The Efficiency of the VSI Exponentially Weighted Moving Average Median Control Chart

Abstract

In the literature, median type control charts have been widely investigated as easy and efficient means to monitor the process mean when observations are from a normal distribution. In this work, a Variable Sampling Interval (VSI) Exponentially Weighted Moving Average (EWMA) median control chart is proposed and studied. A Markov chain method is used to obtain optimal designs and evaluate the statistical performance of the proposed chart. Furthermore, practical guidelines and comparisons with the basic EWMA median control chart are provided. Results show that the proposed chart is considerably more efficient than the basic EWMA median control chart. Finally, the implementation of the proposed chart is illustrated with an example in the food production process.

Full text

The Efficiency of the VSI Exponentially Weighted Moving Average
Median Control Chart
Kim Phuc Tran∗1, Philippe Castagliola2, Thi Hien Nguyen1, and Anne Cuzol1
1Laboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, Université de Bretagne-Sud,
Vannes, France
2Université de Nantes & LS2N UMR CNRS 6004, Nantes, France
May 14, 2018
Abstract
In the literature, median type control charts have
been widely investigated as easy and efficient means to
monitor the process mean when observations are from a
normal distribution. In this work, a Variable Sampling
Interval (VSI) Exponentially Weighted Moving Average
(EWMA) median control chart is proposed and studied.
A Markov chain method is used to obtain optimal de-
signs and evaluate the statistical performance of the pro-
posed chart. Furthermore, practical guidelines and com-
parisons with the basic EWMA median control chart are
provided. Results show that the proposed chart is con-
siderably more efficient than the basic EWMA median
control chart. Finally, the implementation of the pro-
posed chart is illustrated with an example in the food
production process.
Keywords: EWMA, VSI, Median, Control chart, Order statis-
tics.
1 Introduction
Statistical Process Control (SPC) is a method of quality control
which uses statistical methods in achieving process stability
and improving capability through the reduction of variabil-
ity, see Montgomery [1]. It’s well known that control charts
are the fundamental tool for SPC applications. There are nu-
merous types of control charts, the most common ones are
the Shewhart control charts, the cumulative sum (CUSUM)
control charts and the exponentially weighted moving average
(EWMA) control charts. The EWMA control charts have a
“built in” mechanism for incorporating information from all
previous subgroups by means of weights decreasing geomet-
rically with the sample mean age. Thus EWMA type control
charts are very effective for the detection of small or moderate
process shifts, see Tran et al. [2]. Their properties and design
stategies have been thoroughly investigated by many authors.
For further details see, for instance, Robinson and Ho [3],
Hunter [4], Crowder [5], Lucas and Saccucci [6], Tran et al. [2]
to name a few.
In recent years, many researchers have focused on develop-
ing advanced control charts with various applications in man-
ufacturing and service processes, for example, see Castagliola
and Figueiredo [7], Huang [8], Da Costa Quinino et al. [9],
Tran et al. [10], Castagliola et al. [11], Tran [12], Tran et al.
[13] and Tran [14]. Among these control charts, median ( ˜
X)
type charts have been widely investigated as easy and efficient
means to monitor the mean. The main advantages of median
type charts are that they are simpler than mean ( ¯
X) charts and
that they are robust against outliers, contamination or small
deviations from normality, see Castagliola et al. [11].
In the SPC literature, the EWMA median chart was intro-
duced by Castagliola [15] (EWMA- ˜
X) with fast detection of
assignable causes. Then, a generally weighted moving average
median (GWMA- ˜
X) control chart has been studied by Sheu
and Yang [16] as a continuation to improve the statistical per-
formance of median type control charts. When the parameters
are estimated, Castagliola and Figueiredo [7] and Castagliola
et al. [11] developed a Shewhart median chart and a EWMA- ˜
X
chart, respectively, with estimated control limits to monitor the
mean value of a normal process. Very recently, Lin et al. [17]
investigated the performances of the EWMA- ˜
Xcontrol chart
under several distributions. As a result, the EWMA- ˜
Xis always
more efficient than the EWMA- ¯
Xchart in detecting shifts in
the process mean if the data follow a heavy-tailed distribution.
Finally, Tran [18] proposed and studied the Run Rules She-
whart median control charts (RRr,s−˜
Xcharts).
It is known that, the EWMA- ˜
Xcontrol chart suggested
by Castagliola [15] is a Fixed Sampling Interval (FSI) control
chart. By definition, an adaptive control chart involves varying
at least one of the chart’s parameters, such as the sampling
interval or the sample size. Variable Sampling Interval (VSI)
∗kim-phuc.tran@univ-ubs.fr (corresponding author)
Proceedings of the 24th ISSAT International Conference on Reliability and Quality in Design
August 2-4, 2018 - Toronto, ON, Canada
page 203

control charts are adaptive control charts where the sampling
intervals vary as a function of what is observed from the pro-
cess. The VSI control charts are demonstrated to detect process
changes faster than FSI control charts. The idea is that the time
interval until the next sample should be short, if the position
of the last plotted control statistic indicates a possible out-of-
control situation; and long, if there is no indication of a change.
Most work on developing VSI control charts has been done
for the problem of monitoring the mean of the process (see
Reynolds [19], Reynolds et al. [20] and Castagliola et al. [21]).
In this paper, we propose a VSI EWMA- ˜
Xcontrol chart as
a logical extension of the control chart developed by Castagli-
ola [15]. The goal of this paper is to show how the VSI behaves
with respect to the basic EWMA median control chart. The rest
of this paper proceeds as follows: in Section 2, a brief review of
the FSI EWMA- ˜
Xcontrol chart is provided; Section 3provides
a VSI version of the FSI EWMA- ˜
Xcontrol chart; in Section
4, the run length performances of proposed chart are defined by
using the Markov Chain-based approach; in Section 5, the com-
putational results and the tables reporting the optimal design
parameters of the VSI EWMA- ˜
Xchart are presented. Section
6presents an illustrative example and, finally, some concluding
remarks and recommendations are made in Section 7.
2 The FSI EWMA- ˜
Xcontrol chart
Let us assume that, at each sampling period i=1,2,.. ., we col-
lect a sample of nindependent random variables {Xi,1,...,Xi,n}.
We assume that each Xi,jfollows a normal distribution N(µ0+
δ σ0,σ0),j=1,...,n,µ0is the in-control mean value, σ0is
the in-control standard deviation and δis the magnitude of the
standardized mean shift. If δ=0 the process is in-control and,
when δ6=0, the process is out-of-control. Let ˜
Xibe the sample
median of subgroup i, i.e.
˜
Xi=




Xi,((n+1)/2)if nis odd
Xi,(n/2)+Xi,(n/2+1)
2if nis even
(1)
where {Xi,(1),Xi,(2),. ..,Xi,(n)}is the ordered i-th subgroup.
In the rest of this paper, without loss of generality, we as-
sume that the sample size nis an odd value. Let Z1,Z2,...
be the EWMA sequence obtained from ˜
X1,˜
X2,..., i.e. for
i∈ {1,2,.. .},
Zi= (1−λ)Zi−1+λ˜
Xi,(2)
where Z0=µ0and λ∈(0,1]is a smoothing constant. If
the in-control mean value µ0and the standard deviation σ0are
assumed known, the control limits of the EWMA- ˜
Xchart for
the median are simply equal to
LCL =µ0−Krλ
2−λσ0,(3)
UCL =µ0+Krλ
2−λσ0,(4)
where K>0 is a constant that depends on nand on the
desired in-control performance.
3 Implementation of the VSI EWMA- ˜
X
control chart
In this section, a VSI version of the FSI EWMA- ˜
Xcontrol chart
described in the previous section is presented (denoted as VSI
EWMA- ˜
X). The control statistic Zifor the VSI EWMA- ˜
Xcon-
trol chart is given by (2). The upper (UCL) and lower (LCL)
control limits of the VSI EWMA- ˜
Xcontrol chart can be easily
calculated as:
LCL =µ0−Krλ
2−λσ0,(5)
UCL =µ0+Krλ
2−λσ0,(6)
where K≥0 is a constant influencing the width of the control
interval.
For the FSI control chart, the sampling interval is a fixed
value h0. As for the VSI control chart, the sampling interval
depends on the current value of Zi. A longer sampling in-
terval hLis used when the control statistic falls within region
RL= [LW L,UW L]defined as:
LW L =µ0−Wrλ
2−λσ0,(7)
UW L =µ0+Wrλ
2−λσ0,(8)
where Wis the warning limit coefficient of the VSI EWMA-
˜
Xcontrol chart that determines the proportion of times that
the control statistic falls within the long and short sampling
regions. On the other hand, the short sampling interval hS
is used when the control statistic falls within the region RS=
[LCL,LW L)∪(UW L,UCL]. The process is considered out-of-
control and action should be taken whenever Zifalls outside
the range of the control limits [LCL,UCL]. In order to evalu-
ate the ARL and SDRL of the VSI EWMA- ˜
Xcontrol chart, we
follow the discrete Markov chain approach originally proposed
by Brook and Evans [22] . In Appendix, the discrete Markov
chain approach for VSI EWMA- ˜
Xcontrol chart is provided.
4 Optimal design of the VSI EWMA- ˜
X
control chart
In the literature, the Average Run Length (ARL), defined as the
average number of samples before the control chart signals an
out-of-control condition or issues a false alarm, and the Aver-
age Time to Signal (ATS), which is the expected value of the
time between the occurrence of a special cause and a signal
from the chart are used as the performance measures of control
charts, see Castagliola et al. [21]. It is well known that, when
Proceedings of the 24th ISSAT International Conference on Reliability and Quality in Design
August 2-4, 2018 - Toronto, ON, Canada
page 204

the process is in-control, it is better to have a large AT S, since
in this operating condition a signal represents a false alarm (in
this case, the AT S will be denoted as AT S0). On the other hand,
after the parameter of the process under control has shifted, it
is preferable to have an ATS that is as small as possible (in this
case, the AT S will be denoted as AT S1).
For a FSI model, the AT S is a multiple of the ARL since the
sampling interval hFis fixed. Thus, in this case we have the
following expression:
AT SFSI =hF×ARLFSI.(9)
For a VSI model, the ATS is defined as:
AT SVSI =E(h)×ARLVSI.(10)
where E(h)is the expected sampling interval value.
According to Castagliola et al. [21], for VSI type con-
trol charts, we need to define them with the same in-control
ARL =ARL0and the same in-control average sampling interval
E0(h). For FSI-type control charts, the sampling interval is
set equal to hS=hL=hF=1 time units. Then, the in-control
expected sampling interval of the VSI chart is set equal to
E0(h) = 1 time unit to ensure AT S0=ARL0time unit for both
FSI and VSI type control charts. The value of hSrepresents
the shortest feasible time interval between subgroups from the
process, see Castagliola et al. [21] for more details. Then, in
this paper we will consider the impact on the expected time
until detection, using small but non-zero values of hS.
The design procedure of VSI EWMA- ˜
Xcontrol chart is im-
plemented by finding out the optimal combination of parame-
ters λ∗,K∗and h∗
Lwhich minimize the out-of-control AT S for
predefined values of δ,W,hS,nand ATS0, i.e., the optimization
scheme of the VSI EWMA- ˜
Xconsists in finding the optimal
parameters λ∗,K∗and h∗
Lsuch that
(λ∗,K∗,h∗
L) = argmin
(λ,K,hL)
AT S(n,λ,K,W,hL,hS,δ)(11)
subject to the constraint
E0(h) = 1,
AT S(n,λ,K,W,hL,hS,δ=0) = ATS0.(12)
Similar to Tran and Tran [23], the choice of the optimal
combination of parameters generally entails two steps:
1. Find the potential combinations (λ,K,hL)such that
AT S =AT S0and E0(h) = 1.
2. Choose, among these potential combinations (λ,K,hL),
the one (λ∗,K∗,h∗
L)that allows for the best performance,
i.e. the smallest “out-of-control” AT S value for a partic-
ular shift δ.
In this study, like in Tran and Tran [23], in order to find
these optimal combinations (λ∗,K∗,h∗
L)we simultaneously use
a non-linear equation solver coupled to an optimization algo-
rithm (developed with Scicoslab software).
5 Numerical results
Optimal designs were obtained for the FSI and VSI EWMA-
˜
Xcontrol charts, for all combinations of δ∈[0.5,2]and
n={3,5,7,9}. The sampling interval hFof the FSI charts
has been set equal to 1 time unit. The shorter time interval hS
can assume the following values: 0.5 and 0.1 time units. The
optimal combinations of design parameters (λ∗,K∗,h∗
L)have
been selected by constraining the in-control AT S at the value
AT S0=370.4 and the in-control expected sampling interval
of the VSI chart is set equal to E0(h) = 1. To ensure a fair
comparison, the ARL0of EWMA- ˜
Xchart is set as 370.4. The
optimal combinations of design parameters (λ∗,K∗,h∗
L)of the
VSI EWMA- ˜
Xcontrol chart are presented in Tables 1-4. Some
simple conclusions can be drawn from Tables 1-4:
n=3
hS=0.5
δW=0.9W=0.6W=0.3W=0.2W=0.1
0.1(0.0500,1.6686)(0.0500,1.6686) (0.0500,1.6686) (0.0513,1.6750) (0.0514,1.6750)
(1.08,139.5) (1.24,135.9) (1.81,133.8) (2.54,134.7) (4.68,135.9)
0.2(0.0500,1.6686) (0.0500,1.6686) (0.0500,1.6686) (0.0500,1.6686) (0.0514,1.6750)
(1.08,50.0) (1.24,47.3) (1.81,46.2) (2.40,46.4) (4.68,48.6)
0.3(0.0514,1.6750) (0.0518,1.6767) (0.0500,1.6686) (0.0517,1.6763) (0.0535,1.684)
(1.09,26.1) (1.26,24.6) (1.81,24.3) (2.55,24.9) (4.68,26.9)
0.5(0.0989,1.8090) (0.1095,1.8273) (0.1073,1.8234) (0.1046,1.8190) (0.1124,1.8315)
(1.10,11.8) (1.28,11.2) (1.89,11.4) (2.46,11.8) (4.49,13.8)
0.7(0.1605,1.8883) (0.1690,1.8957) (0.1563,1.8844) (0.1742,1.9000) (0.1798,1.9043)
(1.10,6.9) (1.28,6.7) (1.87,7.0) (2.58,7.6) (4.39,9.4)
1.0(0.2743,1.9557) (0.2773,1.9569) (0.2759,1.9563) (0.2783,1.9572) (0.2885,1.9609)
(1.10,4.0) (1.29,3.9) (1.91,4.4) (2.54,5.0) (4.31,6.8)
1.5(0.4746,2.0017) (0.4681,2.0008) (0.4685,2.0008) (0.4745,2.0016) (0.4293,1.9950)
(1.10,2.2) (1.28,2.3) (1.89,2.9) (2.51,3.5) (4.26,5.2)
2.0(0.6883,2.0195) (0.6890,2.0196) (0.5499,2.0100) (0.5498,2.0100) (0.4293,1.9950)
(1.11,1.6) (1.30,1.7) (1.89,2.3) (2.50,2.9) (4.26,4.7)
hS=0.1
δW=0.9W=0.6W=0.3W=0.2W=0.1
0.1(0.0500,1.6686) (0.0500,1.6686) (0.0500,1.6686) (0.0500,1.6686) (0.0500,1.6686)
(1.15,134.2) (1.44,127.7) (2.47,124.0) (3.52,123.9) (6.52,125.5)
0.2(0.0500,1.6686) (0.0500,1.6686) (0.0500,1.6685) (0.0500,1.6686) (0.0500,1.6686)
(1.15,44.8) (1.44,40.0) (2.47,38.0) (3.52,38.5) (6.52,41.0)
0.3(0.0515,1.6752) (0.0515,1.6753) (0.0577,1.701) (0.0502,1.6693) (0.0643,1.7240)
(1.15,22.1) (1.46,19.4) (2.57,19.0) (3.51,19.8) (7.53,23.4)
0.5(0.0987,1.8090) (0.1140,1.8339) (0.1278,1.8530) (0.1325,1.8589) (0.1394,1.8669)
(1.17,9.3) (1.50,8.3) (2.59,8.5) (3.90,9.6) (7.19,12.8)
0.7(0.1737,1.8995) (0.1952,1.9154) (0.2088,1.9241) (0.2142,1.9273) (0.2269,1.9344)
(1.18,5.3) (1.50,4.8) (2.67,5.4)3.81,6.5) (7.02,9.6)
1.0(0.3086,1.9676) (0.3239,1.9721) (0.3326,1.9746) (0.3415,1.9769) (0.3661,1.9829)
(1.19,3.0) (1.52,2.9) (2.63,3.7) (3.74,4.8) (6.90,7.9)
1.5(0.5211,2.0072) (0.5218,2.0072) (0.5366,2.0087) (0.5498,2.0100) (0.4025,1.9903)
(1.18,1.7) (1.50,1.9) (2.59,2.9) (3.70,4.0) (6.88,7.2)
2.0(0.7043,2.0203) (0.5498,2.0100) (0.5498,2.0100) (0.5498,2.0100) (0.4025,1.9903)
(1.19,1.3) (1.50,1.6) (2.59,2.7) (3.70,3.8) (6.88,7.0)
Table 1: Optimal couples (λ∗,K∗)(first row of each block)
and values of (hL,ATS1)(second row of each block) of the VSI
EWMA- ˜
Xcontrol chart for n=3.
•Given the values of δ,nand W, the value of AT S de-
pends on hS. In particular, with smaller values of hS, the
value of ATS1decreases. For example, when δ=0.1,
n=3, W=0.6 we have ATS1=135.9 for hS=0.5 and
AT S1=127.7 for hS=0.1, see Table 1.
•For a defined value of hS, it is obvious that when Wde-
creases the length of the long sampling interval hLin-
creases. For example, when δ=0.1, n=3, hS=0.5 we
have hL=1.08 for W=0.9 and hL=4.68 for W=0.1,
see Table 1.
Proceedings of the 24th ISSAT International Conference on Reliability and Quality in Design
August 2-4, 2018 - Toronto, ON, Canada
page 205

•The VSI EWMA- ˜
Xcontrol chart is directly compared to
the FSI EWMA- ˜
Xcontrol chart, to evaluate the impact
of the adaptive feature on the statistical performance of
the original static chart. As expected, the results in Ta-
bles 1-4clearly indicate that the VSI EWMA- ˜
Xchart is
superior to the FSI EWMA- ˜
Xcontrol chart. For exam-
ple, when δ=0.1, n=3, W=0.6 and hS=0.5 we have
AT S1=135.9 for VSI EWMA- ˜
Xchart and ARL =146.1
for FSI EWMA- ˜
Xcontrol chart, see Table 3 in Castagli-
ola [15].
n=5
hS=0.5
δW=0.9W=0.6W=0.3W=0.2W=0.1
0.1(0.0500,1.3341)(0.0500,1.3341) (0.0500,1.3341) (0.0500,1.3341) (0.0500,1.3341)
(1.03,106.1) (1.14,101.4) (1.60,98.0) (2.05,97.7) (3.60,98.6)
0.2(0.0500,1.3341) (0.0500,1.3341) (0.0500,1.3341) (0.0500,1.3341) (0.0500,1.3341)
(1.03,36.7) (1.14,33.7) (1.60,32.1) (2.05,32.3) (3.60,33.6)
0.3(0.06134,1.3705) (0.0619,1.3719) (0.0689,1.3899) (0.0710,1.3951) (0.0733,1.4002)
(1.04,19.6) (1.15,17.9) (1.63,17.3) (2.11,17.6) (3.53,18.9)
0.5(0.1290,1.4828) (0.1388,1.4921) (0.1467,1.4989) (0.1483,1.5002) (0.1526,1.5036)
(1.04,8.8) (1.16,8.1) (1.63,8.0) (2.15,8.4) (3.83,10.0)
0.7(0.2175,1.5423) (0.2304,1.5479) (0.2325,1.5487) (0.2264,1.5462) (0.2400,1.5517)
(1.05,5.1) (1.17,4.8) (1.65,5.0) (2.12,5.4) (3.76,7.0)
1.0(0.3773,1.5869) (0.3721,1.5860) (0.3662,1.5850) (0.3684,1.5854) (0.3804,1.5874)
(1.05,3.0) (1.17,2.8) (1.64,3.2) (2.21,3.7) (3.71,5.2)
1.5(0.6405,1.6119) (0.6369,1.6117) (0.6400,1.6119) (0.6450,1.6121) (0.6579,1.6127)
(1.05,1.7) (1.17,1.7) (1.68,2.2) (2.19,2.7) (3.67,4.2)
2.0(0.8517,1.6182) (0.8540,1.6182) (0.8594,1.6183) (0.8634,1.6184) (0.8728,1.6185)
(1.05,1.2) (1.17,1.3) (1.67,1.8) (2.19,2.3) (3.66,3.8)
hS=0.1
δW=0.9W=0.6W=0.3W=0.2W=0.1
0.1(0.0500,1.3341) (0.0500,1.3341) (0.0500,1.3341) (0.0500,1.3341) (0.0500,1.3341)
(1.06,102.7) (1.25,94.3) (2.08,88.1) (2.89,87.6,, 370.4) (5.69,89.1)
0.2(0.0500,1.3341 (0.0516,1.3402) (0.0500,1.3341) (0.0500,1.3341) (0.0500,1.3341)
(1.06,33.7) (1.27,28.2,, 370.4) (2.08,25.5) (2.89,25.8,, 370.4) (5.69,28.2)
0.3(0.0695,1.3915) (0.0644,1.3788) (0.0791,1.4124) (0.0837,1.4212) (0.0772,1.4085)
(1.07,17.2) (1.27,14.2) (2.11,13.0) (2.97,13.5) (5.53,15.8)
0.5(0.1573,1.5075) (0.1586,1.5082) (0.1543,1.5050) (0.1799,1.5225) (0.1902,1.5286)
(1.08,7.2) (1.31,6.0) (2.12,5.9) (3.05,6.6) (6.03,9.4)
0.7(0.2618,1.5596) (0.2629,1.5598) (0.2784,1.5647) (0.2264,1.5462) (0.3018,1.5711)
(1.09,4.1) (1.31,3.5) (2.16,3.8) (3.02,4.6) (5.92,7.4)
1.0(0.4059,1.5914) (0.4240,1.5939) (0.3768,1.5868) (0.4404,1.5960) (0.4721,1.5996)
(1.09,2.3) (1.32,2.1) (2.14,2.7) (3.17,3.7) (5.84,6.4)
1.5(0.6616,1.6129) (0.6621,1.6129) (0.3768,1.5868) (0.7056,1.6146) (0.7466,1.6159)
(1.09,1.4) (1.31,1.5) (2.14,2.3) (3.14,3.3) (5.79,5.9)
2.0(0.8505,1.6182) (0.8649,1.6184) (0.3768,1.5868) (0.9103,1.6189) (0.9343,1.6191)
(1.09,1.1) (1.31,1.3) (2.14,2.2) (3.13,3.2) (5.79,5.8)
Table 2: Optimal couples (λ∗,K∗)(first row of each block) and
values of (hL,ATS1)(second row of each block) of the VSI
EWMA- ˜
Xcontrol chart for n=5.
n=7
hS=0.5
δW=0.9W=0.6W=0.3W=0.2W=0.1
0.1(0.0500,1.1427) (0.0500,1.1427) (0.0500,1.1427) (0.0500,1.1427) (0.0500,1.1427)
(1.01,87.0) (1.09,81.8) (1.46,77.5) (1.88,77.0) (3.25,77.7)
0.2(0.0507,1.1449) (0.0500,1.1427) (0.0500,1.1427) (0.0500,1.1427) (0.0505,1.1443)
(1.01,30.2) (1.09,27.0) (1.46,25.3) (1.88,25.4) (3.25,26.5)
0.3(0.0870,1.2223) (0.0767,1.2060) (0.0860,1.2208) (0.0886,1.2247) (0.0913,1.2286)
(1.02,16.2) (1.10,14.4) (1.49,13.6) (1.91,13.9) (3.15,14.9)
0.5(0.1593,1.2921) (0.1723,1.2998) (0.1819,1.3049) (0.1831,1.3055) (0.1873,1.3076)
(1.02,7.2) (1.11,6.5) (1.49,6.3) (1.94,6.6) (3.37,8.0)
0.7(0.2678,1.3370) (0.2845,1.3412) (0.2847,1.3413) (0.2788,1.3398) (0.2912,1.3428)
(1.02,4.2) (1.12,3.8) (1.51,3.9) (1.91,4.3) (3.32,5.7)
1.0(0.4864,1.3705) (0.4590,1.3681) (0.4507,1.3673) (0.4527,1.3675) (0.4646,1.3686)
(1.03,2.4) (1.11,2.3) (1.50,2.6) (1.98,3.0) (3.28,4.3)
1.5(0.7609,1.3831) (0.7604,1.3831) (0.7643,1.3831) (0.7679,1.3832) (0.7766,1.3834)
(1.02,1.4) (1.12,1.4) (1.53,1.8) (1.97,2.3) (3.26,3.5)
2.0(0.9354,1.3853) (0.9377,1.3853) (0.9419,1.3853) (0.9448,1.3853) (0.9513,1.3854)
(1.02,1.1) (1.12,1.2) (1.53,1.6) (1.97,2.0) (3.25,3.3)
hS=0.1
δW=0.9W=0.6W=0.3W=0.2W=0.1
0.1(0.0500,1.1427) (0.0500,1.1427) (0.0500,1.1427) (0.0500,1.1427) (0.0500,1.1427)
(1.02,85.2) (1.17,75.8) (1.82,68.1) (2.59,67.2) (5.05,68.5)
0.2(0.0557,1.1596) (0.0500,1.1427) (0.0505,1.1443) (0.0553,1.1583) (0.0599,1.1702)
(1.03,28.6) (1.17,22.9) (1.82,19.7) (2.57,19.8) (4.99,21.8)
0.3(0.0870,1.2224) (0.0927,1.2306) (0.0986,1.2383) (0.1046,1.2455) (0.0996,1.2395)
(1.03,14.8) (1.19,11.5) (1.87,10.1) (2.61,10.4) (4.84,12.3)
0.5(0.1593,1.2921) (0.1946,1.3111) (0.2161,1.3201) (0.2212,1.3221) (0.2330,1.3264)
(1.04,6.2) (1.20,4.9) (1.94,4.6) (2.67,5.1) (5.22,7.6)
0.7(0.3249,1.3500) (0.3204,1.3491) (0.3368,1.3521) (0.3426,1.3532) (0.3617,1.3563)
(1.04,3.6) (1.21,2.9) (1.91,3.0) (2.79,3.8) (5.14,6.1)
1.0(0.4890,1.3707) (0.5066,1.3721) (0.4709,1.3692) (0.5203,1.3731) (0.5548,1.3753)
(1.05,2.0) (1.21,1.8) (1.90,2.3) (2.76,3.1) (5.08,5.4)
1.5(0.7655,1.3832) (0.7675,1.3832) (0.4709,1.3692 (0.8106,1.3840) (0.8430,1.3845)
(1.04,1.2) (1.21,1.3) (1.90,2.0) (2.74,2.8) (5.06,5.1)
2.0(0.9324,1.3853) (0.9459,1.3853) (0.4709,1.3692) (0.0626,1.1768) (0.9825,1.3854)
(1.04,1.1) (1.21,1.2) (1.90,1.9) (2.56,2.7) (5.06,5.1)
Table 3: Optimal couples (λ∗,K∗)(first row of each block) and
values of (hL,ATS1)(second row of each block) of the VSI
EWMA- ˜
Xcontrol chart for n=7.
n=9
hS=0.5
δW=0.9W=0.6W=0.3W=0.2W=0.1
0.1(0.0500,1.0152)(0.0500,1.0152) (0.0500,1.0152) (0.0500,1.0152) (0.0500,1.0152)
(1.00,74.4) (1.06,69.4) (1.38,64.3) (1.75,63.7) (2.97,64.2)
0.2(0.0548,1.0279) (0.0524,1.0220) (0.0540,1.0258) (0.0569,1.0330) (0.0598,1.0395)
(1.01,26.3) (1.06,23.1) (1.37,21.2) (1.74,21.2) (2.94,22.1)
0.3(0.1013,1.1030)) (0.0954,1.0964) (0.1017,1.1034) (0.1048,1.1066) (0.1077,1.1096)
(1.01,14.1)) (1.07,12.3) (1.39,11.4) (1.76,11.5) (2.85,12.4)
0.5(0.1878,1.1617) (0.2030,1.1679) (0.2141,1.1719) (0.2145,1.1720) (0.2182,1.1733)
(1.01,6.3) (1.08,5.5) (1.42,5.3) (1.78,5.5) (3.03,6.7)
0.7(0.3139,1.1970) (0.3334,1.2003) (0.3318,1.2000 (0.2408,1.1804) (0.3367,1.2008)
(1.01,3.7) (1.08,3.3) (1.41,3.3) (1.77,3.7) (2.99,4.8)
1.0(0.5404,1.2203,) (0.5410,1.2203) (0.5335,1.2199) (0.5354,1.2200) (0.5456,1.2206)
(1.01,2.1) (1.08,2.0) (1.43,2.2) (1.81,2.6) (2.96,3.7)
1.5(0.8434,1.2289) (0.8442,1.2289) (0.8480,1.2289) (0.8507,1.2289) (0.8570,1.2290)
(1.01,1.2) (1.08,1.2) (1.42,1.6) (1.81,2.0) (2.94,3.1)
2.0(0.9757,1.2297) (0.9772,1.2297) (0.9802,1.2297) (0.9822,1.2297) (0.9859,1.2297)
(1.01,1.0) (1.08,1.1) (1.42,1.4) (1.80,1.8) (2.94,3.0)
hS=0.1
δW=0.9W=0.6W=0.3W=0.2W=0.1
0.1(0.0504,1.0163)(0.0500,1.0152) (0.0500,1.0152) (0.0500,1.0152) (0.0500,1.0152)
(1.01,73.6) (1.11,64.7) (1.68,55.6) (2.36,54.4) (4.55,55.4)
0.2(0.0599,1.0397) (0.0598,1.0397) (0.0599,1.0398) (0.0649,1.0501) (0.0711,1.0616)
(1.01,25.4) (1.11,19.9) (1.71,16.3) (2.32,16.2) (4.45,17.9)
0.3(0.1014,1.1031) (0.0964,1.0975) (0.1171,1.1183) (0.1106,1.1124) (0.1027,1.1044)
(1.01,13.2) (1.13,10.0) (1.75,8.3,) (2.36,8.5) (4.35,10.3)
0.5(0.2224,1.1747) (0.2192,1.1737) (0.2524,1.1836) (0.2359,1.1789) (0.2701,1.1880)
(1.02,5.6) (1.13,4.3) (1.75,3.8) (2.39,4.3) (4.61,6.4)
0.7(0.3240,1.1987) (0.3737,1.2060) (0.3496,1.2028) (0.3916,1.2082) (0.4117,1.2104)
(1.02,3.3) (1.15,2.5) (2.6,1.0) (2.48,3.2) (4.55,5.2)
1.0(0.5801,1.2223) (0.5775,1.2221) (0.5783,1.2222) (0.5896,1.2227) (0.6235,1.2240)
(1.02,1.8) (1.14,1.6) (1.77,2.0) (2.46,2.7) (4.51,4.7)
1.5(0.8417,1.2289) (0.8441,1.2289) (0.8686,1.2291) (0.8829,1.2292) (0.9070,1.2294)
(1.02,1.1) (1.14,1.2) (1.76,1.8) (2.45,2.5) (4.50,4.5)
2.0(0.9741,1.2297) (0.9868,1.2297) (0.3496,1.2028) (0.9941,1.2297) (0.9967,1.2297)
(1.02,1.0) (1.14,1.1) (1.73,1.7) (0.92.45,2.5) (4.50,4.5)
Table 4: Optimal couples (λ∗,K∗)(first row of each block) and
values of (hL,ATS1)(second row of each block) of the VSI
EWMA- ˜
Xcontrol chart for n=9.
6 Illustrative example
In this Section, we discuss the implementation of the VSI
EWMA- ˜
Xchart. The context of the example presented here
is similar as the one introduced in Castagliola et al. [11], i.e,
a production process of 500 ml milk bottles where the quality
characteristic Xof interest is the capacity (in ml) of each bot-
tle. Like in Castagliola et al. [11], we have µ0=500.0230 and
σ0=0.9616. In fact, according to the process engineer experi-
ence, a shift δ=0.5 should be interpreted as a signal that some-
thing is going wrong in the production process. Thus, for n=5,
δ=0.5 and AT S0=370.4 the VSI EWMA- ˜
Xparameters are
chosen to be hS=0.5, hL=1.63, λ=0.1467, K=1.4989 and
W=0.3. This yields the following control limits for the VSI
EWMA- ˜
Xchart:
LCL =500.023 −1.4989r0.1467
2−0.1467 ×0.9616 =499.617,
UCL =500.023 +1.4989r0.1467
2−0.1467 ×0.9616 =500.429.
and the warning control limits for the VSI EWMA- ˜
Xchart:
LW L =500.023 −0.3r0.1467
1−0.1467 ×0.9616 =499.942,
UW L =500.023 +0.3r0.1467
1−0.1467 ×0.9616 =500.104.
The first 10 subgroups are supposed to be in-control while the
last 10 subgroups are supposed to have a lower milk capacity,
and thus, to be out-of-control. The corresponding sample me-
dian values ˜
Xiand the EWMA sequence Zifor VSI EWMA- ˜
X
Proceedings of the 24th ISSAT International Conference on Reliability and Quality in Design
August 2-4, 2018 - Toronto, ON, Canada
page 206

control chart are both presented in Table 5and plotted in Fig-
ure 1. This figure confirms that from sample #15 onwards, the
process is clearly out-of-control.
Sampling interval Total time Phase II (Xi,j)˜
XiZi
10.5 0.5 500.01 499.78 498.24 501.29 500.64 500.01 500.021
2 1.63 2.13 499.41 500.95 499.53 498.72 502.81 499.53 499.949
3 1.63 3.76 501.66 500.03 500.23 500.70 500.57 500.57 500.040
4 1.63 5.39 499.67 499.26 501.28 500.21 498.89 499.67 499.986
5 1.63 7.02 499.71 500.36 500.28 499.63 500.45 500.28 500.029
6 1.63 8.65 499.63 499.44 500.94 501.23 501.26 500.94 500.163
7 0.5 9.15 498.32 498.54 499.88 500.58 499.59 499.59 500.079
8 1.63 10.78 500.12 500.62 501.02 499.46 500.09 500.12 500.085
9 1.63 12.41 500.05 499.99 500.64 500.81 501.04 500.64 500.166
10 0.5 12.91 500.79 498.70 501.02 501.04 498.41 500.79 500.258
11 0.5 13.41 500.00 499.07 501.40 499.15 500.70 500.00 500.220
12 0.5 13.91 499.90 500.62 499.81 500.67 501.39 500.62 500.279
13 0.5 14.41 500.04 500.86 501.00 500.15 499.82 500.15 500.260
14 0.5 14.91 501.03 500.42 501.36 502.33 499.83 501.03 500.373
15 0.5 15.41 501.66 501.24 500.26 502.87 501.43 501.43 500.528
16 0.5 15.91 498.44 499.96 500.45 500.47 500.36 500.36 500.503
17 0.5 16.41 498.52 500.45 500.41 501.06 500.54 500.45 500.495
18 0.5 16.91 500.09 500.05 501.02 499.78 500.47 500.09 500.436
19 0.5 17.41 499.88 498.91 500.96 499.65 498.20 499.65 500.321
20 0.5 17.91 500.31 500.48 499.78 499.56 502.04 500.31 500.319
Table 5: Illustrative Phase II dataset
499.6
499.8
500
500.2
500.4
2 4 6 8 10 12 14 16 18 20
t
Zi
U CL
LCL
U W L
LW L
VSI EWMA- ˜
Xchart
Figure 1: VSI EWMA- ˜
Xcontrol chart corresponding to Phase
II data set in Table 5.
7 Concluding remarks
In this paper, we have investigated a VSI EWMA- ˜
Xcontrol
chart for monitoring process median. We have also studied
the statistical properties of the VSI EWMA- ˜
Xcontrol chart
and optimized their parameters for several shift sizes. For
fixed values of the shift size δ, several tables are provided
for presenting the out-of-control AT S1corresponding to many
different scenarios. Also, the numerical comparison with
the performance of the EWMA- ˜
Xcontrol chart proposed
by Castagliola [15] shows that the detection ability of the
proposed control chart are better than the EWMA- ˜
Xcontrol
chart. Thus, the proposed chart can be used as a best alternative
method.
Finally, possible enhancements and future work about VSI
EWMA- ˜
Xcontrol chart include the investigation of the effect
of the parameters estimation and of the measurement errors on
their statistical properties.
Appendix
The Markov chain approach of Brook and Evans [22] and Lucas
and Saccucci [6] is modified to evaluate the Run Length proper-
ties of the VSI EWMA- ˜
Xchart. This procedure involves divid-
ing dividing the interval [LCL,UCL]into 2m+1 subintervals
(Hj−∆,Hj+∆],j∈ {−m,...,0,...,+m}, centered at Hj=
LCL+UCL
2+2j∆where 2∆=UCL−LCL
(2m+1). Each subinterval (Hj−
∆,Hj+∆],j∈ {−m,...,0,...,+m}, represents a transient state
of a Markov chain. If Zi∈(Hj−∆,Hj+∆]then the Markov
chain is in the transient state j∈ {−m,...,0,...,+m}for sam-
ple i. If Zi6∈ (Hj−∆,Hj+∆]then the Markov chain reached
the absorbing state (−∞,LCL]∪[UCL,+∞). We assume that
Hjis the representative value of state j∈ {−m,...,0,...,+m}.
Let Qbe the (2m+1,2m+1)sub-matrix of probabilities Qj,k
corresponding to the 2m+1 transient states defined above, i.e.
Q=













Q−m,−m··· Q−m,−1Q−m,0Q−m,+1· · · Q−m,+m
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Q−1,−m··· Q−1,−1Q−1,0Q−1,+1· · · Q−1,+m
Q0,−m··· Q0,−1Q0,0Q0,+1· · · Q0,+m
Q+1,−m··· Q+1,−1Q+1,0Q+1,+1· · · Q+1,+m
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Q+m,−m··· Q+m,−1Q+m,0Q+m,+1· · · Q+m,+m













.
By definition, we have Qj,k=P(Zi∈(Hk−∆,Hk+∆]|Zi−1=
Hj)or, equivalently, Qj,k=P(Zi≤Hk+∆|Zi−1=Hj)−P(Zi≤
Hk−∆|Zi−1=Hj). Replacing Zi= (1−λ)Zi−1+λ˜
Xi,Zi−1=
Hjand isolating ˜
Xigives
Qj,k=P˜
Xi≤Hk+∆−(1−λ)Hj
λ−P˜
Xi≤Hk−∆−(1−λ)Hj
λ,
=F˜
XHk+∆−(1−λ)Hj
λ



n−F˜
XHk−∆−(1−λ)Hj
λ



n,
where F˜
X(... |n)is the c.d.f. (cumulative distribution function)
of the sample median ˜
Xi,i∈ {1,2,...}, i.e.
F˜
X(y|n) = FβΦy−µ0
σ0
−δ



n+1
2,n+1
2,(13)
where Φ(x)is the c.d.f. of the standard normal distribution and
Fβ(x|a,b)is the c.d.f. of the beta distribution with parameters
(a,b). Here a=b=n+1
2. Let q= (q−m, . . .,q0, . . . , qm)Tbe
the (2m+1,1)vector of initial probabilities associated with the
2m+1 transient states, where
qj=0 if Z06∈ (Hj−∆,Hj+∆]
1 if Z0∈(Hj−∆,Hj+∆].(14)
The AT S1can be evaluated through the following expression:
AT S1=qT(I−Q)−1g(15)
where gis the vector of sampling intervals corresponding to the
discretized states of the Markov chain and the jth element gjof
the vector gis the sampling interval when the control statistic
is in state j(represented by Hj), i.e.
gj=hLif LW L ≤Hj≤UW L,
hSotherwise.(16)
Proceedings of the 24th ISSAT International Conference on Reliability and Quality in Design
August 2-4, 2018 - Toronto, ON, Canada
page 207

The average sampling interval of the VSI EWMA- ˜
Xchart is
given as:
E(h) = qT(I−Q)−1g
qT(I−Q)−11(17)
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