Received: 27 June 2020
Revised: 8 August 2020
Accepted: 10 August 2020
DOI: 10.1002/qre.2752
RESEARCH ARTICLE
A mixed HWMA-CUSUM mean chart with an application
to manufacturing process
Muhammad Abid1,2
Sun Mei1
Hafiz Zafar Nazir3
Muhammad Riaz4
Shahid Hussain1,5
1
Institute of Applied Systems Analysis Jiangsu University, Zhenjiang, Jiangsu, P.R. China
2
Department of Statistics, Government College University, Faisalabad, Pakistan
3
Department of Statistics, University of Sargodha, Sargodha, Pakistan
4
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
5
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, Pakistan
Correspondence
Dr. Muhammad Abid, Institute of Applied
Systems Analysis, Jiangsu University,
Zhenjiang, Jiangsu 212013, P.R. China.
Email: mabid@gcuf.edu.pk
Funding information
National Science Foundation of China,
Grant/Award Number: 71774070
Abstract
Memory-type control charts play a significant role to identify slight changes in
the parameters of the production process. In this article, we have proposed a new
cumulative sum chart that utilizes the statistic of the homogeneously weighted
moving average chart. The performance of the proposed chart is studied using
Monte Carlo simulations. The proposed chart is compared with some existing
charts under different run length profiles. The run length profile comparisons
reveal that the proposed chart performs superior as compared to the existing control charts. A real-life application using a manufacturing process dataset is also
part of this study.
KEYWORDS
average run length, CUSUM, HWMA, manufacturing processes, mixed control chart, run
length profiles
1
INTRODUCTION
The control chart is one of the core techniques from all other techniques of the statistical process control and monitoring
(SPCM). The basic purpose of the control chart is to detect the slight changes in the process production, which occur
due to the special cause variation. Memoryless and memory-type charts are two main branches of the control chart. The
memoryless charts like Shewhart1 charts are popular to identify larger shifts, while the memory-type charts like the cumulative sum (CUSUM) and the exponentially weighted moving average (EWMA) charts proposed by Page2 and Roberts,3
respectively, are famous to identify the smaller shifts in the process parameter(s) (cf. Montgomery4 ).
Nowadays, several types of amendments (such as combining and mixing the design structure of the control charts)
have been recommended in the SPCM literature. Lucas5 suggested the combined Shewhart-CUSUM chart. On the lines
of Lucas,5 Lucas and Saccucci6 developed a combined Shewhart-EWMA chart. A double EWMA chart was suggested by
Shamma and Shamma.7 Abbas et al8 developed a mixed EWMA-CUSUM (MEC) chart and in this chart, they involved the
EWMA statistic in the design structure of the CUSUM chart. Zaman et al9 reversed the design structure of the MEC chart
Qual Reliab Engng Int. 2020;1–14.
wileyonlinelibrary.com/journal/qre
© 2020 John Wiley & Sons Ltd.
1
2
ABID et al.
and suggested a mixed CUSUM-EWMA (MCE). Zaman et al10 offered the design structure of some mixed charts. The incontrol (IC) robustness behavior of the MEC mean and dispersion charts were studied by Abid et al,11,12 respectively. Ali
and Haq13 introduced a mixed GWMA-CUSUM chart, where GWMA is labeled as a generally weighted moving average. A
new adaptive EWMA chart was suggested by Haq.14 A multivariate version of the MEC chart was introduced by Ajadi and
Riaz.15 Haq and Bibi16 proposed a new dual CUSUM (DCUSUM) chart. Haq et al17 suggested the multivariate DCUSUM
mean charts. Recently, the double progressive mean chart was suggested by Abbas et al18
In the SPCM literature, the homogeneously weighted moving average (HWMA) chart proposed by Abbas19 became well
known. The interested readers can see the work of the following researchers: Adegoke et al,20,21 Nawaz and Han,22 Abid
et al,23 and Abbas et al24 on the HWMA chart under various directions. So, taking motivation from the structures of the
HWMA chart, this article suggests a new CUSUM chart by using the statistic of the HWMA chart and from now labeled
as the MHC chart. The contents of this paper are as follows:to give the design structure of the existing and proposed MHC
charts in Sections 2 and 3, respectively; the ARL and graphical comparisons between proposed and existing charts are
offered in Section 4; we provide the real-life application in Section 5; and in Section 6, we make some conclusions and
recommendations of this paper.
2
SOME EXISTING CHARTS
This section presents the design structure of some existing charts.
2.1
The CUSUM chart
Let 𝑌𝑡 be the quality characteristic of interest, which follows a normal distribution with mean 𝜇 and variance 𝜎2 , that is,
𝑌𝑡 ∼ 𝑁(𝜇, 𝜎2 ) for 𝑡 > 1. The known mean and variance for the IC process are 𝜇 = 𝜇0 and 𝜎2 = 𝜎02 . The CUSUM chart
was first suggested by Page.2 As the CUSUM plotting-statistics denoted by 𝐶𝑡+ and 𝐶𝑡− depend both on the present and
past information, that is why it is relatively better to monitor infrequently small variations. The 𝐶𝑡+ and 𝐶𝑡− based on the
sample mean of 𝑌𝑡 , that is., 𝑌̄ 𝑡 are defined as:
] ]
[
+
𝐶𝑡+ = max 0, (𝑌̄ 𝑡 − 𝜇0 ) − K + 𝐶𝑡−1
]
[
−
𝐶𝑡− = 𝑚𝑎𝑥 0, − (𝑌̄ 𝑡 − 𝜇0 ) − K + 𝐶𝑡−1
(1)
√
where 𝐶𝑡+ = 𝐶𝑡− = 0, K = k 𝑉𝑎𝑟(𝑌̄ 𝑡 ) is the reference value for k > 0 and where 𝑉𝑎𝑟 (𝑌̄ 𝑡 ) =
shows an out-of-control (OOC) signal when 𝐶𝑡+
√
or 𝐶𝑡−
2
𝜎𝑌
𝑛
. The CUSUM chart
>𝐻
> 𝐻, else, it is IC. The 𝐻 is called the control limit or decision
̄
interval and it is defined as 𝐻 = ℎ 𝑉𝑎𝑟(𝑌𝑡 ). The values of k and ℎ are selected carefully because the average run length
(𝐴𝑅𝐿) performance of the CUSUM chart is influenced due to these values. We refer Hawkins and Olwell25 for further
details on the CUSUM chart.
2.2
The HWMA chart
The HWMA chart was proposed by Abbas19 which is performed superior to other memory-type charts to monitor the
infrequent variation in the process mean. The plotting-statistic of the HWMA chart is:
𝐻𝑊𝑡 = 𝜑𝑌̄ 𝑡 + (1 − 𝜑) 𝑌̄ 𝑡−1
where 𝜑 ∈ (0, 1], 𝑌̄ 0 = 𝜇𝑜 , and 𝑌̄ 𝑡−1 =
∑𝑡−1
𝑖=1
[(
𝐻𝑊𝑡 = 𝜑𝑌̄ 𝑡 +
𝑌̄ 𝑖
𝑡−1
(2)
. The statistic in (2) can also be written as (cf. Abbas et al19 ):
1−𝜑
𝑡−1
)
(
𝑌̄ 𝑡−1 +
1−𝜑
𝑡−1
)
(
𝑌̄ 𝑡−2 + ⋯ +
1−𝜑
𝑡−1
]
)
𝑌̄ 1
(3)
ABID et al.
3
where 𝜑 denoted the current sample weight and 1 − 𝜑 represented the previous sample weights. The statistic given in (3)
has the following mean and variance for IC process (cf. Abbas19 ):
𝐸 (𝐻𝑊𝑡 ) = 𝜇0
𝑉𝑎𝑟 (𝐻𝑊𝑡 ) =
𝜑2 𝜎02
𝑖𝑓 𝑡 = 1) and
𝑛
𝐸 (𝐻𝑊𝑡 ) = 𝜇0
𝑉𝑎𝑟 (𝐻𝑊𝑡 ) =
𝜑2 𝜎02
𝑛
+ (1 − 𝜑)
𝜎02
2
𝑖𝑓 𝑡 > 1)
𝑛(𝑡−1)
Using 𝐸 (𝐻𝑊𝑡 ) = 𝜇0 and 𝑉𝑎𝑟(𝐻𝑊𝑡 ), the control limits of the HWMA chart are:
√
⎤
𝑖𝑓 𝑡 = 1 ⎥
𝑛
⎥
𝐿𝐶𝐿𝑡 =
√
⎥
2
𝜑2 𝜎02
𝜎
2
0
𝜇0 − 𝐶
+ (1 − 𝜑)
, 𝑖𝑓 𝑡 > 1 ⎥
𝑛
𝑛(𝑡−1)
⎥
⎥
⎥
𝐶𝐿 = 𝜇0
⎥
⎥
√
⎥
𝜑2 𝜎02
𝑖𝑓 𝑡 = 1 ⎥
𝜇0 + 𝐶
𝑛
⎥
𝑈𝐶𝐿𝑡 =
√
⎥
2
𝜑2 𝜎02
2 𝜎0
𝜇0 + 𝐶
+ (1 − 𝜑)
, 𝑖𝑓 𝑡 > 1 ⎥⎦
𝜇0 − 𝐶
𝜑2 𝜎02
𝑛
(4)
𝑛(𝑡−1)
where 𝐶 is nominated to achieve the preferred IC 𝐴𝑅𝐿 and it is represented the width of the control limits.
3
THE PROPOSED CHART
In this study, we attach the HWMA statistic in the design structure of the CUSUM chart. The upper and lower plottingstatistics, that is, 𝐻𝐶𝑡+ and 𝐻𝐶𝑡− , respectively, of the MHC chart are:
] ]
[
+
𝐻𝐶𝑡+ = max 0, (𝐻𝑊𝑡 − 𝜇0 ) − K𝑡 + 𝐻𝐶𝑡−1
]
[
−
𝐻𝐶𝑡− = 𝑚𝑎𝑥 0, − (𝐻𝑊𝑡 − 𝜇0 ) − K𝑡 + 𝐻𝐶𝑡−1
(5)
where 𝐻𝑊𝑡 is given in (3) and 𝐻𝐶𝑡+ = 𝐻𝐶𝑡− = 0. The K𝑡 and H𝑡 are two design parameters of the MHC chart. The 𝐻𝐶𝑡+
and 𝐻𝐶𝑡− are plotted against H𝑡 . The process is stated to be OOC if either 𝐻𝐶𝑡+ or 𝐻𝐶𝑡− exceeds than H𝑡 , otherwise, it is
stated to be IC. The K𝑡 and H𝑡 for the MHC chart are defined as:
√
k𝑡
√
K𝑡 = k𝑡 𝑉𝑎𝑟 (𝐻𝑊𝑡 ) = √
k𝑡
𝑛
𝜑2 𝜎02
𝑛
√
H𝑡 = h𝑡
√
h𝑡
√
𝑉𝑎𝑟 (𝐻𝑊𝑡 ) =
h𝑡
𝜑2 𝜎02
+ (1 − 𝜑)
2
𝜎02
𝑛(𝑡−1)
,
𝜑2 𝜎02
𝑛
𝜑2 𝜎02
𝑛
+ (1 − 𝜑)
2
𝜎02
𝑛(𝑡−1)
,
⎤
𝑖𝑓 𝑡 = 1 ⎥
⎥
⎥
𝑖𝑓 𝑡 > 1⎥
⎦
(6)
⎤
𝑖𝑓 𝑡 = 1 ⎥
⎥
⎥
𝑖𝑓 𝑡 > 1⎥
⎦
(7)
We select the value of k𝑡 = 0.50 by following Abbas et al.8 The value of h𝑡 is selected in such a way that it is reached to
the desired IC 𝐴𝑅𝐿.
In SPCM literature, so many techniques are available to compute the 𝑅𝐿 profiles of a control chart. In this study, we
prefer Monte Carlo simulations due to the complex design structure of the MHC chart. Ali and Haq13 investigated that the
𝐴𝑅𝐿 results computed through Monte Carlo simulations are found to be more accurate as compared with other techniques,
that is, Markov chain and integral equations.
4
ABID et al.
TA B L E 1
Values of ℎ for MHC chart for several combinations of 𝜑 and 𝐴𝑅𝐿𝑜 when k𝑡 = 0.5
𝑨𝑹𝑳𝟎
𝝋
0.1
0.25
0.5
0.75
100
6.47
4.87
4.01
3.71
200
7.401
5.66
4.679
4.32
300
7.95
6.102
5.06
4.7
400
8.303
6.32
5.333
5.04
500
8.575
6.625
5.536
5.187
We select various combinations of 𝜑 and find the values of h𝑡 for different selections of the IC 𝐴𝑅𝐿 (Table 1). In Table 2,
we report the 𝑅𝐿 profile values of the MHC chart using different selections of 𝜑, h𝑡 when the IC 𝐴𝑅𝐿 ≈ 500. We also
|𝜇 −𝜇 |
consider different choices of 𝛿, that is, the amount of shift in the process mean and it is defined as = 1 0 , where
𝜎0
𝜇1 is the out of control (OOC) process mean. The value of h𝑡 is decreased as 𝜑 increases for a fixed value of k𝑡 (like
IC 𝐴𝑅𝐿 = 300, h𝑡 = 7.95, 𝜑 = 0.1, and h𝑡 = 5.06, 𝜑 = 0.5) (Table 1). The OOC 𝐴𝑅𝐿 values of the MHC chart are decreased
with the decrease of 𝜑 (like k𝑡 = 0.5, 𝜑 = 0.25, 𝛿 = 0.1, OOC ARL = 112, and k𝑡 = 0.5, 𝜑 = 0.1, 𝛿 = 0.1, OOC ARL = 62
[Table 2]). For fixed k𝑡 and 𝜑, the OOC 𝐴𝑅𝐿 values of the MHC chart are decreased as 𝛿 increases (Table 2). The IC standard
deviation of the RL (SDRL) and OOC SDRL values are decreased as the value of 𝜑 increases (Table 2). We have also made
the IC RL curve of the MHC chart against various choices of 𝜑 (Figure 1). From Figure 1, it is seen that there is a decrease
in the IC RL values of the MHC chart as the value of 𝜑 is increased.
4
PERFORMANCE COMPARISONS
The performance comparisons between MHC and existing charts (such as CUSUM, EWMA, MEC, MCE, and HWMA)
based on 𝐴𝑅𝐿 are performed in this section. A chart is considered to be an efficient chart if its IC 𝐴𝑅𝐿 is adequately large
and if its OOC 𝐴𝑅𝐿 is adequately small. Another measure called the percentage decrease in 𝐴𝑅𝐿 is also used for compari𝐼𝐶 𝐴𝑅𝐿−𝑂𝑂𝐶 𝐴𝑅𝐿
) × 100.
son and is labeled as the 𝑃𝐷𝐴𝑅𝐿 (cf. Abbas et al18 and Abid et al23 ). The 𝑃𝐷𝐴𝑅𝐿 is calculated as (
𝐼𝐶 𝐴𝑅𝐿
4.1
MHC versus CUSUM
The MHC chart with all selected values of 𝜑 and 𝛿 detects the shifts quickly against the CUSUM chart (like in CSUUM
chart at 𝑘 = 0.5, 𝛿 = 0.1, 0.15, 0.2, 0.5, OOC 𝐴𝑅𝐿 = 369, 271, 201, 39 and in MHC chart at k𝑡 = 0.5, 𝜑 = 0.1, 𝛿 = 0.1,
0.15, 0.2, 0.5, OOC 𝐴𝑅𝐿 = 62, 41, 32, 13, also in MHC chart at k𝑡 = 0.5, 𝜑 = 0.75, 𝛿 = 0.1, 0.15, 0.2, 0.5, OOC 𝐴𝑅𝐿 =
310, 204, 133, 24 [Table 2 vs Table 3]). At 𝛿 = 20%, the 𝑃𝐷𝐴𝑅𝐿 in CUSUM chart is 45.8% but in MHC, the 𝑃𝐷𝐴𝑅𝐿 is 93.6%
and 73.4%, when 𝜑 = 0.1 and 𝜑 = 0.75, respectively.
4.2
MHC versus EWMA
It is noted that the MHC chart performs superior to the EWMA chart (like in EWMA chart at 𝜑 = 0.1, 𝛿 =
0.10, .15, 0.2, 0.5, OOC 𝐴𝑅𝐿 = 62, 41, 32, 13 and in MHC chart OOC 𝐴𝑅𝐿 = 316, 218, 146, 29 [Tables 2 and 4]). Also,
with 𝛿 = 15% and 𝜑 = 0.1, the 𝑃𝐷𝐴𝑅𝐿 in MHC and EWMA charts is 91.8% and 57.4%, respectively.
4.3
MHC versus MEC
From Tables 2 and 5, it is revealed that the MHC chart indicates improved performance against the MEC chart (like in MHC
chart at 𝜑 = 0.1 and 𝛿 = 0.1, 0.15, 0.2, 0.5, OOC 𝐴𝑅𝐿 = 62, 41, 32, 13 and in MEC chart OOC 𝐴𝑅𝐿 = 243, 156, 107, 35).
The MHC chart also performs better for larger values of 𝜑 over the MEC chart (Tables 2 and 5). Furthermore, at 𝛿 = 10%
and 𝜑 = 0.1, the 𝑃𝐷𝐴𝑅𝐿 in the MHC chart is 87.6%, while in the MEC chart is 51.6%.
ABID et al.
5
TA B L E 2
The 𝑅𝐿 profiles for MHC chart at 𝐴𝑅𝐿𝑜 ≅ 500
𝝋
𝐤𝒕
𝐡𝒕
𝜹
Profiles
𝑨𝑹𝑳
𝑺𝑫𝑹𝑳
𝑷𝟓
𝑷𝟐𝟓
𝑷𝟓𝟎
𝑷𝟕𝟓
𝑷𝟗𝟓
0.1
0.5
8.575
0
501
3016
6
12
27
87
1382
0.1
62
107
24
6
11
63
244
0.15
41
57
21
6
11
47
148
0.2
32
35
19
6
10
39
100
0.25
25
25
17
6
10
31
75
0.5
13
8
10
5
7
16
30
0.75
9
4
8
4
6
11
17
1
7
3
6
4
5
8
12
1.5
5
1
5
4
4
6
8
0.25
0.5
0.75
0.5
0.5
0.5
6.625
5.536
5.187
2
4
1
4
3
4
5
6
0
499
1145
5
11
37
265
2636
0.1
112
211
29
5
11
112
508
0.15
65
101
25
5
10
74
266
0.2
41
55
21
5
9
49
145
0.25
31
36
17
5
9
37
103
0.5
12
9
9
4
6
15
30
0.75
8
4
7
4
5
10
16
1
6
3
6
3
4
7
11
1.5
4
1
4
3
4
5
7
2
4
1
4
3
3
4
5
0
501
710
6
30
211
689
1959
0.1
222
302
103
6
23
306
825
0.15
128
166
63
6
18
175
477
0.2
79
96
43
5
15
107
274
0.25
52
60
30
5
13
70
174
0.5
16
13
12
4
7
20
42
0.75
9
5
7
3
5
11
20
1
6
3
5
3
4
8
12
1.5
4
1
4
3
3
5
7
2
3
1
3
2
3
4
5
0
502
544
14
112
322
689
1622
0.1
310
338
199
12
71
430
974
0.15
204
214
133
11
50
282
650
0.2
133
142
86
9
34
186
416
0.25
94
95
62
8
26
129
284
0.5
24
20
18
5
10
31
63
0.75
12
8
10
4
7
15
27
8
4
7
3
5
10
15
1.5
5
2
4
3
3
6
8
2
3
1
3
2
3
4
5
1
4.4
MHC versus MCE
It is observed that the MHC chart shows improved performance than the MEC chart (like in MHC chart at = 0.1,
𝛿 = 0.1, 0.15, 0.25, 0.5, OOC 𝐴𝑅𝐿 = 62, 41, 25, 13 and in MCE chart OOC 𝐴𝑅𝐿 = 350, 244, 126, 36, also, in MHC
chart at 𝜑 = 0.75, 𝛿 = 0.1, 0.15, 0.25, 0.5, 𝑂𝑂𝐶 𝐴𝑅𝐿 = 310, 204, 94, 24 and in MCE chart OOC 𝐴𝑅𝐿 = 369, 269, 144, 37
[Tables 2 and 6]). The 𝑃𝐷𝐴𝑅𝐿 in MHC and MCE charts is 91.8% and 51.2%, respectively, at 𝛿 = 15% and 𝜑 = 0.1.
6
ABID et al.
FIGURE 1
4.5
The IC RL curve of the proposed chart for various choices of φ
MHC versus HWMA
The MHC chart performs efficiently well than to the HWMA chart against all 𝜑 and 𝛿 values (like in MHC chart at 𝜑 = 0.1,
𝛿 = 0.1, 0.15, 0.25, 0.5, OOC 𝐴𝑅𝐿 = 62, 41, 25, 13 and in HWMA chart OOC 𝐴𝑅𝐿 = 250, 161, 81, 29, also, in MHC chart
at 𝜑 = 0.75, 𝛿 = 0.1, 0.15, 0.5, OOC 𝐴𝑅𝐿 = 310, 204, 24 and in HWMA chart OOC 𝐴𝑅𝐿 = 462, 414, 131 [Tables 2 and 7]).
Moreover, at 𝛿 = 15% and 𝜑 = 0.1, the PDARL in MHC and HWMA charts is 91.8% and 50%, respectively.
We have also computed the OOC 𝑆𝐷𝑅𝐿 and OOC percentile point (like 𝑃5 , 𝑃25 , 𝑃50 , 𝑃75 , and 𝑃95 ) values of the proposed
and existing charts (Tables 2–7). The main reason to compute these measures is the skewed distributional behavior of the
𝑅𝐿. From Tables 2–7, it is noted that the MHC chart has a smaller 𝑆𝐷𝑅𝐿, 𝑃5 , 𝑃25 , 𝑃50 , 𝑃75 , and 𝑃95 values as compared to
the CUSUM, EWMA, MEC, MCE, and HWMA charts (like at 𝜑 = 0.1, k𝑡 = 0.5, 𝛿 = 0.1, the values of 𝑆𝐷𝑅𝐿 and 𝑃50 in
MHC chart are 107 and 11 [Table 2]; in CUSUM chart are 363 and 259 [Table 3]; in EWMA chart are 317 and 220 [Table 4];
in MEC chart are 205 and 181 [Table 5]; respectively, in MCE chart are 340 and 246 [Table 6]; and in HWMA chart are
198 and 204 [Table 7]). As the value of 𝑃50 is smaller than the value of 𝐴𝑅𝐿, it indicates that the distribution of the 𝑅𝐿 is
positively skewed.
TA B L E 3
The 𝑅𝐿 profiles for CUSUM chart at 𝐴𝑅𝐿𝑜 ≅ 500
𝒌
𝒉
0.5
5.072
𝜹
Profiles
𝑨𝑹𝑳
𝑺𝑫𝑹𝑳
0
499
0.1
369
𝑷𝟓
𝑷𝟐𝟓
𝑷𝟓𝟎
𝑷𝟕𝟓
𝑷𝟗𝟓
485
31
148
346
681
1467
363
26
112
259
511
1070
0.15
271
264
20
84
189
372
797
0.2
201
193
17
64
142
278
586
0.25
147
139
15
49
105
203
426
0.5
39
31
7
16
29
52
102
0.75
17
11
5
9
14
22
39
1
10
5
4
7
9
13
21
1.5
6
2
3
4
5
7
10
2
4
1
2
3
4
5
6
ABID et al.
7
TA B L E 4
The 𝑅𝐿 profiles for EWMA chart at 𝐴𝑅𝐿𝑜 ≅ 500
𝝋
𝑪
𝜹
Profiles
𝑨𝑹𝑳
𝑺𝑫𝑹𝑳
0.1
2.824
0
499
503
20
0.1
316
317
0.15
218
213
0.2
146
0.25
103
0.5
0.75
0.25
0.5
0.75
3
3.072
3.088
𝑷𝟐𝟓
𝑷𝟓𝟎
𝑷𝟕𝟓
𝑷𝟗𝟓
138
343
694
1519
16
92
220
436
942
12
66
153
298
644
140
11
45
101
204
422
97
8
35
74
138
298
29
23
4
13
23
39
73
14
10
3
7
12
18
32
1
8
5
2
4
7
11
18
1.5
4
2
1
2
4
5
9
2
3
1
1
2
2
3
5
0
498
497
24
144
345
687
1484
0.1
386
385
21
110
269
532
1149
0.15
296
293
16
87
208
409
877
0.2
226
222
14
66
158
317
673
0.25
168
164
12
52
118
232
501
0.5
48
44
5
16
34
65
135
0.75
19
16
3
8
15
25
51
1
10
8
2
5
8
14
25
1.5
5
3
1
3
4
6
10
2
3
2
1
2
3
4
6
0
502
501
26
145
353
695
1510
0.1
437
441
23
123
302
608
1297
0.15
377
372
21
110
259
524
1122
0.2
311
315
17
91
216
425
929
0.25
254
248
14
74
179
356
743
0.5
88
87
6
26
62
121
261
0.75
35
34
3
11
25
48
104
1
17
15
2
6
13
23
47
1.5
6
5
1
3
5
8
15
2
3
2
1
2
3
4
7
0
501
500
26
144
347
699
1508
0.1
465
465
322
24
133
645
1393
0.15
418
419
290
22
121
582
1241
0.2
372
373
259
20
109
514
1114
0.25
320
317
222
17
92
443
956
0.5
140
139
97
8
41
194
417
0.75
63
62
44
4
19
85
186
1
30
29
21
3
10
42
90
1.5
10
9
7
1
4
13
27
4
3
3
1
2
6
11
2
4.6
𝑷𝟓
Graphical comparisons
In this section, we present the graphical comparisons between the MHC and existing charts based on OOC 𝐴𝑅𝐿 value
curves for various choices of 𝜑. It is found that the MEC chart shows enhanced performance against the EWMA, MCE,
and HWMA charts for larger values of 𝜑 and 𝛿 < 1.5 (Figure 2C,D). The HWMA has a better shift detection ability than
8
ABID et al.
TA B L E 5
The 𝑅𝐿 profiles for MEC chart at 𝐴𝑅𝐿𝑜 ≅ 500
𝝋
𝐤𝒕
𝐡𝒕
𝜹
Profiles
𝑨𝑹𝑳
𝑺𝑫𝑹𝑳
𝑷𝟓
𝑷𝟐𝟓
𝑷𝟓𝟎
𝑷𝟕𝟓
𝑷𝟗𝟓
0.1
0.5
37.43
0
501
466
62
171
359
681
1452
0.1
243
205
46
97
181
324
646
0.15
156
121
38
71
120
203
391
0.2
107
74
32
55
86
135
254
0.25
79
48
28
45
66
100
173
0.5
35
13
20
26
33
41
60
0.75
24
6
16
20
23
27
36
1
19
4
14
16
18
21
26
1.5
14
2
11
12
14
15
17
0.25
0.5
0.75
0.5
0.5
0.5
20.19
11.2
7.33
2
11
1
9
10
11
12
14
0
502
480
47
154
357
688
1472
0.1
272
250
35
94
197
372
758
0.15
177
154
29
68
128
236
491
0.2
115
93
23
50
88
154
299
0.25
83
63
21
40
65
108
209
0.5
31
15
14
20
27
38
61
0.75
19
7
11
14
17
22
32
1
14
4
9
11
13
16
21
1.5
10
2
7
8
9
11
13
2
8
1
6
7
7
8
10
0
503
490
37
152
354
705
1509
0.1
312
301
29
99
219
430
910
0.15
208
194
24
70
147
280
603
0.2
139
126
20
50
99
189
392
0.25
100
85
17
40
74
133
265
0.5
30
19
10
17
26
38
68
0.75
17
8
8
11
15
20
31
1
12
4
6
8
11
14
20
1.5
7
2
5
6
7
8
11
2
6
1
4
5
5
6
8
0
499
494
32
150
359
697
1480
0.1
350
340
27
107
249
482
1029
0.15
238
227
22
77
170
322
699
0.2
169
160
18
56
120
232
496
0.25
120
110
15
42
87
161
340
0.5
34
24
9
16
26
44
82
0.75
17
9
6
10
14
21
34
1
11
5
5
7
10
13
20
1.5
6
2
4
5
6
7
10
2
5
1
3
4
4
5
7
EWMA and MCE chart for smaller values of 𝜑 and 𝛿 < 0.75 (Figure 2A,B). Moreover, the MHC chart performs comparatively superior as compared to EWMA, MEC, MCE, and HWMA charts under for all 𝜑 and 𝛿 values (Figure 2A,D). The
dominance zone of the MHC chart becomes closer to the existing charts with an increase in the value of 𝜑 (Figure 2C,D).
From tabulated and graphical comparisons, it is concluded that the MHC shows superior performance against EWMA,
MEC, MCE, and HWMA charts.
ABID et al.
TA B L E 6
9
The 𝑅𝐿 profiles for MCE chart at 𝐴𝑅𝐿𝑜 ≅ 500
𝝋
𝒌
𝑪
𝜹
0.1
0.5
9.66
0
𝑺𝑫𝑹𝑳
𝑷𝟓
𝑷𝟐𝟓
𝑷𝟓𝟎
𝑷𝟕𝟓
𝑷𝟗𝟓
499
482
35
150
352
691
1441
0.1
350
340
27
108
246
482
1026
0.15
244
232
22
78
174
333
708
0.2
177
168
19
59
125
239
514
0.25
126
115
16
44
92
171
355
0.5
36
27
9
17
28
47
90
0.75
18
10
7
11
15
22
37
1
12
5
6
8
11
14
21
7
2
4
6
7
8
11
1.5
0.25
0.5
0.75
0.5
0.5
0.5
9.1
7.42
6.09
2
5
1
3
4
5
6
8
0
500
497
31
149
352
689
1475
0.1
366
359
25
112
255
509
1070
0.15
261
252
20
82
183
355
767
0.2
191
185
17
61
134
262
557
0.25
137
128
14
46
98
189
396
0.5
37
30
8
16
28
49
96
0.75
17
11
6
9
14
21
38
1
10
5
5
7
9
13
20
1.5
6
2
3
5
6
7
10
2
4
1
3
3
4
5
7
0
503
509
26
146
356
699
1505
0.1
371
365
23
112
261
515
1079
0.15
272
265
19
83
191
375
795
0.2
198
193
15
62
139
271
584
0.25
144
138
13
47
102
199
418
0.5
38
32
6
15
28
51
101
0.75
17
11
4
8
14
22
39
1
10
5
3
6
9
12
20
1.5
5
2
2
4
5
6
10
2
4
1
2
3
3
4
6
0
499
498
23
139
347
684
1472
0.1
369
368
21
108
258
511
1072
0.15
269
266
17
81
187
372
797
0.2
198
196
13
60
137
272
588
0.25
144
140
11
46
101
199
422
0.5
37
32
5
14
28
50
101
0.75
16
11
3
8
13
21
38
9
6
2
5
8
12
20
1.5
5
2
2
3
4
6
9
2
3
1
1
2
3
4
6
1
5
Profiles
𝑨𝑹𝑳
A REAL-LIFE APPLICATION OF EXISTING AND PROPOSED CHARTS
This section offers the real-life application of the existing and proposed charts based on the dataset related to the semiconductor manufacturing process (cf. Montgomery4 ). For phase-I analysis, the 25 subgroups each of size 5 have been taken
at every hour to measure the “flow width of the resist,” which is the variable under study. Twenty more subgroups have
10
TA B L E 7
ABID et al.
The 𝑅𝐿 profiles for HWMA chart at 𝐴𝑅𝐿𝑜 ≅ 500
𝝋
𝑪
𝜹
Profiles
𝑨𝑹𝑳
𝑺𝑫𝑹𝑳
𝑷𝟓
𝑷𝟐𝟓
𝑷𝟓𝟎
𝑷𝟕𝟓
𝑷𝟗𝟓
0.1
2.938
0
499
402
44
200
401
696
1274
0.1
250
198
29
105
201
341
633
0.15
161
123
21
71
131
219
403
0.2
110
81
16
51
92
149
266
0.25
81
57
13
40
68
108
193
0.5
29
18
6
15
25
38
61
0.75
15
9
4
8
13
19
31
0.25
0.5
0.75
3.075
3.089
3.09
1
9
5
3
6
8
12
19
1.5
5
2
1
3
5
6
10
2
3
2
1
3
3
4
6
0
500
483
31
151
354
691
1469
0.1
322
305
24
105
231
437
946
0.15
229
213
20
77
167
312
657
0.2
156
140
16
55
117
212
431
0.25
113
98
12
43
86
153
306
0.5
33
25
6
16
27
45
84
0.75
16
11
4
9
14
22
37
1
10
6
3
6
9
13
21
1.5
5
2
1
3
4
6
10
2
3
1
1
2
3
4
6
0
501
500
28
143
343
687
1504
0.1
420
415
23
123
294
582
1224
0.15
351
350
19.95
101
246
484
1049
0.2
280
276
17
85
197
386
828
0.25
217
213
14
65
151
300
645
0.5
68
65
6
23
49
94
194
0.75
28
25
3
10
21
38
78
1
14
12
2
6
11
19
37
1.5
6
4
1
3
5
7
13
2
3
2
1
2
3
4
7
0
502
504
26
146
346
696
1540
0.1
462
462
23
133
320
641
1373
0.15
414
416
23
119
289
575
1216
0.2
369
366
20
107
255
512
1100
0.25
312
314
16
90
213
433
926
0.5
131
129
8
39
92
183
388
0.75
58
58
4
18
41
81
173
1
28
27
2
9
20
38
83
1.5
9
8
1
3
7
12
25
2
4
3
1
2
3
6
10
been taken for phase-II analysis and these subgroups are considered to make the real-life application of the existing and
proposed MHC charts. The scatter plot of these 45 subgroups is provided in Figure 3 and this plot shows a rising drift
after 30 subgroups. To investigate this drift, the values of the selected design parameters are presented in Table 8 by fixing
̂ and standard deviation (𝜎)
̂ for monitoring of the Phase-I
𝐼𝐶 𝐴𝑅𝐿 ≈ 500. The unknown values of the population mean (𝜇)
ABID et al.
11
FIGURE 2
OOC ARL comparisons between proposed and existing charts when (A) φ = 0.1, (B) φ = 0.25, (C) φ = 0.5, and (D) φ = 0.75
FIGURE 3
The scatter plot for all subgroups
TA B L E 8
Design parameter values for the existing and proposed charts
𝐂𝐡𝐚𝐫𝐭𝐬
𝐃𝐞𝐬𝐢𝐠𝐧 𝐩𝐚𝐫𝐚𝐦𝐞𝐭𝐞𝐫𝐬
𝐶𝑈𝑆𝑈𝑀
𝑘 = 0.75, ℎ = 3.538
𝐸𝑊𝑀𝐴
𝜑 = 0.25, 𝐶 = 3
𝑀𝐸𝐶
𝜑 = 0.25, k𝑡 = 0.75, h𝑡 = 13.44
𝐻𝑊𝑀𝐴
𝜑 = 0.75, 𝐶 = 3.09
𝑀𝐻𝐶
𝜑 = 0.25, k𝑡 = 0.75, h𝑡 = 4.125
12
ABID et al.
FIGURE 4
A real-life application of (A) CUSUM, (B) EWMA, (C) MEC, (D) HWMA, and (E) MHC charts
analysis are calculated using the following formulas:
1
𝜇̂ =
𝑙
√
𝑙
∑
𝑌̄ 𝑡 =
𝑡=1
∑𝑙 ∑𝑚
𝑠𝑝 =
𝑡=1
1
25
(37.6403) = 1.5056,
̄ 2
𝑗=1 (𝑌𝑖𝑗 −𝑌𝑖 )
𝑙(𝑚−1)
√
=
1.93424
25(5−1)
= 0.01391 and 𝜎̂ =
𝑠𝑝
𝑐4,𝑙
=
0.019308
0.997503
= 0.13943.
The diagram outputs of the existing and proposed charts are shown in Figure 4A-E. From Figures 4A and 4C, it is seen
that there is no OOC point in the CUSUM and MEC charts because the plotting-statistics of these charts stayed inside
the control limits. From Figures 4B and 4D, it can be noticed that the EWMA and HWMA charts spot shifts at the 20th
sample. On the other hand, the proposed MHC chart spot shifts at the 19th sample (Figure 4E). Hence, the proposed MHC
chart triggers an earlier OOC signal in the process mean over the existing charts included in this section.
6
CONCLUSIONS AND RECOMMENDATIONS
In this study, we have suggested a new CUSUM chart that utilizes the statistic of the HWMA chart and named as the
MHC chart. The 𝑅𝐿 profile comparisons of the MHC chart have been performed with existing charts. From tabular and
graphical comparisons, we have found that the MHC chart offers superior performance than the existing charts under
ABID et al.
13
study. Hence, we suggest the practice of the MHC chart to the SPCM experts for evaluating infrequent changes in the
process mean. The scope of this study may also be extended to assess variations in the process dispersion. Moreover, the
MHC chart might be used to develop efficient nonparametric and multivariate charts.
AC K N OW L E D G M E N T S
The authors are grateful to the editor and referees for their constructive comments that led to substantial improvements
in the article. This research fund was supported by the National Science Foundation of China through project number
71774070.
ORCID
Muhammad Abid https://orcid.org/0000-0001-8996-1120
Sun Mei https://orcid.org/0000-0003-0276-0612
Muhammad Riaz https://orcid.org/0000-0002-7599-6928
Shahid Hussain https://orcid.org/0000-0003-2206-1739
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
Shewhart WA. Economic control of quality of manufactured product. Van Nostrand, New York. 1932;27(178):217-215.
Page ES. Continuous inspection schemes. Biometric. 1954;41(1-2):100-115.
Roberts SW. Control chart tests based on geometric moving averages. Technometrics. 1959;1(3):239–250.
Montgomery DC. Introduction to Statistical Quality Control. 7th ed. New York: John Wiley & Sons;2012.
Lucas JM. Combined Shewhart-CUSUM quality control schemes. J Qual Technol. 1982;14(2):51–59.
Lucas JM, Saccucci MS. Exponentially weighted moving average control schemes: properties and enhancements. Technometrics.
1990;32(1):1–12.
Shamma SE, Shamma AK. Development and evaluation of control charts using double exponentially weighted moving averages. Int J Qual
Reliabil Manage. 1992;9(6):18–25.
Abbas N, Riaz M, Does RJMM. Mixed exponentially weighted moving average-cumulative sum charts for process monitoring. Qual Reliab
Eng Int. 2013;29(3):345–356.
Zaman B, Abbas N, Does RJ. Mixed cumulative sum exponentially weighted moving average control charts. Qual Reliab Eng Int.
2014;31(1):1407–1421.
Zaman B, Abbas N, Lee MH. On the performance of control charts for simultaneous monitorring of location and dispersion parameters.
Qual Reliab Eng Int. 2017;33(1):37–56.
Abid M, Nazir HZ, Riaz M, Lin Z. In-control robustness comparison of different control charts. Trans Inst Meas Control. 2018;40(13):3860–
3871.
Abid M, Nazir HZ, Tahir M, Riaz M, Abbas T. A comparative analysis of robust dispersion control charts with application related to health
care data. J Test Eval. 2020;48(1):247–259.
Ali R, Haq A. A mixed GWMA-CUSUM control chart for monitoring the process mean. Commun Statist Theory Methods. 2018;47(15):3779–
3801.
Haq A. A new adaptive EWMA control chart using auxiliary information for monitoring the process mean. Commun Statist Theory Methods. 2018;47(19):4840–4858.
Ajadi JO, Riaz M. Mixed multivariate EWMA-CUSUM control charts for an improved process monitoring. Commun Statist Theory Methods.
2017;46(14):6980–6993.
Haq A, Bibi L. A new dual CUSUM mean chart. Qual Reliab Eng Int. 2019;35(4):1245–1262.
Haq A, Munir T, Khoo MBC. Dual multivariate CUSUM mean charts. Comput Ind Eng. 2019;137:106028.
Abbas Z, Nazir HZ, Akhtar N, Riaz M, Abid M. An enhanced approach for the progressive mean control charts. Qual Reliab Eng Int.
2019;35(4):1046–1060.
Abbas N. Homogeneously weighted moving average control chart with an application in substrate manufacturing process. Comput Ind
Eng. 2018;120(6):460–470.
Adegoke NA, Abbasi SA, Smith ANH, Anderson MJ, Pawley MDM. A multivariate homogeneously weighted moving average control chart.
IEEE Access. 2019;7:9586–9597.
Adegoke NA, Smith ANH, Anderson MJ, Sanusi RA, Pawley MDM. Efficient homogeneously weighted moving average chart for monitoring process mean using an auxiliary variable. IEEE Access. 2019;7:94021–94032.
Nawaz T, Han D. Monitoring the process location by using new ranked set sampling based memory control charts. Qual Technol Quant
Manage. 2020;17(3):255–284.
Abid M, Shabbir A, Nazir HZ, Sherwani RAK, Riaz M. A double homogeneously weighted moving average control chart for monitoring
of the process mean. Qual Reliab Eng Int. 2020;36(5):1513–1527.
Abbas N, Riaz M, Shabbir A, Abid M, Zaman B. On the efficient monitoring of multivariate processes with unknown parameters. Mathematics. 2020;8:823.
Hawkins DM, Olwell DH. Cumulative Sum Charts and Charting Improvement. New York: Springer: 1998.
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ABID et al.
AU T H O R B I O G R A P H I E S
Muhammad Abid obtained his MSc and MPhil degrees in statistics from Quaid-i-Azam University, Islamabad, Pakistan, in 2008 and 2010, respectively. He did his PhD in statistics from the
Institute of Statistics, Zhejiang University, Hangzhou, China, in 2017. He served as a statistical officer in National Accounts Wing, Pakistan Bureau of Statistics (PBS) during 2010-2011. He is now
serving as an Assistant Professor in the Department of Statistics, Government College University,
Faisalabad, Pakistan, from 2017 to present. His research interests include statistical quality control, Bayesian statistics, nonparametric techniques, and survey sampling. His e-mail address is:
mabid@gcuf.edu.pk.
Sun Mei received her B.S. degree in Mathematics from Yangzhou University and the Ph.D. degree
in System Engineering from Jiangsu University, respectively. She is currently a Professor in the
School of Mathematical Science, Jiangsu University, China. She was the recipient of the Excellent Ph.D. Thesis Award of Jiangsu province of China and the Teaching Achievement Award of
Ministry of Education of China. She has published over 50 peer-reviewed journal papers and four
books. Her research interest includes the theory and application of nonlinear systems, and the
modeling and analysis of complex systems.
Hafiz Zafar Nazir obtained his MSc and M.Phil degrees in Statistics from the Department of
Statistics, QuaidiAzam University, Islamabad, Pakistan, in 2006 and 2008, respectively. He did his
PhD in statistics from the Institute of Business and Industrial Statistics University of Amsterdam,
The Netherlands, in 2014. He served as a lecturer in the Department of Statistics, University of
Sargodha, Pakistan, during 2009-2014. He is now serving as an assistant professor in the Department of Statistics, University of Sargodha, Pakistan, from 2015 to date. His current research interests include statistical process control, nonparametric techniques, and robust methods. His e-mail
address is: hafizzafarnazir@yahoo.com.
Muhammad Riaz obtained his PhD in statistics from the Institute for Business and Industrial
Statistics, University of Amsterdam, The Netherlands, in 2008. He holds the position of professor
in the Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. His current research interests include statistical process control,
nonparametric techniques, and experimental design.
Shahid Hussain obtained his Ph.D. degree in Probability Theory and Mathematical Statistics
from the School of Mathematical Sciences, Dalian University of Technology, Dalian, China in 2018.
He obtained his M.Sc. in statistics (2006) and M.Phil. in statistics (2008) from the Department of
Statistics, Quaid-i-Azam University, Islamabad, Pakistan. He served as a Lecturer during 20062011 and as an Assistant Professor from 2011 to date at the Department of Mathematics, COMSATS
University Islamabad, Attock Campus. Currently, he is serving as a post-doctorate researcher at
the Institute of Applied Systems Analysis, Faculty of Science, Jiangsu University, Zhenjiang, P. R.
China. His current research interests include Statistical Process Control and Application of Sampling Techniques. His email is: shahid_libra82@hotmail.com
How to cite this article: Abid M, Mei S, Nazir HZ, Riaz M, Hussain S. A mixed HWMA-CUSUM mean chart
with an application to manufacturing process. Qual Reliab Engng Int. 2020;1–14. https://doi.org/10.1002/qre.2752