Homogeneously Weighted Moving Average Control Charts: Overview, Controversies, and New Directions

Abstract

The homogeneously weighted moving average (HWMA) chart is a recent control chart that has attracted the attention of many researchers in statistical process control (SPC). The HWMA statistic assigns a higher weight to the most recent sample, and the rest is divided equally between the previous samples. This weight structure makes the HWMA chart more sensitive to small shifts in the process parameters when running in zero-state mode. Many scholars have reported its superiority over the existing charts when the process runs in zero-state mode. However, several authors have criticized the HWMA chart by pointing out its poor performance in steady-state mode due to its weighting structure, which does not reportedly comply with the SPC standards. This paper reviews and discusses all research works on HWMA-related charts (i.e., 55 publications) and provides future research ideas and new directions.

1. Introduction

Control charts are simple graphs used to identify which type of variation exists within a process to determine whether the process is stable or not. The chart shows the value of the quality characteristic versus the time. Thus, it consists of a charting statistic on the vertical axis, the time on the horizontal axis, a centreline representing the charting statistic’s mean, and two horizontal lines, called the lower control limit ($LCL$) and the upper control limit ($UCL$). If the charting statistic is plotted on or beyond the control limits, the process is said to be out-of-control (OOC).

Statistical process control (SPC) distinguishes two main types of charts, namely, memoryless (e.g., Shewhart-type charts) and memory-type (e.g., the cumulative sum (CUSUM) [1], exponentially weighted moving average (EWMA) [2] and homogeneously weighted moving average (HWMA) [3] as well as the CUSUM-, EWMA-, and HWM-related charts (see, e.g., Refs. [4,5,6,7,8,9,10,11])). Memoryless charts only focus on the most recent sample. In contrast, memory-type charts consider past and current samples, which are assigned weights summing up to one; see, e.g., Refs. [9,12]. Memory-type charts are known to be fast in detecting small and moderate shifts in the process parameters and slow in detecting large shifts. To improve the shift detection ability of the classical memory-type charts, many additional memory-type charts have been introduced in the literature, including the following:

(i)

The extended EWMA charts (e.g., the double, triple, and quadruple EWMA (denoted as DEWMA, TEWMA, and QEWMA) charts), the classical extended EWMA (denoted as EEWMA) chart, and the modified EWMA (ModEWMA) chart; see, e.g., Refs. [10,13,14,15];

(ii)

The CUSUM chart, dual CUSUM charts, and the generalization of the CUSUM chart; see, e.g., Refs. [16,17,18,19];

(iii)

The generally weighted moving average (GWMA) chart by [11] and the double GWMA chart by [20] (see also [21]);

(iv)

The extended HWMA charts (e.g., the double and triple HWMA (denoted as DHWMA and THWMA) charts) and the classical extended HWMA (denoted as EHWMA) chart; see, e.g., Refs. [22,23,24];

(v)

The compound EWMA and HWMA charts (e.g., Shewhart–EWMA, Shewhart–CUSUM, mixed EWMA–CUSUM (MEC), mixed CUSUM–EWMA (MCE), mixed HWMA–CUSUM (MHC), and mixed CUSUM–HWMA (MCH) charts); see, e.g., Refs. [25,26,27,28,29].

A literature review is critical in understanding the research concepts on various topics as it gives an overview of the investigations conducted. It also guides researchers by providing possible research ideas that can be pursued concerning research gaps; see, e.g., Refs. [30,31]. This paper gives an overview of the HWMA-related charts and focuses on documenting and categorising all existing research articles discussing HWMA-related charts.

The HWMA chart was first introduced by [3] to improve the sensitivity of the EWMA chart in monitoring small shifts in the process parameter. Since then, there have been about 54 additional publications on HWMA-related charts. These research works were collected from when the first article was published (i.e., 2018). Only research articles and proceedings from accredited sources are considered, and other research documents such as books, theses, and dissertations are not considered. The research articles were collected from multiple scientific sources, publishers, and platforms such as John Wiley, Taylor and Francis, Springer Link, Emerald, Growing Science, Google Scholar, etc., by using keywords such as homogeneously weighted moving average, HWMA chart, statistical process control (SPC), statistical process monitoring (SPM), chart, etc., as well as references within the articles meeting the search criteria.

The rest of this paper is organised as follows: Section 2 discusses the weight functions of memory-type charts. Section 3 gives the basic properties of the HWMA-related charts for monitoring the process mean. In addition, Section 3 presents a detailed outline of how Section 4 categorizes the existing HWMA-related charts based on several criteria. Section 5 focuses on the controversies on the HWMA-related charts, and Section 6 gives some concluding remarks and future research ideas.

2. The Weight Function of the Most Popular Charts

The distribution of weights (or weight structure) amongst samples differs from one chart to the next. It is by this characteristic that the nature of a given chart can be identified. The shift detection ability of memory-type charts mainly depends on the weight structure of the charting statistics. Figure 1 displays the weight structure of the Shewhart, CUSUM, EWMA, and HWMA charts, which depicts the weights assigned to the first 20 observations (assuming samples of size one) used in calculating the charting statistic corresponding to the 20th sample (i.e., weight on the 20th charting statistic) when the smoothing parameter for the HWMA and EWMA charts is equal to 0.5 (i.e., $\lambda =$ 0.5). Note that the 20th sample is designated as the current sample and the rest (i.e., samples 1 to 19) as previous samples. Figure 1 shows that for the EWMA chart, the weight assigned to previous samples decreases exponentially as the samples age; see also [32,33]. For instance, the current and 16th previous observations of the EWMA chart will be assigned weights of 0.5 and 0.03125, respectively. Thus, the older the sample, the smaller the weight. The weight assigned to the last 20 observations of the CUSUM chart from the current to the 20th previous observation is equal to 0.05, i.e., a uniform weight is assigned to the current and previous observations of the CUSUM chart; see also [34]. The HWMA chart assigns a weight of 0.5 to the current observation, and the rest of the observations are assigned a weight of $\frac{1-0.5}{19}$ (=0.026). Note, though, that the Shewhart chart focuses on the current sample with a weight of one, which is equivalent to the weight assigned to the EWMA chart with $\lambda =$ 1. In other words, the weight used to compute the current charting statistic is equal to 1 for the $t^{th}$ observation and zero for all the previous ones. Figure 2 displays the weight structure of the HWMA chart for the first 20 observations for different smoothing parameters. This weight structure allows for the fast detection of small shifts in the process parameter(s). However, the HWMA chart detection ability for moderate and large shifts decreases.

3. The Basic Properties of the HWMA and Enhanced HWMA Charts

3.1. Design of the HWMA and Enhanced HWMA Charts

This section illustrates the HWMA chart’s design and its enhancements for the sample mean. To this end, assume that the quality characteristic of interest is a sequence of observations $X_{tj}$ ($t=$ 1, 2, … and $j=$ 1, 2, …, $n$), which are independent and identically distributed (i.i.d.) $N\left(\mu ,\sigma \right)$ with known process parameters. When the process is in control (IC), the IC mean and standard deviation are given by ${\mu }_0$ and ${\sigma }_0$, respectively. Then, the charting statistic of the HWMA $\overline{X}$ chart (denoted as $H_t$) by [3] is defined by

$$H_t=\lambda {\overline{X}}_t+\left(1-\lambda \right) {\stackrel{=}{X}}_{t-1},$$

(1)

with

$${\stackrel{=}{X}}_{t-1}=\frac{1}{t-1}{\displaystyle {\displaystyle \sum }_{v=1}^{t-1}}{\overline{X}}_v,$$

where the smoothing parameter λ ranges from 0 to 1 and ${\stackrel{=}{X}}_{t-1}$ represents the mean of the previous $t-1$ sample means. Note that ${\stackrel{=}{X}}_0$, the initial value of ${\stackrel{=}{X}}_{t-1}$, is set to equal the IC mean ${\mu }_0$. According to [3], Equation (1) can sometimes be written as

$$H_t=\lambda {\overline{X}}_t+\left(\left(\frac{1-\lambda }{t-1}\right){\overline{X}}_{t-1}+\left(\frac{1-\lambda }{t-1}\right){\overline{X}}_{t-2}+\cdots +\left(\frac{1-\lambda }{t-1}\right){\overline{X}}_2+\left(\frac{1-\lambda }{t-1}\right){\overline{X}}_1 \right).$$

(2)

It can be deduced from Equation (2) that the HWMA $\overline{X}$ statistic assigns weight $\lambda$ to the current sample, and a weight $\left(1-\lambda \right)$ is equally distributed to the previous $t-1$ samples. Furthermore, Abbas [3] indicated that the mean and variance of the charting statistic $H_t$ are given by

$$\begin{gathered} E(H_t)={\mu }_0 \\ \mathrm{and} \\ Var\left(H_t\right)={\sigma }_{H_t}^2=\{\begin{array}{c}\frac{{\lambda }^2{\sigma }_0^2}{n}, t=1\\ \frac{{\lambda }^2{\sigma }_0^2}{n}+\frac{{\left(1-\lambda \right)}^2{\sigma }_0^2}{n\left(t-1\right)}, t>1\end{array}, \end{gathered}$$

(3)

respectively. Therefore, the time-varying lower and upper control limits, i.e., $LC{L}_{H_t}$ and $UC{L}_{H_t}$, of the HWMA $\overline{X}$ chart are defined by

$$\begin{gathered} LC{L}_{H_t}=\{\begin{array}{c}{\mu }_0-L_H\sqrt{{\lambda }^2\frac{ {\sigma }_0^2}{n}}, t=1\\ {\mu }_0-L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}+{\left(1-\lambda \right)}^2\frac{{\sigma }_0^2}{n\left(t-1\right)} }, t>1\end{array} \\ \mathrm{and} \\ UC{L}_{H_t}=\{\begin{array}{c}{\mu }_0+L_H\sqrt{{\lambda }^2\frac{ {\sigma }_0^2}{n}}, t=1\\ {\mu }_0+L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}+{\left(1-\lambda \right)}^2\frac{{\sigma }_0^2}{n\left(t-1\right)} }, t>1\end{array} \end{gathered}$$

(4)

respectively, where the design parameter $L_H>0$ determines the width of the control limits and it is selected such that the chart has an attained IC average run-length ($ARL$) approximately equal to some prespecified nominal IC $ARL$ ($AR{L}_0$) such as 200, 370, or 500. For more details on how to search for the optimal design parameter $L_H$ and compute the characteristics of the run-length distribution of the HWMA $\overline{X}$ chart, readers are referred to Appendix A. Thus, the HWMA $\overline{X}$ chart gives a signal when the charting statistic defined in Equation (1) plots on or outside the control limits defined in Equation (4); that is, if $H_t\ge UC{L}_{H_t}$ or $H_t\le LC{L}_{H_t}$. In cases where the process is considered to have been running for a considerable time (i.e., $t\to \infty$), the term $\frac{{\left(1-\lambda \right)}^2{\sigma }_0^2}{n\left(t-1\right)}\to 0$. Therefore, the control limits in Equation (4) reduce to the following asymptotic ones:

$$\begin{gathered} LCL={\mu }_0-L_H\frac{\lambda {\sigma }_0}{\sqrt{n}} \\ \mathrm{and} \\ UCL={\mu }_0+L_H\frac{\lambda {\sigma }_0}{\sqrt{n}}. \end{gathered}$$

(5)

The above expressions are summarised in Table 1 along with the charting statistics, design parameters, and control limits of the HHWMA, THWMA, HWMA–CUSUM, and CUSUM–HWMA $\overline{X}$ charts. Note that all starting values of the charting statistics of these charts are typically set to be equal to the IC process mean (i.e., ${\mu }_0$) except the one for the CUSUM-related chart, which is set to be equal to zero. When the smoothing parameters ${\lambda }_1$ and ${\lambda }_2$ of the HHWMA chart are equal, the HHWMA chart reduces to the DHWMA chart. For more details on the properties summarised in Table 1, readers are referred to [3,22,25,35,36,37,38,39,40].

Note that in Table 1, the underlying process parameters are assumed to be known (i.e., Case K), and the process monitoring can start immediately after computing the control limits that yield a desired attained IC $ARL$ value; see, e.g., Refs. [13,26,38,41]. Typically, the process parameters are unknown (i.e., Case U). In this case, the process monitoring is implemented in two phases; the parameters are estimated in Phase I when the process is considered IC, and then, in Phase II, online monitoring is performed.

For more details on the design and implementation of HWMA-type charts, readers are referred to [3,25,42,43].

3.2. Performance Evaluation

The performance of a chart can be evaluated using the characteristics of the run-length distribution such as the $ARL,$ the standard deviation of the run-length ($SDRL$), the median run-length ($MRL$), etc. The most popular performance measure is the $ARL$. This measure represents the average number of charting statistics to be plotted on a chart before the first OOC signal is observed. The run-length characteristics are often investigated using two different modes, namely, the zero- and steady-state modes; see, e.g., Ref. [44]. The zero- and steady-state run-length modes are used to investigate the short-term and the long-term run-length properties of a chart, respectively. The zero-state run-length is defined as the number of samples at which the chart first signals given that it begins in some specific initial state. However, the steady-state run-length is the number of samples at which the chart first signals given that the process begins and stays IC for a long period, and then at some random time, an OOC is observed; see, e.g., Refs. [44,45].

Table 1. A summary of the charting statistics, design parameters, and control limits of the HWMA chart and its existing enhancements when the characteristic of interest is the process mean.

Chart	Charting Statistics	Design Parameters	Time-Varying Control Limits	Asymptotic Control Limits
HWMA	$H_t=\lambda {\overline{X}}_t+\left(1-\lambda \right) {\stackrel{=}{X}}_{t-1}$ $\mathrm{where} {\stackrel{=}{X}}_0={\mu }_0$ and ${\stackrel{=}{X}}_{t-1}=\frac{1}{t-1}{\displaystyle {\displaystyle {\displaystyle \sum }_{v=1}^{t-1}}}{\overline{X}}_v.$	$L_H$ > 0, $0<\lambda \le 1$	$LC{L}_{H_t}=\{\begin{array}{c}{\mu }_0-L_H\sqrt{{\lambda }^2\frac{ {\sigma }_0^2}{n}}, t=1\\ {\mu }_0-L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}+{\left(1-\lambda \right)}^2\frac{{\sigma }_0^2}{n\left(t-1\right)} }, t>1\end{array}$ and $UC{L}_{H_t}=\{\begin{array}{c}{\mu }_0+L_H\sqrt{{\lambda }^2\frac{ {\sigma }_0^2}{n}}, t=1\\ {\mu }_0+L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}+{\left(1-\lambda \right)}^2\frac{{\sigma }_0^2}{n\left(t-1\right)} } t>1\end{array}.$	$LC{L}_H={\mu }_0-L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}}$ and $UC{L}_H={\mu }_0+L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}}.$
HHWMA	$H{H}_t={\lambda }_2{H}_t+\left(1-{\lambda }_2\right) {\overline{H}}_{t-1}$ where $H_t={\lambda }_1{\overline{X}}_t+\left(1-{\lambda }_1\right) {\overline{\overline{X}}}_{t-1},$ ${\overline{H}}_0={\stackrel{=}{X}}_0={\mu }_0$ and ${\overline{H}}_{t-1}=\frac{1}{t-1}{\displaystyle {\displaystyle {\displaystyle \sum }_{v=1}^{t-1}}}{H}_v.$	$L_{HH}>$ 0, $0<{\lambda }_1,{\lambda }_2\le 1.$	$LC{L}_{H{H}_t}={\mu }_0-L_{HH}\sqrt{Var\left(H{H}_t\right)}$ and $UC{L}_{H{H}_t}={\mu }_0+L_{HH}\sqrt{Var\left(H{H}_t\right)}$ where $Var\left({HH}_t\right)=\left\{\begin{array}{cc}{\lambda }_1^2{\lambda }_2^2\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t=1\hfill \\ \left[{\lambda }_1^2{\lambda }_2^2+{\left({\lambda }_1+{\lambda }_2-2{\lambda }_1{\lambda }_2\right)}^2\right]\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t=2\hfill \\ \left[{\lambda }_1^2{\lambda }_2^2+term1+term2\right]\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t>2.\hfill \end{array}\right.$	$LC{L}_{HH}={\mu }_0-L_{HH}\sqrt{Var\left(H{H}_t\right)}$ and $UC{L}_{HH}={\mu }_0+L_{HH}\sqrt{Var\left(H{H}_t\right)}$ with $Var\left({HH}_t\right)=\left\{\begin{array}{cc}{\lambda }_1^2{\lambda }_2^2\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t=1\hfill \\ \left[{\lambda }_1^2{\lambda }_2^2+{\left({\lambda }_1+{\lambda }_2-2{\lambda }_1{\lambda }_2\right)}^2\right]\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t=2\hfill \\ \left[{\lambda }_1^2{\lambda }_2^2\right]\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t>2.\hfill \end{array}\right.$
THWMA	$T{H}_t=\lambda D{H}_t+\left(1-\lambda \right) {\stackrel{=}{X}}_{t-1}$	$L_{TH}$ > 0, $0<\lambda \le 1$	$LC{L}_{T{H}_t}={\mu }_0-L_{TH}\sqrt{Var\left(T{H}_t\right)}$ and ${UCL}_{{TH}_t}={\mu }_0+L_{TH}\sqrt{Var\left({TH}_t\right)}$ where $Var\left(T{H}_t\right)=\{\begin{array}{c}\sqrt{{\lambda }^4\frac{ {\sigma }_0^2}{n}}, t=1\\ \sqrt{{\lambda }^4\frac{{\sigma }_0^2}{n}+{\left(1-{\lambda }^2\right)}^2\frac{{\sigma }_0^2}{n\left(t-1\right)} }, t>1\end{array}$	$LC{L}_{TH}={\mu }_0-L_{TH}\sqrt{{\lambda }^4\frac{ {\sigma }_0^2}{n}}$ and $UC{L}_{TH}={\mu }_0+L_{TH}\sqrt{{\lambda }^4\frac{ {\sigma }_0^2}{n}}.$
HWMA–CUSUM	$H{C}_t^{+}=\max \left[0, (H_t-{\mu }_0-k_H\sqrt{Var\left(H_t\right)})+H{C}_{t-1}^{+}\right],$ and $H{C}_t^{-}=\max \left[0, \left({\mu }_0-H_t-k_H\sqrt{Var\left(H_t\right)}\right)+H{C}_{t-1}^{-}\right],$ $\mathrm{with} H{C}_0^{+}=H{C}_0^{-}=0$	$k_H>$ 0, $0<\lambda \le 1$, $h>$ 0	$UC{L}_{H{C}_t}=h\sqrt{Var\left(H_t\right)}$	$UC{L}_{HC}=h\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}}$
CUSUM–HWMA	$C{H}_t^{-}=\lambda C_t^{-}+\left(1-\lambda \right) {\overline{C}}_{t-1}^{-}$ and $C{H}_t^{+}=\lambda C_t^{+}+\left(1-\lambda \right) {\overline{C}}_{t-1}^{+},$ where $C_t^{-}=\max \left[0, \left({\mu }_0-{\overline{X}}_t-k\frac{{\sigma }_0}{\sqrt{n}}\right)+C_{t-1}^{-}\right]$ and $C_t^{+}=\max \left[0, ({\overline{X}}_t-{\mu }_0-k\frac{{\sigma }_0}{\sqrt{n}})+C_{t-1}^{+}\right]$ $\mathrm{with} C_0^{+}=C_0^{-}=0.$	$L_{CH}$ > 0, $0<\lambda \le 1$,	$LC{L}_{C{H}_t} =\{\begin{array}{cc}{\mu }_{C_t}- L_{CH} \left(\lambda {\sigma }_{C_t}\right),\hfill & t=1\hfill \\ {\mu }_{C_t}-L_{CH} {\sigma }_{C_t}\sqrt{{\lambda }^2+\frac{{\left(1-\lambda \right)}^2}{\left(t-1\right)} }\hfill & t>1\hfill \end{array}$ and $UC{L}_{C{H}_t} =\{\begin{array}{cc}{\mu }_{C_t}+ L_{CH} \left(\lambda {\sigma }_{C_t}\right),\hfill & t=1\hfill \\ {\mu }_{C_t}+L_{CH} {\sigma }_{C_t}\sqrt{{\lambda }^2+\frac{{\left(1-\lambda \right)}^2}{\left(t-1\right)} },\hfill & t>1\hfill \end{array}$ $\mathrm{with} {\mu }_{C_t}=E\left(C_t^{+}\right)=E\left(C_t^{-}\right)\mathrm{and} {\sigma }_{C_t}=Var\left(C_t^{+}\right)=Var\left(C_t^{-}\right).$	$LC{L}_{CH} ={\mu }_{C_t}- L_{CH} \left(\lambda {\sigma }_{C_t}\right)$ and $UC{L}_{CH} ={\mu }_{C_t}+L_{CH} \left(\lambda {\sigma }_{C_t}\right).$

Note: $term1=\frac{{\left({\lambda }_1+{\lambda }_2-2{\lambda }_1{\lambda }_2\right)}^2}{{\left(t-1\right)}^2}$ and $term2=\frac{1}{{\left(t-1\right)}^2}{\displaystyle {\sum }_{u=1}^{t-2}}{\left({\lambda }_1+{\lambda }_2-2{\lambda }_1{\lambda }_2+\left(1-{\lambda }_{1 }\right)\left(1-{\lambda }_2\right){\displaystyle {\sum }_{k=u}^{t-2}}\frac{1}{k}\right)}^2.$

Table 1. A summary of the charting statistics, design parameters, and control limits of the HWMA chart and its existing enhancements when the characteristic of interest is the process mean.

Chart	Charting Statistics	Design Parameters	Time-Varying Control Limits	Asymptotic Control Limits
HWMA	$H_t=\lambda {\overline{X}}_t+\left(1-\lambda \right) {\stackrel{=}{X}}_{t-1}$ $\mathrm{where} {\stackrel{=}{X}}_0={\mu }_0$ and ${\stackrel{=}{X}}_{t-1}=\frac{1}{t-1}{\displaystyle {\displaystyle {\displaystyle \sum }_{v=1}^{t-1}}}{\overline{X}}_v.$	$L_H$ > 0, $0<\lambda \le 1$	$LC{L}_{H_t}=\{\begin{array}{c}{\mu }_0-L_H\sqrt{{\lambda }^2\frac{ {\sigma }_0^2}{n}}, t=1\\ {\mu }_0-L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}+{\left(1-\lambda \right)}^2\frac{{\sigma }_0^2}{n\left(t-1\right)} }, t>1\end{array}$ and $UC{L}_{H_t}=\{\begin{array}{c}{\mu }_0+L_H\sqrt{{\lambda }^2\frac{ {\sigma }_0^2}{n}}, t=1\\ {\mu }_0+L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}+{\left(1-\lambda \right)}^2\frac{{\sigma }_0^2}{n\left(t-1\right)} } t>1\end{array}.$	$LC{L}_H={\mu }_0-L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}}$ and $UC{L}_H={\mu }_0+L_H\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}}.$
HHWMA	$H{H}_t={\lambda }_2{H}_t+\left(1-{\lambda }_2\right) {\overline{H}}_{t-1}$ where $H_t={\lambda }_1{\overline{X}}_t+\left(1-{\lambda }_1\right) {\overline{\overline{X}}}_{t-1},$ ${\overline{H}}_0={\stackrel{=}{X}}_0={\mu }_0$ and ${\overline{H}}_{t-1}=\frac{1}{t-1}{\displaystyle {\displaystyle {\displaystyle \sum }_{v=1}^{t-1}}}{H}_v.$	$L_{HH}>$ 0, $0<{\lambda }_1,{\lambda }_2\le 1.$	$LC{L}_{H{H}_t}={\mu }_0-L_{HH}\sqrt{Var\left(H{H}_t\right)}$ and $UC{L}_{H{H}_t}={\mu }_0+L_{HH}\sqrt{Var\left(H{H}_t\right)}$ where $Var\left({HH}_t\right)=\left\{\begin{array}{cc}{\lambda }_1^2{\lambda }_2^2\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t=1\hfill \\ \left[{\lambda }_1^2{\lambda }_2^2+{\left({\lambda }_1+{\lambda }_2-2{\lambda }_1{\lambda }_2\right)}^2\right]\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t=2\hfill \\ \left[{\lambda }_1^2{\lambda }_2^2+term1+term2\right]\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t>2.\hfill \end{array}\right.$	$LC{L}_{HH}={\mu }_0-L_{HH}\sqrt{Var\left(H{H}_t\right)}$ and $UC{L}_{HH}={\mu }_0+L_{HH}\sqrt{Var\left(H{H}_t\right)}$ with $Var\left({HH}_t\right)=\left\{\begin{array}{cc}{\lambda }_1^2{\lambda }_2^2\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t=1\hfill \\ \left[{\lambda }_1^2{\lambda }_2^2+{\left({\lambda }_1+{\lambda }_2-2{\lambda }_1{\lambda }_2\right)}^2\right]\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t=2\hfill \\ \left[{\lambda }_1^2{\lambda }_2^2\right]\frac{{\sigma }_0^2}{n},\hfill & \mathrm{f}\mathrm{o}\mathrm{r} t>2.\hfill \end{array}\right.$
THWMA	$T{H}_t=\lambda D{H}_t+\left(1-\lambda \right) {\stackrel{=}{X}}_{t-1}$	$L_{TH}$ > 0, $0<\lambda \le 1$	$LC{L}_{T{H}_t}={\mu }_0-L_{TH}\sqrt{Var\left(T{H}_t\right)}$ and ${UCL}_{{TH}_t}={\mu }_0+L_{TH}\sqrt{Var\left({TH}_t\right)}$ where $Var\left(T{H}_t\right)=\{\begin{array}{c}\sqrt{{\lambda }^4\frac{ {\sigma }_0^2}{n}}, t=1\\ \sqrt{{\lambda }^4\frac{{\sigma }_0^2}{n}+{\left(1-{\lambda }^2\right)}^2\frac{{\sigma }_0^2}{n\left(t-1\right)} }, t>1\end{array}$	$LC{L}_{TH}={\mu }_0-L_{TH}\sqrt{{\lambda }^4\frac{ {\sigma }_0^2}{n}}$ and $UC{L}_{TH}={\mu }_0+L_{TH}\sqrt{{\lambda }^4\frac{ {\sigma }_0^2}{n}}.$
HWMA–CUSUM	$H{C}_t^{+}=\max \left[0, (H_t-{\mu }_0-k_H\sqrt{Var\left(H_t\right)})+H{C}_{t-1}^{+}\right],$ and $H{C}_t^{-}=\max \left[0, \left({\mu }_0-H_t-k_H\sqrt{Var\left(H_t\right)}\right)+H{C}_{t-1}^{-}\right],$ $\mathrm{with} H{C}_0^{+}=H{C}_0^{-}=0$	$k_H>$ 0, $0<\lambda \le 1$, $h>$ 0	$UC{L}_{H{C}_t}=h\sqrt{Var\left(H_t\right)}$	$UC{L}_{HC}=h\sqrt{{\lambda }^2\frac{{\sigma }_0^2}{n}}$
CUSUM–HWMA	$C{H}_t^{-}=\lambda C_t^{-}+\left(1-\lambda \right) {\overline{C}}_{t-1}^{-}$ and $C{H}_t^{+}=\lambda C_t^{+}+\left(1-\lambda \right) {\overline{C}}_{t-1}^{+},$ where $C_t^{-}=\max \left[0, \left({\mu }_0-{\overline{X}}_t-k\frac{{\sigma }_0}{\sqrt{n}}\right)+C_{t-1}^{-}\right]$ and $C_t^{+}=\max \left[0, ({\overline{X}}_t-{\mu }_0-k\frac{{\sigma }_0}{\sqrt{n}})+C_{t-1}^{+}\right]$ $\mathrm{with} C_0^{+}=C_0^{-}=0.$	$L_{CH}$ > 0, $0<\lambda \le 1$,	$LC{L}_{C{H}_t} =\{\begin{array}{cc}{\mu }_{C_t}- L_{CH} \left(\lambda {\sigma }_{C_t}\right),\hfill & t=1\hfill \\ {\mu }_{C_t}-L_{CH} {\sigma }_{C_t}\sqrt{{\lambda }^2+\frac{{\left(1-\lambda \right)}^2}{\left(t-1\right)} }\hfill & t>1\hfill \end{array}$ and $UC{L}_{C{H}_t} =\{\begin{array}{cc}{\mu }_{C_t}+ L_{CH} \left(\lambda {\sigma }_{C_t}\right),\hfill & t=1\hfill \\ {\mu }_{C_t}+L_{CH} {\sigma }_{C_t}\sqrt{{\lambda }^2+\frac{{\left(1-\lambda \right)}^2}{\left(t-1\right)} },\hfill & t>1\hfill \end{array}$ $\mathrm{with} {\mu }_{C_t}=E\left(C_t^{+}\right)=E\left(C_t^{-}\right)\mathrm{and} {\sigma }_{C_t}=Var\left(C_t^{+}\right)=Var\left(C_t^{-}\right).$	$LC{L}_{CH} ={\mu }_{C_t}- L_{CH} \left(\lambda {\sigma }_{C_t}\right)$ and $UC{L}_{CH} ={\mu }_{C_t}+L_{CH} \left(\lambda {\sigma }_{C_t}\right).$

Note: $term1=\frac{{\left({\lambda }_1+{\lambda }_2-2{\lambda }_1{\lambda }_2\right)}^2}{{\left(t-1\right)}^2}$ and $term2=\frac{1}{{\left(t-1\right)}^2}{\displaystyle {\sum }_{u=1}^{t-2}}{\left({\lambda }_1+{\lambda }_2-2{\lambda }_1{\lambda }_2+\left(1-{\lambda }_{1 }\right)\left(1-{\lambda }_2\right){\displaystyle {\sum }_{k=u}^{t-2}}\frac{1}{k}\right)}^2.$

The zero- and steady-state run-lengths of the HWMA $\overline{X}$ chart denoted as $L_Z$ and $L_S$ are mathematically defined by (see, Refs. [45,46,47])

$$\begin{gathered} L_Z=min\left\{t\ge 1|H_t\le LC{L}_{H_t}or H_t\ge UC{L}_{H_t}\right\} \\ \mathrm{and} \\ {L}_S=min\left\{t\ge \tau +1|H_t<LC{L}_{H_t}or H_t>UC{L}_{H_t}\right\}, \tau =1,2,\dots \end{gathered}$$

(6)

respectively; where τ is the change point applied in the change point model:

$$\mu =\{\begin{array}{c}{\mu }_0=0, if t<\tau \\ {\mu }_1=\delta , if t\ge \tau \end{array},$$

(7)

where $\delta$ represents the change in the process mean.

For more details on how to compute the characteristics of the run-length distribution, readers are referred to Appendix A.

3.3. Outline of the Review

The structure of the review in Section 4 is outlined in

Table 2. Outline of the review in Section 4.

	Chart	Distribution	Section	Process Characteristic	Uni	Multi
Section 4.1	HWMA	Parametric	Section 4.1.1	Location	🗸	🗸
			Section 4.1.2	Variability	🗸	$\times$
			Section 4.1.3	Joint Location and Variability	🗸	$\times$
			Section 4.1.4	Profile	🗸	🗸
		Nonparametric	Section 4.1.5	Location	🗸	🗸
				Scale	$\times$	×
			Section 4.1.6	Joint Location and Scale	🗸	$\times$
				Profile	$\times$	$\times$
Section 4.2	DHWMA and HHWMA	Parametric	Section 4.2.1	Location	🗸	$\times$
				Variability	$\times$	×
			Section 4.2.2	Joint Location and Variability	🗸	$\times$
				Profile	$\times$	$\times$
		Nonparametric	Section 4.2.3	Location	🗸	$\times$
				Scale	$\times$	$\times$
				Joint Location and Scale	$\times$	$\times$
				Profile	×	$\times$
Section 4.3	THWMA	Parametric	Section 4.3.1	Location	🗸	$\times$
			Section 4.3.2	Variability	🗸	$\times$
				Joint Location and Variability	×	×
				Profile	$\times$	$\times$
		Nonparametric	Section 4.3.3	Location	🗸	$\times$
				Scale	×	$\times$
				Joint Location and Scale	×	$\times$
				Profile	×	$\times$
Section 4.4	HWMA-CUSUM & CUSUM-HWMA	Parametric	Section 4.4.1	Location	🗸	$\times$
				Variability	$\times$	$\times$
				Joint Location and Variability	$\times$	$\times$
				Profile	$\times$	$\times$
		Nonparametric		Location	$\times$	$\times$
				Scale	$\times$	$\times$
				Joint Location and Scale	$\times$	×
				Profile	$\times$	$\times$

Note: Uni = univariate process; Multi = multivariate process.

. The “tick symbol” (i.e., “🗸”) indicates topics already treated, and the “cross symbol” (i.e., “$\times$”) indicates topics not yet published in the SPC literature.

For ease in identifying the different HWMA-type charts and their enhancements, in Table 3, the charts have been summarized with respect to the following characteristics:

(i)

The process characteristic being monitored, i.e., the location (L), variability (V), both the location and variability (LV), or profile monitoring (P), e.g., for monitoring regression parameters and/or the regression error variance;

(ii)

Whether the charts are parametric (Par) or nonparametric (NPar);

(iii)

The design of the chart, i.e., economic (E), statistical (S), or economic statistical design (ESD);

(iv)

Whether the underlying process parameters are known or unknown, i.e., Case K or Case U;

(v)

The dimension of the process, i.e., univariate (Uni) or multivariate (Multi);

(vi)

Whether the observations are i.i.d (IID) or autocorrelated (AC);

(vii)

With measurement error (ME);

(viii)

The mode, i.e., either zero-state (ZS) or steady-state (SS);

(ix)

The type of data used in the illustrative examples, i.e., simulated (Sim) or real-life (Real) data;

(x)

The number of citations (NC);

(xi)

The performance metrics as discussed and used in the design;

(xii)

The type of chart;

(xiii)

The journal where the paper was published.

Note that Table 3 contains information about the HWMA-related articles that are the focus of this review paper. In this article, readers are also referred to other types of charts for general information. The list of acronyms used in this paper is given in Appendix B.

A summary of the journals and conference proceedings that published research on HWMA-type charts and their enhancements are provided in

Table 4. Types and JCR rank of the journals/conference proceedings that published research on HWMA-type charts.

Journal/Conference Proceedings	Open Access/Hybrid	JCR Quartile	Number of Publications
Computers & Industrial Engineering (CIE)	Hybrid	1	2
Mathematics	Open access	1	2
Quality and Reliability Engineering International (QREI)	Hybrid	2	18
IEEE Access	Open Access	1	4
Journal of Applied Statistics (JAS)	Hybrid	2	1
IEEE International Conference on Industrial Engineering and Engineering Management (IEEE ICIEEM)	Open access	2	2
Computer Modeling in Engineering & Sciences (CMES)	Open access	3	1
Mathematical Problems in Engineering (MPE)	Open access	2	5
Journal of Statistical Computation and Simulation (JSCS)	Hybrid	2	2
Chemometrics and Laboratory Systems (CLS)	Hybrid	2	1
Symmetry	Open access	2	4
PLoS ONE	Open access	1	2
AIMS Mathematics	Open access	2	1
Quality Technology & Quantitative Management (QTQM)	Hybrid	1	1
Communications in Statistics-Simulation and Computation (CS-SC)	Hybrid	1	2
Journal Testing and Evaluation (JTE)	Open access	3	1
Processes	Open access	2	1
Scientific Reports	Open access	1	1
Transactions of the Institute of Measurement and Control (TIMC)	Hybrid	2	1
International Journal of Industrial Engineering Computations (IJIEC)	Hybrid	1	1
Journal of Advanced Manufacturing Systems (JAMS)	Hybrid	3	1
Journal of Quality Technology (JQT)	Hybrid	1	1

; this includes the type of journal, i.e., open access or hybrid, the ranking of the journal according to the journal citation reports (JCR) or proceedings, and the total number of publications. Table 4 shows that HWMA-type articles were published in 21 journals, and two were published as conference proceedings papers. Fifty percent of the journals in which these articles were published are open access, and the rest are hybrid journals. Eighteen out of fifty-five articles (i.e., 32.7%) were published in Quality and Reliability Engineering International (QREI), a quartile 2 journal. Note that according to the JCR ranking criteria, the top 25% of journals are classified in quartile 1, followed by the next 25% top journals in quartile 2, etc. From Table 4 and Figure 3, it can be noticed that 52% of the articles were published in quartile 2 journals and 35% were published in quartile 1 journals, followed by quartile 3 journals that published 13% of articles. No article was published in quartile 4; see Figure 3.

Table 3. Classification of the articles discussing the HWMA-related charts.

Articles	L, V, LV, P	Par or NPar	E, SD, ESD	K	U	Dimension	IID	AC	ME	NC	Data	Mode	Performance Metrics	Type of HWMA Chart	Journal
Abbas [3]	L	Par	SD	🗸	🗸	Uni	🗸			94	Real	ZS	ARL, SDRL	HWMA $\overline{X}$	CIE
Adegoke et al. [43]	L	Par	SD	🗸	🗸	Uni		🗸		52	Sim	ZS	ARL, SDRL	HWMA $\overline{X}$ with AIB	IEEE Access
Adegoke et al. [48]	L	Par	SD	🗸		Multi	🗸			43	Sim, Real	ZS	ARL, SDRL	$\mathrm{MHWMA} T^2$	IEEE Access
Nawaz and Han [49]	L	Par	SD	🗸		Uni	🗸			32	Real	ZS	ARL	HWMA $\overline{X}$ using RSS	QTQM
Abbas et al. [42]	L	Par	SD		🗸	Multi	🗸	🗸		12	Real	ZS	ARL	$\mathrm{MHWMA} T^2$	Mathematics
Abid et al. [25]	L	Par	SD	🗸	🗸	Uni	🗸			35	Real	ZS	ARL, SDRL, MRL	DHWMA $\overline{X}$	QREI
Abid et al. [26]	L	Par	SD	🗸		Uni	🗸			25	Real	ZS	ARL, SDRL, PRL	HWMA–CUSUM $\overline{X}$	QREI
Abid et al. [50]	L	Par	SD	🗸		Uni	🗸			8	Real	ZS	ARL, SDRL, PRL	CUSUM–HWMA $\overline{X}$	QREI
Adeoti and Koleoso [51]	L	Par	SD	🗸		Uni	🗸			24	Sim	ZS	ARL, SDRL	HHWMA $\overline{X}$	QREI
Raza et al. [52]	L	NPar	SD	🗸		Uni	🗸			19	Real	ZS	ARL, SDRL, MRL, EQL	HWMA SN and SR	JTE
Riaz et al. [53]	L	NPar	SD	🗸		Uni	🗸			16	Real	ZS	ARL	DHWMA SN	QREI
Riaz et al. [41]	V	Par	SD	🗸		Uni	🗸			17	Real	ZS	ARL, SDRL, MDRL	$\mathrm{HWMA} S^2$	Mathematics
Dawod et al. [54]	P	Par	SD		🗸	Multi		🗸		14	Real	ZS	ARL	HWMA profiles	CLS
Thanwane et al. [36]	L	Par	SD	🗸		Uni	🗸		🗸	11	Real	ZS	ARL, EARL	HWMA $\overline{X}$	TIMC
Thanwane et al. [40]	L	Par	SD		🗸	Uni	🗸	🗸	🗸	7	Real	ZS	ARL, SDRL, EARL, ESDRl	HWMA $\overline{X}$	IEEE Access
Thanwane et al. [37]	L	Par	SD		🗸	Uni	🗸			6	Real	ZS	ARL, SDRL, EARL, ESDRl	HWMA $\overline{X}$ with FIR	QREI
Thanwane et al. [38]	L	Par	SD	🗸		Uni		🗸		2	Real	ZS	ARL, SDRL, EARL, ESDRl, PCI	HWMA $\overline{X}$	IJIEC
Thanwane et al. [39]	L	Par	SD		🗸	Uni	🗸		🗸	13	Real	ZS	ARL, EARL	HWMA $\overline{X}$	JSCS
Alevizakos et al. [22]	L	Par	SD	🗸		Uni	🗸			13	Real, Sim	ZS	ARL, SDRL, AEQL, PCI, RMI	DHWMA & HHWMA $\overline{X}$	QREI
Alevizakos et al. [23]	L	NPar	SD	🗸		Uni	🗸			2	Real, Sim	ZS	ARL, SDRL, MRL, AEQL	DHWMA SN	QREI
Riaz et al. [24]	L	Par	SD		🗸	Uni	🗸			14	Real	ZS,SS	ARL, SDRL, MDRL	THWMA	Symmetry
Malela-Majika et al. [35]	L	Par	SD	🗸		Multi	🗸			3	-	ZS	ARL, SDRL, MRL	HHWMA	QREI
Adegoke et al. [55]	L	Par	SD		🗸	Uni	🗸			0	Real	ZS	ARL, SDRL	DSHWMA	QREI
Adegoke et al. [56]	L	Par	SD		🗸	Multi	🗸			2	Real	ZS,SS	ARL, SDRL	MHWMA	IEEE Access
Noor-ul-Amin and Noor [57]	L	Par	SD		🗸	Uni	🗸			1	Real	ZS	ARL, SDRL	HWMA $\overline{X}$ BTB	QREI
Adeoti et al. [58]	LV	Par	SD	🗸		Uni	🗸			10	Real, Sim	ZS	ARL, SDRL, MRL	HWMA (COM-Poisson)	JAS
Knoth et al. [44]	L	Par	SD	🗸		Uni	🗸			24	-	ZS,SS	ARL (CED)	HWMA with PM	QREI
Chan et al. [59]	LV	NPar	SD	🗸		Uni	🗸			16	Real	ZS	ARL, EARL, PRL	DEWMA–HWMA	CIE
Abbasi et al. [60]	V	Par	SD		🗸	Uni	🗸			3	Real		ARL, SDRL, MRL, EQL, RARL	HWMA dispersion	QREI
Abbasi et al. [61]	P	Par	SD	🗸		Uni	🗸			3	Real	ZS	ARL	HWMA profile	QREI
Rasheed et al. [62]	L	Par	SD	🗸		Multi	🗸			10	Real	ZS	ARL, EQL, RARL, PCI	MHWHC, MHCHW	MPE
Riaz et al. [63]	L	Par	SD	🗸		Uni	🗸			8	-	ZS,SS	ARL	HWMA	Processes
Aslam et al. [64]	L	Par	SD	🗸		Uni	🗸			1	Real	ZS	ARL, SDRL, EQL, RARL, PCI	HWMA TBE	QREI
Anwar et al. [65]	L	Par	SD	🗸		Uni	🗸			14	Real	ZS	ARL, RARL, EQL, PCI	DHWMA	QREI
Letshedi et al. [7]	L	NPar	SD		🗸	Uni	🗸			0	Real	ZS	ARL, SDRL, EARL, ESDRL	HWMA and HHWMA	PLoS one
Anwar et al. [66]	L	Par	SD	🗸		Uni		🗸		0	Real	ZS	ARL, SDRL, MRL, EQL, RARL, PCI	THWMA AIB	MPE
Al-Sayed et al. [67]	LV	Par	SD		🗸	Uni		🗸		0	Real	ZS	ARL, RMI	PR–HWMA, PR–DHWMA (ZIP)	IEEE ICIEEM
Rasheed et al. [68]	L	NPar	SD	🗸		Uni	🗸			8	Real	ZS	ARL, SDRL, MRL	DHWMA SR (RSS)	MPE
Chong et al. [69]	L	Par	SD	🗸		Uni	🗸			0	-	ZS	ARL, SDRL, PRL, EARL, EMRL	HWMA, DHWMA	IEEE ICIEEM
Iqbal et al. [70]	LV	Par	SD		🗸	Uni		🗸		13	Real	ZS	ARL, SDRL	HWMA and DHWMA DR, SDR	Symmetry
Zhang et al. [71]	L	NPar	SD	🗸		Uni	🗸			2	Real	ZS	ARL, SDRL, MRL	THWMA SR	MPE
Lone et al. [72]	L	Par	SD	🗸		Uni	🗸			3	Real	ZS	ARL, SDRL, EQL, RARL, PCI	THWMA TBE	AIMS Mathematics
Arslan et al. [73]	L	Par	SD	🗸		Multi		🗸		3	Real, Sim	ZS	ARL, EQL, RMI	TAHWMA	Symmetry
Yousefi et al. [74]	L	Par	SD	🗸		Multi		🗸	🗸	4	Real	ZS	ARL, SDRL	MHWMA	JAMS
Almanjahie et al. [75]	L	NPar	SD	🗸		Uni	🗸			1	Real	ZS	ARL, SDRL, MRL, EQL, RARL, PCI	HWMA SR	CMES
Noor-ul-Amin and Arshad [76]	L	Par	SD	🗸		Uni		🗸		0	Real	ZS	ARL, SDRL	HWMA AIB	CS-SC
Anwer et al. [77]	L	Par	SD	🗸		Uni		🗸		1	Real	ZS	ARL, PDARL	HWMA–CUSUM AIB	JSCS
Knoth [78]	L	Par	SD	🗸		Uni	🗸			0	-	SS	ARL, CED	HWMA	QREI
Knoth et al. [47]	L	Par	SD	🗸		Uni	🗸			16	-	ZS,SS	CED	Memory-type	JQT
Khan et al. [79]	V	Par	SD	🗸		Uni	🗸			0	Real	ZS	ARL, SDRL, EQL, RARL, PCI	THWMAV	MPE
Pieters et al. [80]	L	Par	SD	🗸		Multi	🗸			1	Real	ZS,SS	ARL, SDRL	MEHWMA	QREI
Ghasemi et al. [81]	P	Par	SD	🗸		Multi		🗸		0	Sim	ZS	ARL	MHWMA	CS-SC
Zubair et al. [82]	L	Par	SD	🗸		Uni		🗸		0	Sim	ZS	ARL, SDRL, MRL	HWMA–CUSUM: AIB	PLoS one
Shafqat et al. [83]	L	NPar	SD	🗸	🗸	Uni		🗸		0	Sim	ZS	ARL, SDRL, MRL	EWMA and HWMA SR: RS and AIB	Scientific Report
Sunthornwat et al. [84]	L	Par	SD	🗸		Uni		🗸		0	Real	ZS	ARL, SDRL, MRL, EARL, EMRL	Max HWMA	Symmetry

L = location; V = variability; LV = location and variability/scale; P = profile; Par = parametric; NPar = nonparametric; E = economic, SD = statistical design, ESD = economic statistical design; K = known; U = unknown; Uni = univariate process; Multi = multivariate process; IID = independent and identically distributed; AC = autocorrelated; ME = measurement error; AIB = auxiliary information based; SN = sign; SR = signed-rank; SDR = standardized residuals; RS = repetitive sampling; NC = number of citations; Real = real-life; Sim = simulation; CED = conditional expected delay; ZS = zero-state; SS = steady-state; ARL = average run-length; SDRL = standard deviation of the run-length; MRL = median run-length; PRL = percentile of the run-length; PDARL = percentage decrease ARL; EARL = expected ARL; ESDRL = expected SDRL; EMRL = expected MRL; RARL = relative average run-length; EQL = extra quadratic loss; AEQL = average EQL; RMI = relative mean index; PCI = performance comparison index; CMP = Conway–Maxwell–Poisson.

Table 3. Classification of the articles discussing the HWMA-related charts.

Articles	L, V, LV, P	Par or NPar	E, SD, ESD	K	U	Dimension	IID	AC	ME	NC	Data	Mode	Performance Metrics	Type of HWMA Chart	Journal
Abbas [3]	L	Par	SD	🗸	🗸	Uni	🗸			94	Real	ZS	ARL, SDRL	HWMA $\overline{X}$	CIE
Adegoke et al. [43]	L	Par	SD	🗸	🗸	Uni		🗸		52	Sim	ZS	ARL, SDRL	HWMA $\overline{X}$ with AIB	IEEE Access
Adegoke et al. [48]	L	Par	SD	🗸		Multi	🗸			43	Sim, Real	ZS	ARL, SDRL	$\mathrm{MHWMA} T^2$	IEEE Access
Nawaz and Han [49]	L	Par	SD	🗸		Uni	🗸			32	Real	ZS	ARL	HWMA $\overline{X}$ using RSS	QTQM
Abbas et al. [42]	L	Par	SD		🗸	Multi	🗸	🗸		12	Real	ZS	ARL	$\mathrm{MHWMA} T^2$	Mathematics
Abid et al. [25]	L	Par	SD	🗸	🗸	Uni	🗸			35	Real	ZS	ARL, SDRL, MRL	DHWMA $\overline{X}$	QREI
Abid et al. [26]	L	Par	SD	🗸		Uni	🗸			25	Real	ZS	ARL, SDRL, PRL	HWMA–CUSUM $\overline{X}$	QREI
Abid et al. [50]	L	Par	SD	🗸		Uni	🗸			8	Real	ZS	ARL, SDRL, PRL	CUSUM–HWMA $\overline{X}$	QREI
Adeoti and Koleoso [51]	L	Par	SD	🗸		Uni	🗸			24	Sim	ZS	ARL, SDRL	HHWMA $\overline{X}$	QREI
Raza et al. [52]	L	NPar	SD	🗸		Uni	🗸			19	Real	ZS	ARL, SDRL, MRL, EQL	HWMA SN and SR	JTE
Riaz et al. [53]	L	NPar	SD	🗸		Uni	🗸			16	Real	ZS	ARL	DHWMA SN	QREI
Riaz et al. [41]	V	Par	SD	🗸		Uni	🗸			17	Real	ZS	ARL, SDRL, MDRL	$\mathrm{HWMA} S^2$	Mathematics
Dawod et al. [54]	P	Par	SD		🗸	Multi		🗸		14	Real	ZS	ARL	HWMA profiles	CLS
Thanwane et al. [36]	L	Par	SD	🗸		Uni	🗸		🗸	11	Real	ZS	ARL, EARL	HWMA $\overline{X}$	TIMC
Thanwane et al. [40]	L	Par	SD		🗸	Uni	🗸	🗸	🗸	7	Real	ZS	ARL, SDRL, EARL, ESDRl	HWMA $\overline{X}$	IEEE Access
Thanwane et al. [37]	L	Par	SD		🗸	Uni	🗸			6	Real	ZS	ARL, SDRL, EARL, ESDRl	HWMA $\overline{X}$ with FIR	QREI
Thanwane et al. [38]	L	Par	SD	🗸		Uni		🗸		2	Real	ZS	ARL, SDRL, EARL, ESDRl, PCI	HWMA $\overline{X}$	IJIEC
Thanwane et al. [39]	L	Par	SD		🗸	Uni	🗸		🗸	13	Real	ZS	ARL, EARL	HWMA $\overline{X}$	JSCS
Alevizakos et al. [22]	L	Par	SD	🗸		Uni	🗸			13	Real, Sim	ZS	ARL, SDRL, AEQL, PCI, RMI	DHWMA & HHWMA $\overline{X}$	QREI
Alevizakos et al. [23]	L	NPar	SD	🗸		Uni	🗸			2	Real, Sim	ZS	ARL, SDRL, MRL, AEQL	DHWMA SN	QREI
Riaz et al. [24]	L	Par	SD		🗸	Uni	🗸			14	Real	ZS,SS	ARL, SDRL, MDRL	THWMA	Symmetry
Malela-Majika et al. [35]	L	Par	SD	🗸		Multi	🗸			3	-	ZS	ARL, SDRL, MRL	HHWMA	QREI
Adegoke et al. [55]	L	Par	SD		🗸	Uni	🗸			0	Real	ZS	ARL, SDRL	DSHWMA	QREI
Adegoke et al. [56]	L	Par	SD		🗸	Multi	🗸			2	Real	ZS,SS	ARL, SDRL	MHWMA	IEEE Access
Noor-ul-Amin and Noor [57]	L	Par	SD		🗸	Uni	🗸			1	Real	ZS	ARL, SDRL	HWMA $\overline{X}$ BTB	QREI
Adeoti et al. [58]	LV	Par	SD	🗸		Uni	🗸			10	Real, Sim	ZS	ARL, SDRL, MRL	HWMA (COM-Poisson)	JAS
Knoth et al. [44]	L	Par	SD	🗸		Uni	🗸			24	-	ZS,SS	ARL (CED)	HWMA with PM	QREI
Chan et al. [59]	LV	NPar	SD	🗸		Uni	🗸			16	Real	ZS	ARL, EARL, PRL	DEWMA–HWMA	CIE
Abbasi et al. [60]	V	Par	SD		🗸	Uni	🗸			3	Real		ARL, SDRL, MRL, EQL, RARL	HWMA dispersion	QREI
Abbasi et al. [61]	P	Par	SD	🗸		Uni	🗸			3	Real	ZS	ARL	HWMA profile	QREI
Rasheed et al. [62]	L	Par	SD	🗸		Multi	🗸			10	Real	ZS	ARL, EQL, RARL, PCI	MHWHC, MHCHW	MPE
Riaz et al. [63]	L	Par	SD	🗸		Uni	🗸			8	-	ZS,SS	ARL	HWMA	Processes
Aslam et al. [64]	L	Par	SD	🗸		Uni	🗸			1	Real	ZS	ARL, SDRL, EQL, RARL, PCI	HWMA TBE	QREI
Anwar et al. [65]	L	Par	SD	🗸		Uni	🗸			14	Real	ZS	ARL, RARL, EQL, PCI	DHWMA	QREI
Letshedi et al. [7]	L	NPar	SD		🗸	Uni	🗸			0	Real	ZS	ARL, SDRL, EARL, ESDRL	HWMA and HHWMA	PLoS one
Anwar et al. [66]	L	Par	SD	🗸		Uni		🗸		0	Real	ZS	ARL, SDRL, MRL, EQL, RARL, PCI	THWMA AIB	MPE
Al-Sayed et al. [67]	LV	Par	SD		🗸	Uni		🗸		0	Real	ZS	ARL, RMI	PR–HWMA, PR–DHWMA (ZIP)	IEEE ICIEEM
Rasheed et al. [68]	L	NPar	SD	🗸		Uni	🗸			8	Real	ZS	ARL, SDRL, MRL	DHWMA SR (RSS)	MPE
Chong et al. [69]	L	Par	SD	🗸		Uni	🗸			0	-	ZS	ARL, SDRL, PRL, EARL, EMRL	HWMA, DHWMA	IEEE ICIEEM
Iqbal et al. [70]	LV	Par	SD		🗸	Uni		🗸		13	Real	ZS	ARL, SDRL	HWMA and DHWMA DR, SDR	Symmetry
Zhang et al. [71]	L	NPar	SD	🗸		Uni	🗸			2	Real	ZS	ARL, SDRL, MRL	THWMA SR	MPE
Lone et al. [72]	L	Par	SD	🗸		Uni	🗸			3	Real	ZS	ARL, SDRL, EQL, RARL, PCI	THWMA TBE	AIMS Mathematics
Arslan et al. [73]	L	Par	SD	🗸		Multi		🗸		3	Real, Sim	ZS	ARL, EQL, RMI	TAHWMA	Symmetry
Yousefi et al. [74]	L	Par	SD	🗸		Multi		🗸	🗸	4	Real	ZS	ARL, SDRL	MHWMA	JAMS
Almanjahie et al. [75]	L	NPar	SD	🗸		Uni	🗸			1	Real	ZS	ARL, SDRL, MRL, EQL, RARL, PCI	HWMA SR	CMES
Noor-ul-Amin and Arshad [76]	L	Par	SD	🗸		Uni		🗸		0	Real	ZS	ARL, SDRL	HWMA AIB	CS-SC
Anwer et al. [77]	L	Par	SD	🗸		Uni		🗸		1	Real	ZS	ARL, PDARL	HWMA–CUSUM AIB	JSCS
Knoth [78]	L	Par	SD	🗸		Uni	🗸			0	-	SS	ARL, CED	HWMA	QREI
Knoth et al. [47]	L	Par	SD	🗸		Uni	🗸			16	-	ZS,SS	CED	Memory-type	JQT
Khan et al. [79]	V	Par	SD	🗸		Uni	🗸			0	Real	ZS	ARL, SDRL, EQL, RARL, PCI	THWMAV	MPE
Pieters et al. [80]	L	Par	SD	🗸		Multi	🗸			1	Real	ZS,SS	ARL, SDRL	MEHWMA	QREI
Ghasemi et al. [81]	P	Par	SD	🗸		Multi		🗸		0	Sim	ZS	ARL	MHWMA	CS-SC
Zubair et al. [82]	L	Par	SD	🗸		Uni		🗸		0	Sim	ZS	ARL, SDRL, MRL	HWMA–CUSUM: AIB	PLoS one
Shafqat et al. [83]	L	NPar	SD	🗸	🗸	Uni		🗸		0	Sim	ZS	ARL, SDRL, MRL	EWMA and HWMA SR: RS and AIB	Scientific Report
Sunthornwat et al. [84]	L	Par	SD	🗸		Uni		🗸		0	Real	ZS	ARL, SDRL, MRL, EARL, EMRL	Max HWMA	Symmetry

L = location; V = variability; LV = location and variability/scale; P = profile; Par = parametric; NPar = nonparametric; E = economic, SD = statistical design, ESD = economic statistical design; K = known; U = unknown; Uni = univariate process; Multi = multivariate process; IID = independent and identically distributed; AC = autocorrelated; ME = measurement error; AIB = auxiliary information based; SN = sign; SR = signed-rank; SDR = standardized residuals; RS = repetitive sampling; NC = number of citations; Real = real-life; Sim = simulation; CED = conditional expected delay; ZS = zero-state; SS = steady-state; ARL = average run-length; SDRL = standard deviation of the run-length; MRL = median run-length; PRL = percentile of the run-length; PDARL = percentage decrease ARL; EARL = expected ARL; ESDRL = expected SDRL; EMRL = expected MRL; RARL = relative average run-length; EQL = extra quadratic loss; AEQL = average EQL; RMI = relative mean index; PCI = performance comparison index; CMP = Conway–Maxwell–Poisson.

4. HWMA and Enhanced HWMA Charts

4.1. HWMA Charts

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Parametric HWMA Charts

4.1.1. Location

The HWMA chart for the process mean, i.e., the HWMA $\overline{X}$ chart, was introduced by Abbas (2018) to improve the detection ability of the EWMA $\overline{X}$ chart for monitoring small shifts. The charting statistic and control limits of the HWMA $\overline{X}$ chart are defined in Equations (1) and (4), respectively. In his paper, Abbas [3] compared the HWMA chart to the EWMA, CUSUM, and mixed EWMA–CUSUM (MEC)$\overline{X}$ charts in terms of their zero-state ARL ($ZSARL$) profiles. He reported that for small values of $\lambda$, the HWMA $\overline{X}$ chart outperforms the EWMA $\overline{X}$ chart for small shifts. However, for large $\lambda$ values, the EWMA $\overline{X}$ chart performs better than the HWMA $\overline{X}$ chart for small to large shifts. Compared to the CUSUM $\overline{X}$ chart, the HWMA $\overline{X}$ chart performs better regardless of the magnitude of the shift. Compared to the MEC $\overline{X}$ chart, the HWMA $\overline{X}$ chart performs better for small $\lambda$ values. However, for large λ values, the MEC $\overline{X}$ chart performs better.

Adegoke et al. [48] proposed an auxiliary-information-based (AIB) HWMA $\overline{X}$ chart for monitoring small shifts in the process mean with correlated auxiliary variables. They reported that the AIB HWMA $\overline{X}$ chart performs better than the classical HWMA $\overline{X}$ chart and other competing charts (see [83]). Nawaz et al. (2020) designed the HWMA $\overline{X}$ chart using ranked set sampling (RSS) [49]. Thanwane et al. (2020) investigated the combined effect of autocorrelation and measurement error on the performance of the HWMA $\overline{X}$ chart [40].

4.1.2. Variability

Riaz et al. [41] proposed a one-sided HWMA chart for monitoring small shifts in the process dispersion. They discussed its performance in terms of the ARL, $SDRL,$ and $MRL$. They found that the proposed HWMA chart performs better than the existing counterparts for small shifts.

Abbasi et al. [60] proposed an HWMA S chart to detect small shifts in the process standard deviation to monitor disturbances in the process dispersion. Their chart is used for the normal, heavy-tailed, symmetrical, and skewed underlying process distributions. Noor-ul-Amin and Arshad [76] proposed the HWMA $S^2$ chart for the fast detection of shifts in the variance. It was found that the proposed HWMA $S^2$ chart is sensitive to process shifts by using the auxiliary information with a high degree of correlation.

4.1.3. Joint Location and Variability

Adeoti et al. [58] proposed an HWMA chart based on the Conway–Maxwell–Poisson distribution, denoted as the CMP–HWMA chart, to monitor both the mean and dispersion when dealing with underspread and overspread count data. They investigated the performance of the CMP–HWMA chart for a range of shifts (including the overall performance) in terms of the ARL, SDRL, and MRL profiles along with the expected ARL, SDRL, and MRL metrics. They found that the CMP–HWMA chart performs better in many cases compared to other CMP memory-type charts. Al-Sayed et al. [67] proposed the HWMA chart based on the Pearson residuals (PR) of zero-inflated Poisson (ZIP) model, denoted as the PR–HWMA chart, to monitor small shifts in the location and variance simultaneously.

4.1.4. Profile

Dawod et al. [54] applied a Bayesian framework to construct HWMA-type charts for linear profiles to monitor the intercept, slope, and error variance for a linear regression applied in the pharmaceutical industry. Compared to the Shewhart, EWMA, CUSUM, and Hotelling’s $T^2$ counterparts in terms of individual and overall performance metrics, the proposed HWMA charts outperform the competing charts, especially for small shifts in the regression parameters and error variance. A real-life example was used to illustrate the application of the proposed chart in shrinking the variations in the quality of a pharmaceutical product.

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Nonparametric HWMA Charts

4.1.5. Location

Raza et al. [52] proposed two nonparametric HWMA charts based on the sign (SN) and Wilcoxon signed-rank (SR) tests for monitoring deviations in the process location from the target value under skewed and symmetric distributions, respectively. These charts are denoted as the NPHWMA SN and NPHWMA SR charts. In their paper, Raza et al. [52] compared the proposed charts to the existing nonparametric EWMA and CUSUM SN and SR charts denoted as the NPEWMA and NPCUSUM SN and SR charts. They reported that the NPHWMA SN and NPHWMA SR perform better than their nonparametric counterparts. Recently, Shafqat et al. [83] proposed an AIB HWMA SR chart to improve the existing NPHWMA SR chart in monitoring small shifts in the process location parameter. Letshedi et al. [7] used a different nonparametric test, namely, the Wilcoxon rank-sum (WRS) test, to construct a nonparametric HWMA chart for monitoring very small shifts in the location parameter.

4.1.6. Joint Location and Scale

Chan et al. [59] proposed two nonparametric process monitoring schemes based on the double EWMA (DEWMA) and HWMA Lepage statistic for jointly monitoring a process’s location and scale parameters. These charts’ implementation and designs were based on the time-varying and asymptotic upper control limits. It was found that the HWMA Lepage (HL) chart with a time-varying control limit is better than the HL chart with the asymptotic control limit in reducing the rate of early false alarms. The DEWMA Lepage (DL) chart performs well in detecting small to moderate shifts in the process. The HL chart with the time-varying UCL performs better than the EWMA Lepage (EL) chart. In addition, the EL and DL charts are good with both time-varying and asymptotic control limits, but the HL chart works better only with the time-varying control limit.

To improve the HWMA-type chart, the literature in SPC proposes using the double or hybrid HWMA (i.e., DHWMA or HHWMA) chart. This is discussed in the next section.

4.2. DHWMA and HHWMA Charts

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Parametric DHWMA and HHWMA Charts

4.2.1. Location

Abid et al. [26] proposed the double HWMA (DHWMA) $\overline{X}$ chart to detect small disturbances in the process mean efficiently. Their results reveal that the DHWMA $\overline{X}$ chart is not IC robust, especially for large values of $\lambda$ where, for instance, the attained IC $ARL$ value can vary from 483.53 to 158.76 under the Gamma distribution and is much smaller under the t and Logistic distributions. Compared to the existing EWMA and HWMA $\overline{X}$ charts, they found that the DHWMA $\overline{X}$ chart is superior to the EWMA and HWMA $\overline{X}$ charts in many cases for various values of $\lambda$. Note that the DHWMA by Abid et al. [26] is just a reparameterisation of the HWMA chart where ${\lambda }^2$ is used instead of $\lambda$. Thus, Alevizakos et al. [22] extended the HWMA chart by imitating precisely the double EWMA (DEWMA) technique. Their chart is also denoted the DHWMA chart. A comparison study against the EWMA, DEWMA, HWMA, MEC, CUSUM, and the generally weighted moving average (GWMA) $\overline{X}$ charts showed that the DHWMA $\overline{X}$ chart is more effective in detecting small to moderate shifts. At the same time, it performs well against its competitors for large shifts. Alevizakos et al. [22] also studied the robustness of the DHWMA $\overline{X}$ chart under several non-normal distributions, and they found that the DHWMA $\overline{X}$ chart is IC robust for small values of $\lambda$. They also reported that the DHWMA $\overline{X}$ chart is more IC robust than the HWMA $\overline{X}$ chart regardless of the nature or type of the underlying process distribution. In addition, they showed that under the t($\nu$) and Gamma($\alpha$,1) distributions, the DHWMA $\overline{X}$ chart is more IC robust as $\nu$ and $\alpha$ increase. Adeoti and Koleoso [51] proposed the hybrid HWMA (HHWMA) $\overline{X}$ chart for monitoring the process mean. Malela-Majika et al. [35] corrected the variance used in the HHWMA $\overline{X}$ chart design proposed by [51]. For more details regarding the parametric DHWMA and HWMA charts for monitoring the location parameter, readers are referred to Alevizakos et al. [22,23,56,65].

4.2.2. Joint Location and Variability

Al-Sayed et al. [67] proposed the DHWMA chart based on the Pearson residuals (PR) of the ZIP model, denoted as the PR–DHWMA chart, to monitor small shifts in the location and variance simultaneously. They reported that the PR–HWMA chart performs relatively better than the PR–HWMA chart. Iqbal et al. [70] proposed the generalized linear model (GLM)-based HWMA and DHWMA charts based on the deviance residuals (DR) and standardized residuals (hereafter, SDR) of the Poisson regression model to monitor shifts in the location and variance simultaneously. These HWMA (DHWMA)-type charts are denoted the DR–HWMA (DR–DHWMA) and SDR–HWMA (SDR–DHWMA) charts. The authors concluded that the SDR–HWMA and SDR–DEWMA charts perform better than the DR–HWMA and DR–DHWMA charts. In addition, the SDR–DHWMA and DR–DHWMA charts outperform the SDR–HWMA and DR–HWMA charts, respectively.

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Nonparametric DHWMA and HHWMA Charts

4.2.3. Location

Riaz et al. [53] proposed the DHWMA SN chart, which is a reparameterisation of the HWMA SN chart. At the same time, Alevizakos et al. [23] mimicked the DEWMA SN chart and proposed the DHWMA SN chart for monitoring shifts in the location parameter. Compared to the HWMA, GWMA, double GWMA (DGWMA), triple exponentially weighted moving average (TWMA) SN charts, and classical DHWMA, it was found that the DHWMA SN chart performs better than the competing charts in many cases, especially for small shifts in the process parameter. Rasheed et al. [68] used the SR test to construct the DHWMA chart using RSS. Letshedi et al. [7] proposed the HWMA, DHWMA, and HHWMA charts based on the WRS W statistic, denoted as the W–HWMA, W–DHWMA, and W–HHWMA charts, for monitoring the location parameter. Their simulation results reveal the superiority and flexibility of the W–HHWMA chart over the W–HWMA, W–EWMA, and W–CUSUM charts.

4.3. THWMA Charts

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Parametric THWMA Charts

4.3.1. Location

Riaz et al. [24] extended the reparameterised DHWMA chart by introducing the triple HWMA (THWMA) chart. Anwar et al. [66] used auxiliary information to improve the THWMA chart proposed by [24]. Their results revealed that the AIB THWMA chart outperforms the classical THWMA chart, and both charts (i.e., THWMA and AIB THWMA) perform better than the DHWMA and AIB DHWMA, respectively. The THWMA chart for the time between events (TBE) was proposed by Lone et al. [72]. In their paper, Lone et al. [72] reported that the THWMA TBE outperforms the DHWMA TBE, HWMA TBE, EWMA TBE, and DEWMA TBE charts over a range of shifts.

Arslan et al. [73] proposed an HWMA chart using two supplementary variables; this chart is denoted as the TAHWMA chart. In their paper, they considered two supplementary variables that are correlated with the variable of interest in the form of a regression estimator, which is an efficient and unbiased estimator for the process location. The TAHWMA charting structure is studied and compared in terms of multicollinearity amidst the two additional variables. It was reported that the TAHWMA chart performs effectively when the two supplementary variables have no collinearity. A comparison between the TAHWMA and existing charts revealed the supremacy of the THWMA chart over its existing counterparts.

4.3.2. Variability

Khan et al. [79] proposed a one- and two-sided reparameterised THWMA chart for monitoring shifts in the process dispersion, and they reported the superiority of the THWMA chart over the DHWMA, HWMA, TEWMA, DEWMA, and EWMA charts in detecting shifts in the dispersion parameter.

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Nonparametric THWMA and HHWMA Charts

4.3.3. Location

Rasheed et al. [68] proposed a nonparametric THWMA SR chart using RSS, denoted as the NPTHWMA chart, to monitor shifts in the process location. The performance of the THWMA SR chart was investigated in terms of the ARL, SDRL, and MRL profiles. It was found that the THWMA SR chart is very sensitive to small shifts. The performance of this chart was compared to that of the NPTEWMA SR, TEWMA $\overline{X}$, and NPTEWMA SN charts and the NPTEWMA SR and NPDHWMA U charts using RSS. The comparison results revealed that the proposed NPTHWMA SR chart with RSS outperforms all competing charts considered in their paper, particularly for small to moderate shifts in the process location.

4.4. HWMA–CUSUM and CUSUM–HWMA Charts

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Parametric HWMA Charts

4.4.1. Location

The mixed HWMA–CUSUM (MHC) and mixed CUSUM–HWMA (MCH) $\overline{X}$ charts were proposed by Abid et al. [25] and Abid et al. [50], respectively, to enhance the shift detection ability of the CUSUM $\overline{X}$ and HWMA $\overline{X}$ charts in detecting small shifts in the process mean. The MHC and MCH $\overline{X}$ charts were compared to the CUSUM, EWMA, MEC, and HWMA $\overline{X}$ charts. In addition, the MHC $\overline{X}$ chart was compared to the MCE $\overline{X}$ chart. It was found that the MCH chart is quick in identifying any shifts in the process mean when compared to the CUSUM chart, while the MHC chart was found to be fast in detecting any shifts for all values of $\lambda$ considered in the paper by Abid et al. [25]. The MHC chart performs better than the EWMA chart when $\lambda =$ 0.1 for specific shifts. However, the MCH chart outperforms the EWMA chart in detecting small shifts when $\lambda =$ 0.1. They also reported that as $\lambda$ increases, the MCH exhibits a great sensitivity in detecting small to moderate shifts as compared to the EWMA and HWMA charts. It was also reported that the MCH performs relatively better than the MEC chart regardless of the magnitude of the shift in the process mean when $\lambda =$ 0.1. When the shifts in the process mean are greater than 0.25 (i.e., $\delta >$ 0.25), the MCH chart outperforms the MEC chart for large values of $\lambda$. The MHC chart is superior to the MEC and MCE charts when λ = 0.1 for small shifts. For large shifts, the MHC chart performs better than the MEC chart.

Anwer et al. [77] and Zubair et al. [82] proposed the AIB mixed HWMA–CUSUM (AIB–MHC) for monitoring the process mean. The results from these two papers showed that the AIB–MHC chart performs better than the AIB–HWMA chart for all choices of $\lambda$ and correlation between the two variables ($\rho$) when $\delta <$ 0.75. Compared to the AIB–MEC chart, the AIB–MHC chart presents a reasonably superior performance against the AIB–MEC chart for all choices of $\lambda$, $\rho ,$ and $\delta$.

4.5. Multivariate Classical and Enhanced HWMA Charts

4.5.1. Location

Adegoke et al. [43] proposed a multivariate HWMA (MHWMA) chart for monitoring the process mean vector. The performance of this chart in terms of the ARL profile was evaluated and compared with the multivariate ${\chi }^2$, MEWMA, and MCUSUM charts considering a variety of charting parameters. The comparison results revealed that the proposed MHWMA chart is superior to the competing charts, particularly in detecting small shifts in the process mean vector. Abbas et al. [42] investigated the effect of an estimated variance covariance matrix on the performance of the MHWMA $T^2$ chart and studied its performance in terms of the ARL, SDRL, EQL, and RARL for a wide range of the number of quality characteristics and varying sample sizes. Their performance analysis revealed that the MHWMA T² chart outperforms its multivariate counterparts under known and estimated process parameters. Later, Adegoke et al. [56] proposed one and two one-sided MHWMA charts for monitoring small shifts in the process mean vector, denoted as the OMHWMAI and OMHWMAII, respectively. The OMHWMAI chart is a one-sided chart for monitoring upward shifts by transforming the observation $X_i$ into positive values. The OMHWMAII chart uses two one-sided MHWMA charts to monitor both upward and downward shifts in the process mean vector. The performances of these charts were evaluated in terms of their run-length properties for different shift sizes in the process mean vector. It was found that the OMHWMAII chart for monitoring upward shifts detects smaller shifts in the mean vector quicker than larger shifts. In contrast, the OMHWMAI chart, for monitoring upward shifts, detects larger shifts in the mean vector quicker than smaller shifts. In addition, the OMHWMAI and OMHWMA charts’ sensitivities to non-normal distributions showed that they are remarkably robust to non-normality when a smaller value of $\lambda$ is used. Thus, these charts can be designed to have an IC run-length performance that is very close to that of the charts based on normal distribution when a small value of $\lambda$ is used. Recently, Pieters et al. [80] proposed the multivariate extended HWMA (MEHWMA) chart by imitating the multivariate extended EWMA (MEEWMA) chart. This chart is designed to monitor small shifts in a multivariate process mean vector. To enable the identification of the variable that caused the OOC signal, a support vector machine was incorporated into the MEHWMA chart.

4.5.2. Profile

Ghasemi et al. [81] proposed three charts, namely, the MHWMA, MHWMA ${\chi }^2$, and the combination of the MHWMA2 and mixed multivariate EWMA–CUSUM (denoted as MWHMA_2/MMECD) charts to monitor multivariate simple linear profiles efficiently. The performances of these three charts were evaluated in terms of their ARL profiles. Following a comprehensive comparison of the performance of these three charts, it was found that the MHWMA and MHWMA ${\chi }^2$ charts perform better than MHWMA_2/MMECD chart for all shifts. Also, the MHWMA chart outperforms the MHWMA ${\chi }^2$ chart in detecting small shifts, and the performance of these two charts is similar in detecting larger shifts.

For more details on the design of MHWMA-type charts, readers are referred to [42,56,74,80].

4.6. Other HWMA-Type Charts

Several other charts based on the HWMA statistics have been developed. Rasheed et al. [62] combined the HWMA chart’s features with the existing mixed memory (MCE and MEC) charts to improve the shift detection ability of the mixed memory-type charts. The resulting charts are called the mixed HWMA homogeneously CUSUM (denoted as MHWHC) and mixed homogeneously CUSUM–HWMA (denoted as MHCHW) charts, respectively. The MHWHC and MHCHW charts’ performances were compared to those of the classical CUSUM and EWMA, MEC, MCE, and HWMA charts. The comparison results revealed that the MHWHC and MHCHW charts are superior to their counterparts, specifically in detecting small and moderate shifts.

5. Controversies on the Practicality of the HWMA Chart

Several authors have criticized the weight structure of the HWMA chart. Knoth et al. [44] advised against the HWMA chart because it allocates equal weight to past samples, which greatly affects its performance, especially for small values of $\lambda$ and at the start of any process monitoring regardless of the value of $\lambda$; see also Knoth [78]. They insisted that the zero-state performance of the HWMA chart that has been reported in the literature is deceptive since, in general, the steady-state kicks in from time $t>$ 1, meaning that a memory-type chart practically runs in steady-state most of the time. After a thorough investigation of the steady-state performance of the HWMA chart using the conditional expected delay (CED) metric, they found that its performance was inferior to that of the other classical memory-type charts, i.e., the EWMA and CUSUM charts. In passing, they also advised against extending the HWMA and EWMA charts to more complex schemes for the same reasons; see also Knoth et al. [47]. Knoth et al. [47] reported that some of these charts, including compound charts such as the MEC, MCE, MCH, and MHC are found to be flawed and unnecessary.

Contrary to Knoth et al. [44] and Knoth [78], many authors encouraged the use of the HWMA chart because it has appealing zero-state properties and outperforms its counterparts in many cases in steady-state mode; see, e.g., Riaz et al. [63] and Anwar et al. [65]. Riaz et al. [63] reassessed the zero- and steady-state performances of the HWMA chart, following the criticism by [44], and reported that the weight structure of the HWMA chart can safeguard the detection ability and the run-length properties under various delays in process shifts. In addition, they showed that the HWMA chart is superior to the EWMA chart in many cases in the zero-state mode and can maintain its dominance if the process experiences a shift delay. However, in the steady-state mode, the performance of the HWMA chart depends on the suitable choice of design parameters.

5.1. Mathematical Background of the CED Metric

Kenett and Pollak [46] defined the CED as the delay from the first opportunity to detect a change and not from the time of the change itself. The CED, denoted as $D_{\tau }$, is mathematically defined by

$$D_{\tau }=E_{\tau }\left(L-\tau +1|L\ge \tau \right), \tau =1,2,3, \dots ,$$

(8)

where $L$ represents the number of samples until an alarm is raised and τ is defined in Equation (6).

5.2. CED Performance Comparison

In this section, we present and discuss some results of the comparison analysis between the HWMA, EWMA, and CUSUM charts. These results are retrieved from the two important articles by Abbas [3] and Knoth et al. [44].

Table 5. Zero-state performance comparison of the HWMA, EWMA, and CUSUM charts.

Shift	$\mathbf{HWMA} (\mathit{\lambda }=$ 0.05)	EWMA (λ = 0.05)	CUSUM (k = 0.125)	HWMA (λ = 0.1)	EWMA (λ = 0.1)	CUSUM (k = 0.225)
0.25	73.01	77.76	83.39	81.48	103.32	91.16
0.5	24.92	23.71	34.65	28.61	28.81	31.19
0.75	12.78	11.87	21.68	14.85	13.61	18.01
1	8.04	7.31	15.78	9.35	8.21	12.63
1.5	4.4	3.77	10.26	4.98	4.17	7.94
2	2.98	2.43	7.64	3.32	2.66	5.83
2.5	2.2	1.77	6.12	2.45	1.92	4.65
3	1.67	1.41	5.13	1.87	1.51	3.89
5	1.01	1.01	3.19	1.03	1.02	2.41

compares the performances of the HWMA ($\lambda =$ 0.05), HWMA ($\lambda =$ 0.1), EWMA ($\lambda =$ 0.05), EWMA ($\lambda =$ 0.1), CUSUM ($k=$ 0.125), and CUSUM ($k=$ 0.225) charts in terms of the $ZSARL$ profile, where $k$ is the reference value of the CUSUM chart. From Table 5, it can be seen that the HWMA chart outperforms the EWMA chart for very small shifts, i.e., 0 $<\delta \le$ 0.25. However, for moderate to large shifts, the EWMA chart performs better than the HWMA chart. In addition, Table 5 reveals the superiority of the HWMA chart over the CUSUM chart regardless of the magnitude of the shift.

Figure 4 compares the performances of the HWMA and EWMA charts in terms of the CED profile when $\lambda =$ 0.1. Note that when τ = 1, the CED profile is equivalent to the conditional ZSARL profile denoted by $D_1$, and when $\tau >1$, it is equivalent to the conditional steady-state $ARL \left(SSARL\right)$ profile denoted by $D_{\tau }$ with $\tau \ne$ 1. From Figure 4, it can be seen that when $\delta =$ 0.25, in zero-state mode, the HWMA chart performs better than the EWMA chart. However, in steady-state mode, the HWMA outperforms the EWMA for $\tau <53$. When $0<\delta \le 1$, the EWMA chart performs better than the HWMA chart regardless of the value of $\tau$. Note that for small shifts, the $D_{\tau }$ profile of the HWMA chart is an increasing function of $\tau$, while the $D_{\tau }$ profile of the EWMA chart converges towards fixed values for different shifts. This clearly shows that the steady-state performance of the HWMA chart is worse than that of the EWMA chart (see Figure 4a). For large shifts, the $D_{\tau }$ profile of the EWMA chart is smaller than that of the HWMA chart when $0\le \delta <3$. When $\delta =3$ and 5, the EWMA chart performs better than the HWMA chart for $\tau \le 220$ and $\tau \le 50$, respectively. As τ increases, when δ = 3 and 5, the HWMA chart outperforms the EWMA chart for $\tau >220$ and $\tau >50$, respectively (see Figure 4b).

From the above discussion, it can be seen that the EWMA chart performs better than the HWMA chart in many cases. However, it must not be ignored that the HWMA chart performs better in some of the cases.

6. Conclusions, Recommendation, Future Research Work, and New Directions

In this paper, we reviewed the HWMA chart and its associated enhancements. The HWMA chart is a memory-type chart that assigns a specific weight to the most recent sample and the rest is distributed equally between previous samples. A thorough review of the HWMA and HWMA-related charts, i.e., DHWMA, THWMA, HWMA–CUSUM, CUSUM–HWMA, etc., was documented, and based on the observations from the published literature, it was seen that these charts have excellent detection ability for small shifts when compared to the corresponding CUSUM- and EWMA-type charts (including their enhancements).

The HWMA and memory-type charts received criticism; see, e.g., Refs. [44,47]. Knoth et al. [44] advised against the use of the HWMA chart in practice because of its incapacity to detect changes that occur after the start-up period of the process due to its incoherent weight structure. We acknowledge that this constructive criticism fosters a healthy debate and improves the research quality. This review aimed to consolidate the recent developments and provide future research ideas based on the gaps identified, not conclude the debate. Thus, below, we provide a summary of topics that have not yet been addressed:

(i)

Most quality characteristics are dependent on one or more independent variables. In this case, profile monitoring can be used. However, only three articles on profile monitoring are available in the literature; hence, researchers interested in profile monitoring can construct charts using different regression models and investigate their performance.

(ii)

Parametric and nonparametric HWMA charts with unknown process parameters under the assumption of i.i.d. and correlated observations are scarce. More research and thorough investigations regarding the performances of these types of charts are required.

(iii)

No publications in nonparametric statistical monitoring for compound HWMA-type charts exist; hence, more research needs to be conducted to address this gap.

(iv)

To enhance the literature on HWMA charts, researchers are encouraged to conduct additional research focusing on parametric and nonparametric HWMA charts with unknown process parameters, particularly under the assumptions of i.i.d. and correlated observations. A comprehensive exploration study about the zero- and steady-state performances of these types of charts would significantly contribute to the existing body of knowledge.

(v)

The literature has various parametric and nonparametric synthetic charts using structured and unstructured sampling techniques (e.g., RSS, VSS, variable sample size and interval (VSSI), etc.). However, there is no work on synthetic HWMA and HWMA-related charts. Interested researchers can investigate this topic using different sampling structures.

(vi)

Measurement errors (or imperfect measurements) are widespread in SPC applications and have been shown to harm the performance of any chart. Only the performance of the standard HWMA chart has been investigated under imperfect measurements. Thus, researchers are encouraged to consider this when designing extended and compound HWMA-type charts.

(vii)

There are few parametric and nonparametric HWMA-type charts for joint monitoring of the process mean and variability (or process location and scale). Researchers are encouraged to use different test statistics to construct new HWMA-type charts.

(viii)

Evaluating the characteristics of the run-length of charts using exact formulae, is critical; hence, more research on these topics is needed to simplify the evaluation of the run-length properties of the HWMA and HWMA-related charts.

(ix)

The application of the HWMA-type charts in practice in various disciplines such as healthcare services and engineering is recommended.

(x)

Only a few attribute HWMA charts have been proposed in the literature. More investigations on attribute HWMA charts as well as high-yield processes are needed.

(xi)

There is no study on the economic and economic–statistical designs of the HWMA-type charts. Therefore, researchers are encouraged to investigate these topics under the assumption of i.i.d, serial dependency, perfect and imperfect measurements, and the combined effect of autocorrelation and measurement errors.

(xii)

Only three publications on multivariate HWMA charts are available in the literature. Given the relevance of multivariate charts in applications, there is a lot of research on HWMA-type charts that can be conducted based on parametric and nonparametric settings for monitoring the location, variability, profile, joint location, and variability of multivariate processes.

(xiii)

Researchers are advised to consider the weight structure when designing memory-type charts. The best weight structure will likely be the one that allocates small weights to previous samples.

(xiv)

In practice, most applications are characterized by several quality characteristics that depend on one or several variables; in such cases, charts for multivariate profiles are more appropriate. Table 2 and Table 3 show that there is a significant gap in the literature concerning the HWMA-related charts for monitoring multivariate processes. Thus, researchers are encouraged to design efficient classical multivariate charts as well as linear and nonlinear profile HWMA charts considering the effect of multicollinearity and dependence structure.

(xv)

Recent advances in information technology (IT) have enabled researchers to collect, store, and easily access large volumes of data at a modest cost. Researchers are encouraged to look at the design of multivariate HWMA-related charts for monitoring high-dimensional profile data.

(xvi)

Researchers are encouraged to continue the collaborative dialogue between critics and proponents of the HWMA chart, fostering a comprehensive understanding of its strengths and limitations. This inclusive approach will contribute to addressing research gaps and improve the overall quality of research in the field of SPC.