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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˜

XHk+∆−(1−λ)Hj

λ

n−F˜

XHk−∆−(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) = FβΦy−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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