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International Journal of Systems Science
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Multiple-process performance analysis chart based on
process loss indices
W. L. Pearn a , Y. C. Chang b & Chien-Wei Wu c
a
Department of Industrial Engineering and Management , National Chiao Tung University , Taiwan
b
Department of Industrial Engineering and Management , Ching Yun University , Taiwan
c
Department of Industrial Engineering and Systems Management , Feng Chia University , Taiwan
Published online: 02 Sep 2006.
To cite this article: W. L. Pearn , Y. C. Chang & Chien-Wei Wu (2006) Multiple-process performance analysis chart based on process loss indices, International Journal of Systems Science, 37:7, 429-435, DOI: 10.1080/00207720600566263
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Vol. 37, No. 7, 10 June 2006, 429–435
Multiple-process performance analysis chart
based on process loss indices
W. L. PEARNy, Y. C. CHANGz and CHIEN-WEI WUx*
yDepartment of Industrial Engineering and Management, National Chiao Tung University, Taiwan zDepartment of Industrial Engineering and Management, Ching Yun University, Taiwan xDepartment of Industrial Engineering and Systems Management, Feng Chia University, Taiwan
(Received 7 April 2005; in final form 3 May 2005)
Control chart techniques have been widely used in the manufacturing industry for controlling and monitoring process performance and are practical tools for quality improvement. When dealing with variable data, one usually employs the Xchart and R chart (or S chart) to detect the process mean and process variance change. These charts are easy to understand and effectively communicate critical process information without using words and formulae. In this paper, we develop a new multiple-process performance analysis chart (MPPAC), using the process loss index Leto control the product quality and/or reliability for multiple manufacturing processes. Upper confidence bounds are applied to the LeMPPAC to ensure the capability groupings are accurate, which is essential to product quality assurance. The Le MPPAC displays the multiple-process relative inconsistency and process relative off-target degree on one single chart in order to provide simultaneous capability control for multiple processes. We demonstrate the applicability of the proposed Le MPPAC incorporating the upper confidence bounds by presenting a case study on some liquid-crystal display module manufacturing processes, to evaluate the factory performance.
Keywords: Multiple-process performance analysis chart; Process capability indices; Process loss indices; Upper confidence bound
1. Introduction
Process capability indices (PCIs), including Cp, Ca, Cpk,
Cpmand Cpmk(Kane 1986, Chan et al. 1988, Pearn et al.
1992, 1998), have been widely used in the manufacturing industry to measure whether the product quality meets the preset specifications, particularly, in automated, semiconductor and integrated-circuit (IC) assembly manufacturing industries. Those indices provide the manufacturers the means for monitoring their quality levels. By analysing the PCIs, a production department
can improve and enhance a poor process to meet their customers’ need. Those indices have been defined as
Cp¼ USL LSL 6 , Ca¼1 j Tj d , ð1Þ Cpk¼min USL 3 , LSL 3 , Cpm¼ USL LSL 6 2þ ð TÞ21=2, ð2Þ Cpmk¼min USL 3 2þ ð TÞ21=2, LSL 3 2þ ð TÞ21=2 ( ) , ð3Þ
*Corresponding author. Email: cweiwu@fcu.edu.tw
International Journal of Systems Science
ISSN 0020–7721 print/ISSN 1464–5319 onlineß 2006 Taylor & Francis
http://www.tandf.co.uk/journals DOI: 10.1080/00207720600566263
where is the process mean, is the process standard deviation, USL is the upper specification limit, LSL is the lower specification limit, T is the target value and d ¼(USL – LSL)/2 is the half-length of the specification interval.
The index Cp considers the overall process
vari-ability relative to the manufacturing tolerance, which reflects product consistency (process precision). The index Ca measures the degree of process centring (the
ability to cluster around the centre), which has been regarded as the process accuracy index. The index Cpk
takes the process mean into consideration but can fail to distinguish between on-target and off-target processes. The index Cpk is a yield-based index which
provides lower bounds on process yield. The index Cpm takes the proximity of process mean from the
target value into account, which is more sensitive to process departure than Cpk. The design of Cpm is
based on the average process loss relative to the manufacturing tolerance, providing an upper bound on the average process loss, which has been alterna-tively called the Taguchi index. The index Cpmk is
constructed from combining the modifications to Cp
that produced Cpk and Cpm, which inherits the merits
of both indices.
Hsiang and Taguchi (1985) first used the loss function to improve process quality, focusing on reducing the process variation around the target value. Johnson (1992) introduced the relative expected loss index Lefor
processes with symmetric tolerances. Tsui (1997) rewrote Le¼LpeþLot to provide an uncontaminated
separa-tion between informasepara-tion concerning process relative inconsistency loss Lpeand process relative off-target loss
Lot. The index Leis defined as the ratio of the ‘expected
quadratic loss’ to the ‘square of the half-specification-width’: Le¼ Z 1 1 ðx TÞ2 d2 dFðxÞ ¼ d 2 þ T d 2 , ð4Þ
where F(x) is the cumulative distribution function of the measured characteristic. If we denote the first term (/d)2 as Lpe and the second term [( T)/d2] as Lot,
then Le can be rewritten as Le¼LpeþLot. We note
that the mathematical relationships Le¼(3Cpm)2,
Lot¼(1 Ca) 2
and Lpe¼(3Cp) 2
can be established. The advantage of using Leover Cpmis that the estimator
of the former has better statistical properties than that of the latter, as the former does not involve a recipro-cal transformation of the process mean and variance. Also it provides an uncontaminated separation between information concerning the process precision and process accuracy. The process accuracy reflects the departure of the process mean from the target
value, and the process precision reflects the overall process variability. The separation suggests which parameters practitioners may consider to improve process quality. Some commonly used values of Le,
namely 1.00 (process is incapable), 0.44 (process is incapable), 0.11 (process is normally called capable), 0.06 (process is called satisfactory), 0.05 (process is normally called good), 0.04 (process is normally called excellent) and 0.03 (process is normally called super), and the corresponding Cpm values are listed in
table 1.
The subindex Lot measures the relative process
departure, which has been referred to as the relative off-target loss index. On the other hand, the subindex Lpemeasures process variation relative to the
specifica-tion tolerance, which has been referred to as the relative inconsistency loss index. Some commonly used values of Lpe, namely 0.11, 0.06, 0.05, 0.04 and 0.03, and the
corresponding quality conditions are listed in table 2. Note that those values of Lpe are equivalent to
Cp¼1.00, 1.33, 1.50, 1.67 and 2.00 respectively, covering
a wide range of the precision requirements used for most real-world applications.
Ever since Shewhart introduced control charts, it has become a common practice for practitioners to use various control charts to monitor different pro-cesses on a routine basis. For example, when dealing with variable data, one usually employs a chart (such as the X chart) to monitor the process mean, and a chart (such as the R chart or S chart) to monitor
Table 1. Some commonly used Leand equivalent Cpmvalues.
Condition Le Cpm Incapable 1.00 0.33 Incapable 0.44 0.50 Capable 0.11 1.00 Satisfactory 0.06 1.33 Good 0.05 1.50 Excellent 0.04 1.67 Super 0.03 2.00
Table 2. Some commonly used precision requirements.
Quality condition Precision requirement Incapable 0.11 < Lpe Capable 0:06 < Lpe0:11 Satisfactory 0:05 < Lpe0:06 Good 0:04 < Lpe0:05 Excellent 0:03 < Lpe0:04 Super Lpe0:03 430 W. L. Pearnet al.
process spread. Those charts are essential tools for quality control. In the multiple-manufacturing-lines environment where a group of processes need to be controlled, it could be difficult and time consuming for factory engineers or supervisors to analyse each individual chart in order to evaluate overall factory performance. To evaluate the performance of a group of multiple processes with symmetric specifications, Singhal (1990, 1991) introduced a multiple-process performance analysis chart (MPPAC) using process capability index Cpm.
Pearn and Chen (1997) proposed a modification to the CpkMPPAC combining the more-advanced process
capability indices Cpm or Cpmk, to identify problems
causing the processes that fail to centre around the target. Chen et al. (2001) considered an extension of the MPPAC to processes with multiple characteristics. In current practice of implementing those charts, practitioners simply plot the estimated index values on the chart and then draw conclusions on whether processes meet the capability requirement and modifi-cations need to be made for capability improvement. Unfortunately, those approaches (Singhal 1990, 1991, Pearn and Chen 1997, Chen et al. 2001) are highly unreliable since the estimated index values are random variables and sampling errors are ignored. Therefore, process information conveyed from those charts is often misleading.
Traditional ( X, R) or ( X, S) control charts are online statistical process control techniques for mon-itoring and surveillance of the process. However, process capability analysis is a vital part of an overall quality-improvement program. In this paper, we introduce the Le MPPAC based on the subindices
Lpe and Lot. The Le MPPAC chart is an offline
technique for evaluating the performance of multiple processes, which sets priority activity to be taken for process improvement (reducing process variability
or process departure). The Le MPPAC displays
process variability relative to their specification toler-ances (in relative inconsistency Lpe) and process
departure (in relative off-target loss Lot) for multiple
processes on one single chart. We propose a reliable approach by first converting the estimated index values to the upper confidence bounds and then plotting the corresponding upper confidence bounds
on the Le MPPAC. The upper confidence bounds
not only provide us with a clue on the minimal actual process performance but are also useful in making decisions for capability testing. We demonstrate the applicability of the Le MPPAC by presenting a case
study of a group of liquid-crystal display module manufacturing processes, to evaluate the factory performance.
2. Estimations and upper confidence bounds of Lpe, Lot
and Le
2.1 Estimations of Lpe, Lotand Le
To estimate the process relative inconsistency loss, we consider the estimator ^Lpe defined as follows, where
S2n¼Pni¼1ðXi XÞ2=n is the maximum-likelihood
estimator (MLE) of the process variance 2:
^ Lpe¼ 1 n Xn i¼1 ðXi XÞ2 d2 ¼ S2n d2: ð5Þ
The estimator ^Lpe can be rewritten as
^ Lpe¼ Lpe n n ^Lpe Lpe ¼Lpe n Xn i¼1 ðXi XÞ2 2 : ð6Þ
If the process follows the normal distribution, the estimator ^Lpeis distributed as ðLpe=nÞ2n1, where 2n1is
a chi-squared distribution with n 1 degrees of freedom. Pearn et al. (2004) showed that the estimator ^Lpe is the
MLE of Lpe, which is consistent, asymptotically
unbi-ased and efficient. To estimate the relative off-target loss, we consider the natural estimator ^Lot defined as
follows, where X ¼Pni¼1Xi=n is the conventional
esti-mator of the process mean :
^ Lot¼
ð X TÞ2
d2 : ð7Þ
We note that the estimator ^Lot can also be written as
a function of Lpe: Lot¼ Lpe n n ^Lot Lpe ¼Lpe n nð X TÞ2 2 : ð8Þ
If the process characteristic is normally distributed, Pearn et al. (2004) showed that the estimator ^Lot is
distributed as ðLpe=nÞ21ðÞ, where 21ðÞ is a non-central
chi-squared distribution with one degree of freedom and non-centrality parameter ¼ nð TÞ2=2¼nL
ot=Lpe.
Since X is the MLE of , then, by the invariance property of MLE, the natural estimator ^Lotis the MLE
of Lot. To estimate the expected relative loss of the
process (a combined measure of the relative inconsis-tency loss of the process and the relative off-target loss of the process), we consider the natural estimator ^Le
defined as follows: ^ Le¼ 1 n Xn i¼1 ðXiTÞ2 d2 ¼ 1 n Xn i¼1 ðXi XÞ2 d2 þ ð X TÞ2 d2 : ð9Þ
We note that the estimator ^Lecan also be written as a function of Lpe: ^ Le¼ Lpe n n ^Le Lpe ¼Lpe n Xn i¼1 ðXiTÞ2 2 : ð10Þ
If the process characteristic is normally distributed, then the estimator ^Le is distributed as ðLpe=nÞ2nðÞ,
where 2
nðÞ is a non-central chi-squared distribution
with n degrees of freedom and non-centrality parameter . Pearn et al. (2004) showed that ^Leis the MLE, which
is also the uniformly minimum variance unbiased estimator of Le. The statistic L^e is consistent and
asymptotically efficient. Since the estimator has all the desired statistical properties, in practice using ^Le
to estimate the expected relative loss of the process would be reasonable. Some distributional and inferential properties of the process loss indices have been provided by Pearn et al. (2004).
2.2 Upper confidence bounds of Lpe, Lotand Le
In the following, we derive the upper confidence bounds of the three process loss indices Lpe, Lot and Le
respectively. We note the expression ^ Le¼ 1 n Xn i¼1 ðXiTÞ2 d2 ¼1 n Xn i¼1 ðXi XÞ2 d2 þ ð X TÞ2 d2 ¼ ^Lpeþ ^Lot, ð11Þ
where L^pe and L^ot are the MLEs of Lpe and Lot
respectively. By estimating the mean of the unknown parameters by the sample mean X, and the variance 2 by S2
n, the relationship ^Le¼ ^Lpeþ ^Lot may be
estab-lished. This expression provides an uncontaminated separation between calculated information concerning the relative inconsistency loss ^Lpe of the process and
the relative off-target loss ^Lot of the process.
Under the normality assumption, n ^Lpe=Lpe is
dis-tributed as 2
n1, a chi-squared distribution with n 1
degrees of freedom. A 100(1 )% upper confidence bound for Lpecan be expressed, in terms of ^Lpe, as
Upe¼
n ^Lpe
2 n1ðÞ
, ð12Þ
where 2n1ðÞ is the (lower) th percentile of the 2n1 distribution. Under the normality assumption, ^Lot=Lot
is distributed as 2
1ðÞ, a non-central chi-squared
distri-bution with one degree of freedom and non-centrality
parameter . We note that Pð ^Lot=Lot21ð,ÞÞ ¼ 1 .
A 100(1 )% upper confidence bound on Lot can be
expressed, in terms of ^Lot, as Uot¼ ^Lot 2 1ð,Þ , ð13Þ where 2
1ð,Þ is the (lower) th percentile of the
21ðÞ distribution. Under the normality assumption, ðn þ Þ ^Le=Le is distributed as 2nðÞ, a non-central
chi-squared distribution with n degrees of freedom
and non-centrality parameter . We note that
Pððn þ Þ ^Le=Le2nð,ÞÞ ¼ 1 . A 100(1 )%
upper confidence bound for Le can be expressed,
in terms of ^Le, as Ue¼ ðn þ Þ ^Le 2 nð,Þ , ð14Þ where 2
nð,Þ is the (lower) th percentile of the 2nðÞ
distribution.
3. The LeMPPAC
Statistical process control charts, such as the X, R, S2, S and MR charts, have been widely used in the manu-facturing industries for controlling and/or monitoring process performance and are essential tools for any quality improvement activities. These charts are easy to understand and effectively communicate critical process information without using words and formulae. However, they are applicable only for a single process (one process at a time). Thus, using these charts in a multiple-process environment can be a difficult and time-consuming task for the supervisor or shop engineer to analyse each individual chart to evaluate the overall status of shop process control activity.
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, to indicate whether reducing the variabil-ity or the departure of the process mean should be the focus and to provide an easy way to quantify the process improvement by comparing the locations on the chart of the processes before and after the improvement effort. The MPPAC is an efficient tool for communicating between the product designer, the manufacturers and the quality engineers, and between the management departments.
Based on the definition Le¼( T)2/d2þ2/d2,
we first set Le¼k, for various k values, and then
a set of (, ) values satisfying the equation ( T)2þ2¼kd2 can be plotted on the contour
(a curve) of Le¼k. These contours are semicircles
432 W. L. Pearnet al.
centred at (, ) ¼ (T, 0) with radius k1/2d. The more capable the process, the smaller the semicircle is. We plot the seven contours on the LeMPPAC for the seven
Levalues listed in table 1, as shown in figure 1. On the
LeMPPAC, we note the following.
(i) As the point gets closer to (, ) ¼ (T, 0), the value of Le becomes smaller, and the process
perfor-mance is better.
(ii) For the points inside the semicircle of contour Le¼k, the corresponding Le values are smaller
than k. For the points outside the semicircle of contour Le¼k, the corresponding Le values are
greater than k.
(iii) When the processes have fixed values of Le, for
points within the envelope of the two 45 lines,
the variability is contributed mainly by the variance of the process.
(iv) When processes have fixed values of Le, for points
outside the envelope of the two 45 lines, the
variability is contributed mainly by the departure of the process mean from the target.
(v) The perpendicular line and parallel line through the plotted point intersecting the horizontal axis and vertical axis at points represent its Lot and
Lperespectively. For example, the point (0.11, 0.44)
represents Lot¼0.11 and Lpe¼0.44.
(vi) The distance between T and the point at which the perpendicular line through the plotted point inter-sects the horizontal axis denotes the departure of process mean from target. For example, the point (0.11, 0.44) represents the departure of the process mean from target given by T ¼ ð0:11Þ1=2d. (vii) The distance between T and the point at which the
parallel line through the plotted point intersects the vertical axis denotes the departure of the standard deviation of the process. For example, the point (0.11, 0.44) represents the departure of the standard deviation of the process given by ¼ ð0:44Þ1=2d.
4. An application example
In the following, we consider a liquid-crystal display module manufacturing process. Three key components make the liquid-crystal display module function prop-erly. Those include the liquid-crystal display, the back lighting and the peripheral (interface) system. The liquid-crystal display module is one of the key ponents used in many high-technology electronic com-mercial devices, such as cellular phones, the personal digital assistants, pocket calculators, digital watches and automobile accessory visual displays. Currently, the mounting technology of the chip on glass, which makes the exposed particle overturned with the side of the circuits facing downwards, is the best manufacturing technology for the liquid-crystal display module in terms of the mounting density. Conduction of electricity occurs between the IC and the panel of the liquid-crystal display through the mounting material. We note that different mounting materials requires different mounting technologies of the chip on glass.
We consider the following case taken from a manu-facturing factory making liquid-crystal display modules and located at the Science-Based Industrial Park in Taiwan. With a focus on the main bonding process (i.e. the stages of manufacturing the chip on the glass), the bonding precision is essentially a process key parameter. We investigated eight specific types of liquid-crystal display module requiring different bond-ing precision standards.
A random sample of size 100 is taken from each of the eight main bonding processes. With the target value T set to zero (i.e. T ¼ 0), their required tolerances, bonding precision specifications are displayed in table 3. If the characteristic data do not fall within the tolerance (LSL, USL), the lifetime or reliability of the liquid-crystal display module product will be discounted. The calcu-lated sample mean, standard deviation, the index values and 95% upper confidence bounds of Le, Lotand Lpeare
shown in table 4. Figure 2 plots the LeMPPAC for the
eight processes listed in table 4. We analyse the process
Figure 1. The LeMPPAC.
Table 3. The bonding specifications of the liquid-crystal display module.
Code Tolerance (mm) LSL T USL
A 25 25 0 25 B 25 25 0 25 C 15 15 0 15 D 15 15 0 15 E 20 20 0 20 F 5 5 0 5 G 10 10 0 10 H 30 30 0 30
points in figure 2, and obtain the following summary of quality conditions.
(i) The plotted point A is very close to the contour Le¼0.44; this indicates that the process has a low
capability. Since the point A is close to the target line, it demonstrates that the poor capability is mainly contributed by the process variation. Thus, it calls for immediate quality improvement action to reduce the variance of the process.
(ii) The plotted points B and C lie outside the contour Le¼0.11; this indicates that their Le values are
higher than 0.11. Since these points lie inside the envelope of the two 45 lines, it demonstrates that
their Lpe values must be higher than their Lot
values. Thus, reducing their process variances has higher priority than reducing the departures of the process means.
(iii) The plotted points D and E lie outside the contour Le¼0.11 and the envelope of the two 45 lines;
therefore their Lotvalues must be higher than their
Lpe values. Quality improvement efforts for these
processes should be first focused on reducing the departures of process means from the target value. (iv) The plotted point F is close to the 45 line and is
outside the contour of Le¼0.11. This indicates that
the variability of the process is contributed equally by the mean departure and the process variance. (v) The plotted point G lies inside the contour
Le¼0.11; this means that its Le value is lower
than 0.11. The capability of this process is consid-ered to be satisfactory, but it will be a candidate for lower-priority quality improvement efforts. (vi) Process H is very close to (, ) ¼ (T, 0) and its Le
is small; so the process H is considered to perform well.
5. Conclusions
Existing research on manufacturing capability control ignores sampling errors. In this paper, we develop a MPPAC, using the process loss index Leto control the
product quality for multiple manufacturing processes. Taking into account the sampling errors, we obtained the upper confidence bounds. We applied the upper confidence bounds to the Le MPPAC to ensure that
the capability groupings are accurate, which is essential to product quality assurance. The Le MPPAC is
an effective tool for multiple-process control, which displays multiple processes with the relative inconsis-tency of the process and relative off-target degree of the process on one single chart. An application example is given to demonstrate the applicability of the proposed Le MPPAC, which incorporates the upper confidence
bounds, to evaluate the factory performance.
References
L.K. Chan, S.W. Cheng and F.A. Spiring, ‘‘A new measure of process capability: Cpm’’, J. Quality Technol., 20, pp. 162–175, 1988.
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Table 4. The calculated statistics and upper confidence bounds.
Process x sn L^pe L^ot L^e Upe Uot Ue
A 0.542 12.711 0.259 0.001 0.259 0.336 0.018 0.332 B 0.731 8.785 0.124 0.001 0.124 0.160 0.075 0.160 C –0.627 6.824 0.207 0.002 0.209 0.269 0.161 0.268 D 4.502 3.554 0.056 0.090 0.146 0.073 0.119 0.178 E –5.921 4.644 0.054 0.088 0.142 0.070 0.116 0.172 F 1.118 1.175 0.055 0.050 0.105 0.072 0.073 0.131 G –1.057 2.561 0.066 0.011 0.077 0.085 0.031 0.098 H 1.271 3.947 0.017 0.002 0.019 0.023 0.008 0.025
Figure 2. The LeMPPAC for the example.
434 W. L. Pearnet al.
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W. L. Pearnis a professor of operations research and quality assurance at National Chiao Tung University, Taiwan. He received his PhD degree in operations research from the University of Maryland, College Park, Maryland, USA. He worked at AT&T Bell Laboratories as a quality research scientist before joining National Chiao Tung University. His research interests include process capability, network optimization and production management. His publications have appeared in the Journal of the Royal Statistical Society, Journal of Quality Technology, Journal of Applied Statistics, Statistics, Journal of the Operational Research Society, European Journal of Operations Research, Operations Research Letters, Omega, Networks, International Journal of Production Research, and others.
Y. C. Chang received his PhD degree in industrial engineering and management from National Chiao Tung University, Taiwan. Currently, he is an assistant professor of quality management and operations research at the Department of Industrial Engineering and Management, Ching Yun University, Taiwan. His research interests include process capability analysis and queueing systems management.
Chien-Wei Wuis currently an assistant professor at the Department of Industrial Engineering and Systems Management of the Feng Chia University, Taiwan. He received his PhD degree in industrial engineering and management from National Chiao Tung University, and an MS degree in statistics from National Tsing Hua University, Taiwan. His primary research focuses on the area of statistical quality control, process capability analysis and data analysis.
