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Generally weighted moving average monitoring schemes:
Overview and perspectives
Kutele Mabude, Jean-claude Malela-majika, Philippe Castagliola, Sandile
Shongwe
To cite this version:
Kutele Mabude, Jean-claude Malela-majika, Philippe Castagliola, Sandile Shongwe. Generally
weighted moving average monitoring schemes: Overview and perspectives. Quality and Reliability
Engineering International, Wiley, 2021, 37 (2), pp.409-432. �10.1002/qre.2765�. �hal-03321174�
1
Generally Weighted Moving Average monitoring schemes – Overview and Perspectives
1
Mabude K.,
1
Malela-Majika J.-C.,
2
Castagliola P. and
1*
Shongwe S.C.
Abstract
An overview of monitoring schemes from a class called generally weighted moving average
(GWMA) is provided. A GWMA scheme is an extended version of the exponentially weighted
moving average (EWMA) scheme with an additional adjustment parameter that introduces more
flexibility in the GWMA model as it adjusts the kurtosis of the weighting function so that the GWMA
scheme can be designed such that it has an advantage over the corresponding EWMA scheme in the
detection of certain shift values efficiently. The parametric and distribution-free GWMA schemes to
monitor various quality characteristics and its existing enhanced versions (i.e. double GWMA,
composite Shewhart-GWMA, mixed GWMA-CUSUM and mixed CUSUM-GWMA) have better
performance than their corresponding EWMA counterparts in many situations; hence, all such
existing research works discussing GWMA-related schemes (i.e. 61 publications in total) are
documented and categorized in such a manner that it is easy to identify research gaps. Finally, a
number of possible future research ideas are provided.
Keywords: Generally weighted moving average, GWMA, Memory-type scheme, Run-length, Monte
Carlo Simulation.
1. Introduction
In any field, a literature review is essential as it gives an overview of what investigations have been
conducted and what are some of the possible research ideas that can be pursued. In statistical process
monitoring (SPM) there have been a number of literature reviews that have shed some light on the
current state of a particular topic and in doing so, made way for many publications thereafter; see for
instance
1,2,3,4,5
. Thus, the tradition emphasized by the latter articles is continued with the generally
weighted moving average (GWMA) monitoring schemes in this paper. The exponentially weighted
moving average (EWMA) monitoring scheme with an additional adjustment parameter was discussed
in Sheu and Griffith
6
, Sheu
7,8
as an enhancement procedure to further improve the sensitivity of the
EWMA scheme towards small shifts in the process mean. Thereafter, Sheu and Lin
9
formally
introduced this enhancement procedure as a stand-alone monitoring scheme which is simply called the
GWMA control chart; and they defined it as a moving average of past data where a specific weight is
assigned to each data point. Moreover, the moving average tends to be a representation of the more
recent process performance, as larger weights are allocated to the most recent observations. In
addition, Sheu and Lin
9
derived the properties that are required to compute the run-length distribution
and they discussed its importance as well as illustrated its implementation.
The purpose of this literature review is to acquaint SPM researchers as well as practitioners about
GWMA schemes which were supposed to be, in essence, replacements of the EWMA-type schemes;
however, this is not really the case in the literature or in practice; because there are actually way more
*
Corresponding author. S.C. Shongwe. E-mail: sandile@tuks.co.za.
1
Department of Statistics, College of Science, Engineering and Technology, University of South Africa;
Pretoria, South Africa;
2
Département Qualité Logistique Industrielle et Organisation, Université de Nantes &
LS2N UMR CNRS 6004, Nantes, France.
2
research output that have been reported on EWMA schemes than on the GWMA schemes from 2003
to mid-year 2020. Based on different authors who have commented on the latter predicament, they
have indicated that the implementation of the EWMA scheme is easier as compared to that of the
GWMA one. By conducting this review, it is also meant to convince researchers and practitioners
alike that GWMA schemes are not as complex as they are thought to be and to further explain that
they provide fascinating results when compared to other well-known monitoring schemes, more
especially, the EWMA monitoring scheme.
Since 2003, the year of publication of the first article, there have been a total of 61 publications on the
GWMA-related monitoring schemes and their enhancements. So far, the existing known
enhancements of the GWMA-related scheme are:
the double GWMA scheme – denoted as DGWMA scheme;
the composite Shewhart-GWMA scheme;
the mixed GWMA-CUSUM scheme and its reverse version, the mixed CUSUM-GWMA
scheme.
A DGWMA scheme is a weighted moving average of a weighted moving average; which implies that
the smoothing process is done twice (this concept was first introduced by Shamma and Shamma
10
for
the double EWMA (DEWMA) scheme). A composite Shewhart-GWMA scheme is a combination of
the Shewhart and GWMA schemes which is an efficient way of harnessing the benefits of these two
schemes (this concept was first introduced by Lucas
11
and Lucas and Saccucci
12
for the Shewhart-
CUSUM and Shewhart-EWMA schemes, respectively). The mixed GWMA-CUSUM scheme is a
combination of the GWMA and CUSUM schemes where the GWMA statistic is used as input in the
CUSUM scheme; however, the CUSUM-GWMA scheme uses the CUSUM statistic as input in the
GWMA scheme (this concept was first introduced by Abbas et al
13
for the EWMA-CUSUM scheme
and the reverse version was introduced by Zaman et al
14
).
The basic properties of the GWMA scheme and its existing enhancements as well as a detailed outline
of how the review is structured are provided in Section 2. Thereafter, the publications discussing
research works on GWMA, DGWMA, Shewhart-GWMA and GWMA-CUSUM (as well as its
reverse version) schemes are reviewed in Sections 3, 4, 5 and 6, respectively. Section 7 provides
concluding remarks and some possible future research ideas. Finally, Appendices A and B provide an
illustration of how the weight function kurtosis varies for different design parameters and an outline
of how the run-length properties are determined for GWMA-related monitoring schemes,
respectively.
3
2. Basic propeties of GWMA schemes and their enhancements
2.1 Operation of the GWMA scheme and its enhancements
The weight structure of the GWMA
scheme as compared to the corresponding EWMA scheme for
various design parameters are discussed in Appendix A. Assume that the quality characteristic of
interest is a subgroup mean, then
are i.i.d. (independent and identically distributed) normally
distributed random variables with in-control (IC) mean
and standard deviation
. In this section,
the operation of the GWMA scheme and its enhancements are illustrated for the subgroup mean. The
charting statistic of the GWMA
scheme is given by
(1)
where
is defined in Appendix A,
and
> 0. The expected value and standard
deviation of
are given by
and
(2)
where
(3)
Hence, the time-varying upper and lower control limits (denoted as
and
) of the GWMA
monitoring scheme are calculated as (with
> 0, i.e. a width parameter):
(4)
When , Equation (3) reduces to the following limiting constant
Thus, the asymptotic upper and lower control limits (denoted as
and
):
(5)
The above derivations given in Equations (1) to (5) are summarized in Table 1. Following a similar
line of argument, it follows that the charting statistics, design parameters and control limits of the
DGWMA, Shewhart-GWMA, GWMA-CUSUM and CUSUM-GWMA
schemes are as shown in
Table 1. For more details on the properties summarized in Table 1, refer to Sheu and Lin
9
, Lin
15
, Lu
16
and, Ali and Haq
17
.
4
Table 1: An illustration of different charting statistics, design parameters and control limits of the GWMA scheme and its existing enhancements when the
characteristic of interest is the process mean
Scheme
Charting statistics
Design
parameters
Time-varying
control limits
Asymptotic
control limits
GWMA
> 0,
,
> 0
with
with
DGWMA
with
0,
,
0
with
Shewhart-GWMA
OR
> 0,
,
> 0,
> 0
OR
OR
GWMA-CUSUM
0,
,
> 0
0
CUSUM-GWMA
> 0,
,
> 0
