCppis consistent, asymptotically efficient, and that pffiffiffinð ^CCpp CppÞ converges to Nð0; 2CipCiaþ 2CipCppÞ in distribution. Since the estimator has all the
desired statistical properties, in practice using ^CCpp to estimate process incapability would be reasonable.
3. The CppMPPAC
Many statistical control charts, such as XX, R, S2, S, and MRcharts, have been widely used in monitoring and controlling process quality. Those charts, however, are applicable only for single processes (one process at a time). Thus, using those charts in multi-process envi-ronment can be a difficult and time-consuming task for the supervisor or shop engineer to analyze each indi-vidual chart to evaluate the overall status of shop pro-cess control activity.
The MPPAC can be used to evaluate the perfor-mance of a single process as well as multi-processes; to set the priorities among multiple processes for quality improvement, and indicate if reducing the variability, or the departure of the process mean should be the focus; to provide an easy way to quantify the process im-provement by comparing the locations on the chart of the processes before and after the improvement effort. The MPPAC is an effective tool for communicating between the product designer, the manufacturers, the quality engineers, and among the management depart-ments.
Based on the definition, Cpp¼ ðl T Þ 2
=D2þ r2=D2, we first set Cpp¼ k, for various k values, then a set of (l; r) values satisfying the equation:ðl T Þ2þ r2¼ kD2 can be plotted on the contour (a curve) of Cpp¼ k. These contours are semicircles centered at l¼ T with radiuspffiffiffikD. The more capable the process, the smaller the semicircle is. We plot the six contours on the Cpp MPPAC for the six Cpp values listed in Table 1, as shown in Fig. 1. On the CppMPPAC, we note that:
1. As the point gets closer to the target, the value of the Cppbecomes smaller, and the process performance is better.
2. For the points inside the semicircle of contour Cpp¼ k, the corresponding Cpp values are smaller than k. For the points outside the semicircle of con-tour Cpp¼ k, the corresponding Cppvalues are great-er than k.
3. For processes with fixed values of Cpp, the points within the two 45° lines envelop, the variability is contributed mainly by the process variance.
4. For processes with fixed values of Cpp, the points out-side the two 45° lines envelop, the process variability is contributed mainly by the process departure. 5. The perpendicular line and parallel line through the
plotted point intersecting the horizontal axis and ver-tical axis at points represent its Cia and Cip, respec-tively.
6. The distance between T and the point, which the per-pendicular line through the plotted point intersecting the horizontal axis, denote the departure of process mean from target.
7. The distance between T and the point, which the par-allel line through the plotted point intersecting the vertical axis, denote the process variance.
4. An application
In the following, we consider a resistor manufactur-ing process. Resistor is an electronic passive component commonly used on electronic circuits, providing the function of reducing the current, voltage, as well as re-leasing the heat. Eight standard precisions, with their required tolerances, are displayed in Table 3 (see Chen [6], Chen [7], and Wang [8]).
We consider the following case taken from Cinetech, a factory located on an industrial park making chip resistors. We investigated 15 specific types of chip re-sistors widely used on the personal computers, televi-sions, and other audio and video electronic devices with different resistance specifications. A random sample of size 100 is taken from each of the fifteen resistor
man-ufacturing processes. Their resistance specifications are displayed in Table 4. The calculated sample mean, sam-ple variance, and the index values of Cpp, Cia, and Cipare shown as Table 5.
Fig. 2 plots the CppMPPAC for the fifteen processes listed in Table 5. We analyze the process points in Fig. 2, and obtain the following summary of the quality con-dition.
1. The plotted point H is very close to the contour Cpp¼ 4, it indicates that the process has a low capa-bility. Since the point H is close to the target line, it indicates that the poor capability is mainly contrib-uted by the process variation. Thus, it calls for an im-mediate quality improvement action to reduce the process variance.
Table 3
Code and tolerance
Level Tolerance (%) I 0.1 II 0.25 III 0.5 IV 1.0 V 2.0 VI 5.0 VII 10 VIII 20 Table 4
The resistance specifications
Code X Tolerance (%) USL LSL A 220 5 231.00 209.00 B 10 5 10.50 9.50 C 1 1 1.01 0.99 D 5 K 2 5.10 4.90 E 1.5 K 2 1.53 1.47 F 2 M 1 2.02 1.98 G 10 M 2 10.20 9.80 H 100 0.1 100.10 99.90 I 10 0.5 10.05 9.95 J 470 K 2 479.40 460.60 K 180 0.25 180.45 179.55 L 22 K 1 22.22 21.78 M 0.3 10 0.33 0.27 N 68 5 71.40 64.60 O 33 K 2 33.66 32.34 Table 5
The calculated statistics
Process XX S Cia Cip Cpp A 223.031 3.252 0.68 0.79 1.47 B 10.102 0.126 0.38 0.57 0.95 C 0.996 0.003 1.82 0.92 2.74 D 5.011 0.040 0.10 1.43 1.54 E 1.505 0.008 0.27 0.71 0.98 F 1.992 0.003 1.44 0.20 1.64 G 10.011 0.030 0.02 0.20 0.23 H 100.012 0.060 0.13 3.24 3.37 I 10.009 0.012 0.29 0.52 0.81 J 468.058 3.492 0.38 1.24 1.63 K 180.200 0.120 1.78 0.64 2.42 L 21.905 0.045 1.68 0.38 2.05 M 0.298 0.009 0.04 0.81 0.85 N 68.958 0.906 0.71 0.64 1.35 O 32.850 0.250 0.46 1.29 1.76
2. The plotted points D, J, and O lie outside of the con-tour Cpp¼ 1, it indicates their Cppare higher than 1. Since these points all lie inside the two 45° line enve-lope range, it indicates that their Cip must be higher than their Cia. Thus, reducing their process variance has higher priority than reducing the process depar-ture.
3. The plotted points C, F, K, and L lie outside of the contour Cpp¼ 1 and the two 45° line envelope range, their Cia must be higher than their Cip. Quality im-provement efforts for these processes should be first focused on reducing the departure of process mean from the target value.
4. The plotted points A and N are close to the two 45° lines, and are outside the contour of Cpp¼ 1. It indi-cates that the variability of those processes is contrib-uted equally by the mean departure and process variance.
5. The plotted points B, E, I, and M lie inside the con-tour of Cpp¼ 1, it means their Cppare lower than 1. Capabilities of these processes are considered to be satisfactory. But they will be the candidates for lower priority quality improvement efforts.
6. Process G is very close to T and its Cppis small, so the process G is considered performing well.
5. Conclusions
In this paper, we introduced a new control chart, called the CppMPPAC, using the incapability index Cpp. The Cpp MPPAC displays multiple processes with the mean departure, and process variability relative to the specification tolerances, on one single chart. We dem-onstrated the use of the Cpp MPPAC by presenting a case study on some resistor manufacturing processes, to evaluate the factory performance. The Cpp MPPAC chart is an efficient tool for the shop supervisors and engineers to evaluate the overall status of shop process control activity. The Cpp MPPAC provides critical in-formation regarding process conditions and useful to quality improvement activity.
References
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[3] Chan LK, Cheng SW, Spring FA. A new measure of process capability: Cpm. J Qual Technol 1988;20(3):162–75.
[4] Greenwich M, Jahr-Schaffrath BL. A process incapability index. Int J Qual Reliab Mgmt 1995;12(4):58–71.
[5] Pearn WL, Lin GH. On the reliability of the estimated process incapability index. Qual Reliab Eng Int 2001;17: 279–90.
[6] Chen CH. Introduction of chip resistor. Industr Mater 1996;109:91–5.
[7] Chen WS. The explore of resistor and exact resistor. Industr Electron Automat Control Dev Des 1996;2:134–44. [8] Wang CL. Electronic materials. 1st ed. Chan Wen
Com-pany; 1992. Fig. 2. The CppMPPAC for the example.
