Control Chart for Monitoring the Coefficient of Variation with an Exponentially Weighted Moving Average Procedure Jiujun Zhang 1,2 , Zhonghua Li 2 , and Zhaojun Wang 2∗ 1 Department of Mathematics, Liaoning University, Shenyang 110036, P.R.China 2 Institute of Statistics and LPMC, Nankai University, Tianjin 300071, P.R.China Abstract The coefficient of variation (CV) of a population is defined as the ratio of the population standard deviation to the population mean, which can be regarded as a measure of stability or uncertainty, and can also indicate the relative dispersion of data to the population mean. This paper proposes a new exponentially weighted moving average (EWMA) chart for monitoring CV, which is constructed by truncating those negative normalized observations to zero in the traditional EWMA CV statistics. The implementation and optimization procedures of the proposed chart are presented. The new chart is compared with some existing CV charts by means of average run length (ARL), and the comparison results show that the new chart outperforms other charts in most cases. Two examples illustrate the use of this chart on real data gathered from a metal sintering process and from a die casting hot chamber process. Key words: control charts; statistical process control; coefficient of variation; expo- nentially weighted moving average; reflecting boundary. 1 Introduction Ever since Shewhart introduced the term of control charts, it has become a common practice for practitioners to use various control charts to monitor different processes ([1],[2]). When we deal with variable data, the charting technique usually employs one chart to monitor the process mean and another chart to monitor the process variance. The Shewhart and (or ) charts are industry standards for quality control applications where the mean and the standard deviation of a process must be statistically controlled at the nominal values 0 and 0 . The baseline assumption is that the nominal values are fixed constants, ∗ Corresponding author, email: [email protected]1
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Control Chart for Monitoring the Coefficient ofVariation with an Exponentially Weighted Moving
Average Procedure
Jiujun Zhang1,2, Zhonghua Li2, and Zhaojun Wang2∗1Department of Mathematics, Liaoning University, Shenyang 110036, P.R.China
2Institute of Statistics and LPMC, Nankai University, Tianjin 300071, P.R.China
Abstract
The coefficient of variation (CV) of a population is defined as the ratio of thepopulation standard deviation to the population mean, which can be regarded as ameasure of stability or uncertainty, and can also indicate the relative dispersion of datato the population mean. This paper proposes a new exponentially weighted movingaverage (EWMA) chart for monitoring CV, which is constructed by truncating thosenegative normalized observations to zero in the traditional EWMA CV statistics. Theimplementation and optimization procedures of the proposed chart are presented. Thenew chart is compared with some existing CV charts by means of average run length(ARL), and the comparison results show that the new chart outperforms other chartsin most cases. Two examples illustrate the use of this chart on real data gathered froma metal sintering process and from a die casting hot chamber process.
Key words: control charts; statistical process control; coefficient of variation; expo-nentially weighted moving average; reflecting boundary.
1 Introduction
Ever since Shewhart introduced the term of control charts, it has become a common practice
for practitioners to use various control charts to monitor different processes ([1],[2]). When
we deal with variable data, the charting technique usually employs one chart to monitor
the process mean and another chart to monitor the process variance. The Shewhart 𝑋 and
𝑆 (or 𝑅) charts are industry standards for quality control applications where the mean 𝜇
and the standard deviation 𝜎 of a process must be statistically controlled at the nominal
values 𝜇0 and 𝜎0. The baseline assumption is that the nominal values are fixed constants,
1.1, 1.2, 1.25, 1.50, 2.0 are considered for purpose of comparison. Note that 𝜏 ∗=0.4, 0.5,0.6,
0.7, 0.8, 0.9, 0.95 and 𝜏 ∗=1.1, 1.2, 1.25, 1.50, 2.0 represent downward and upward shifts in
the CV respectively. Table 6 lists ARL1 for the RES and SSGR charts, where both charts
have an ARL0 of 370.4.
[Insert Table 6 about here]
From Table 6, we observed that when the sample size 𝑛 = 5, the RES chart outperforms
the SSGR chart when 0.5 ≤ 𝜏 < 1 or 1 < 𝜏 ≤ 1.3. For instance, concerning the increasing
case, when 𝛾0 = 0.1 and 𝜏 ∗ = 1.25, [25] suggested (𝐿∗, 𝐿𝐶𝐿∗, 𝑈𝐶𝐿∗) = (18, 0.0296, 0.1787),
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and Table 2 suggests (𝜆∗, 𝐾∗) = (0.05, 3.047). With these parameters, the ARL1 value of RES
chart is 13.6, which is about 11% less than 15.3 of SSGR chart. When the shift size is small.
i.e., 𝜏 ∗ = 1.1, the advantage of our chart is more significant. In this case, the ARL1 value of
RES chart is 46.5, which is about 61% less than that of RES chart. Concerning the decreasing
case, when 𝛾0 = 0.1 and 𝜏 ∗ = 0.8, [25] suggested (𝐿∗, 𝐿𝐶𝐿∗, 𝑈𝐶𝐿∗) = (7, 0.0348, 0.1682),
and Table 2 suggests (𝜆∗, 𝐾∗) = (0.05, 1.293). With these parameters, the ARL1 value of
RES chart is 17.6, which is about 88.8% less than 157.7 of SSGR chart. When the shift size
is small. i.e., 𝜏 ∗ = 0.9, the advantage of our chart is more significant. In this case, the ARL1
value of RES chart is 51.3, which is about 61% less than 370.9 of RES chart. Also, we can
see that the SSGR chart is ARL-biased when 0.9 < 𝜏 < 1 , i.e., the ARL1 values of the
SSGR charts are all larger than ARL0=370. With the increase of sample size 𝑛, i.e., 𝑛 = 10,
the SSGR chart performs slightly better than the RES chart if the CV shift size is large,
i.e., 𝜏 ≥ 1.3, while the RES chart performs much better than the SSGR chart for small CV
shifts.
We also conducted some simulations for other choices of sample size and ARL0, the pre-
ceding findings still hold. Generally speaking, the new scheme provides quite a satisfactory
performance for various types of shifts including the increase and decrease in CV. By taking
the consideration of its easy design and implementation, we believe our new proposed scheme
is a serious alternative in practical applications.
5 Real Data Applications
In this section, we demonstrate the application of the proposed methodology by two real
data examples.
Example ♯ 1
The first example considers real industrial data from a sintering process manufacturing
mechanical parts. This example has been introduced in [18] for the implementation of their
EWMA𝛾2 chart and it has also been used in [21], [22] and [25]. As introduced in [18],
production of gears or mechanical components having complex shapes by means of powder
metallurgy technological processes is spreading in industry due to the potential cost savings
achievable by this technology relative to traditional machining operations. Sintering is an
operation of powder metallurgy whereby compressed metal powder is heated to a temperature
that allows bonding of the individual particles. Proper control of the furnace temperature
is essential for successful sintering to obtain optimum properties. Among the many factors
influencing the strength of the bond between particles, the pore shrinkage plays an important
role. The process manufactures parts which are required to guarantee a pressure test drop
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time 𝑇𝑝𝑑 from 2 bar to 1.5 bar larger than 30 sec as a quality characteristic related to the
pore shrinkage. Using molten copper to fill pores during the sintering process allows the drop
time to be significantly extended. Generally, the larger the quantity 𝑄𝐶 of molten copper
absorbed within the sintered compact during cooling, the larger is the expected pressure
drop time 𝑇𝑝𝑑.
A preliminary regression study ([18]) relating 𝑇𝑝𝑑 to the quantity 𝑄𝐶 of molten copper has
demonstrated the presence of a constant proportionality 𝜎𝑝𝑑 = 𝛾𝑝𝑑×𝜇𝑝𝑑 between the standard
deviation of the pressure drop time and its mean. According to process engineers, the most
important special cause that leads to an anomalous increase in 𝜎𝑝𝑑 is when the sintering
steel has a heterogeneous microstructure and an irregular grain size, which strongly affects
the way copper is adsorbed within each sintered part and its pore filling. The consequence
is that data dispersion within a sample can be larger than expected.
To perform statistical process control (SPC) by means of control charts, the quality
practitioner decided to monitor the coefficient of variation 𝛾𝑝𝑑 = 𝜎𝑝𝑑/𝜇𝑝𝑑 in order to detect
changes in the process variability. Given the nominal quantity of copper 𝑄𝐶 , a Phase I
dataset of 𝑚 = 20 sample data, each having sample size 𝑛 = 5, have been collected; they
are listed in Table 7 (top) of [18]. The analysis of the Phase I data resulted in an estimate
𝛾0 = 0.417 based on a root-mean-square computation and proved that the sintering process
is perfectly in-control.
In order to be consistent with [18], [21], [22] and [25], 𝜏 ∗ is set to 1.25, which implies a
shift of 25% in the CV should be considered to be as a signal that something is wrong in
the production process of the parts. The parameters of the new chart which is optimal for
detecting a shift from 𝛾0 = 0.417 to 𝛾1 = 𝛾0×1.25 = 0.521 (i.e. increase of 25%) when 𝑛 = 5
are found by the optimizing algorithm to be (𝜆∗, 𝐾∗) = (0.05, 4.9648). Using Equations (3)
and (4), we have 𝜇0(𝛾2) = 0.1557, 𝜎0(𝛾
2) = 0.1643, and the UCL is 0.464 when ARL0=370.
A set of data collected during Phase II of the chart implementation are presented in
Table 7 (bottom) of [18]. These data consist of 20 new samples taken from the process after
the occurrence of a special cause increasing process variability. The charting statistics and
the control limit UCL=0.464 are plotted in Figure 1. From Figure 1, it is observed that the
new chart gives an OC signal at the 10𝑡ℎ observation. It is interesting to note the MOSE
and OSE charts detect an OC signal at the 13𝑡ℎ sample, the SRR and SSGR charts detect
an OC signal at the 15𝑡ℎ sample, respectively.
[Insert Figure 1 about here]
Example ♯ 2
The following example has been introduced in [20] for the implementation of a VSI control
chart monitoring the CV in a long production run context and also has been used in [23]
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in short production runs. For more information concerning this example, refer to [20]. This
example considers actual data from a die casting hot chamber process kindly provided by
a Tunisian company manufacturing zinc alloy (ZAMAK) parts for the sanitary sector. The
quality characteristic 𝑋 of interest is the weight (in grams) of scrap zinc alloy material
to be removed between the molding process and the continuous plating surface treatment.
A regression analysis based on past historical data estimated a constant proportionality
𝜎 = 𝛾 × 𝜇 between the standard-deviation 𝜎 and the mean 𝜇 of the weight of scrap alloy.
With the regression study, the in-control CV 𝛾0 has been estimated to 0.01.
According to the process engineer, the most important special cause that leads to an
anomalous increase in 𝜎 is due to the shift from the nominal value of the injection pressure
of the zinc alloy into the die. In fact, the injection pressure holds the molten metal into the
die during solidification. As a consequence, its variation can lead to an uncontrolled item
solidification leading to excessive scrap material.
In order to be consistent with [20] and [23], 𝜏 ∗ is set to 1.25. The parameters of the
new chart which is optimal for detecting a shift from 𝛾0 = 0.01 to 𝛾1 = 𝛾0 × 1.25 = 0.0125
(i.e. increase of 25%) when 𝑛 = 5 are found by the optimizing algorithm to be (𝜆∗, 𝐾∗) =(0.05, 2.93). The corresponding 𝜇0(𝛾
2), 𝜎0(𝛾2) and the control limit are 10−4, 7.07 × 10−5
and 0.274 respectively, when ARL0=370.
A second set of data collected during Phase II of the chart implementation are presented
in Table 6 of [20]. These data consist of 30 new samples taken from the process after the
occurrence of a special cause increasing process variability. The charting statistics and the
control limit are plotted in Figure 2. It can be observed that the new chart gives an OC
signal at the 18𝑡ℎ observation and this result is consistent with charts of [20] and [23]. Again,
it shows that the RES chart is quite a useful alternative tool for practitioners by taking into
account its performance of detecting CV shifts.
[Insert Figure 2 about here]
6 Conclusions and Considerations
This paper presents a new control charting technique to monitor the CV, i.e., the RES
CV chart by truncating negative normalized observations to zero in the traditional EWMA
CV statistic. Monitoring the CV using control charts is essential as in many situations
even though the sample mean and sample standard deviation changes, the sample standard
deviation is proportional to the sample mean. Under such circumstances, it is difficult to
implement control charts for the mean and the variance in process monitoring.
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In this paper, the implementation and the optimal design procedures of the RES CV chart
have been explained in detail. The ARL and SDRL are employed to measure the performance
of the RES CV chart. The findings show that RES CV chart is generally superior to all CV-
type charts under comparison, except large increasing CV shifts compared with the SSGR
chart. The construction of the RES CV chart is demonstrated with two examples using real
life data.
As the RES CV and other existing CV charts are constructed based on the assumption
that the Phase I process parameters, i.e. 𝜇 and 𝜎 are both known, further studies can consider
the case when these parameters are estimated [42]. Note that all of the CV charts are based
on the assumption that each random variable follows a normal distribution. However, the
underlying process is not normal in many applications [43]-[45], and as a result the statistical
properties of CV charts can be highly affected in such situations. Hence, it is necessary to
check how the proposed methodology performs when the underlying distribution is violated.
Furthermore, the nonparametric CV chart also warrants future research.
Acknowledgements
The authors are grateful to the editor and the anonymous referee for their valuable comments
that have greatly improved this paper. This paper is supported by the National Natural
Science Foundation of China Grants 11571191, 11431006 and 11371202, the Science and
Technology Project of Hebei Science and Technology Department of China 162176489.
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Authors’ Biographies
Dr. Jiujun Zhang is Associate Professor of the Department of Mathematics, Liaoning
University. He obtained his B.Sc, M.Sc degree in statistics from Liaoning Normal University
and PhD degree in statistics from Nankai University. His research interests include statistical
process control, applied statistics and related applications. His research has been published
in various refereed journals including Quality and Reliability Engineering International, In-
ternational Journal of Advanced Manufacturing Technology, Computers and Industrial En-
gineering, etc.
Dr. Zhonghua Li is Associate Professor of the Institute of Statistics, Nankai University.
He received his PhD degree in statistics from Nankai University. His research interests in-
clude statistical process control and quality engineering. His research has been published in
various refereed journals including Technometrics, Journal of Quality Technology, Interna-
tional Journal of Production Research, etc.
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Dr. Zhaojun Wang is Distinguished Professor and Vice Dean of the Institute of Statistics,
Nankai University. His primary research interests include statistical process control, quality
improvement, and high-dimensional data analysis. His research has been published in various
refereed journals including Journal of the American Statistical Association, Technometrics,
Journal of Quality Technology, IIE Transactions, Statistica Sinica, Naval Research Logistic,
etc.
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Table 1: 𝐾+ and 𝐾− values of the RES chart when ARL0=370.