# Monitoring the Multivariate Coefficient of Variation using Run Rules Type
Control Charts

P. H. Tran  Institute of Artificial Intelligence and Data Science, Dong A
University, Da Nang, Vietnam  HEC - Management School, University of Liège,
Liège 4000, Belgium, *Corresponding author. Email<EMAIL_ADDRESS>A. C.
Rakitzis  Department of Statistics and Actuarial - Financial Mathematics,
Karlovasi, Samos, Greece  H. D. Nguyen  Institute of Artificial Intelligence
and Data Science, Dong A University, Da Nang, Vietnam  Q. T. Nguyen  Institute
of Artificial Intelligence and Data Science, Dong A University, Da Nang,
Vietnam  K. P. Tran  Ecole Nationale Supérieure des Arts et Industries
Textiles, GEMTEX Laboratory, BP 30329 59056 Roubaix Cedex 1, France  C.
Heuchenne  HEC - Management School, University of Liège, Liège 4000, Belgium,
*Corresponding author. Email<EMAIL_ADDRESS>
###### Abstract

In practice, there are processes where the in-control mean and standard
deviation of a quality characteristic is not stable. In such cases, the
coefficient of variation (CV) is a more appropriate measure for assessing
process stability. In this paper, we consider the statistical design of Run
Rules based control charts for monitoring the CV of multivariate data. A
Markov chain approach is used to evaluate the statistical performance of the
proposed charts. The computational results show that the Run Rules based
charts outperform the standard Shewhart control chart significantly. Moreover,
by choosing an appropriate scheme, the Run Rules based charts perform better
than the Rum Sum control chart for monitoring the multivariate CV. An example
in a spring manufacturing process is given to illustrate the implementation of
the proposed charts.

Keyword Run rules; Multivariate Coefficient of Variation; Control chart;
Markov chain

##  1 Introduction

Representing the ratio of the standard deviation to the mean, the coefficient
of variation (CV) is a useful measure of relative dispersion of a random
variable. It has the meaning that the higher the CV, the greater the level of
dispersion around the mean. The CV is widely used in a large number of areas
such as clinical chemistry, materials engineering, agricultural experiments
and medicine, see, for example Castagliola et al. 1. In laboratory medicine
for comparing the reproducibility of assay techniques, a lower CV leads to a
better analytical precision. In many manufacturing processes, keeping the CV
in-control means ensuring the product quality. Monitoring the CV is then an
important task red that had attracted the interest of many authors and the
literature of monitoring the CV is abundant. Kang et al. 2 are the first to
use the Shewhart chart to monitor and detect changes in the CV. Their work is
developed with a number of advantage-type control charts or adaptive
strategies such as the synthetic control chart (Calzada and Scariano 3), EWMA
control chart (Castagliola et al. 1), CUSUM control chart (Tran and Tran 4),
Run Rules based control chart (Castagliola et al. 5), variable sampling
interval (VSI) control chart (Castagliola et al. 6), side-sensitive group runs
(SSGR) control chart (You et al. 7), and Run Sum control chart (Teoh et al.
8). The EWMA chart designed by Castagliola et al. 1 was further improved by
Zhang et al. 9 (based on a modified EWMA charting statistic), Yeong et al. 10
(which integrates the VSI feature into the EWMA chart) and Zhang et al. 11 (by
applying the resetting technique in Shu and Jiang 12).

The aforementioned studies focus on the univariate CV. In fact, there are
various situations where the multivariate coefficient of variation (MCV) is
the main concern. For example, in biometry and genetics, it is quite often to
measure multiple characteristics on individuals from several populations and
the problem is to assess the relative variability of each population. The
single calculation of the univariate CV of each characteristic is obviously
insufficient because it does not consider the correlation between these
features, see Albert and Zhang 13. However, a literature search reveals that
not much attention has been paid to the CV for multivariate data despite its
potential importance. Very recently, Yeong et al. 14 have suggested a Shewhart
control chart for monitoring the MCV (denoted as Shewhart$-$MCV chart in this
paper). It is well-known that Shewhart-type charts are only efficient in
detecting large and sudden process shifts and it is not the case for moderate
or small shifts. To enhance the performance of the Shewhart chart monitoring
MCV in detecting small shifts, Lim et al. 15 suggested to use a Rum Sum
control chart. In this paper, we propose to apply supplementary Run Rules with
the Shewhart MCV control chart.

In the literature, several Run Rules charts have been studied and suggested by
a number of authors. Champ and Woodall 16 were the first to obtain the exact
formula evaluating the run length distribution and then calculated its average
value, i.e. the average run length (ARL) which is the expected number of
points plotted on chart until an out-of-control signal is given. They have
also showed a disadvantage of the use of supplementary runs rules in the
Shewhart chart as it reduces the in-control $ARL$ (denoted as $ARL_{0}$). In
order to overcome this downside, alternative methods for the statistical
design of control charts with Run Rules have been proposed: Klein 17
considered the 2-of-2 and 2-of-3 schemes while Khoo 18 considered the 2-of-4,
3-of-3 and 3-of-4 schemes. A modified version of $r$-out-of-$m$ control chart
was studied in Antzoulakos and Rakitzis 19. An overview of control charts with
supplementary runs rules until 2006 is presented in Koutras et al. 20.
Recently, the Run Rules control chart are applied to monitor the coefficient
of variation, the ratio of two normal variables as well as other non-normal
processes; see, for example Acosta and Pignatiello 21; Amdouni et al. 22;
Castagliola et al. 5; Faraz et al. 23, Tran et al. 24. Recently, Chew et al.
25 studied upper and lower one-sided run rules control charts, based on the
2-of-3 and the 3-of-4 runs rules, for the MCV. However, in this work we
consider additionally the 4-of-5 runs rule, a rule that it is well-know for
its efficiency (see, for example, Tran 26). Therefore, in this study we
provide additional numerical comparisons between the most frequently used one-
sided Shewhart-MCV charts with or without run rules. Numerical simulations
shows that our proposed charts are efficient in detecting the process shifts.
Moreover, the implementation of Run Rules control charts is also less complex
compared to the Run Sum chart for MCV.

The paper is organized as follows. In Section 2, we provide a brief review of
distribution of the sample MCV as discovered by Yeong et al. 14. In Section 3,
we present the design and implementation of the Run Rules control charts for
monitoring the MCV. Section 4 is devoted to assessing the performance of the
proposed charts. An example is illustrated in Section 5 and some concluding
remarks are given in Section 6.

##  2 A brief review on the distribution of the sample multivariate
coefficient of variation

We present in this Section a brief review of the distribution of the sample
multivariate coefficient of variation (abbr. MCV). From the literature, there
are different point of views about the definitions of the MCV. A formal
definition for the multivariate coefficient of variation was firstly proposed
by Reyment 27. Valen 28 has pointed out a few shortcomings of this definition
and suggested a more appropriate definition. Another definition of the MCV was
given by Nikulin and Voinov 29 based on the Mahalanobis distance. Recently,
Albert and Zhang 13 proposed a novel approach of defining the MCV to overcome
the limitations of previous definitions. We use in this paper the definition
of the MCV suggested by Nikulin and Voinov 29 which is considered as a natural
generalization for the CV. This definition was also adopted by Yeong et al. 14
to monitor the MCV. Let $\mathbf{X}$ denote a random vector from a $p$-variate
normal distribution with mean vector $\boldsymbol{\mu}$ and covariance matrix
$\boldsymbol{\Sigma}$. The MCV is defined as

$\gamma=(\boldsymbol{\mu}^{T}\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu})^{-\frac{1}{2}}.$
(1)

Suppose that a random sample of size $n$, say
$\mathbf{X}_{1},\mathbf{X}_{2},\ldots,\mathbf{X}_{n}$, is taken from this
distribution, i.e., $\mathbf{X}_{i}=(x_{i,1},x_{i,2},\ldots,x_{i,p})\sim
N(\boldsymbol{\mu},\boldsymbol{\Sigma})$, $i=1,\ldots,n$. Let
$\bar{\mathbf{X}}$ and $\mathbf{S}$ be the sample mean and the sample
covariance matrix of $\mathbf{X}_{1},\mathbf{X}_{2},\ldots,\mathbf{X}_{n}$,
i.e.,

$\bar{\mathbf{X}}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{X}_{i},$

and

$\mathbf{S}=\frac{1}{n-1}\sum_{i=1}^{n}(\mathbf{X}_{i}-\bar{\mathbf{X}})(\mathbf{X}_{i}-\bar{\mathbf{X}})^{T}.$

The sample multivariate coefficient of variation $\hat{\gamma}$ is then
defined as

$\hat{\gamma}_{i}=(\bar{\mathbf{X}}^{T}\mathbf{S}^{-1}\bar{\mathbf{X}})^{-\frac{1}{2}}.$
(2)

The $c.d.f$ (cumulative distribution function) and the inverse $c.d.f$ of
$\hat{\gamma}$ are given in Yeong et al. 14 as

$F_{\hat{\gamma}}(x|n,p,\delta)=1-F_{F}\left(\frac{n(n-p)}{(n-1)px^{2}}|p,n-p,\delta\right),$
(3)

and

$F^{-1}_{\hat{\gamma}}(\alpha|n,p,\delta)=\sqrt{\frac{n(n-p)}{(n-1)p}\left(\frac{1}{F^{-1}_{F}(1-\alpha|p,n-p,\delta)}\right)},$
(4)

where $F_{F}(.|p,n-p,\delta)$ and $F^{-1}_{F}(.|p,n-p,\delta)$ are the non-
central $F$ distribution and the inverse of the non-central $F$ distribution
with $p$ and $n-p$ degrees of freedom, respectively, and the non-centrality
parameter is
$\delta=n\boldsymbol{\mu}^{T}\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}=\frac{n}{\gamma^{2}}$.

##  3 Implementation of the RRr,s-MCV control charts

Similar to the one-sided Run Rules control charts presented in Tran et al. 24
and Castagliola et al. 5, we suggest the definition of two one-sided Run Rules
control charts for monitoring the MCV as follows:

  * •

A lower-sided $r$-out-of-$s$ Run Rules control chart (denoted as
RR${}^{-}_{r,s}-$ MCV) to detect a decrease in $\hat{\gamma}_{i}$ with a lower
control limit $LCL^{-}$ and a corresponding upper control limit
$UCL^{-}=+\infty$.

  * •

An upper-sided $r$-out-of-$s$ Run Rules control chart (denoted as
RR${}^{+}_{r,s}-$ MCV) to detect an increase in $\hat{\gamma}_{i}$ with an
upper control limit $UCL^{+}$ and a corresponding lower control limit
$LCL^{+}=0$.

Given the value of the control limits for each chart, an out-of-control signal
is given at time $i$ if $r$-out-of-$s$ consecutive $\hat{\gamma}_{i}$ values
are plotted outside the control interval, i.e. $\hat{\gamma}_{i}<LCL^{-}$ in
the lower-sided chart and $\hat{\gamma}_{i}>UCL^{+}$ in the upper-sided chart.
The control chart designed above is called $pure$ Run Rules type chart.
Compared to the $composite$ Run Rules type charts which require both control
and warning limits, these $pure$ type charts are more simple to implement and
interpret, see Klein 17. In this study, we only consider the 2-out-of-3,
3-out-of-4 and 4-out-of-5 Run Rules charts. More complex Run Rules schemes
with larger values of $(r,s)$ are possible to design in a similar manner.
However, their efficiency should be taken into consideration in terms of the
increased complexity of implementation.

The performance of the proposed one-sided RR${}_{r,s}-$MCV control charts is
measured by the out-of-control $ARL$, denoted as $ARL_{1}$. We utilize a
Markov chain method, similar to the one initially proposed by Brook and Evans
30, to calculate the $ARL_{1}$ value. Further details on this method can be
found in Fu et al. 31, Castagliola et al. 5 and Li et al. 32. Let us now
suppose that the occurrence of an unexpected condition shifts the in-control
MCV value $\gamma_{0}$ to the out-of-control value
$\gamma_{1}=\tau\times\gamma_{0}$, where $\tau>0$ is the shift size. Values of
$\tau\in(0,1)$ correspond to a decrease of the $\gamma_{0}$, while values of
$\tau>1$ correspond to an increase of the in-control MCV. It is worth
mentioning that a decrease (resp. increase) in $\gamma_{0}$ is related to
process improvement (resp. deterioration). The probability $p$ of the event
that a sample falls into an in-control interval is equal to:

  * •

for the RR${}_{r,s}^{-}-$MCV chart:

$p=P(\hat{\gamma}_{i}\geq
LCL^{-})=1-F_{\hat{\gamma}}(LCL^{-}|n,p,\delta_{1}),$ (5)

  * •

for the RR${}_{r,s}^{+}-$MCV chart:

$p=P(\hat{\gamma}_{i}\leq UCL^{+})=F_{\hat{\gamma}}(UCL^{+}|n,p,\delta_{1}),$
(6)

where $\delta_{1}=\frac{n}{{(\tau\gamma_{0})}^{2}}$.

The Transition Probability Matrix (TPM) matrix $\mathbf{P}$ of the embedded
Markov chain for the two RR${}_{2,3}-$MCV control charts is

$\mathbf{P}=\left(\begin{array}[]{cc}\mathbf{Q}&\mathbf{r}\\\ &\\\
\mathbf{0}^{T}&1\end{array}\right)=\left(\begin{array}[]{ccc|c}0&0&p&1-p\\\
p&0&0&1-p\\\ 0&1-p&p&0\\\ \hline\cr 0&0&0&1\end{array}\right),$ (7)

where $\mathbf{Q}$ is the $(3,3)$ matrix of transient probabilities,
$\mathbf{r}$ is the $(3,1)$ vector satisfied
$\mathbf{r}=\mathbf{1}-\mathbf{Q}\mathbf{1}$ with $\mathbf{1}=(1,1,1)^{T}$ and
$\mathbf{0}=(0,0,0)^{T}$. The corresponding $(3,1)$ vector $\mathbf{q}$ of
initial probabilities associated with the transient states is equal to
$\mathbf{q}=(0,0,1)^{T}$ , i.e. the third state is the initial state.

Extended to Run Rules charts with larger $(r,s)$ values, the matrix
$\mathbf{Q}_{(7\times 7)}$ of transient probabilities for the two
RR${}_{3,4}-$MCV control charts is given by

$\mathbf{Q}=\left(\begin{array}[]{ccccccc}0&0&p&0&0&0&0\\\ 0&0&0&0&p&0&0\\\
0&0&0&0&0&1-p&p\\\ p&0&0&0&0&0&0\\\ 0&1-p&p&0&0&0&0\\\ 0&0&0&1-p&p&0&0\\\
0&0&0&0&0&1-p&p\\\ \end{array}\right).$ (8)

In this case, the seventh state in the vector $\mathbf{q}=(0,0,0,0,0,0,1)^{T}$
is the initial state. The (15,15) matrix $\mathbf{Q}$ of transient
probabilities for the two RR4,5-MCV control charts is not presented here due
to space constraints, but it can be seen in, for example, Tran et al. 24. Once
matrix $\mathbf{Q}$ and vector $\mathbf{q}$ have been determined, the $ARL$
and $SDRL$ (standard deviation of run length) are given by

$\displaystyle ARL$ $\displaystyle=$ $\displaystyle\nu_{1},$ (9)
$\displaystyle SDRL$ $\displaystyle=$ $\displaystyle\sqrt{\mu_{2}},$ (10)

with

$\displaystyle\nu_{1}$ $\displaystyle=$
$\displaystyle\mathbf{q}^{T}(\mathbf{I}-\mathbf{Q})^{-1}\mathbf{1},$ (11)
$\displaystyle\nu_{2}$ $\displaystyle=$ $\displaystyle
2\mathbf{q}^{T}(\mathbf{I}-\mathbf{Q})^{-2}\mathbf{Q}\mathbf{1},$ (12)
$\displaystyle\mu_{2}$ $\displaystyle=$
$\displaystyle\nu_{2}-\nu_{1}^{2}+\nu_{1}.$ (13)

A control chart is considered to be better than its competitors if it gives
smaller value of the $ARL_{1}$ while the $ARL_{0}$ is the same. Therefore, the
control limit of the RR${}_{r,s}-$MCV control charts should be found out as a
solution of the following equations:

  * •

for the RR${}_{r,s}^{-}-$MCV chart:

$ARL(LCL^{-},n,p,\gamma_{0},\tau=1)=ARL_{0},$ (14)

  * •

for the RR${}_{r,s}^{+}-$MCV chart:

$ARL(UCL^{+},n,p,\gamma_{0},\tau=1)=ARL_{0},$ (15)

where $ARL_{0}$ is the predetermined in-control $ARL$ value.

##  4 The Performance of the RRr,s-MCV control charts

In this Section, we investigate the performance of the RR ${}_{r,s}-$ MCV
control charts. The desired in-control $ARL$ value, say $ARL_{0}$ is set at
370.4, for all the considered IC cases. The control limit $LCL^{-}$ of lower-
sided chart and $UCL^{+}$ of upper-sided chart, which are the solution of
equations (14) and (15), are shown in Table 1 for different combinations of
$n\in\\{5,10,15\\}$, $p\in\\{2,3,4\\}$ and $\gamma_{0}\in\\{0.1$, $0.2$,
$0.3$, $0.4$, $0.5\\}$.

| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
---|---|---|---|---|---
$\gamma_{0}$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $p=2$
0.10 | (0.027, 0.146) | (0.053, 0.135) | (0.063, 0.129) |  | (0.039, 0.124) | (0.063, 0.121) | (0.071, 0.119) |  | (0.048, 0.111) | (0.070, 0.113) | (0.077, 0.112)
0.20 | (0.053, 0.296) | (0.104, 0.274) | (0.125, 0.261) |  | (0.077, 0.251) | (0.125, 0.245) | (0.142, 0.239) |  | (0.095, 0.223) | (0.139, 0.227) | (0.154, 0.224)
0.30 | (0.079, 0.457) | (0.155, 0.419) | (0.185, 0.399) |  | (0.114, 0.382) | (0.185, 0.372) | (0.211, 0.362) |  | (0.142, 0.337) | (0.206, 0.343) | (0.229, 0.339)
0.40 | (0.104, 0.633) | (0.203, 0.574) | (0.242, 0.544) |  | (0.150, 0.520) | (0.243, 0.504) | (0.277, 0.490) |  | (0.186, 0.455) | (0.272, 0.462) | (0.302, 0.456)
0.50 | (0.127, 0.831) | (0.248, 0.744) | (0.297, 0.700) |  | (0.184, 0.667) | (0.299, 0.643) | (0.342, 0.623) |  | (0.229, 0.576) | (0.335, 0.584) | (0.373, 0.577)
| $p=3$
0.10 | (0.014, 0.128) | (0.047, 0.129) | (0.059, 0.125) |  | (0.024, 0.106) | (0.057, 0.115) | (0.067, 0.115) |  | (0.031, 0.093) | (0.063, 0.106) | (0.073, 0.108)
0.20 | (0.028, 0.259) | (0.092, 0.260) | (0.117, 0.253) |  | (0.047, 0.213) | (0.112, 0.231) | (0.134, 0.231) |  | (0.062, 0.185) | (0.126, 0.213) | (0.146, 0.216)
0.30 | (0.041, 0.395) | (0.137, 0.397) | (0.173, 0.385) |  | (0.069, 0.322) | (0.166, 0.350) | (0.199, 0.349) |  | (0.092, 0.279) | (0.187, 0.321) | (0.217, 0.326)
0.40 | (0.054, 0.539) | (0.179, 0.541) | (0.227, 0.524) |  | (0.090, 0.434) | (0.218, 0.472) | (0.262, 0.470) |  | (0.121, 0.372) | (0.246, 0.431) | (0.286, 0.438)
0.50 | (0.065, 0.692) | (0.219, 0.695) | (0.278, 0.670) |  | (0.110, 0.547) | (0.267, 0.599) | (0.322, 0.596) |  | (0.148, 0.465) | (0.303, 0.543) | (0.352, 0.551)
| $p=4$
0.10 | (0.002, 0.104) | (0.040, 0.122) | (0.055, 0.121) |  | (0.007, 0.081) | (0.050, 0.108) | (0.063, 0.111) |  | (0.011, 0.067) | (0.057, 0.099) | (0.069, 0.104)
0.20 | (0.005, 0.208) | (0.080, 0.246) | (0.109, 0.245) |  | (0.013, 0.162) | (0.099, 0.217) | (0.126, 0.222) |  | (0.023, 0.133) | (0.113, 0.199) | (0.138, 0.208)
0.30 | (0.007, 0.314) | (0.118, 0.373) | (0.161, 0.371) |  | (0.019, 0.242) | (0.146, 0.327) | (0.187, 0.335) |  | (0.033, 0.199) | (0.167, 0.299) | (0.204, 0.313)
0.40 | (0.009, 0.421) | (0.154, 0.506) | (0.211, 0.503) |  | (0.025, 0.321) | (0.192, 0.440) | (0.245, 0.451) |  | (0.044, 0.262) | (0.219, 0.399) | (0.269, 0.418)
0.50 | (0.011, 0.529) | (0.188, 0.645) | (0.259, 0.641) |  | (0.031, 0.399) | (0.234, 0.553) | (0.301, 0.569) |  | (0.053, 0.324) | (0.268, 0.500) | (0.331, 0.525)

Table 1:  The control limits of RRr,s-MCV control charts, $LCL^{-}$ on left
side and $UCL^{+}$ on right side, for different values of $n,p$ and
$\gamma_{0}$.

It can be seen from Table 1 that given $n$ and $p$ the values of $LCL^{-}$ and
$UCL^{+}$ depend (in general) on $\gamma_{0}$ . In particular, small values of
$\gamma_{0}$ lead to small values of $LCL^{-}$ and $UCL^{+}$, as well. For
example, in RR${}_{2,3}-$MCV chart with $n=5$ and $p=2$, we have
$LCL^{-}=0.027$ and $UCL^{+}=0.146$ when $\gamma_{0}=0.10$, while
$LCL^{-}=0.127$ and $UCL^{+}=0.831$ when $\gamma_{0}=0.50$. Also, given $n$
and $\gamma_{0}$, the values of $LCL^{-}$ and $UCL^{+}$ depend on $p$. The
larger the value of $p$, the smaller the value of $LCL^{-}$ and $UCL^{+}$. For
example, in RR${}_{3,4}-$MCV chart with $n=10$ and $\gamma_{0}=0.2$, we have
$LCL^{-}=0.125$ and $UCL^{+}=0.245$ when $p=2$, while $LCL^{-}=0.099$ and
$UCL^{+}=0.217$ when $p=4$.

Using the control limits in Table 1, the values of $ARL_{1}$ and $SDRL_{1}$
(out-of-control) of the RR${}_{r,s}-$MCV control charts are provided in Tables
2-4. We set different combinations of $\tau\in\\{0.50$,
$0.75,0.9,1.10,1.25,1.5\\}$, $p\in\\{2,3,4\\}$, $n\in\\{5,10,15\\}$ and
$\gamma_{0}\in\\{0.1,0.2,0.3,0.4,0.5\\}$. Some simple conclusions can be drawn
from these tables as follows:

| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
---|---|---|---|---|---
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
0.50 | (14.2, 12.6) | (2.8, 1.3) | (2.1, 0.4) |  | (8.5, 6.1) | (3.3, 0.7) | (3.0, 0.2) |  | (7.1, 3.9) | (4.1, 0.4) | (4.0, 0.1)
0.75 | (84.0, 82.2) | (21.6, 20.0) | (10.4, 8.9) |  | (55.6, 52.9) | (14.8, 12.4) | (8.0, 5.7) |  | (42.2, 38.9) | (12.4, 9.3) | (7.5, 4.4)
0.90 | (211.8, 210.0) | (116.2, 114.4) | (77.7, 75.9) |  | (177.0, 174.2) | (90.3, 87.6) | (58.9, 56.3) |  | (154.8, 151.1) | (76.6, 73.1) | (49.8, 46.4)
1.10 | (109.6, 107.7) | (67.0, 65.2) | (48.9, 47.1) |  | (109.3, 106.5) | (63.4, 60.8) | (45.2, 42.6) |  | (111.1, 107.5) | (62.6, 59.1) | (44.1, 40.7)
1.25 | (32.5, 30.8) | (14.7, 13.0) | (9.4, 7.9) |  | (33.5, 30.9) | (14.7, 12.3) | (9.6, 7.2) |  | (35.3, 32.0) | (15.3, 12.2) | (10.2, 7.1)
1.50 | (10.5, 8.9) | (4.6, 3.1) | (3.3, 1.7) |  | (11.7, 9.3) | (5.4, 3.0) | (4.0, 1.6) |  | (13.1, 10.0) | (6.3, 3.1) | (4.9, 1.6)
| $\gamma_{0}=0.2$
0.50 | (14.6, 13.0) | (2.9, 1.3) | (2.1, 0.4) |  | (8.7, 6.3) | (3.3, 0.7) | (3.0, 0.2) |  | (7.2, 4.0) | (4.1, 0.4) | (4.0, 0.1)
0.75 | (85.6, 83.8) | (22.5, 20.9) | (10.9, 9.3) |  | (56.8, 54.2) | (15.4, 12.9) | (8.3, 6.0) |  | (43.3, 39.9) | (12.9, 9.8) | (7.8, 4.7)
0.90 | (213.7, 211.8) | (118.9, 117.1) | (80.2, 78.4) |  | (179.0, 176.2) | (92.8, 90.0) | (61.0, 58.4) |  | (156.9, 153.2) | (78.8, 75.3) | (51.7, 48.3)
1.10 | (111.9, 110.1) | (69.5, 67.8) | (51.1, 49.4) |  | (111.6, 108.8) | (65.8, 63.1) | (47.3, 44.7) |  | (113.3, 109.7) | (64.9, 61.4) | (46.1, 42.7)
1.25 | (33.8, 32.1) | (15.6, 13.9) | (10.1, 8.5) |  | (34.8, 32.2) | (15.5, 13.1) | (10.1, 7.8) |  | (36.6, 33.3) | (16.1, 13.0) | (10.7, 7.6)
1.50 | (11.1, 9.5) | (4.9, 3.4) | (3.4, 1.9) |  | (12.3, 9.9) | (5.7, 3.3) | (4.2, 1.8) |  | (13.7, 10.5) | (6.5, 3.4) | (5.1, 1.8)
| $\gamma_{0}=0.3$
0.50 | (15.2, 13.6) | (3.0, 1.5) | (2.2, 0.5) |  | (9.0, 6.6) | (3.4, 0.8) | (3.0, 0.2) |  | (7.4, 4.3) | (4.1, 0.5) | (4.0, 0.1)
0.75 | (88.2, 86.4) | (24.0, 22.3) | (11.7, 10.2) |  | (58.9, 56.3) | (16.3, 13.9) | (8.9, 6.5) |  | (45.0, 41.6) | (13.7, 10.5) | (8.2, 5.1)
0.90 | (216.6, 214.8) | (123.1, 121.3) | (84.1, 82.3) |  | (182.3, 179.5) | (96.6, 93.9) | (64.5, 61.8) |  | (160.2, 156.5) | (82.4, 78.9) | (54.7, 51.3)
1.10 | (116.0, 114.2) | (73.8, 72.0) | (54.9, 53.2) |  | (115.5, 112.7) | (69.8, 67.1) | (50.7, 48.1) |  | (117.2, 113.5) | (68.7, 65.2) | (49.4, 46.0)
1.25 | (36.2, 34.4) | (17.1, 15.5) | (11.1, 9.6) |  | (37.0, 34.4) | (16.9, 14.5) | (11.1, 8.7) |  | (38.9, 35.5) | (17.5, 14.3) | (11.6, 8.5)
1.50 | (12.1, 10.5) | (5.4, 3.9) | (3.7, 2.2) |  | (13.3, 10.9) | (6.1, 3.8) | (4.5, 2.1) |  | (14.7, 11.6) | (7.0, 3.8) | (5.3, 2.1)
| $\gamma_{0}=0.4$
0.50 | (16.0, 14.4) | (3.1, 1.6) | (2.2, 0.5) |  | (9.4, 7.1) | (3.4, 0.9) | (3.1, 0.2) |  | (7.7, 4.6) | (4.2, 0.5) | (4.0, 0.1)
0.75 | (91.7, 89.9) | (26.0, 24.3) | (12.9, 11.3) |  | (61.8, 59.2) | (17.7, 15.3) | (9.6, 7.3) |  | (47.4, 44.0) | (14.7, 11.6) | (8.8, 5.7)
0.90 | (220.5, 218.7) | (128.5, 126.7) | (89.3, 87.5) |  | (186.6, 183.8) | (101.8, 99.0) | (69.0, 66.3) |  | (164.6, 160.9) | (87.2, 83.7) | (58.8, 55.3)
1.10 | (121.9, 120.1) | (79.6, 77.8) | (60.1, 58.4) |  | (121.0, 118.2) | (75.2, 72.5) | (55.4, 52.8) |  | (122.5, 118.9) | (73.9, 70.4) | (53.9, 50.4)
1.25 | (39.7, 38.0) | (19.4, 17.8) | (12.7, 11.2) |  | (40.3, 37.7) | (18.9, 16.5) | (12.5, 10.1) |  | (42.1, 38.7) | (19.4, 16.2) | (12.9, 9.8)
1.50 | (13.7, 12.1) | (6.2, 4.7) | (4.2, 2.7) |  | (14.8, 12.4) | (6.8, 4.5) | (4.9, 2.5) |  | (16.2, 13.1) | (7.6, 4.5) | (5.7, 2.5)
| $\gamma_{0}=0.5$
0.50 | (17.1, 15.5) | (3.3, 1.8) | (2.3, 0.6) |  | (10.0, 7.7) | (3.5, 1.0) | (3.1, 0.3) |  | (8.1, 5.0) | (4.2, 0.6) | (4.0, 0.2)
0.75 | (96.1, 94.3) | (28.6, 26.9) | (14.4, 12.8) |  | (65.5, 62.8) | (19.5, 17.0) | (10.6, 8.3) |  | (50.5, 47.1) | (16.1, 12.9) | (9.6, 6.5)
0.90 | (225.1, 223.3) | (134.9, 133.1) | (95.5, 93.7) |  | (191.7, 188.9) | (107.8, 105.1) | (74.5, 71.8) |  | (170.0, 166.3) | (93.0, 89.4) | (63.8, 60.3)
1.10 | (129.7, 127.9) | (87.0, 85.2) | (66.7, 64.9) |  | (128.1, 125.3) | (81.9, 79.2) | (61.3, 58.7) |  | (129.2, 125.6) | (80.3, 76.8) | (59.4, 55.9)
1.25 | (44.7, 43.0) | (22.5, 20.9) | (14.9, 13.3) |  | (44.8, 42.2) | (21.6, 19.2) | (14.3, 11.9) |  | (46.4, 43.0) | (22.0, 18.7) | (14.7, 11.5)
1.50 | (16.0, 14.4) | (7.3, 5.7) | (4.8, 3.3) |  | (16.9, 14.5) | (7.8, 5.4) | (5.4, 3.1) |  | (18.3, 15.1) | (8.5, 5.4) | (6.2, 3.0)

Table 2: The values of $ARL_{1}$ and $SDRL_{1}$ using the RR${}_{r,s}-$MCV
control charts for $p=2$ and different values of $\tau,n$ and $\gamma_{0}$.

| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
---|---|---|---|---|---
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
0.50 | (32.7, 31.0) | (3.3, 1.8) | (2.2, 0.5) |  | (17.4, 15.0) | (3.5, 0.9) | (3.0, 0.2) |  | (12.4, 9.2) | (4.2, 0.6) | (4.0, 0.1)
0.75 | (129.4, 127.6) | (26.5, 24.8) | (11.7, 10.2) |  | (90.5, 87.8) | (17.7, 15.3) | (8.8, 6.5) |  | (69.5, 66.0) | (14.6, 11.4) | (8.1, 5.0)
0.90 | (250.7, 248.9) | (128.2, 126.4) | (83.4, 81.6) |  | (217.2, 214.4) | (100.5, 97.8) | (63.5, 60.8) |  | (193.7, 190.0) | (85.5, 81.9) | (53.7, 50.2)
1.10 | (128.4, 126.6) | (72.4, 70.7) | (51.7, 49.9) |  | (130.8, 128.0) | (69.1, 66.4) | (48.0, 45.3) |  | (134.6, 130.9) | (68.4, 64.9) | (46.9, 43.5)
1.25 | (43.1, 41.3) | (16.5, 14.9) | (10.1, 8.6) |  | (45.6, 43.0) | (16.5, 14.1) | (10.3, 7.9) |  | (48.8, 45.4) | (17.2, 14.0) | (10.9, 7.8)
1.50 | (14.8, 13.2) | (5.1, 3.6) | (3.4, 1.9) |  | (16.7, 14.3) | (5.9, 3.6) | (4.2, 1.8) |  | (18.8, 15.6) | (6.8, 3.7) | (5.1, 1.8)
| $\gamma_{0}=0.2$
0.50 | (33.4, 31.7) | (3.4, 1.8) | (2.2, 0.5) |  | (17.8, 15.4) | (3.5, 1.0) | (3.0, 0.2) |  | (12.6, 9.5) | (4.2, 0.6) | (4.0, 0.1)
0.75 | (131.2, 129.4) | (27.5, 25.8) | (12.3, 10.7) |  | (92.1, 89.4) | (18.4, 15.9) | (9.2, 6.8) |  | (70.9, 67.4) | (15.1, 11.9) | (8.4, 5.3)
0.90 | (252.3, 250.4) | (131.0, 129.1) | (86.0, 84.2) |  | (219.0, 216.2) | (103.0, 100.3) | (65.7, 63.0) |  | (195.7, 192.0) | (87.8, 84.3) | (55.6, 52.2)
1.10 | (130.9, 129.0) | (75.1, 73.3) | (54.0, 52.3) |  | (133.2, 130.4) | (71.6, 68.9) | (50.1, 47.5) |  | (137.0, 133.3) | (70.8, 67.3) | (49.0, 45.6)
1.25 | (44.7, 43.0) | (17.5, 15.9) | (10.8, 9.3) |  | (47.2, 44.6) | (17.4, 15.0) | (10.9, 8.5) |  | (50.5, 47.0) | (18.1, 14.9) | (11.5, 8.3)
1.50 | (15.6, 14.0) | (5.5, 4.0) | (3.6, 2.1) |  | (17.5, 15.1) | (6.2, 3.9) | (4.4, 2.0) |  | (19.7, 16.5) | (7.1, 4.0) | (5.3, 2.0)
| $\gamma_{0}=0.3$
0.50 | (34.5, 32.8) | (3.5, 2.0) | (2.2, 0.6) |  | (18.5, 16.1) | (3.6, 1.1) | (3.1, 0.3) |  | (13.1, 9.9) | (4.3, 0.7) | (4.0, 0.1)
0.75 | (134.2, 132.4) | (29.2, 27.5) | (13.2, 11.6) |  | (94.8, 92.1) | (19.6, 17.1) | (9.8, 7.4) |  | (73.3, 69.8) | (16.0, 12.8) | (8.9, 5.8)
0.90 | (254.8, 252.9) | (135.3, 133.4) | (90.1, 88.3) |  | (222.0, 219.1) | (107.1, 104.3) | (69.3, 66.6) |  | (198.9, 195.2) | (91.6, 88.1) | (58.8, 55.4)
1.10 | (135.1, 133.2) | (79.5, 77.7) | (57.9, 56.2) |  | (137.2, 134.4) | (75.7, 73.0) | (53.7, 51.1) |  | (141.0, 137.3) | (74.9, 71.3) | (52.4, 49.0)
1.25 | (47.6, 45.8) | (19.2, 17.6) | (12.0, 10.4) |  | (50.1, 47.4) | (19.0, 16.6) | (11.9, 9.5) |  | (53.3, 49.9) | (19.7, 16.4) | (12.5, 9.3)
1.50 | (17.0, 15.4) | (6.1, 4.6) | (4.0, 2.5) |  | (19.0, 16.5) | (6.8, 4.4) | (4.7, 2.3) |  | (21.2, 18.0) | (7.7, 4.5) | (5.5, 2.3)
| $\gamma_{0}=0.4$
0.50 | (36.2, 34.5) | (3.7, 2.2) | (2.3, 0.6) |  | (19.5, 17.0) | (3.7, 1.2) | (3.1, 0.3) |  | (13.7, 10.6) | (4.3, 0.7) | (4.0, 0.2)
0.75 | (138.2, 136.4) | (31.6, 29.9) | (14.5, 12.9) |  | (98.5, 95.8) | (21.2, 18.7) | (10.6, 8.3) |  | (76.6, 73.0) | (17.3, 14.1) | (9.5, 6.4)
0.90 | (258.1, 256.2) | (140.9, 139.0) | (95.5, 93.7) |  | (225.9, 223.1) | (112.5, 109.7) | (74.0, 71.3) |  | (203.1, 199.4) | (96.7, 93.1) | (63.1, 59.7)
1.10 | (141.2, 139.3) | (85.6, 83.8) | (63.4, 61.6) |  | (143.0, 140.2) | (81.4, 78.7) | (58.6, 56.0) |  | (146.5, 142.9) | (80.3, 76.8) | (57.1, 53.6)
1.25 | (51.9, 50.2) | (21.8, 20.2) | (13.7, 12.2) |  | (54.2, 51.6) | (21.3, 18.8) | (13.4, 11.0) |  | (57.5, 54.0) | (21.9, 18.6) | (13.9, 10.7)
1.50 | (19.3, 17.6) | (7.0, 5.4) | (4.5, 3.0) |  | (21.2, 18.7) | (7.6, 5.3) | (5.1, 2.8) |  | (23.5, 20.2) | (8.5, 5.3) | (6.0, 2.8)
| $\gamma_{0}=0.5$
0.50 | (38.3, 36.6) | (4.0, 2.5) | (2.4, 0.7) |  | (20.8, 18.3) | (3.8, 1.3) | (3.1, 0.4) |  | (14.6, 11.4) | (4.4, 0.9) | (4.0, 0.2)
0.75 | (143.2, 141.4) | (34.6, 32.9) | (16.2, 14.6) |  | (103.2, 100.5) | (23.3, 20.8) | (11.8, 9.4) |  | (80.8, 77.2) | (18.9, 15.7) | (10.4, 7.3)
0.90 | (262.1, 260.3) | (147.4, 145.6) | (101.9, 100.1) |  | (230.7, 227.9) | (118.8, 116.1) | (79.8, 77.1) |  | (208.4, 204.7) | (102.8, 99.2) | (68.4, 64.9)
1.10 | (149.3, 147.5) | (93.4, 91.6) | (70.2, 68.4) |  | (150.5, 147.7) | (88.5, 85.8) | (64.8, 62.1) |  | (153.7, 150.0) | (87.1, 83.6) | (62.9, 59.4)
1.25 | (58.0, 56.3) | (25.3, 23.6) | (16.1, 14.5) |  | (59.9, 57.2) | (24.4, 21.9) | (15.5, 13.0) |  | (63.0, 59.5) | (24.8, 21.5) | (15.8, 12.6)
1.50 | (22.6, 21.0) | (8.2, 6.7) | (5.2, 3.7) |  | (24.4, 21.9) | (8.7, 6.4) | (5.8, 3.4) |  | (26.7, 23.4) | (9.5, 6.4) | (6.6, 3.4)

Table 3: The values of $ARL_{1}$ and $SDRL_{1}$ using the RR${}_{r,s}-$MCV
control charts for $p=3$ and different values of $\tau,n$ and $\gamma_{0}$.

| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
---|---|---|---|---|---
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
0.50 | (101.5, 99.6) | (4.0, 2.5) | (2.3, 0.6) |  | (61.8, 59.1) | (3.8, 1.3) | (3.1, 0.3) |  | (41.7, 38.3) | (4.4, 0.8) | (4.0, 0.1)
0.75 | (215.1, 213.3) | (33.3, 31.6) | (13.4, 11.8) |  | (172.2, 169.3) | (21.9, 19.4) | (9.8, 7.4) |  | (141.9, 138.2) | (17.6, 14.4) | (8.8, 5.7)
0.90 | (303.4, 301.5) | (142.7, 140.9) | (90.0, 88.2) |  | (278.9, 276.1) | (113.0, 110.3) | (68.7, 66.0) |  | (258.8, 255.1) | (96.5, 92.9) | (58.1, 54.7)
1.10 | (161.2, 159.4) | (78.9, 77.1) | (54.8, 53.1) |  | (169.6, 166.8) | (75.8, 73.1) | (51.1, 48.5) |  | (178.0, 174.3) | (75.5, 71.9) | (50.0, 46.6)
1.25 | (65.9, 64.2) | (18.8, 17.2) | (11.0, 9.4) |  | (73.5, 70.8) | (18.9, 16.4) | (11.1, 8.7) |  | (81.1, 77.5) | (19.7, 16.5) | (11.7, 8.6)
1.50 | (25.8, 24.1) | (5.8, 4.3) | (3.6, 2.1) |  | (30.7, 28.1) | (6.6, 4.3) | (4.4, 2.0) |  | (35.6, 32.2) | (7.6, 4.4) | (5.3, 2.0)
| $\gamma_{0}=0.2$
0.50 | (102.7, 100.9) | (4.1, 2.6) | (2.3, 0.6) |  | (62.8, 60.1) | (3.8, 1.4) | (3.1, 0.3) |  | (42.5, 39.1) | (4.4, 0.9) | (4.0, 0.2)
0.75 | (216.7, 214.9) | (34.5, 32.8) | (14.0, 12.4) |  | (173.9, 171.1) | (22.7, 20.2) | (10.2, 7.8) |  | (143.7, 140.0) | (18.2, 15.0) | (9.2, 6.0)
0.90 | (304.3, 302.4) | (145.5, 143.7) | (92.6, 90.8) |  | (280.2, 277.3) | (115.7, 112.9) | (71.1, 68.4) |  | (260.3, 256.6) | (98.9, 95.4) | (60.2, 56.7)
1.10 | (163.7, 161.8) | (81.7, 79.9) | (57.3, 55.5) |  | (172.0, 169.2) | (78.5, 75.8) | (53.4, 50.7) |  | (180.4, 176.7) | (78.0, 74.5) | (52.2, 48.8)
1.25 | (68.0, 66.3) | (20.0, 18.3) | (11.7, 10.2) |  | (75.6, 72.9) | (19.9, 17.5) | (11.7, 9.4) |  | (83.3, 79.8) | (20.7, 17.5) | (12.3, 9.2)
1.50 | (27.1, 25.4) | (6.2, 4.7) | (3.9, 2.4) |  | (32.1, 29.5) | (7.0, 4.7) | (4.6, 2.2) |  | (37.1, 33.7) | (7.9, 4.8) | (5.5, 2.2)
| $\gamma_{0}=0.3$
0.50 | (104.8, 103.0) | (4.3, 2.8) | (2.3, 0.7) |  | (64.5, 61.8) | (3.9, 1.5) | (3.1, 0.3) |  | (43.8, 40.4) | (4.5, 1.0) | (4.0, 0.2)
0.75 | (219.3, 217.5) | (36.5, 34.8) | (15.1, 13.4) |  | (176.8, 174.0) | (24.1, 21.6) | (10.9, 8.5) |  | (146.6, 143.0) | (19.3, 16.1) | (9.7, 6.6)
0.90 | (305.9, 304.0) | (149.8, 148.0) | (96.9, 95.1) |  | (282.3, 279.4) | (119.9, 117.1) | (74.8, 72.1) |  | (262.7, 259.0) | (102.9, 99.4) | (63.6, 60.1)
1.10 | (167.8, 166.0) | (86.2, 84.5) | (61.3, 59.6) |  | (176.1, 173.2) | (82.8, 80.1) | (57.1, 54.4) |  | (184.3, 180.6) | (82.3, 78.7) | (55.8, 52.3)
1.25 | (71.7, 69.9) | (21.9, 20.3) | (13.0, 11.4) |  | (79.4, 76.7) | (21.8, 19.3) | (12.9, 10.5) |  | (87.1, 83.6) | (22.5, 19.2) | (13.4, 10.3)
1.50 | (29.4, 27.7) | (6.9, 5.4) | (4.2, 2.7) |  | (34.5, 32.0) | (7.7, 5.3) | (5.0, 2.6) |  | (39.8, 36.4) | (8.6, 5.5) | (5.8, 2.6)
| $\gamma_{0}=0.4$
0.50 | (107.7, 105.8) | (4.6, 3.1) | (2.4, 0.8) |  | (66.9, 64.2) | (4.1, 1.7) | (3.1, 0.4) |  | (45.7, 42.2) | (4.5, 1.1) | (4.0, 0.2)
0.75 | (222.9, 221.0) | (39.4, 37.6) | (16.5, 14.9) |  | (180.8, 178.0) | (26.1, 23.6) | (11.9, 9.5) |  | (150.7, 147.1) | (20.9, 17.6) | (10.5, 7.3)
0.90 | (308.0, 306.1) | (155.5, 153.7) | (102.5, 100.7) |  | (285.0, 282.2) | (125.4, 122.7) | (79.8, 77.1) |  | (266.0, 262.3) | (108.3, 104.7) | (68.1, 64.6)
1.10 | (173.9, 172.0) | (92.6, 90.8) | (66.9, 65.2) |  | (181.7, 178.9) | (88.8, 86.1) | (62.2, 59.6) |  | (189.8, 186.1) | (88.1, 84.5) | (60.7, 57.2)
1.25 | (77.2, 75.4) | (24.8, 23.2) | (14.9, 13.3) |  | (84.8, 82.1) | (24.4, 21.9) | (14.6, 12.1) |  | (92.6, 89.1) | (25.0, 21.7) | (15.0, 11.8)
1.50 | (33.0, 31.3) | (8.0, 6.5) | (4.8, 3.3) |  | (38.3, 35.7) | (8.7, 6.3) | (5.5, 3.1) |  | (43.8, 40.4) | (9.6, 6.4) | (6.3, 3.1)
| $\gamma_{0}=0.5$
0.50 | (111.3, 109.5) | (5.0, 3.5) | (2.5, 0.9) |  | (69.9, 67.3) | (4.3, 1.9) | (3.2, 0.5) |  | (48.1, 44.7) | (4.7, 1.3) | (4.1, 0.3)
0.75 | (227.3, 225.4) | (42.9, 41.2) | (18.4, 16.8) |  | (185.8, 183.0) | (28.7, 26.1) | (13.2, 10.8) |  | (155.9, 152.3) | (22.9, 19.6) | (11.5, 8.3)
0.90 | (310.6, 308.8) | (162.2, 160.4) | (109.1, 107.3) |  | (288.5, 285.6) | (132.1, 129.3) | (85.8, 83.1) |  | (270.0, 266.3) | (114.7, 111.1) | (73.6, 70.1)
1.10 | (181.9, 180.0) | (100.7, 98.9) | (74.0, 72.3) |  | (189.1, 186.3) | (96.3, 93.6) | (68.7, 66.0) |  | (196.7, 193.0) | (95.3, 91.7) | (66.8, 63.3)
1.25 | (84.8, 83.0) | (28.8, 27.1) | (17.5, 15.9) |  | (92.1, 89.4) | (27.9, 25.4) | (16.8, 14.3) |  | (99.9, 96.3) | (28.4, 25.1) | (17.1, 13.9)
1.50 | (38.3, 36.5) | (9.5, 8.0) | (5.6, 4.1) |  | (43.6, 41.0) | (10.0, 7.7) | (6.2, 3.9) |  | (49.3, 45.9) | (10.9, 7.8) | (7.0, 3.8)

Table 4:  The values of $ARL_{1}$ and $SDRL_{1}$ using the RR${}_{r,s}-$MCV
control charts for $p=4$ and different values of $\tau,n$ and $\gamma_{0}$.

  * •

The in-control value $\gamma_{0}$ of MCV and the value of multivariate level
$p$ have strong influence on the performance of RR${}_{r,s}-$MCV control
charts. In particular, the increase of $\gamma_{0}$ and $p$ results in the
increase of the $ARL_{1}$. For example, in Table 2 with $\tau=0.5,n=5$ and
$(r,s)=(2,3)$, we have $ARL_{1}=14.2$ when $\gamma_{0}=0.1$ and $ARL_{1}=17.1$
when $\gamma_{0}=0.5$. Also, with the same $\tau=0.5,n=5,\gamma_{0}=0.2$ and
$(r,s)=(2,3)$, we have $ARL_{1}=14.6$ when $p=2$ in Table 2 and
$ARL_{1}=102.7$ when $p=4$ in Table 4. That is to say, the RR${}_{r,s}-$MCV
charts are more efficient for processes with small values of in-control MCV
and multivariate levels.

  * •

The sample size $n$ has positive impact on the power of proposed charts: the
larger the sample size, the smaller the average number of samples needed to
detect the out-of-control status. For instance, with
$\gamma_{0}=0.3,\tau=1.25$ and $p=3$ (Table 3), we have $ARL_{1}=50.1$ when
$n=5$ but this value dropped significantly to $ARL_{1}=11.9$ when $n=15$ on
RR${}_{3,4}-$MCV chart.

  * •

Larger values for $(r,s)$ do not necessarily deliver better performance for
the Run Rules based control charts; it depends on the value of the sample size
$n$, the shift size $\tau$ and especially the type of control chart. In
general, using larger values for $(r,s)$ results in better performance for the
lower-sided chart but worse performance for upper-sided chart. For example,
with $n=5,\gamma_{0}=0.5$ in Table 4, the RR${}_{2,3}-$MCV charts results in
$ARL_{1}=227.3$ for $\tau=0.75$ (lower-sided) and $ARL_{1}=181.9$ for
$\tau=1.1$ (upper-sided) compared to $ARL_{1}=155.9$ for $\tau=0.75$ (lower-
sided) and $ARL_{1}=196.7$ for $\tau=1.1$ (upper-sided) in the
RR${}_{4,5}-$MCV charts.

To compare the performance of the RR${}_{r,s}-$MCV control charts with the
Shewhart$-$ MCV control chart, we define the index $\Delta_{A}$ as

$\Delta_{A}=100\times\frac{ARL_{\mathrm{Shewhart}}-ARL_{\mathrm{RR}_{r,s}}}{ARL_{\mathrm{RR}_{r,s}}}.$
(16)

In this definition, $ARL_{\mathrm{Shewhart}}$ and $ARL_{\mathrm{RR}_{r,s}}$
represent the $ARL$ value of the Shewhart-MCV chart and
RR${}_{r,s}-\mathrm{MCV}$ chart, respectively. Values $\Delta_{A}>0$ indicate
that the RR${}_{r,s}-$MCV charts outperform the Shewhart-MCV chart;
conversely, values $\Delta_{A}<0$ indicate that the Shewhart-MCV chart
outperforms the RRr,s-MCV charts. Tables 5-7 present the rounded results (to
the nearest integer) of $\Delta_{A}$. It can be seen from these tables that
the RRr,s-MCV charts outperforms the Shewhart-MCV chart in most cases.

The above conclusions can be seen more clearly in Figure 1, where we present
the $ARL$ profiles for both the Shewhart chart (designed by Yeong et al. 14)
and the Run Rules charts for a number of different in-control scenarios. Since
the $ARL$ curves for the upper case overlap each other, we include the ARL
curves for $\tau\geqslant 1.2$ as an inset plot. The Figure 1 shows that for
the lower case (decrease shifts), the 4-of-5 Run Rule chart remarkably
outperforms the Shewhart chart and the other Run Rules charts, especially when
$n=5,p=2$ and $\gamma_{0}=0.1$. As $n,p,\gamma_{0}$ increase, the improvement
is not as much as in the first case but still, it is substantial. For upper
case (increase shifts), we have also an improvement with Run Rules charts but
it is not as much as in the lower case. In addition, a part of the $ARL$ curve
of the Shehwart chart corresponding to very large shifts (i.e., 1.50 or 0.50)
is below $ARL$ curves of Run Rules charts. We deduce that the Shewhart chart
becomes more efficient than the proposed Run Rule based charts in detecting
very large shifts. This is also confirmed by the negative values of
$\Delta_{A}$ index with the large values of $\tau$ in the Tables 5-7.

| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
---|---|---|---|---|---
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
0.50 | 71 | 46 | -10 |  | 83 | 38 | -56 |  | 85 | 22 | -107
0.75 | 47 | 59 | 58 |  | 65 | 72 | 68 |  | 73 | 76 | 70
0.90 | 22 | 35 | 40 |  | 35 | 49 | 55 |  | 43 | 57 | 62
1.10 | 8 | 17 | 22 |  | 8 | 22 | 28 |  | 6 | 23 | 30
1.25 | 9 | 18 | 18 |  | 6 | 18 | 17 |  | 1 | 14 | 12
1.50 | -1 | -5 | -17 |  | -13 | -22 | -45 |  | -26 | -42 | -77
| $\gamma_{0}=0.2$
0.50 | 71 | 47 | -7 |  | 83 | 40 | -51 |  | 85 | 25 | -100
0.75 | 47 | 59 | 58 |  | 65 | 72 | 68 |  | 73 | 76 | 70
0.90 | 22 | 34 | 39 |  | 34 | 49 | 54 |  | 42 | 56 | 61
1.10 | 8 | 17 | 22 |  | 8 | 22 | 28 |  | 6 | 23 | 30
1.25 | 9 | 18 | 19 |  | 6 | 18 | 19 |  | 1 | 15 | 14
1.50 | 0 | -2 | -13 |  | -11 | -18 | -38 |  | -24 | -36 | -67
| $\gamma_{0}=0.3$
0.50 | 70 | 49 | -2 |  | 82 | 43 | -43 |  | 86 | 29 | -88
0.75 | 46 | 58 | 58 |  | 64 | 71 | 68 |  | 72 | 76 | 71
0.90 | 21 | 33 | 39 |  | 34 | 48 | 53 |  | 42 | 55 | 60
1.10 | 8 | 17 | 22 |  | 8 | 22 | 28 |  | 7 | 23 | 30
1.25 | 10 | 19 | 21 |  | 7 | 20 | 21 |  | 3 | 17 | 17
1.50 | 2 | 2 | -7 |  | -7 | -11 | -28 |  | -19 | -27 | -53
| $\gamma_{0}=0.4$
0.50 | 70 | 51 | 4 |  | 82 | 46 | -32 |  | 86 | 35 | -74
0.75 | 45 | 57 | 58 |  | 63 | 71 | 68 |  | 72 | 76 | 71
0.90 | 21 | 32 | 37 |  | 33 | 46 | 52 |  | 41 | 54 | 59
1.10 | 8 | 17 | 22 |  | 9 | 22 | 28 |  | 8 | 23 | 30
1.25 | 11 | 20 | 23 |  | 10 | 22 | 24 |  | 6 | 20 | 21
1.50 | 6 | 7 | -1 |  | -2 | -3 | -17 |  | -12 | -15 | -36
| $\gamma_{0}=0.5$
0.50 | 69 | 53 | 11 |  | 82 | 50 | -21 |  | 85 | 40 | -58
0.75 | 44 | 56 | 57 |  | 62 | 70 | 68 |  | 71 | 75 | 71
0.90 | 20 | 31 | 36 |  | 32 | 45 | 50 |  | 39 | 53 | 57
1.10 | 9 | 17 | 21 |  | 10 | 22 | 28 |  | 10 | 24 | 30
1.25 | 14 | 22 | 25 |  | 14 | 25 | 27 |  | 11 | 24 | 26
1.50 | 12 | 12 | 7 |  | 7 | 6 | -5 |  | -1 | -3 | -20

Table 5: The values of index $\Delta_{A}$ for $p=2$ and different values of
$\tau,\gamma_{0}$ and $n$.

| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
---|---|---|---|---|---
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
0.50 | 65 | 55 | 2 |  | 81 | 52 | -37 |  | 87 | 42 | -80
0.75 | 38 | 58 | 59 |  | 57 | 72 | 69 |  | 67 | 77 | 71
0.90 | 17 | 33 | 39 |  | 28 | 47 | 54 |  | 36 | 55 | 61
1.10 | 4 | 16 | 21 |  | 2 | 20 | 27 |  | -1 | 21 | 29
1.25 | 3 | 17 | 19 |  | -2 | 17 | 18 |  | -9 | 13 | 13
1.50 | -4 | -3 | -14 |  | -18 | -18 | -40 |  | -33 | -36 | -69
| $\gamma_{0}=0.2$
0.50 | 65 | 56 | 5 |  | 81 | 54 | -32 |  | 87 | 44 | -73
0.75 | 38 | 58 | 58 |  | 56 | 72 | 69 |  | 66 | 77 | 72
0.90 | 16 | 33 | 39 |  | 27 | 47 | 53 |  | 35 | 55 | 60
1.10 | 4 | 16 | 21 |  | 2 | 20 | 27 |  | -1 | 21 | 29
1.25 | 4 | 17 | 19 |  | -2 | 18 | 19 |  | -9 | 14 | 15
1.50 | -4 | -1 | -10 |  | -17 | -14 | -34 |  | -31 | -31 | -60
| $\gamma_{0}=0.3$
0.50 | 64 | 57 | 9 |  | 81 | 56 | -24 |  | 86 | 47 | -63
0.75 | 37 | 57 | 58 |  | 56 | 71 | 69 |  | 66 | 77 | 72
0.90 | 16 | 32 | 38 |  | 27 | 46 | 52 |  | 34 | 54 | 59
1.10 | 4 | 16 | 21 |  | 2 | 20 | 27 |  | -0 | 21 | 29
1.25 | 4 | 18 | 21 |  | -1 | 19 | 21 |  | -7 | 16 | 18
1.50 | -2 | 3 | -5 |  | -14 | -8 | -24 |  | -27 | -22 | -47
| $\gamma_{0}=0.4$
0.50 | 64 | 58 | 15 |  | 80 | 58 | -15 |  | 86 | 51 | -50
0.75 | 36 | 56 | 58 |  | 55 | 71 | 69 |  | 65 | 76 | 72
0.90 | 15 | 31 | 37 |  | 26 | 45 | 51 |  | 33 | 52 | 58
1.10 | 4 | 16 | 21 |  | 3 | 20 | 27 |  | 1 | 21 | 29
1.25 | 6 | 19 | 22 |  | 2 | 21 | 24 |  | -4 | 19 | 22
1.50 | 1 | 8 | 1 |  | -8 | -1 | -13 |  | -20 | -12 | -31
| $\gamma_{0}=0.5$
0.50 | 63 | 59 | 20 |  | 80 | 61 | -5 |  | 86 | 55 | -36
0.75 | 35 | 55 | 57 |  | 53 | 70 | 69 |  | 63 | 76 | 73
0.90 | 15 | 30 | 35 |  | 25 | 43 | 49 |  | 32 | 51 | 57
1.10 | 6 | 16 | 21 |  | 5 | 20 | 27 |  | 3 | 21 | 29
1.25 | 8 | 21 | 24 |  | 6 | 24 | 27 |  | 1 | 23 | 26
1.50 | 7 | 13 | 8 |  | -0 | 8 | -2 |  | -10 | -1 | -16

Table 6:  The values of $\Delta_{A}$ index for $p=3$ and different values of
$\tau,\gamma_{0}$ and $n$.

| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
---|---|---|---|---|---
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
0.50 | 45 | 62 | 14 |  | 67 | 64 | -17 |  | 78 | 59 | -53
0.75 | 23 | 57 | 59 |  | 38 | 72 | 70 |  | 49 | 77 | 73
0.90 | 9 | 31 | 38 |  | 16 | 45 | 53 |  | 22 | 53 | 60
1.10 | -2 | 14 | 21 |  | -8 | 18 | 26 |  | -13 | 18 | 27
1.25 | -7 | 16 | 19 |  | -19 | 16 | 18 |  | -31 | 12 | 13
1.50 | -15 | -1 | -11 |  | -37 | -15 | -35 |  | -59 | -31 | -62
| $\gamma_{0}=0.2$
0.50 | 45 | 62 | 17 |  | 66 | 65 | -12 |  | 77 | 60 | -47
0.75 | 22 | 57 | 59 |  | 38 | 72 | 70 |  | 49 | 77 | 73
0.90 | 9 | 31 | 38 |  | 16 | 45 | 52 |  | 22 | 53 | 60
1.10 | -2 | 14 | 20 |  | -8 | 18 | 26 |  | -13 | 18 | 27
1.25 | -7 | 16 | 19 |  | -19 | 16 | 19 |  | -31 | 13 | 15
1.50 | -15 | 1 | -8 |  | -36 | -12 | -29 |  | -57 | -26 | -53
| $\gamma_{0}=0.3$
0.50 | 45 | 63 | 20 |  | 66 | 66 | -6 |  | 77 | 62 | -38
0.75 | 22 | 56 | 58 |  | 37 | 71 | 70 |  | 48 | 77 | 73
0.90 | 9 | 30 | 37 |  | 16 | 44 | 51 |  | 22 | 52 | 59
1.10 | -2 | 14 | 20 |  | -7 | 18 | 26 |  | -12 | 18 | 27
1.25 | -6 | 17 | 20 |  | -17 | 18 | 21 |  | -29 | 15 | 18
1.50 | -13 | 4 | -3 |  | -33 | -6 | -21 |  | -53 | -19 | -41
| $\gamma_{0}=0.4$
0.50 | 44 | 64 | 25 |  | 65 | 68 | 3 |  | 76 | 64 | -26
0.75 | 21 | 55 | 58 |  | 36 | 70 | 70 |  | 47 | 76 | 73
0.90 | 8 | 29 | 36 |  | 15 | 43 | 50 |  | 21 | 50 | 57
1.10 | -1 | 14 | 20 |  | -6 | 18 | 26 |  | -11 | 18 | 27
1.25 | -4 | 18 | 22 |  | -15 | 20 | 24 |  | -25 | 17 | 22
1.50 | -10 | 8 | 3 |  | -28 | 1 | -10 |  | -46 | -10 | -27
| $\gamma_{0}=0.5$
0.50 | 43 | 64 | 30 |  | 64 | 69 | 11 |  | 75 | 66 | -14
0.75 | 21 | 54 | 57 |  | 35 | 69 | 70 |  | 46 | 75 | 73
0.90 | 8 | 28 | 34 |  | 15 | 41 | 48 |  | 20 | 49 | 56
1.10 | -0 | 14 | 20 |  | -4 | 18 | 26 |  | -9 | 19 | 28
1.25 | -2 | 20 | 24 |  | -11 | 22 | 27 |  | -20 | 21 | 26
1.50 | -5 | 13 | 9 |  | -20 | 9 | 0 |  | -35 | 1 | -12

Table 7: The values of $\Delta_{A}$ index for $p=4$ and different values of
$\tau,\gamma_{0}$ and $n$.

(a) $p=2,n=5,\gamma_{0}=0.1$ (b) $p=2,n=5,\gamma_{0}=0.1$ (c)
$p=3,n=10,\gamma_{0}=0.3$ (d) $p=3,n=10,\gamma_{0}=0.3$ (e)
$p=4,n=15,\gamma_{0}=0.5$ (f) $p=4,n=15,\gamma_{0}=0.5$

Figure 1: The $ARL$ profiles of Shewhart chart and Run Rules charts for
various in-control settings; left side: lower-sided charts, right side: upper-
sided charts

The analysis presented above is only for the case of specific shift size. In
practice, however, it is hard for quality practitioners to predetermine a
specific shift without any previous experience. Thus, they usually have an
interest in detecting a range of shifts $\tau\in[a,b]$ rather than preference
for any particular size of the process shift. The use of the uniform
distribution has been proposed to account for the unknown shift size by some
authors (for instance, see Chen and Chen 33 and Celano et al. 34). The
statistical performance of the corresponding chart can be evaluated through
the $EARL$ (Expected Average Run Length) given by

$EARL=\int_{a}^{b}ARL(\tau)\times f_{\tau}(\tau)d\tau,$ (17)

where $ARL(\tau)$ is the $ARL$, as a function of shift $\tau$, of the proposed
Run Rules charts, with $f_{\tau}(\tau)=\frac{1}{b-a}$ for $\tau\in[a,b]$. In
the following section, we will consider a specific range of shift
$[a,b]=[0.5,1)$ (decreasing case, denoted as (D)) for lower-sided
RR${}_{r,s}-$MCV control chart and $[a,b]=(1,2]$ (increasing case, denoted as
(I)) for upper-sided RR${}_{r,s}-$MCV control chart.

| $p=2$
---|---
| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
| $p=2$
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
(D) | (101.8, 100.1) | (48.1, 46.5) | (33.0, 31.3) |  | (79.4,76.7) | (38.5,36.0) | (26.9,24.3) |  | (67.8,64.4) | (34.0,30.6) | (24.2,20.7)
(I) | (29.4, 27.8) | (17.4, 15.8) | (13.2, 11.5) |  | (30.3,27.9) | (17.4,15.0) | (13.1,10.6) |  | (31.7,28.5) | (18.0,14.7) | (13.7,10.2)
| $\gamma_{0}=0.2$
(D) | (103.1, 101.3) | (49.2, 47.5) | (33.9, 32.2) |  | (80.4,77.8) | (39.4,36.9) | (27.6,25.1) |  | (68.8,65.4) | (34.8,31.4) | (24.9,21.4)
(I) | (30.3, 28.6) | (18.1, 16.5) | (13.7, 12.1) |  | (31.2,28.7) | (18.1,15.6) | (13.6,11.1) |  | (32.6,29.3) | (18.6,15.3) | (14.1,10.7)
| $\gamma_{0}=0.3$
(D) | (105.1, 103.3) | (50.9, 49.2) | (35.3, 33.7) |  | (82.2,79.6) | (40.9,38.4) | (28.8,26.3) |  | (70.4,67.0) | (36.1,32.7) | (26.0,22.5)
(I) | (31.8, 30.1) | (19.2, 17.7) | (14.6, 13.0) |  | (32.6,30.1) | (19.1,16.7) | (14.5,12.0) |  | (34.0,30.8) | (19.6,16.3) | (14.9,11.5)
| $\gamma_{0}=0.4$
(D) | (107.8, 106.1) | (53.2, 51.5) | (37.2, 35.6) |  | (84.7,82.0) | (42.9,40.3) | (30.5,27.9) |  | (72.6,69.2) | (37.9,34.5) | (27.4,24.0)
(I) | (34.0, 32.4) | (20.9, 19.3) | (15.9, 14.3) |  | (34.7,32.3) | (20.6,18.1) | (15.6,13.1) |  | (36.1,32.9) | (21.1,17.8) | (16.0,12.6)
| $\gamma_{0}=0.5$
(D) | (111.2, 109.4) | (55.9, 54.3) | (39.5, 37.9) |  | (87.7,85.0) | (45.2,42.7) | (32.4,29.9) |  | (75.4,72.0) | (40.1,36.7) | (29.2,25.8)
(I) | (37.2, 35.5) | (23.0, 21.4) | (17.5, 16.0) |  | (37.6,35.1) | (22.5,20.1) | (17.1,14.6) |  | (38.9,35.6) | (22.9,19.6) | (17.4,14.1)
| $p=3$
| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
(D) | (136.9, 135.1) | (53.3, 51.7) | (35.1, 33.5) |  | (107.0,104.3) | (42.5,40.0) | (28.5,26.0) |  | (90.6,87.2) | (37.3,34.0) | (25.6,22.1)
(I) | (36.0, 34.3) | (18.8, 17.2) | (13.8, 12.2) |  | (37.9,35.4) | (18.8,16.4) | (13.8,11.2) |  | (40.2,36.9) | (19.5,16.2) | (14.3,10.9)
| $\gamma_{0}=0.2$
(D) | (138.3, 136.5) | (54.5, 52.8) | (36.1, 34.4) |  | (108.3,105.6) | (43.5,41.0) | (29.3,26.8) |  | (91.8,88.3) | (38.2,34.8) | (26.3,22.9)
(I) | (37.0, 35.4) | (19.5, 17.9) | (14.4, 12.8) |  | (38.9,36.4) | (19.5,17.1) | (14.3,11.8) |  | (41.2,38.0) | (20.1,16.8) | (14.8,11.4)
| $\gamma_{0}=0.3$
(D) | (140.6, 138.8) | (56.3, 54.7) | (37.6, 35.9) |  | (110.3,107.6) | (45.1,42.5) | (30.6,28.0) |  | (93.7,90.2) | (39.6,36.3) | (27.4,24.0)
(I) | (38.8, 37.2) | (20.7, 19.2) | (15.3, 13.7) |  | (40.7,38.2) | (20.7,18.2) | (15.2,12.7) |  | (43.1,39.8) | (21.2,18.0) | (15.6,12.3)
| $\gamma_{0}=0.4$
(D) | (143.6, 141.8) | (58.8, 57.1) | (39.6, 37.9) |  | (113.2,110.5) | (47.2,44.7) | (32.3,29.8) |  | (96.4,92.9) | (41.5,38.2) | (29.0,25.6)
(I) | (41.6, 40.0) | (22.5, 20.9) | (16.7, 15.1) |  | (43.4,40.9) | (22.3,19.8) | (16.4,13.9) |  | (45.8,42.5) | (22.8,19.6) | (16.8,13.4)
| $\gamma_{0}=0.5$
(D) | (147.5, 145.7) | (61.8, 60.1) | (42.0, 40.4) |  | (116.8,114.1) | (49.8,47.2) | (34.4,31.8) |  | (99.7,96.2) | (43.9,40.5) | (30.9,27.4)
(I) | (45.6, 43.9) | (24.8, 23.3) | (18.4, 16.9) |  | (47.2,44.6) | (24.4,22.0) | (18.0,15.5) |  | (49.5,46.2) | (24.8,21.6) | (18.3,15.0)
| $p=4$
| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
(D) | (208.8, 206.9) | (60.1, 58.4) | (37.6, 35.9) |  | (171.7,168.9) | (47.6,45.1) | (30.4,27.9) |  | (147.1,143.5) | (41.6,38.2) | (27.2,23.8)
(I) | (50.0, 48.3) | (20.5, 18.9) | (14.5, 12.9) |  | (55.1,52.6) | (20.6,18.1) | (14.5,12.0) |  | (60.3,57.0) | (21.3,18.0) | (15.0,11.6)
| $\gamma_{0}=0.2$
(D) | (210.1, 208.2) | (61.3, 59.6) | (38.6, 36.9) |  | (173.1,170.3) | (48.7,46.2) | (31.3,28.7) |  | (148.5,144.9) | (42.5,39.2) | (27.9,24.5)
(I) | (51.4, 49.7) | (21.3, 19.7) | (15.1, 13.5) |  | (56.6,54.1) | (21.3,18.9) | (15.0,12.5) |  | (61.9,58.5) | (22.0,18.7) | (15.6,12.2)
| $\gamma_{0}=0.3$
(D) | (212.3, 210.4) | (63.3, 61.6) | (40.2, 38.5) |  | (175.4,172.6) | (50.4,47.9) | (32.6,30.1) |  | (150.8,147.2) | (44.1,40.7) | (29.2,25.7)
(I) | (53.9, 52.2) | (22.6, 21.0) | (16.1, 14.5) |  | (59.2,56.7) | (22.6,20.2) | (16.0,13.5) |  | (64.7,61.2) | (23.3,20.0) | (16.4,13.1)
| $\gamma_{0}=0.4$
(D) | (215.2, 213.4) | (66.0, 64.3) | (42.3, 40.6) |  | (178.6,175.8) | (52.7,50.2) | (34.4,31.9) |  | (154.0,150.4) | (46.2,42.8) | (30.8,27.4)
(I) | (57.7, 56.0) | (24.5, 22.9) | (17.6, 16.0) |  | (63.1,60.5) | (24.4,22.0) | (17.3,14.8) |  | (68.7,65.3) | (25.0,21.8) | (17.7,14.4)
| $\gamma_{0}=0.5$
(D) | (218.9, 217.1) | (69.3, 67.6) | (44.8, 43.2) |  | (182.6,179.8) | (55.5,53.0) | (36.6,34.1) |  | (158.0,154.4) | (48.8,45.4) | (32.8,29.4)
(I) | (63.1, 61.4) | (27.1, 25.5) | (19.5, 17.9) |  | (68.5,65.8) | (26.8, 24.3) | (19.0,16.5) |  | (74.1,70.7) | (27.3,24.0) | (19.3,16.0)

Table 8: The values $(EARL_{1},ESDRL_{1})$ for $RR_{r,s}^{-}-$MCV control
chart when $[a,b]=[0.5,1)$ and for RR${}^{+}_{r,s}-$MCV control chart when
$[a,b]=(1,2]$ with different values of $p,n,\gamma_{0}$ and $\tau$.

Table 8 presents the values of $EARL$ and $ESDRL$ (Expected Standard Deviation
Run Length) for various combinations of $n=\\{5,10,15\\}$,
$\gamma_{0}=\\{0.1,0.2,0.3,0.4,0.5\\}$ and $p=\\{2,3,4\\}$. The same trends as
the case of specific shift size are observed from this table. The value of
$EARL$ in the upper-sided Run Rule control chart corresponding to smaller
values of $(r,s)$ is significantly smaller than that corresponding to larger
values of $(r,s)$ In the contrary, the values of $EARL$ decrease from smaller
$(r,s)$ scheme to larger $(r,s)$ scheme of Run Rules for lower-sided chart.
Therefore, the choice of using RR${}_{2,3}-$MCV, RR${}_{3,4}-$MCV or
RR${}_{4,5s}-$MCV control charts depends on the goal of practitioners: if they
want to detect increasing shifts, they are advised to choose smaller $(r,s)$
scheme of Run Rules (say RR${}_{2,3}-$MCV in this paper); otherwise, the
larger $(r,s)$ scheme of Run Rules (say RR${}_{4,5}-$MCV in this paper) should
be used.

Similar to the specific shift size case, a comparison between the performance
of RRr,s-MCV control charts and the performance of Shewhart-MCV control chart
is provided in Table 9. It has been undertaken by defining the index

$\Delta_{E}=100\times\frac{EARL_{\mathrm{Shewhart}}-EARL_{\mathrm{RR}_{r,s}}}{EARL_{\mathrm{RR}_{r,s}}},$
(18)

where $EARL_{\mathrm{Shewhart}}$ and $EARL_{\mathrm{RR}_{r,s}}$ are the $EARL$
value for the Shewhart-MCV and RR${}_{r,s}-$MCV chart. If $\Delta_{E}>0$, the
RR${}_{r,s}-$MCV charts give better performance than the Shewhart-MCV chart;
if $\Delta_{E}<0$, the Shewhart-MCV chart is better. Once again, the obtained
results show that the RR${}_{r,s}-$MCV charts outperforms the Shewhart-MCV
chart in most cases.

In comparison with the Run Sum MCV control chart suggested by Lim et al. 15,
the Run Rules based charts also have some outstanding advantages. Indeed, to
compare the performance between the Run Sum MCV chart and the Run Rules MCV
chart, we present in Tables 10-11 the values of

$\displaystyle\Delta^{\prime}_{A}$ $\displaystyle=$ $\displaystyle
100\times\frac{ARL_{RS}-ARL_{RR_{r,s}}}{ARL_{RR_{r,s}}},$ (19)
$\displaystyle\Delta^{\prime}_{E}$ $\displaystyle=$ $\displaystyle
100\times\frac{EARL_{RS}-EARL_{RR_{r,s}}}{EARL_{RR_{r,s}}},$ (20)

where $ARL_{RS}$ and $EARL_{RS}$ are the $ARL$ value and the $EARL$ value of
the Run Sum chart. We consider only situations where the values of
$n,\gamma_{0}$ and $p$ in the design of the two kinds of control chart are the
same. In these tables, the Run Sum chart is compared to the lower–sided
RR${}_{4,5}-$MCV chart for decreasing shift size ($\tau<1$) and the upper-
sided RR${}_{2,3}-$MCV chart for increasing shift size ($\tau>1$).

The negative values of $\Delta^{\prime}_{A}$ in Table 10 show that the Run Sum
chart may lead to better performance for the case a specific shift size is
predetermined exactly. In contrast, the positive values of
$\Delta^{\prime}_{E}$ in Table 11 show the better performance of the Run Rules
control chart. It should be considred that in this case, the Run Sum chart was
desinged based on optimizing parameters over an anticipated interval, while
the Run Rules charts do not require any prediction. Even so, the global
performance of the Run Rules charts is still better than that of the Run Sum
chart. In particular, with $n=5,\gamma_{0}=0.1$ and $p=2$, for the upward
chart we have $EARL=29.4$ in the RR${}^{+}_{2.3}-$MCV chart (Table 8 in this
paper) while $EARL=31.37$ in the Run Sum control chart (Table 1 in Lim et al.
15); for the downward we have $EARL=67.9$ in the RR${}^{-}_{4.5}-$MCV chart
(Table 8 in this paper) while $EARL=70.49$ in the Run Sum control chart (Table
3 in Lim et al. 15).

As mentioned above, the use of the Run Sum control chart requires to optimize
the score vectors over a range of shifts that is difficult to predetermine
exactly in practice. When the predetermined value of the shift size $\tau$ is
different from the true shift size, the run-length properties of the designed
control chart could be seriously affected (Tran et al. 35). Meanwhile, the Run
Rules charts only need the determination of a single control limit value for
all shift sizes. This makes the Run Rules MCV chart more easier to implement.

| $p=2$
---|---
| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
(D) | 38 | 41 | 41 |  | 53 | 54 | 53 |  | 61 | 60 | 58
(I) | 5 | 12 | 14 |  | 2 | 12 | 13 |  | -4 | 8 | 8
| $\gamma_{0}=0.2$
(D) | 38 | 41 | 41 |  | 53 | 54 | 53 |  | 60 | 60 | 58
(I) | 5 | 12 | 14 |  | 2 | 12 | 14 |  | -3 | 9 | 9
| $\gamma_{0}=0.3$
(D) | 38 | 41 | 41 |  | 52 | 54 | 53 |  | 60 | 60 | 58
(I) | 6 | 13 | 15 |  | 3 | 13 | 15 |  | -2 | 10 | 11
| $\gamma_{0}=0.4$
(D) | 37 | 40 | 40 |  | 52 | 53 | 52 |  | 59 | 59 | 57
(I) | 7 | 14 | 16 |  | 5 | 15 | 17 |  | 1 | 12 | 14
| $\gamma_{0}=0.5$
(D) | 36 | 39 | 39 |  | 51 | 52 | 51 |  | 59 | 58 | 57
(I) | 10 | 16 | 17 |  | 9 | 17 | 19 |  | 5 | 15 | 17
| $p=3$
| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
(D) | 34 | 41 | 41 |  | 49 | 55 | 53 |  | 58 | 61 | 59
(I) | 1 | 11 | 14 |  | -5 | 11 | 13 |  | -12 | 7 | 8
| $\gamma_{0}=0.2$
(D) | 34 | 41 | 41 |  | 49 | 54 | 53 |  | 57 | 60 | 58
(I) | 1 | 12 | 14 |  | -5 | 11 | 14 |  | -11 | 8 | 9
| $\gamma_{0}=0.3$
(D) | 33 | 41 | 41 |  | 48 | 54 | 53 |  | 57 | 60 | 58
(I) | 2 | 12 | 15 |  | -4 | 12 | 15 |  | -10 | 9 | 11
| $\gamma_{0}=0.4$
(D) | 33 | 40 | 40 |  | 48 | 53 | 52 |  | 56 | 59 | 58
(I) | 3 | 13 | 16 |  | -1 | 14 | 17 |  | -8 | 11 | 14
| $\gamma_{0}=0.5$
(D) | 32 | 39 | 39 |  | 47 | 52 | 52 |  | 55 | 59 | 57
(I) | 6 | 15 | 17 |  | 2 | 16 | 19 |  | -3 | 14 | 17
| $p=4$
| RR2,3-MCV chart |  | RR3,4-MCV chart |  | RR4,5-MCV chart
$\tau$ | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$ |  | $n=5$ | $n=10$ | $n=15$
| $\gamma_{0}=0.1$
(D) | 23 | 41 | 41 |  | 37 | 55 | 54 |  | 46 | 61 | 59
(I) | -8 | 11 | 14 |  | -20 | 10 | 13 |  | -32 | 6 | 8
| $\gamma_{0}=0.2$
(D) | 23 | 41 | 41 |  | 37 | 54 | 53 |  | 46 | 61 | 59
(I) | -7 | 11 | 14 |  | -19 | 10 | 13 |  | -31 | 6 | 9
| $\gamma_{0}=0.3$
(D) | 22 | 40 | 41 |  | 36 | 54 | 53 |  | 45 | 60 | 58
(I) | -7 | 11 | 15 |  | -18 | 11 | 15 |  | -30 | 8 | 11
| $\gamma_{0}=0.4$
(D) | 22 | 40 | 40 |  | 35 | 53 | 52 |  | 45 | 60 | 58
(I) | -5 | 12 | 16 |  | -16 | 13 | 16 |  | -27 | 10 | 14
| $\gamma_{0}=0.5$
(D) | 21 | 39 | 39 |  | 35 | 52 | 52 |  | 44 | 59 | 57
(I) | -3 | 14 | 17 |  | -12 | 15 | 19 |  | -22 | 13 | 17

Table 9: The $\Delta_{E}$ index values for different values of $p,n$ and
$\gamma_{0}$.

| $p=2$ |  | $p=3$
---|---|---|---
$\tau$ | $n=5$ | $n=10$ |  | $n=5$ | $n=10$
| $\gamma_{0}=0.1$
0.50 | -10 | -36 |  | -22 | -30
0.75 | -32 | -15 |  | -37 | -17
0.90 | -21 | -20 |  | -21 | -20
1.10 | -19 | -27 |  | -17 | -26
1.25 | -27 | -29 |  | -25 | -29
1.50 | -26 | -22 |  | -27 | -22
| $\gamma_{0}=0.3$
0.50 | -10 | -33 |  | -24 | -28
0.75 | -32 | -16 |  | -37 | -18
0.90 | -21 | -20 |  | -20 | -20
1.10 | -19 | -26 |  | -16 | -24
1.25 | -27 | -29 |  | -25 | -29
1.50 | -26 | -22 |  | -27 | -23
| $\gamma_{0}=0.5$
0.50 | -11 | -29 |  | -26 | -23
0.75 | -31 | -18 |  | -36 | -20
0.90 | -19 | -19 |  | -19 | -19
1.10 | -17 | -23 |  | -14 | -23
1.25 | -25 | -30 |  | -23 | -30
1.50 | -25 | -24 |  | -25 | -24

Table 10: The values of $\Delta^{\prime}_{A}$ index for $p\in\\{2,3\\}$,
$n\in\\{5,10\\}$ and $\gamma_{0}\in\\{0.1,0.3,0.5\\}$.

| $p=2$ |  | $p=3$
---|---|---|---
$\tau$ | $n=5$ | $n=10$ |  | $n=5$ | $n=10$
| $\gamma_{0}=0.1$
(D) | 4 | 30 |  | -5 | 27
(I) | 6 | 18 |  | 3 | 16
| $\gamma_{0}=0.3$
(D) | 3 | 29 |  | -6 | 25
(I) | 5 | 16 |  | 2 | 14
| $\gamma_{0}=0.5$
(D) | 2 | 25 |  | -6 | 22
(I) | 3 | 11 |  | 0 | 9

Table 11: The $\Delta^{\prime}_{E}$ index values for $p\in\\{2,3\\}$,
$n\in\\{5,10\\}$ and $\gamma_{0}\in\\{0.1,0.3,0.5\\}$.

##  5 Illustrative example

An illustrative example of RR${}_{r,s}-$MCV control chart is given in this
Section. Let us consider a sintering process in an Italian company that
manufactures sintered mechanical parts, which is introduced in Lim et al. 15.
The data are recorded from a spring manufacturing process, for which the
quality characteristics are the spring inner diameter ($X_{1}$) and the spring
elasticity ($X_{2}$). From Phase I, we have the estimated value of
$\gamma_{0}=0.089115$, while (according to Lim et al. 15) the assumption that
the MCV during Phase I is constant holds. The data collected during the Phase
II process with sample size $n=5$ are shown in Table 12. Further details on
the process can be found in Lim et al. 15. From the obtained results in
Section 3, the control limits for the different control charts are as follows.

  * •

For the upper-sided Shewhart$-$MCV chart, $UCL_{\mathrm{Shewhart}}=0.1691.$

  * •

For the upper-sided RR${}^{+}_{2,3}-\mathrm{MCV}$ chart,
$UCL_{\mathrm{RR}_{2,3}}=0.1296.$

  * •

For the upper-sided RR${}^{+}_{3,4}-\mathrm{MCV}$ chart,
$UCL_{\mathrm{RR}_{3,4}}=0.1106.$

  * •

For the upper-sided RR${}^{+}_{4,5}-\mathrm{MCV}$ chart,
$UCL_{\mathrm{RR}_{4,5}}=0.0986.$

Sample number $t$ | $\bar{X}_{1,t}$ | $\bar{X}_{2t}$ | $S^{2}_{1t}$ | $S^{2}_{2t}$ | $S_{12t}$ | $\hat{\gamma}_{i}$
---|---|---|---|---|---|---
1 | 7.781 | 1.592 | 1.164 | 0.734 | 0.35645 | 0.113710
2 | 7.385 | 1.804 | 1.006 | 1.667 | 0.96049 | 0.104890
3 | 7.988 | 2.260 | 0.762 | 0.359 | 0.17373 | 0.108870
4 | 8.189 | 2.100 | 1.885 | 0.470 | 0.13026 | 0.156790
5 | 7.436 | 2.061 | 1.404 | 0.519 | 0.08280 | 0.139290
6 | 6.746 | 2.289 | 0.846 | 0.811 | 0.43835 | 0.133240
7 | 7.356 | 1.917 | 0.197 | 2.587 | 0.01597 | 0.059996
8 | 8.492 | 1.845 | 1.460 | 1.746 | 1.42051 | 0.055093
9 | 7.272 | 1.580 | 1.353 | 0.345 | 0.27988 | 0.117710
10 | 7.585 | 1.568 | 1.098 | 0.788 | 0.41252 | 0.109610
11 | 7.734 | 1.709 | 0.952 | 0.228 | 0.11462 | 0.102440
12 | 8.160 | 1.498 | 1.598 | 1.178 | 1.00757 | 0.122950
13 | 7.102 | 2.661 | 1.508 | 0.945 | 0.73607 | 0.101260
14 | 8.392 | 1.883 | 0.536 | 0.706 | 0.23234 | 0.085637
15 | 7.592 | 2.531 | 0.256 | 0.563 | 0.24827 | 0.043489
16 | 8.141 | 2.093 | 0.394 | 0.603 | 0.25584 | 0.072202
17 | 7.883 | 2.490 | 1.321 | 1.179 | 0.65037 | 0.142430
18 | 7.886 | 2.877 | 0.883 | 1.431 | 0.22524 | 0.106680
19 | 7.830 | 1.008 | 0.878 | 0.558 | 0.14223 | 0.112090
20 | 8.196 | 1.482 | 0.791 | 0.220 | 0.13724 | 0.088460

Table 12: Illustrative example of Phase II dataset. Figure 2:
RR${}^{+}_{2,3}-$MCV control chart corresponding to Phase II data set in Table
12 Figure 3:  RR${}^{+}_{3,4}-$MCV control chart corresponding to Phase II
data set in Table 12 Figure 4:  RR${}^{+}_{4,5}-$MCV control chart
corresponding to Phase II data set in Table 12

The corresponding values of $\hat{\gamma}_{i}$ are presented in the rightmost
column of Table 12 and plotted in Figures 2-4, respectively. Each figure
consists of the $UCL$ for the upper-sided Shewhart - MCV chart along with the
$UCL$ of the $RR^{+}_{r,s}-$MCV control chart, for
$(r,s)\in\\{(2,3),(3,4),(4,5)\\}$. It is not difficult to see that the
RR${}^{+}_{2,3}-$MCV control chart signals the occurrence of the out-of-
control condition by two out of three successive plotting points #4 and #5
above the control limit $UCL_{\mathrm{RR}_{2,3}}$, see Figure 2. The
RR${}_{3,4}-$MCV control chart signals the occurrence of the out-of-control
condition by three out of four successive plotting points #4, #5 and #6 above
the control limit $UCL_{\mathrm{RR}_{3,4}}$, see Figure 3. The
RR${}_{4,5}-$MCV control chart signals the occurrence of the out-of-control
condition by four out of five successive plotting points #1, #2, #3 and #4
above the control limit $UCL_{\mathrm{RR}_{4,5}}$, see Figure 4. The figures
2-4 also show that all the points are plotted below the $UCL_{Shewhart}$. That
is to say, in this situation, the upper Shewhart - MCV control chart (designed
in Yeong et al. 14) fails to detect the out-control condition detected by the
RR${}_{r,s}-$MCV control charts. In addition, it is worth noting that similar
conclusions have been reached by Lim et al. 15 with the Run Sum MCV control
chart. Specifically, their Run Sum chart gives an out-of-control signal at
sample 4. However, the design of this run sum control chart requires the a
priori selection for a shift of interest so as to determine the optimal scores
for the chart. This means that the run sum chart is designed so as to be the
optimal one in the selection of a specific shift. Their choice, for
illustrative purposes, was a 25% increase in the IC value of the MCV. Clearly,
this is not required here where only the value of the $UCL$ is needed for the
application of the Run Rules chart. Before closing this section we mention
that the lower-sided $RR^{-}_{r,s}-$ MCV control charts can be constructed in
a similar manner. The respective limits are 0.010029, 0.02403, 0.03464,
0.04275 . As in the case of the lower-sided run sum chart of Lim et al. 15 no
out-of-control signal is given. However, due to space constraints, we do not
provide the respective figures.

##  6 Concluding remarks

In this paper, we have investigated the one-sided control charts with Run
Rules for monitoring the coefficient of variation of multivariate data. Two
one-sided charts were considered to detect separately both increases and
decreases in the multivariate CV. The performance of proposed charts is
evaluated through $ARL$ for deterministic shift size and $EARL$ for unknown
shift size. The numerical results showed that the Run Rules control charts
enhance the performance of Shewhart control chart significantly. For purpose
of optimizing the performance of Run Rules charts, it is recommended to use
the RR${}^{-}_{4,5}-$MCV for detecting decreasing process shifts and
RR${}^{+}_{2,3}-$MCV for detecting increasing process shifts. Moreover, under
certain conditions, this careful choice of the Run Rules charts also lead to
an improved efficiency compared to the Run Sum control chart for MCV.

## References

  * Castagliola et al. 2011 P. Castagliola, A. Amdouni, H. Taleb, and G. Celano.  Monitoring the coefficient of variation using ewma charts.  _Journal of Quality Technology_ , 43(3):249–265, 2011.
  * Kang et al. 2007 C.W. Kang, M.S. Lee, Y.J. Seong, and D.M. Hawkins.  A control chart for the coefficient of variation.  _Journal of Quality Technology_ , 39(2):151–158, 2007.
  * Calzada and Scariano 2013 M.E. Calzada and S. M. Scariano.  A synthetic control chart for the coefficient of variation.  _Journal of Statistical Computation and Simulation_ , 83(5):853–867, 2013.
  * Tran and Tran 2016 P.H. Tran and K. P. Tran.  The efficiency of CUSUM schemes for monitoring the coefficient of variation.  _Applied Stochastic Models in Business and Industry_ , 32(6):870–881, 2016.
  * Castagliola et al. 2013a P. Castagliola, A. Achoure, H. Taleb, G. Celano, and S. Psarakis.  Monitoring the coefficient of variation using control charts with run rules.  _Quality Technology & Quantitative Management_, 10:75–94, 2013a.
  * Castagliola et al. 2013b P. Castagliola, A. Achouri, H. Taleb, G. Celano, and S. Psarakis.  Monitoring the coefficient of variation using a variable sampling interval control chart.  _Quality and Reliability Engineering International_ , 29(8):1135–1149, 2013b.
  * You et al. 2016 H.W. You, B.C.M. Michael Khoo, P. Castagliola, and A. Haq.  Monitoring the coefficient of variation using the side sensitive group runs chart.  _Quality and Reliability Engineering International_ , 32(5):1913–1927, 2016.
  * Teoh et al. 2017 W.L. Teoh, M.B. Khoo, P. Castagliola, W.C. Yeong, and S.Y. Teh.  Run-sum control charts for monitoring the coefficient of variation.  _European Journal of Operational Research_ , 257(1):144–158, 2017.
  * Zhang et al. 2014 J. Zhang, Z. Li, B. Chen, and Z. Wang.  A new exponentially weighted moving average control chart for monitoring the coefficient of variation.  _Computers & Industrial Engineering_, 78:205–212, 2014\.
  * Yeong et al. 2017 W.C. Yeong, M.B.C. Khoo, L.K. Teoh, and M.A. Rahim.  Monitoring the coefficient of variation using a variable sampling interval ewma chart.  _Journal of Quality Technology_ , 49(3):380–401, 2017.
  * Zhang et al. 2018 Jiujun Zhang, Zhonghua Li, and Zhaojun Wang.  Control chart for monitoring the coefficient of variation with an exponentially weighted moving average procedure.  _Quality and Reliability Engineering International_ , 34(2):188–202, 2018.
  * Shu and Jiang 2008 Lianjie Shu and Wei Jiang.  A new ewma chart for monitoring process dispersion.  _Journal of Quality Technology_ , 40(3):319–331, 2008.
  * Albert and Zhang 2010 A. Albert and L. Zhang.  A novel definition of the multivariate coefficient of variation.  _Biometrical Journal_ , 52(5):667–675, 2010.
  * Yeong et al. 2016 W.C. Yeong, M. B. C. Khoo, W. L.Teoh, and P. Castagliola.  A control chart for the multivariate coefficient of variation.  _Quality and Reliability Engineering International_ , 32(3):1213–1225, 2016.
  * Lim et al. 2017 A. Lim, M.B.C Khoo, W.L. Teoh, and A. Haq.  Run sum chart for monitoring multivariate coefficient of variation.  _Computers & Industrial Engineering_, 109:84–95, 2017\.
  * Champ and Woodall 1987 C.W. Champ and W.H. Woodall.  Exact results for shewhart control charts with supplementary runs rules.  _Technometrics_ , 29(4):393–399, 1987.
  * Klein 2000 M. Klein.  Two Alternatives to the Shewhart $\bar{X}$ Control Chart.  _Journal of Quality Technology_ , 32:427–431, 2000.
  * Khoo 2004 M.B.C. Khoo.  Design of Runs Rules Schemes.  _Quality Engineering_ , 16(2):27–43, 2004.
  * Antzoulakos and Rakitzis 2008 D.L. Antzoulakos and A.C. Rakitzis.  The Modified $r$ out of $m$ Control Chart.  _Communications in Statistics – Simulation and Computation_ , 37(2):396–408, 2008.
  * Koutras et al. 2007 M.V. Koutras, S. Bersimis, and P. Maravelakis.  Statistical process control using shewhart control charts with supplementary runs rules.  _Methodology and Computing in Applied Probability_ , 9:207–224, 2007.
  * Acosta and Pignatiello 2009 C. Acosta and J. Pignatiello.  Arl-design of s charts with k-of-k runs rules.  _Communications in Statistics - Simulation and Computation_ , 38:1625–1639, 2009.
  * Amdouni et al. 2016 A. Amdouni, P. Castagliola, H. Taleb, and G. Celano.  One-sided run rules control charts for monitoring the coefficient of variation in short production runs.  _European Journal of Industrial Engineering_ , 10(5):639–663, 2016.
  * Faraz et al. 2014 A. Faraz, G. Celano, E. Saniga, C. Heuchenne, and S. Fichera.  The variable parameters t2 chart with run rules.  _Statistical Papers_ , 55(4):933–950, 2014.
  * Tran et al. 2016 K.P. Tran, P. Castagliola, and G. Celano.  Monitoring the Ratio of Two Normal Variables Using Run Rules Type Control Charts.  _International Journal of Production Research_ , 54(6):1670–1688, 2016.
  * Chew et al. 2019 X. Y. Chew, K. W. Khaw, and W. C. Yeong.  The efficiency of run rules schemes for the multivariate coefficient of variation: a markov chain approach.  _Journal of Applied Statistics_ , pages 1–21, 2019.
  * Tran 2016 K.P. Tran.  The efficiency of the 4-out-of-5 Runs Rules scheme for monitoring the Ratio of Population Means of a Bivariate Normal distribution.  _International Journal of Reliability, Quality and Safety Engineering_ , 2016.  In press, DOI: 10.1142/S0218539316500200.
  * Reyment 1960 R.A. Reyment.  Studies on nigerian upper cretaceous and lower tertiary ostracoda. part 1. senonian and maestrichtian ostracoda 1960.  _Stockholm Contributions in Geology_ , 7:1?238, 1960.
  * Valen 1974 L. Van Valen.  Multivariate structural statistics in natural history.  _Journal of Theoretical Biology_ , 45(1):235–247, 1974.
  * Nikulin and Voinov 2011 M. Nikulin and V. Voinov.  Unbiased estimators and their applications.  In _International Encyclopedia of Statistical Science_ , pages 1619–1621. Springer, 2011.
  * Brook and Evans 1972 D. Brook and D.A. Evans.  An approach to the probability distribution of CUSUM run length.  _Biometrika_ , 59(3):539–549, 1972.
  * Fu et al. 2003 J. Fu, F. Spiring, and H. Xie.  On the average run lengths of quality control schemes using a markov chain approach.  _Statistics & Probability Letters_, 56:369–380., 2003\.
  * Li et al. 2014 Z. Li, C. Zou, Z. Gong, and Z. Wang.  The computation of average run length and average time to signal: an overview.  _Journal of Statistical Computation and Simulation_ , 84(8):1779–1802, 2014.
  * Chen and Chen 2007 A. Chen and Y. K. Chen.  Design of EWMA and CUSUM control charts subject to random shift sizes and quality impacts.  _IIE Transactions_ , 39(12):1127–1141, 2007.
  * Celano et al. 2014 G. Celano, P. Castagliola, A. Faraz, and S. Fichera.  Statistical Performance of a Control Chart for Individual Observations Monitoring the Ratio of two Normal Variables.  _Quality and Reliability Engineering International_ , 30(8):1361–1377, 2014.
  * Tran et al. 2019 K. P. Tran, H. D. Nguyen, P. H. Tran, and C. Heuchenne.  On the performance of cusum control charts for monitoring the coefficient of variation with measurement errors.  _The International Journal of Advanced Manufacturing Technology_ , pages 1–15, 2019.