control chart for monitoring the coefficient of variation...
TRANSCRIPT
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,
∗Corresponding author, email: [email protected]
1
and there are many applications for which this assumption is reasonable. To this end, it is
reasonable to monitor the process mean and variance simultaneously by a single chart, see
[3]-[10]. [11] presented an overview of joint monitoring of mean and variance. However, it
is important to point out that not all processes have constant means or constant standard
deviations that enable control charts for the mean or the standard deviation to be used for
process monitoring. A common relationship is that the standard deviation is proportional
to the mean so that the ratio of the standard deviation to the mean is a constant. This ratio
is referred to as the coefficient of variation (CV).
The CV, which is the ratio of the standard deviation to the mean, is a dimensionless
measure of dispersion found to be very useful in many situations. In chemical experiments,
the CV is often used as a yardstick of precision of measurements; two measurement methods
may be compared on the basis of their respective CVs. In finance, it is interpreted as a
measure of the risk faced by investors, by relating the volatility of the return on an asset
to the expected value of the return. For example, it can be used as a measure of relative
risks ([12]) and a test of the equality of the CVs for two stocks can help determine whether
the two stocks possess the same risk or not. [13] used the CV to assess the homogeneity of
bone test samples produced from a particular method to help assess the effect of external
treatments, such as irradiation, on the properties of bones. [14] used the CV in the analysis
of fault trees. The CV was also employed by [15] in assessing the strength of ceramics. It can
also be used in the fields of materials engineering and manufacturing, where some quality
characteristics related to the physical properties of products constituted by metal alloys or
composite materials often have standard-deviations that are proportional to their population
means. These properties are usually related to the way atoms of a metal diffuse into another.
Tool cutting life and several properties of sintered materials are typical examples from this
setting.
Thus, for these quality characteristics, monitoring the CV using a control chart has
gained remarkable attention in recent years. The first control chart for monitoring the CV,
i.e., the Shewhart CV chart was developed by [16]. It was shown that this chart is useful
for monitoring the process CV. Subsequently, the exponentially weighted moving average
(EWMA) chart for the CV was proposed by [17] to improve the performance of Shewhart
CV chart for detecting small shifts in the CV. Furthermore, the two one-sided EWMA
CV squared charts were proposed by [18] (denoted as OSE chart), respectively. The OSE
chart consists of a downward and an upward one-sided EWMA CV squares charts to detect
decreases and increases in the CV, respectively. In general, it was shown that the OSE chart
produces slightly smaller out-of-control (OC) average run length (ARL) than that of the
EWMA CV chart proposed by [17].
2
Recently, [19] developed a synthetic control chart for monitoring the CV (denoted as
SYN chart). The results showed that the synthetic chart performed better than that of [16],
but worse than [18] as long as the increasing shift in the CV is not too large. [20] evaluated
an adaptive Shewhart control chart implementing variable sampling interval (VSI) strategy
to monitor the process CV. [21] proposed a Shewhart chart with supplementary run rules to
monitor the process CV (denoted as SRR chart). However, as they pointed out, the SRR
chart does not outperform more advanced strategies as the OSE chart or the SYN chart.
[22] developed a modified OSE CV chart (denoted as MOSE chart) based on the preliminary
work of [18]. The comparisons showed that the MOSE chart has an ARL performance that
is superior to some other competing procedures. In short production runs, [23] proposed a
Shewhart chart and [24] proposed a variable sample size (VSS) control chart. [25] developed
a side sensitive group runs chart (denoted as SSGR chart) and compared with some of the
existing CV charts in terms of ARL and expected ARL (EARL). [26] proposed a one-sided
run rules control charts for monitoring the CV in short production runs and used Markov
chains to get their main statistical properties. [27] developed a procedure to monitor the CV
using run-sum control charts. The run-length properties of the charts are characterized by
the Markov chain approach. In addition, concerning the studies on the control charts that
monitor the multivariate CV, the first control chart for the multivariate CV is Shewhart-type
chart, which was presented by [28]. Next, [29] proposed the run sum chart for monitoring
multivariate CV in the Phase-II process.
In this paper, a new chart is proposed to further improve the performance of EWMA
chart by applying a resetting scheme for monitoring the CV. This resetting technique was
originally proposed by [30] for monitoring the process variance. With resetting boundaries,
[31] obtained necessary and sufficient conditions for non-interaction of a pair of one-sided
EWMA schemes. For our proposed chart, sets of optimal design parameters are provided
for different values of the in-control CV, for different sample sizes, and for a wide range
of deterministic shifts, including both decreasing and increasing cases. The new chart is
compared with most of the existing competing charts, including the EWMA type charts,
i.e., the OSE and MOSE charts, and the SYN, SRR and SSGR charts, respectively. The
underlying process is assumed to follow a normal distribution.
The rest of this paper is structured as follows: Section 2 gives an overview of the existing
competing CV charts, as these charts are also considered in the performance comparisons.
In Section 3, our proposed resetting EWMA chart (denoted as RES chart) is presented and
the statistical performance of the new chart is investigated. The numerical comparisons with
some other procedures are carried out in Section 4. The application of our proposed method
is illustrated in Section 5 by two real data examples from a metal sintering process control
3
and from a die casting hot chamber process. Several remarks conclude this paper in Section
6.
2 An Overview of Some Existing CV Charts
Suppose that we observe subgroups 𝑋𝑘 = {𝑋𝑘1, 𝑋𝑘2, . . . , 𝑋𝑘𝑛} of size 𝑛 at times 𝑘 = 1, 2, . . ..
We also assume that there is independence within and between these subgroups and each
random variable 𝑋𝑘𝑗 follows a normal 𝑁(𝜇𝑘, 𝜎𝑘) distribution, where the parameters 𝜇𝑘 and
𝜎𝑘 are constrained by the relation 𝛾𝑘 =𝜇𝑘
𝜎𝑘= 𝛾0 when the process is in control. This implies
that, from one subgroup to another, the values of 𝜇𝑘 and 𝜎𝑘 may change, but the coefficient
of variation 𝛾𝑘 = 𝜇𝑘
𝜎𝑘must be equal to some predefined in-control value 𝛾0, common to all
the subgroups.
Let 𝑋𝑘 and 𝑆𝑘 be the sample mean and the sample standard deviation of 𝑋𝑘, i.e., 𝑋𝑘 =1𝑛
∑𝑛𝑖=1 𝑋𝑘𝑖 and 𝑆𝑘 =
√1
𝑛−1
∑𝑛𝑖=1(𝑋𝑘𝑖 −𝑋𝑘)2. The sample CV 𝛾𝑘 is defined as 𝛾𝑘 = 𝑆𝑘
𝑋𝑘
and its distribution has been extensively studied in the literature. [32] noted that when
the parent population distribution is normal,√𝑛
𝛾𝑘has a non-central 𝑡-distribution with 𝑛− 1
degrees of freedom and non-centrality parameter√𝑛𝛾.
Next, we give a brief review of some existing CV charts, including two competing EWMA
type charts, i.e., OSE and MOSE charts, and then the SYN, SRR and SSGR charts, as these
five charts are also considered in the performance comparisons given in Section 4.
2.1 The OSE chart [18]
[18] proposed a method to monitor the process CV by means of two one-sided EWMA charts
of the CV squared. First, an upper-sided OSE chart aims to detect an increase in the CV
and is defined as
𝑍+𝑘 = max(𝜇0(𝛾
2), (1− 𝜆)𝑍+𝑘−1 + 𝜆𝛾𝑘
2),
with 𝑍+0 = 𝜇0(𝛾
2) as the initial value and with the asymptotic corresponding upper control
limit (UCL)
𝑈𝐶𝐿 = 𝜇0(𝛾2) +𝐾
√𝜆
2− 𝜆𝜎0(𝛾
2). (1)
Second, a downward OSE chart is defined as
𝑍−𝑘 = min(𝜇0(𝛾
2), (1− 𝜆)𝑍−𝑘−1 + 𝜆𝛾𝑘
2),
with 𝑍−0 = 𝜇0(𝛾
2) and with the asymptotic corresponding lower control limit (LCL)
𝐿𝐶𝐿 = 𝜇0(𝛾2)−𝐾
′√
𝜆
2− 𝜆𝜎0(𝛾
2), (2)
4
where 𝜇0(𝛾2) and 𝜎0(𝛾
2) are the mean and standard deviation of 𝛾2 when the process is in
control, and 𝜆, 𝐾 and 𝐾′are the smoothing constant and chart coefficient of the upward
and downward OSE charts. [33] provided guidelines for selecting 𝜆 and it was shown that
smaller values of 𝜆 should be used to detect smaller shifts and larger values should be used
to detect larger shifts. In general, values of 𝜆 in the interval 0.05 ≤ 𝜆 ≤ 0.25 work well in
practice.
Approximations for 𝜇0(𝛾2) and 𝜎0(𝛾
2) are provided by [34] as
𝜇0(𝛾2) = 𝛾2
0(1−3𝛾2
0
𝑛), (3)
and
𝜎0(𝛾2) =
{𝛾40
( 2
𝑛− 1+ 𝛾2
0(4
𝑛+
20
𝑛(𝑛− 1)+
75𝛾20
𝑛2))− (
𝜇0(𝛾2)− 𝛾2
0
)2} 12 . (4)
Using a Markov chain approach to compute ARLs, [18] discussed optimization procedures
for their OSE charts and compared them with the original CV chart proposed by [16] and a
modified version of a two-sided EWMA chart proposed by [17]. They concluded that their
OSE charts show significant improvement in detecting changes in 𝛾 when compared with
the original CV chart of [16] and that they almost always yielded smaller OC ARLs when
compared with the modified chart of [17].
2.2 The MOSE chart [22]
[22] proposed a modified one-sided EWMA CV procedure based on the work of [18] in order
to further enhance the sensitivity of the OSE chart in monitoring the process CV. First, an
upward MOSE EWMA chart was defined as follows:
𝑍+𝑘 = max(𝜇0(𝛾
2), 𝑈+𝑘 ),
where 𝑈+𝑘 is defined as
𝑈+𝑘 = (1− 𝜆)𝑈+
𝑘−1 + 𝜆𝛾𝑘2
with 𝑈+0 = 𝜇0(𝛾
2) as the initial value. The asymptotic corresponding UCL is the same
as defined in Equation (1). Analogously, a downward MOSE EWMA chart was defined as
follows:
𝑍−𝑘 = max(𝜇0(𝛾
2), 𝑈−𝑘 ),
where 𝑈−𝑘 is defined as
𝑈−𝑘 = (1− 𝜆)𝑈−
𝑘−1 + 𝜆𝛾𝑘2,
5
with 𝑈−0 = 𝜇0(𝛾
2) as the initial value. The asymptotic corresponding LCL is the same as
defined in Equation (2). A combination of the two one-sided MOSE charts can be imple-
mented to detect both increase and decrease shifts in CV. The MOSE chart performs better
than the OSE chart in most cases, especially for detecting small to moderate shifts.
2.3 The SYN chart [19]
The SYN CV chart developed by [19] consists of two sub-charts: a standard Shewhart ��
CV chart and a conforming run length (CRL) CV chart. According to [19], a sample that
causes the monitoring statistic to traverse the control limits of the CV sub-chart is only
regarded as a non-conforming sample. Whenever a non-conforming sample occurs, a count
is initiated to check the number of samples between the current non-conforming sample and
the previous non-conforming sample. Only when this count variable is less than a threshold
value, say L, is the process deemed OC. Otherwise, reset the count variable and restart the
underlying control charting procedure.
The ARL and standard deviation run length (SDRL) of the SYN CV chart given by [19]
are
𝐴𝑅𝐿𝑆𝑌 𝑁 = 𝐴𝑅𝐿𝑆𝑌 𝑁(𝜏, 𝐿𝐶𝐿,𝑈𝐶𝐿,𝐿∣𝑛, 𝛾0) = (1
1− (1− 𝑝)𝐿)(1
𝑝)
and
𝑆𝐷𝑅𝐿𝑆𝑌 𝑁 = 𝑆𝐷𝑅𝐿𝑆𝑌 𝑁(𝜏, 𝐿𝐶𝐿,𝑈𝐶𝐿,𝐿∣𝑛, 𝛾0)
={ 2− 𝑝
(1− (1− 𝑝)𝐿)𝑝2+ [
1
(1− (1− 𝑝)𝐿)2][1
𝑝2− 2
𝐿∑𝑡=1
𝑡(1− 𝑝)𝑡−1]} 1
2 ,
where
𝑝 = 𝑝(𝜏, 𝐿𝐶𝐿,𝑈𝐶𝐿,𝐿∣𝑛, 𝛾0)) = 1− 𝑃𝑟(𝐿𝐶𝐿 < 𝛾𝑡 < 𝑈𝐶𝐿)
and 𝛾 = 𝜏𝛾0. Given significance level p, the LCL and UCL respectively, are given by
𝐿𝐶𝐿 = 𝐹−1𝛾𝑘
(𝑝
2, 1∣𝑛, 𝛾0) =
√𝑛
𝐹−1𝑡 (1− 𝑝
2∣𝑛− 1,
√𝑛/𝛾)
(5)
and
𝑈𝐶𝐿 = 𝐹−1𝛾𝑘
(1− 𝑝
2, 1∣𝑛, 𝛾0) =
√𝑛
𝐹−1𝑡 (𝑝
2∣𝑛− 1,
√𝑛/𝛾)
, (6)
where 𝐹−1𝑡 is the inverse cumulative distribution function (c.d.f.) of the non-central 𝑡-
distribution with 𝑛− 1 degrees of freedom and non-centrality parameter√𝑛/𝛾.
6
2.4 The SRR chart [21]
[21] considered pure run rules type charts for monitoring the CV, where only warning limits
are required by these charts. Then for a c-out-of-d SRR CV chart, an OC signal is issued
by the chart when (i) c out of d successive 𝛾 points are plotted below the lower warning
limit (LWL) or (ii) c out of d successive 𝛾 points are plotted above the upper warning limit
(UWL). The LWL and UWL are
𝐿𝑊𝐿 = 𝜇0(𝛾)−𝐾𝜎0(𝛾), 𝑈𝑊𝐿 = 𝜇0(𝛾) +𝐾𝜎0(𝛾)
respectively. Here, 𝐾 is the charting parameter for the SRR CV chart. Note that 𝜇0(𝛾) and
𝜎0(𝛾) are the mean and standard deviation of the sample CV when the process is in-control
respectively. The approximations of 𝜇0(𝛾) and 𝜎0(𝛾) are as follows:
𝜇0(𝛾) = 𝛾0{1 +
1
𝑛(𝛾2
0 −1
4) +
1
𝑛2(3𝛾4
0 −𝛾20
4− 7
32) +
1
𝑛3(15𝛾6
0 −3𝛾4
0
4− 7𝛾2
0
32− 19
128)}
and
𝜎0(𝛾) = 𝛾0{ 1𝑛(𝛾2
0 +1
2) +
1
𝑛2(8𝛾4
0 − 𝛾20 +
3
8) +
1
𝑛3(69𝛾6
0 +7𝛾4
0
2+
3𝛾20
4+
3
16)} 1
2 .
[21] considered the 2-out-of-3 SRR, 3-out-of-4 SRR and 4-out-of-5 SRR charts and found
that depending on the shift size 𝜏 , the 2-out-of-3 SRR CV and 4-out-of-5 SRR CV charts
have the best performances, and in this paper, 4-out-of-5 SRR CV chart is used to compare
with our new chart. The ARL and SDRL of the SRR CV chart are
𝐴𝑅𝐿𝑆𝑅𝑅 = 𝑣1 (7)
and
𝑆𝐷𝑅𝐿𝑆𝑅𝑅 =√𝑣2 − 𝑣21 + 𝑣1 (8)
where
𝑣𝑚 = 𝑚!q𝑇 (I-Q)−𝑚Q𝑚−11
for 𝑚 = 1, 2. Note that I is the identity matrix and q is a vector of initial probabilities. The
ARL and SDRL for the 4-out-of-5 SRR CV chart are computed using Equations (7) and (8),
and for this case, the size of matrix Q of transient probabilities is (79× 79), while the size
of vector q of initial probabilities is (79× 1) with the initial state as the 40𝑡ℎ state.
7
2.5 The SSGR CV chart [25]
The SSGR chart proposed by [25] consists of a CV sub-chart and an extended version of a
CRL sub-chart. Here, CRL is defined as the number of conforming CV samples between two
nonconforming ones, including the ending nonconforming CV sample. The LCL and UCL of
the CV sub-chart are computed using Equations (5) and (6) respectively, with 𝑝 replaced by
𝛼∗. Let 𝐿𝑆𝑆𝐺𝑅 represent the lower limit of the CRL subchart. Both 𝛼∗ and 𝐿𝑆𝑆𝐺𝑅 control
the in-control ARL (ARL0) of the SSGR chart.
The operation of the SSGR CV chart of [25] is outlined as follows:
1. Compute the optimal control limits LCL, UCL and L using the optimization procedure.
2. Take a sample of n observations and compute the sample CV 𝛾.
3. If 𝛾 ∈[LCL,UCL], the current sample is classified as conforming and the process is
in-control. Then the control flow returns to Step 2. Otherwise, the current sample is
classified as nonconforming and the control flow moves to the next step.
4. Count the number of samples between two nonconforming samples (inclusive of the
ending nonconforming sample) and take this count as the CRL value.
5. (i) For the first CRL, if CRL1 ≤L, the process is declared as OC and the control flow
advances to Step 6. Otherwise, the process is in-control and the control flow returns
to Step 2. (ii) For the second CRL onwards, if CRL𝑟 ≤L, for 𝑟 = 2, 3 . . . , the process
is not immediately classified as OC, and the control flow returns to Step 2. However,
if CRL𝑟 ≤L and CRL𝑟+1 ≤L, for 𝑟 = 2, 3 . . ., and that both CRL𝑟 and CRL𝑟+1 are
having shifts on the same side of the CV sub-chart, the process is declared as OC and
the control flow proceeds to Step 6.
6. Investigate and remove the assignable cause(s). Then return to Step 2.
Finally, the ARL𝑆𝑆𝐺𝑅 and SDRL𝑆𝑆𝐺𝑅 are computed using Equations (7) and (8) re-
spectively, but by replacing 𝑄 with its own transition probability matrix 𝑄∗, where 𝑞 =
(1, 0, . . . , 0)𝑇 is a (4L𝑆𝑆𝐺𝑅 +1)× 1 vector. Here, the first element of vector 𝑞 corresponds to
the initial state. For more details of the computation of 𝑄∗ of the SSGR chart, see [25].
3 The Proposed RES Procedure
[30] considered a resetting rule in the one-sided EWMA scheme, particularly for non-normal
data. They suggested resetting the current observation or normalized observation to the
8
target rather than the EWMA statistic. This scheme has been shown to have better per-
formance than the direct resetting of the EWMA statistic. To this end, we adapt this idea
to the monitoring of CV in this paper. First, we will describe the construction of our new
chart and then explain the optimization procedures.
3.1 The construction of the RES chart
Clearly, the reset of negative EWMA statistic to zero can ameliorate the inertia problem
of the EWMA statistic. Besides the traditional resetting scheme, another possible way of
resetting in the EWMA procedure is to truncate the EWMA statistics itself to the target
whenever it is less than the target. Define the standardized 𝛾𝑘2 as
𝑍𝑘 =𝛾𝑘
2 − 𝜇0(𝛾2)
𝜎0(𝛾2)∣𝛾𝑘 = 𝛾0,
where 𝜇0(𝛾2) and 𝜎0(𝛾
2) is computed from Equations (3) and (4), respectively. The pro-
posed procedure first winsorizes 𝑍𝑘 and then applies a conventional EWMA scheme to the
winsorized data. That is, an upper-sided EWMA chart can be defined as
𝑊′𝑘 = 𝜆𝑍+
𝑘 + (1− 𝜆)𝑊′𝑘−1, (9)
where 𝑍+𝑘 = max(0, 𝑍𝑘), and 𝑊
′0 = 𝐸[𝑍+
𝑘 ∣𝛾 = 𝛾0]. Note that if 𝑍𝑘 ∼ 𝑁(0, 1), the mean and
variance of the winsorized normal variable, 𝑍+𝑘 = max(0, 𝑍𝑘), are given by [35]:
𝜇𝑍+𝑘=
1√2𝜋
, 𝜎2𝑍+𝑘= 1− 1
2𝜋.
Therefore, the asymptotic mean of𝑊′𝑘 is not equal to zero but
1√2𝜋
in the in-control situation.
To make the mean of𝑊′𝑘 be zero, it is convenient to rewrite the EWMA recursion in Equation
(9) as
𝑊+𝑘 = 𝜆(𝑍+
𝑘 − 1√2𝜋
) + (1− 𝜆)𝑊+𝑘−1, (10)
where 𝑊+𝑘 = 𝑊
′𝑘 − 1√
2𝜋. This new chart triggers an OC signal when 𝑊+
𝑘 exceeds the UCL,
𝑈𝐶𝐿 = ℎ𝑈𝑊 = 𝐾+
√𝜆
2− 𝜆𝜎𝑍+
𝑘, (11)
where 𝐾+ can be chosen to achieve the desired ARL0. The start value of the RES chart is
usually set to the target, i.e., 𝑊0 = 0, while other positive head-start values can be chosen
for fast initial response (FIR) features [36].
9
Analogously, if our interest focuses on detecting a decrease in the process CV, a lower-
sided RES EWMA chart can be similarly defined as
𝑅′𝑘 = 𝜆𝑍−
𝑘 + (1− 𝜆)𝑅′𝑘−1, (12)
where 𝑍−𝑘 = min(0, 𝑍𝑘), and 𝑅
′0 = 𝐸[𝑍−
𝑘 ∣𝛾 = 𝛾0]. Then, we rewrite the EWMA recursion in
Equation (12) as
𝑅−𝑘 = 𝜆(𝑍−
𝑘 +1√2𝜋
) + (1− 𝜆)𝑅−𝑘−1. (13)
Subsequently, the LCL of the chart is given by
𝐿𝐶𝐿 = ℎ𝐿𝑊 = −𝐾−
√𝜆
2− 𝜆𝜎𝑍−
𝑘, (14)
where 𝐾− can be determined to achieve the desired ARL0 and 𝜎𝑍+𝑘= 𝜎𝑍−
𝑘.
A combination of the two one-sided RES charts can be implemented to detect both
increase and decrease shifts in CV. The 𝐾+, 𝐾− values of the upper and lower charts for
some combinations of 𝜆, 𝛾0 and 𝑛 are presented in Table 1 when the ARL0 is 370. Upon
request, Fortran programs that optimize our new chart for other parameter conditions will
be provided.
[Insert Table 1 about here]
From Table 1, we see that for any fixed combination of (𝜆, 𝛾0) values, as 𝑛 increases,
𝐾+ decreases and 𝐾− increases. Further, for any fixed combination of (𝜆, 𝑛) values, as 𝛾0
increases, 𝐾+ increases and 𝐾− decreases. However, for any fixed combination of (𝑛, 𝛾0)
values, as 𝜆 increases, both 𝐾+ and 𝐾− increase.
As a summary, the RES chart differs from the OSE chart in that the OSE chart resets
the EWMA statistic to the target value whenever it is less than the target value while the
RES CV chart resets the current negative normalized observations, 𝑍𝑘, to zero. Moreover,
when 𝜆 = 1, both OSE and RES charts reduce to a one-sided Shewhart chart of 𝑍𝑘 and
would perform the same. In addition, for the new chart, only the parameters of 𝐾 and 𝜆
are needed, so it is more convenient in the practical application than some other charts.
3.2 ARL optimization for the new chart
To achieve certain properties of the control procedure, one must select the corresponding
parameters. The design approach recommended by [18] and [22] will be followed in this
paper for the RES CV chart. This approach involves the joint choice of 𝜆 and 𝐾 that yields
10
a desired zero-state ARL0 when the process is in-control 𝛾 = 𝛾0 and also yields the smallest
zero-state OC ARL (ARL1) for a specified 𝛾 = 𝛾1 = 𝜏𝛾0. Next, consideration is given to the
magnitude of a shift in the CV, 𝛾 = 𝜏𝛾0, regarded as most detrimental to process quality,
which must be identified and eliminated as soon as possible.
An optimal RES chart would minimize the ARL at this shift, 𝜏 ∗, subject to the chosen
ARL0 constraint. That is, optimal values (𝜆∗, 𝐾∗) are given by
(𝜆∗, 𝐾∗) = arg min(𝜆,𝐾)
𝐴𝑅𝐿(𝛾0, 𝛾1, 𝜆,𝐾, 𝑛), (15)
subject to the constraint
𝐴𝑅𝐿(𝛾0, 𝛾0, 𝜆∗, 𝐾∗, 𝑛) = 𝐴𝑅𝐿0. (16)
The following design procedure is implemented:
– when the process is in control, 𝛾 = 𝛾0, then ARL=ARL0.
– for a specified value 𝛾 = 𝜏 ∗𝛾0 ∕= 𝛾0, the couple (𝜆∗, 𝐾∗) yields the smallest possible
ARL1.
Optimization programs are coded in Fortran to search the optimal limits of the RES
chart using the procedures presented above when the process shift size is deterministic. The
optimization results are given in the next section and compared with the other two EWMA
type charts.
It should be noted that the numerical method used above can only find the approximate
optimal values because the true optimal 𝜆 may occur at the value different from the prespec-
ified interval (0.05, 1). For a fixed ARL0, smaller 𝜆 gives smaller ARL1, but when too small
a 𝜆 is used, the SDRL is usually very large for a control chart. For this reason, the value of
𝜆∗ is always kept larger than 0.05.
4 Numerical Results and Comparisons
There are different statistical measures available in the literature. They are used to judge
the performance of statistical control schemes. Some of them are used to evaluate the
performance of chart for a specific value (single amount) of shift while others are calculated
for a range of shifts ([37]). The most famous and commonly used statistical measure is ARL.
The ARL value evaluates the performance of a charting structure at a single shift point. The
performance is assessed by two types of ARLs, i.e., ARL0 and ARL1. ARL0 is the expected
number of samples before an OC point is detected when the process is actually in control
while ARL1 is the expected number of samples before an OC signal is received when the
process is actually shifted to an OC state ([1]). For the fixed value of ARL0, a chart is
11
considered to be more effective than other charts if it has a smaller ARL1 value for the same
amount of shift. Similarly, the SDRL is also considered as a good supportive measure along
with ARLs to judge the amount of variation in the run length values. In this section, the
performance of our new chart is compared with some competing charts, including the OSE,
MOSE, SYN, SRR and SSGR charts in terms of ARL and SDRL, respectively.
4.1 Comparison with the OSE and MOSE charts
First, we compare three upper-sided EWMA-type charts, i.e., the RES chart, the MOSE
chart and the OSE chart. Each chart is calibrated so that the ARL0 is approximately
equal to 370. For simplicity, the ARL values are obtained using at least 50,000 run length
simulations although the Markov chain method can be used, as [38] showed simulation is
also a popular method. A Fortran program is coded for these simulations. If 𝜆∗ is chosen to
be very small then the chart will be ineffective for detecting large shifts. So, the values of 𝜆∗
are always kept larger than 0.05 for the three charts. The simulation results are tabulated in
Table 2 for n=5 and n=10, respectively, where all charts have been optimized to minimize
ARL1 at shift 𝜏 ∗.
[Insert Table 2 about here]
The optimal couples (𝜆∗, 𝐾∗) for the new charts are presented in the first row of each
block, for 𝛾0 = {0.05, 0.1, 0.15, 0.2}, and 𝜏 ∗ = {0.5, 0.65, 0.8, 0.9} (i.e. decreasing case),
𝜏 ∗ = {1.1, 1.25, 1.5, 2} (i.e. increasing case), while the ARL1 values of the RES chart (left
side), MOSE chart (midst), and OSE charts (right side) are presented in the second row of
each block. From Table 2, we can see that, whatever the values of 𝑛, 𝛾0 or 𝜏∗, the performance
of the RES chart is similar to the MOSE chart. The RES chart performs slightly better for
detecting moderate to large decreasing shifts while the MOSE chart performs slightly better
for detecting small decreasing shifts. Both of the RES and MOSE charts perform much
better than the OSE chart. For instance, concerning the increasing case, if 𝑛 = 5, 𝛾0 = 0.1
and the critical shift is 𝜏 ∗ = 1.1, then the ARL1 values in this case are 46.5, 44.5 and 51.5
for the RES, MOSE and OSE charts, respectively. Concerning the decreasing case, if the
critical shift is 𝜏 ∗ = 0.65, then the ARL1 values are 7.5, 7.9 and 8.8, respectively. For very
large increasing shifts (e.g., 𝜏 ∗ = 1, 5, 2.0), these three charts have similar performances. The
same conclusion can be obtained for other 𝑛, 𝜏 ∗ and 𝛾0.
The SDRL is usually used as another measure to evaluate the performance of control
charts. The smaller the values of SDRL, the better the performance of a control chart.
Computation of SDRLs for the three charts demonstrates that the run-length distribution
of the RES and MOSE charts is always more under-dispersed than the one corresponding
12
to the OSE chart. For instance, concerning the two examples described above, the SDRL
values corresponding to the increasing case are 40.4, 35.9 and 41.2 for the RES, MOSE and
OSE charts, and the SDRL values corresponding to the decreasing case are 3.1, 2.9 and 3.6
respectively (not shown in this paper, available from the authors). Again, the RES chart is
always more under dispersed than the OSE chart and similar to the MOSE chart.
In addition, it is observed from Table 2 that the run length profiles for the two charts are
highly influenced by the sample size but not strongly influenced by the size of the in-control
𝛾0. For example, for the RES chart, when n=5 and 𝜏 = 1.25, the ARL values are 13.4,
13.6, 14.0 and 14.6 when 𝛾0=0.05, 0.1, 0.15 and 0.2, respectively. However when 𝑛 = 10,
the corresponding values are 7.8, 7.9, 8.1 and 8.4, respectively. Under the fixed sample
size rational sub-grouping model, practitioners using these charts should choose the largest
sample size that resources allow.
The overall conclusion that can be obtained from Table 2 is that the RES chart, generally,
has the satisfactory detection performance for various changes in the process CV. It can
be seen that compared with the MOSE chart, in most cases, the ARL1 performances are
comparable. The former does slightly better for small shifts (i.e. 0.8≤ 𝜏 <1.25) and the
latter performs better for moderate to large shifts (i.e. 𝜏 <0.8 and 𝜏 ≥1.25). In addition, the
RES chart significantly performs better than the OSE chart. This shows that the RES chart
is quite a useful alternative tool for practitioners by taking into account its performance of
detecting various CV shifts.
As noted by [18], specifying the shift a priori is often too restrictive because the quality
practitioners may not have historical knowledge of the process, or because shifts are not
deterministic but follow some unknown distribution. If the practitioner pre-specifies a shift
𝜏 ∗, and uses the corresponding optimal parameters but experiences a different shift in the CV,
then the run length performance of the chart may be seriously undermined. [18] suggested an
alternate optimization procedure in order to cope with the random shift-size problem in the
design of control charts monitoring the sample CV. Similar approaches have been proposed
by [39-41].
According to [18], the optimal value of 𝜆 is 0.05 when the sample size 𝑛 ≤ 10. In this case,
we will not consider the optimization procedure, instead, we make a comparison among the
RES, MOSE and OSE charts with 𝜆 = 0.05, 𝛾0 = 0.1, 0.2 and 𝑛 = 5, 7, 10, 15, respectively.
Numerical computations based on 50,000 runs are used to determine the ARL values. The
simulation results are displayed in Table 3.
[Insert Table 3 about here]
From Table 3 we can see that the charts with fixed 𝛾0 and 𝜏 perform better for larger
sample size. For example, when 𝛾0 = 0.2 and 𝜏 = 1.1, the ARL1 values of RES chart are
13
49.5, 38.5, 29.6 and 22.2 when 𝑛 = 5, 7, 10 and 15, respectively. Also we can see that the
RES chart always does better than the OSE chart and has similar performance to the MOSE
chart. The MOSE chart does better for small shifts, but for moderate and large shifts, RES
chart performs better than the other two charts. Take 𝛾0 = 0.2 and 𝑛 = 10 as example,
when the process CV increases by 10% (i.e. 𝜏 = 1.1), the ARL1 values of RES, MOSE and
OSE charts are 29.6, 28.8 and 31.7, respectively. However, when the process CV increases
by 25%, (i.e. 𝜏 = 1.25), the corresponding ARL1 values are 8.7, 9.3 and 10.0, respectively.
Similar results can be found for other parameter combinations.
4.2 Comparison with the SYN chart
Since the original SYN chart is designed to monitor increases in 𝛾, we will only compare the
upward RES chart with the SYN chart. We compared the behavior of the proposed chart
for two sizes of rational groups, 𝑛 = 5 and 𝑛 = 10. Both charts have an ARL0 of 370 and
have been optimized to minimize ARL1 at shift 𝜏 ∗ = 1.25, 1.50 and 2.00, respectively. The
results are tabulated in Table 4.
[Insert Table 4 about here]
From Table 4, we observed the following results:
∙ When the sample size 𝑛 = 5, the RES chart outperforms the SYN chart in almost all
cases except when the shift size is very large (e.g., 𝜏 = 2). The RES chart performs
much better than the SYN chart for small to moderate shifts. For instance, when 𝛾0 =
0.1, 𝜏 = 1.25 and 𝜏 ∗ = 1.25, [19] suggested (𝐿∗, 𝐿𝐶𝐿∗, 𝑈𝐶𝐿∗) = (31, 0.02271, 0.19499),
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 44% less than 24.3 of SYN 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 46.6, which is about 61% less than 119.4 of SYN
chart. In addition, the computation of SDRLs (not shown in this paper, available from
the authors) for both the SYN and the RES charts also demonstrates that the RES
chart run-length distribution is always more under-dispersed than the SYN chart. For
instance, concerning the example described above, the SDRL is 9.3 for the RES chart,
while it is 30.6 for the SYN chart.
∙ When the sample size 𝑛 = 10, the SYN chart does better when 𝜏 ≥ 1.5, but the
difference is negligible. In other cases, the RES chart performs better than the SYN
chart.
14
The overall conclusion that can be obtained is that our new chart generally has satisfac-
tory detection performance for various changes in CV. This, again, shows that the new chart
is quite a useful tool for practitioners to monitor the CV.
4.3 Comparison with the SRR chart
According to [21], the 4-out-of-5 chart has better performance in most cases, so, we choose
this chart as a benchmark in this comparison. Because the SRR chart is two-sided, in
order to make a fair comparison between the two charts, we have computed the OC ARLs
corresponding to two-sided RES CV chart. The smoothing parameter 𝜆 is set to 0.05 and
the ARL0 of each of the one-sided chart when used alone is approximately 720 such that
the combined chart produces an ARL0 of 370. Such chart is designed to protect in balance
against both increasing and decreasing shifts in CV. For comparison purposes, the value of
𝛾0 and 𝜏 considered here are the same as those considered in [21]. The simulation results
are tabulated in Table 5.
[Insert Table 5 about here]
From Table 5 we can see that, for most of the OC cases, the ARL1 value of the RES
chart is usually much smaller than that of the SRR chart except in a few cases where 𝜏
is less than 0.6. For instance, concerning the increasing case, when 𝑛 = 10, 𝛾0 = 0.1 and
𝜏 = 1.1, the ARL1 value of RES chart is 34.7, which is about 60% less than 87.7 of SYN
chart. Concerning the decreasing case, when 𝑛 = 10, 𝛾0 = 0.1 and 𝜏 = 0.6, the ARL1 value
of RES chart is 6.6 and 5.0 of SYN chart.
4.4 Comparison with the SSGR chart
We will compare the upward and downward RES charts with the SSGR charts. In this study,
ARL0=370.4, 𝑛 = 5, 10, 𝛾0 = 0.05, 0.10, 0.15, 0.20 and 𝜏 ∗=0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95,
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),
15
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
16
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]
17
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.
18
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.
21
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.
22
Table 1: 𝐾+ and 𝐾− values of the RES chart when ARL0=370.
𝜆
0.05 0.1 0.2 0.3 0.5
n 𝛾0 𝐾+ 𝐾− 𝐾+ 𝐾− 𝐾+ 𝐾− 𝐾+ 𝐾− 𝐾+ 𝐾−
5 0.05 2.965 1.411 3.742 1.685 4.631 1.872 5.220 1.921 6.021 1.882
0.10 3.047 1.292 3.821 1.585 4.714 1.794 5.315 1.853 6.115 1.823
0.15 3.174 1.112 3.951 1.438 4.848 1.675 5.460 1.743 6.285 1.734
0.20 3.341 0.868 4.114 1.241 5.050 1.511 5.668 1.598 6.548 1.613
7 0.05 2.895 1.598 3.595 1.892 4.396 2.106 4.920 2.175 5.625 2.173
0.10 2.960 1.498 3.661 1.809 4.465 2.038 4.995 2.115 5.705 2.125
0.15 3.075 1.342 3.768 1.681 4.579 1.934 5.113 2.021 5.853 2.041
0.20 3.236 1.135 3.926 1.506 4.739 1.785 5.293 1.891 6.049 1.928
10 0.05 2.828 1.755 3.477 2.058 4.199 2.305 4.665 2.395 5.305 2.435
0.10 2.875 1.675 3.525 1.995 4.256 2.248 4.741 2.348 5.356 2.388
0.15 2.978 1.536 3.623 1.883 4.348 2.156 4.839 2.264 5.475 2.315
0.20 3.121 1.362 3.741 1.741 4.479 2.028 4.980 2.148 5.645 2.216
15 0.05 2.752 1.881 3.351 2.214 4.021 2.483 4.440 2.599 5.010 2.673
0.10 2.802 1.825 3.400 2.153 4.061 2.440 4.485 2.561 5.052 2.639
0.15 2.891 1.705 3.479 2.069 4.146 2.364 4.573 2.489 5.152 2.576
0.20 3.000 1.566 3.584 1.949 4.251 2.260 4.688 2.396 5.269 2.489
23
Table 2: Optimal couples (𝜆∗, 𝐾∗) and ARL1 of the RES, MOSE and OSE charts(ARL0=370).
𝜏 ∗ 𝛾0 = 0.05 𝛾0 = 0.1 𝛾0 = 0.15 𝛾0 = 0.2
n=5
0.50 (0.30,1.921) (0.31,1.853 ) (0.28,1.734 ) (0.05,0.869 )
(4.1,4.5,4.8 ) (4.1,4.5,4.8 ) (4.1,4.4,4.8 ) (3.7,4.4,4.8 )
0.65 (0.13,1.766) (0.09,1.547 ) (0.05,1.112 ) (0.05,0.869 )
(7.6,7.9,8.7 ) (7.5,7.9,8.8 ) (6.9,7.9,8.8 ) (6,7.8,8.8 )
0.80 (0.05,1.411) (0.05,1.293 ) (0.05,1.112 ) (0.05,0.869 )
(18.3,17.8,20.6 ) (17.6,17.6,20.6 ) (16.7,17.2,20.7) (15.0,16.3,20.9 )
0.90 (0.05,1.411) (0.05,1.291) (0.05,1.112) (0.05,0.869 )
(52.6,43.9,52.8 ) (51.3,43.4,54.1) (49.9,42.5,54.3 ) (47.5,41.1,55.4 )
1.10 (0.05,2.965) (0.05,3.047) (0.05,3.174) (0.05,3.341 )
(45.8,44.1,51.2 ) (46.5,44.5,51.5 ) (47.7,46.3,51.9 ) (49.4,48.5,52.4 )
1.25 (0.05,2.965) (0.05,3.047 ) (0.05 ,3.174) (0.05,3.341 )
(13.4,13.5,15 ) (13.6,13.7,15.2 ) (14,14.2,15.4) (14.6,14.8,15.9 )
1.50 (0.135,4.113) (0.12,4.043 ) (0.13,4.258 ) (0.11,4.235 )
(5.3,5.3,5.7 ) (5.4,5.4,5.8 ) (5.5,5.6,5.9 ) (5.7,5.8,6.1 )
2.00 (0.29,5.164) (0.30,5.315 ) (0.24,5.112 ) (0.27,5.518 )
(2.3,2.3,2.4 ) (2.3,2.3,2.4 ) (2.4,2.4,2.5 ) (2.4,2.5,2.6 )
n=10
0.50 (0.64,2.402) (0.57,2.377) (0.45,2.310) (0.42,2.207)
(2.3,2.4,2.5) (2.3,2.4,2.5) (2.3,2.4,2.5) (2.3,2.4,2.5)
0.65 (0.29,2.395) (0.28,2.332) (0.27,2.241) (0.26,2.114)
(4.1,4.3,4.6) (4.1,4.3,4.6) (4.1,4.3,4.7) (4.1,4.4,4.7)
0.80 (0.10,2.061) (0.08,1.897) (0.05,1.541) (0.05,1.362)
(10.4,10.1,11.3) (10.2,10.1,11.4) (10,10.1,11.5) (9.5,10.0,11.6)
0.90 (0.05,1.757) (0.05,1.674) (0.05,1.541) (0.05,1.362 )
(29.7,25.6,30.6) (29.3,25.6,30.9) (28.6,25.2,31) (28.0,24.8,31.7)
1.10 (0.05,2.832,) (0.05,2.890) (0.05,2.99) (0.05,3.125 )
(27.6,26.0,30.2) (27.9,26.5,30.4) (28.7,27.5,31.0) (29.6,28.7,31.4)
1.25 (0.08,3.259) (0.09,3.434) (0.12,3.806) (0.10,3.749)
(7.8,7.7,8.4) (7.9,7.8,8.5) (8.1,8.0,8.7) (8.4,8.4,9)
1.50 (0.27,4.544) (0.28,4.656) (0.26,4.659) (0.25,4.752)
(3.0,3.0,3.2) (3.0,3.1,3.2) (3.1,3.2,3.3) (3.2,3.3,3.4)
2.00 (0.61,5.534) (0.61,5.606) (0.41,5.228) (0.53,5.716)
(1.4,1.4,1.4) (1.4,1.4,1.4) (1.4,,1.5,1.5) (1.5,1.5,1.5)
24
Table 3: ARL1 values of the RES, MOSE and OSE charts (ARL0=370, 𝜆 = 0.05).
n=5 n=7 n=10 n=15
𝛾0 𝜏 RES MOSE OSE RES MOSE OSE RES MOSE OSE RES MOSE OSE
0.1 1.00 370 370 370 370 370 370 370 370 370 370 370 370
1.10 46.6 44.8 51.2 36.2 34.5 39.8 27.8 26.7 30.3 20.6 20.0 22.7
1.15 26.9 26.7 30.2 21.0 20.4 23.3 15.9 15.9 17.8 11.9 12.0 13.5
1.20 18.3 18.5 20.8 14.1 14.3 16.0 10.8 11.1 12.4 8.1 8.5 9.6
1.25 13.5 13.9 15.6 10.5 10.8 12.1 8.1 8.5 9.5 6.1 6.6 7.3
1.50 5.6 6.1 6.7 4.3 4.8 5.3 3.4 3.9 4.3 2.6 3.1 3.4
2.00 2.5 2.8 3.1 2.0 2.3 2.5 1.7 1.9 2.1 1.4 1.6 1.7
0.2 1.10 49.5 48.1 52.3 38.5 37.2 40.9 29.6 28.8 31.7 22.2 21.7 23.7
1.15 29.0 29.1 31.2 22.4 22.4 24.2 17.1 17.3 18.7 12.7 13.1 14.3
1.20 19.5 20.3 21.7 15.1 15.6 16.9 11.6 12.1 13.1 8.7 9.3 10.1
1.25 14.5 15.3 16.3 11.2 11.9 12.8 8.7 9.3 10.0 6.5 7.2 7.8
1.50 5.9 6.7 7.1 4.7 5.3 5.7 3.7 4.2 4.5 2.8 3.3 3.6
2.00 2.7 3.1 3.3 2.2 2.5 2.7 1.8 2.1 2.2 1.4 1.7 1.8
0 5 10 15 20
−0.2
0.00.2
0.40.6
0.8
t
RES t
UCL=0.464
Figure 1: RES CV chart for real data of Example 1.
25
Table 4: ARL1 values of the RES and SYN charts (ARL0=370).
n=5 n=10
𝜏 ∗ = 1.25 𝜏 ∗ = 1.5 𝜏 ∗ = 2.0 𝜏 ∗ = 1.25 𝜏 ∗ = 1.5 𝜏 ∗ = 2.0
𝛾0 𝜏 RES SYN RES SYN RES SYN RES SYN RES SYN RES SYN
0.05 1.00 370 370 370 370 370 370 370 370 370 370 370 370
1.05 103.5 220.2 127.4 228.9 152.6 239.2 75.7 195.6 107.8 208.2 139.5 218.8
1.10 45.6 118.9 58.1 128.3 74.1 141.1 29.2 84.4 42.6 95.9 61.7 106.7
1.15 26.8 65.0 32.2 71.5 41.7 82.0 16.1 38.0 21.3 44.5 31.9 51.6
1.20 17.9 37.9 20.6 41.7 26.2 49.1 10.6 19.5 12.7 22.6 18.5 26.8
1.25 13.3 24.0 14.6 25.9 17.6 30.8 7.8 11.5 8.6 12.8 11.9 15.3
1.30 10.4 16.5 11 17.2 12.8 20.4 6.1 7.6 6.4 8.1 8.2 9.5
1.50 5.5 6.3 5.3 5.8 5.6 6.3 3.2 3.0 3 2.7 3.2 2.9
2.00 2.5 2.2 2.3 2.1 2.3 2.0 1.6 1.3 1.4 1.3 1.4 1.2
0.10 1.05 105.1 220.5 126.1 229.3 155.1 239.7 78.8 196.2 111.0 209.4 139.8 219.7
1.10 46.6 119.4 56.5 129.0 75.9 141.8 30.4 85.3 43.9 97.1 62.4 107.8
1.15 27.1 65.5 31.6 72.2 42.7 82.7 16.6 38.6 22.1 45.3 32.4 52.4
1.20 18.4 38.3 20.4 42.2 26.7 49.6 10.9 19.8 13.2 23.1 18.8 27.3
1.25 13.6 24.3 14.6 26.2 18.2 31.2 7.9 11.7 8.9 13.1 12.0 15.6
1.30 10.7 16.7 11 17.4 13.3 20.7 6.1 7.8 6.5 8.3 8.4 9.8
1.50 5.6 6.4 5.4 5.9 5.7 6.4 3.2 3.0 3 2.8 3.3 3.0
2.00 2.5 2.3 2.4 2.1 2.3 2.0 1.6 1.3 1.4 1.3 1.4 1.2
26
Table 5: ARL1 values of the RES and SRR charts (ARL0=370, 𝜆 = 0.05).
𝛾0 = 0.05 𝛾0 = 0.1 𝛾0 = 0.15 𝛾0 = 0.2n 𝜏 RES SRR RES SRR RES SRR RES SRR5 0.5 8.4 6.2 8.2 6.2 7.9 6.3 7.5 6.3
0.6 10.2 11.8 10.0 11.8 9.7 12.0 9.1 12.20.7 13.7 28.4 13.4 28.6 13.0 29.0 12.3 29.50.8 22.3 80.0 21.9 80.6 21.4 81.6 20.5 83.00.9 60.8 236.5 60.3 237.5 59.8 239.3 58.8 241.61.1 61.5 144.5 62.7 145.3 64.4 146.6 67.2 148.41.2 22.7 54.4 23.2 54.9 23.9 55.8 25.3 57.01.5 7.0 11.8 7.1 12.0 7.4 12.2 7.8 12.62.0 3.1 5.6 3.2 5.7 3.3 5.7 3.5 5.9
10 0.5 5.6 4.1 5.5 4.1 5.4 4.1 5.2 4.10.6 6.7 5.0 6.6 5.0 6.5 5.0 6.2 5.10.7 8.7 8.9 8.6 9.0 8.4 9.1 8.1 9.30.8 13.4 26.1 13.3 26.4 13.1 26.9 12.7 27.60.9 33.2 119.1 33.1 120.3 32.9 122.1 32.5 124.61.1 34.0 86.9 34.7 87.7 35.6 89.2 37.3 91.21.2 13.1 25.6 13.4 26.0 13.9 26.6 14.6 27.51.5 4.4 6.2 4.4 6.3 4.6 6.4 4.8 6.62.0 2.1 4.2 2.1 4.3 2.2 4.3 2.3 4.3
0 5 10 15 20 25 30
−0.2
0.00.2
0.40.6
0.81.0
1.2
t
RES t
UCL=0.274
Figure 2: RES CV chart for real data of Example 2.
27
Table 6: ARL1 values of the RES and SSGR charts (ARL0=370).
𝛾0 = 0.05 𝛾0 = 0.1 𝛾0 = 0.15 𝛾0 = 0.2𝜏 ∗ RES SSGR RES SSGR RES SSGR RES SSGR
n=5 0.4 2.9 1.8 2.9 1.8 2.9 1.9 2.9 1.90.5 4.1 4.7 4.1 4.7 4.1 4.8 3.7 4.90.6 6.0 14.8 6.0 14.9 5.7 15.2 4.9 15.50.7 9.8 49.2 9.4 49.7 8.6 50.4 7.6 51.60.8 18.4 156.6 17.6 157.7 16.7 159.5 15.0 162.00.9 52.6 370.2 51.3 370.9 49.9 371.9 47.5 373.40.95 118.4 419.4 118.5 419.4 118.5 419.4 117.9 419.31.1 45.8 84.5 46.5 85.2 47.7 86.4 49.4 88.01.2 18.0 23.6 18.3 23.9 18.8 24.4 19.7 25.11.25 13.4 15.1 13.6 15.3 14.0 15.6 14.6 16.01.5 5.3 4.0 5.4 4.1 5.5 4.1 5.7 4.32.0 2.3 1.6 2.3 1.6 2.4 1.7 2.4 1.7
n=10 0.4 1.5 1.0 1.5 1.0 1.5 1.0 1.5 1.00.5 2.3 1.2 2.3 1.2 2.3 1.2 2.3 1.20.6 3.2 2.2 3.2 2.3 3.2 2.3 3.2 2.40.7 5.3 7.1 5.3 7.2 5.3 7.3 5.3 7.60.8 10.4 33.1 10.2 33.6 10.0 34.4 9.5 35.50.9 29.7 186.9 29.3 188.5 28.6 191.0 28.0 194.40.95 76.0 357.4 76.4 358.0 75.2 358.9 76.0 360.21.1 27.6 51.3 27.9 52.0 28.7 53.1 29.6 54.81.2 11.5 11.6 10.8 11.8 11.1 12.2 11.1 12.71.25 7.8 7.1 7.9 7.3 8.1 7.5 8.4 7.81.5 3.0 2.0 3.0 2.1 3.1 2.1 3.2 2.22.0 1.4 1.1 1.4 1.1 1.4 1.2 1.5 1.2
28