A double generally weighted moving average control chart for monitoring the process variability

Abstract

In the present article, a double generally weighted moving average (DGWMA) control chart based on a three-parameter logarithmic transformation is proposed for monitoring the process variability, namely the $S^2$-DGWMA chart. Monte-Carlo simulations are utilized in order to evaluate the run-length performance of the $S^2$-DGWMA chart. In addition, a detailed comparative study is conducted to compare the performance of the $S^2$-DGWMA chart with several well-known memory-type control charts in the literature. The comparisons indicate that the proposed one is more efficient in detecting small shifts, while it is more sensitive in identifying upward shifts in the process variability. A real data example is given to present the implementation of the new $S^2$-DGWMA chart.

1. Introduction

There are two types of variation that are present in a production process, namely, the common causes and assignable causes of variation. The process is declared as in-control (IC) when it operates with the common causes of variation. Nevertheless, when assignable causes of variation arise from external sources, they lead to an out-of-control (OOC) process [34]. The control charts constitute an essential part of the Statistical Process Control (SPC) in detecting assignable causes of variation that may affect either the mean or variance of the process. They are also classified into location and dispersion charts, where the first are used to identify shifts in the process mean, while the latter are suitable for detecting shifts in the process variance.

The Shewhart-type charts, like $\overline{X}$, R and S charts, are efficient in detecting large shifts due to their memoryless property [34]. However, the memory-type charts are more sensitive in detecting small and moderate shifts because they take into consideration both the current and past information. The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts are the first memory-type charts, which were developed by Page [35,36] and Roberts [38], respectively. Shamma and Shamma [39] developed the Double EWMA (DEWMA) chart, Haq [21,22] proposed the Hybrid EWMA (HEWMA) chart, and Alevizakos et al. [4] introduced the Triple EWMA (TEWMA) chart. Sheu and Lin [41] proposed the generally weighted moving average (GWMA) chart as an extension of the EWMA chart, while Sheu and Hsieh [40] extended the GWMA chart to a Double GWMA (DGWMA) chart. Particularly, the DGWMA charting scheme is a weighted moving average of a weighted moving average, that is, the smoothing process is performed twice. More information about the DGWMA scheme can be found in Chiu and Sheu [19], Tai et al. [43], Kang and Baik [28], Huang et al. [25], Chiu and Lu [18], Lu [31], Alevizakos et al. [7], Karakani et al. [29], Alevizakos et al. [6], Mabude et al. [32] and Chatterjee et al. [16].

In many industrial applications, it is important to monitor the presence of shifts in the process dispersion rather than the process mean. Therefore, many researchers have developed memory-type control charts for monitoring the process variability. For instance, Castagliola [11] and Castagliola et al. [12] used a three-parameter logarithmic transformation to sample variance ( $S^2$) in order to construct the $S^2$-EWMA and $S^2$-CUSUM charts, respectively. Furthermore, Castagliola et al. [14] introduced a Variable Sampling Interval (VSI) version of the $S^2$-EWMA [11] chart, referred to as VSI $S^2$-EWMA chart. Taking into account the aforementioned logarithmic transformation, Abbas, Riaz and Does [1] developed the CS-EWMA chart, Tariq et al. [44] introduced a HEWMA chart for monitoring the process variability (named as HEWMTn chart, but hereafter, for simplicity purposes, it is denoted by $S^2$-HEWMA), Chatterjee et al. [15] proposed the $S^2$-TEWMA chart for monitoring the process dispersion and Alevizakos et al. [5] developed the $S^2$-GWMA chart as an expansion of the $S^2$-EWMA chart. Other works about the control charting technique for monitoring the process variability are those of Castagliola et al. [13], Huwang et al. [26], Sheu and Lu [42], Abbasi et al. [3], Ali and Haq [8,9], Haq [23,24], Riaz et al. [37], Mahadik et al. [33], Abbasi et al. [2], Arshad et al. [10], Li and Mukherjee [30], and Zaman [45,46], to name a few.

To the best of our knowledge, as well as the research of Mabude et al. [32], most of the works on the DGWMA scheme are related with monitoring the process location or jointly monitoring the process location and variability. Therefore, motivated by Sheu and Hsieh [40], and Castagliola [11], the current article extends the work of Alevizakos et al. [5] to a DGWMA chart based on a three-parameter logarithmic transformation to $S^2$, named as $S^2$-DGWMA chart for monitoring the process variability. The proposed chart is compared with several dispersion control charts such as the $S^2$-EWMA, CS-EWMA, $S^2$-HEWMA, $S^2$-TEWMA, $S^2$-GWMA and VSI $S^2$-EWMA charts. The remainder of this article is structured as follows. In Section 2, we present the proposed $S^2$-DGWMA chart. A simulation study is conducted in Section 3 to evaluate its efficiency through the run-length distribution, while its performance is compared with those of the previously mentioned charts in Section 4. A real data example is presented in Section 5 and some concluding remarks are summarized in Section 6.

2. The proposed $S^2$-DGWMA control chart

Assume that $X_{ij}$, $i=1,2,\dots$, $j=1,2,\dots ,n$, is the jth observation in the ith random sample of size n (>1) and $X_{ij}\stackrel{\mathrm{i}\mathrm{i}\mathrm{d}}{\sim }N({\mu }_0,\tau {\sigma }_0)$, where ${\mu }_0$ and ${\sigma }_0$ are the corresponding IC values of the process mean and standard deviation. The process is considered to be IC, if $\tau =1$; otherwise, the process is declared as OOC and consequently, $\tau \ne 1$. Here, we are interested in monitoring the shifts in the process variance from an IC value ${\sigma }_0^2$ to an OOC ${\sigma }_1^2=(\tau {\sigma }_0{)}^2$, where $\tau \ne 1$, considering that the process mean ${\mu }_0$ remains stable. The sample variance $S_i^2$ is given by

$$S_i^2=\frac{1}{n-1}\sum _{j=1}^n(X_{ij}-{\overline{X}}_i{)}^2,i=1,2,\dots$$ (1)

where ${\overline{X}}_i=\frac{1}{n}\sum _{j=1}^n{X}_{ij}$, $i=1,2,\dots$ is the sample mean. In order to monitor the process variability, the three-parameter logarithmic transformation applied to $S_i^2$ [27], is utilized, i.e.

$$T_i=a+b\ln (S_i^2+c),i=1,2,\dots$$ (2)

where $a=A\left(n\right)-2B\left(n\right)\ln \left({\sigma }_0\right)$, $b=B\left(n\right)$, $c=C\left(n\right){\sigma }_0^2$ and, the functions $A\left(n\right)$, $B\left(n\right)$ and $C\left(n\right)$ depend only on n. Castagliola [11] first showed that, for a fixed n, if the constants a, b and c are suitably selected, then the statistic $T_i$ is approximately a normal random variable with mean ${\mu }_T\left(n\right)$ and standard deviation ${\sigma }_T\left(n\right)$. Table 1 shows the values of $A\left(n\right)$, $B\left(n\right)$, $C\left(n\right)$, ${\mu }_T\left(n\right)$ and ${\sigma }_T\left(n\right)$ for $n=3,4,\dots ,15$, that were originally presented in Table I of Castagliola [11].

Table 1.

Values of $A\left(n\right),B\left(n\right),C\left(n\right),{\mu }_T\left(n\right),{\sigma }_T\left(n\right)$ and $D{G}_0$ for $n=3,4,\dots ,15$.

n	$A\left(n\right)$	$B\left(n\right)$	$C\left(n\right)$	${\mu }_T\left(n\right)$	${\sigma }_T\left(n\right)$	$D{G}_0$
3	−0.6627	1.8136	0.6777	0.02472	0.9165	0.276
4	−0.7882	2.1089	0.6261	0.01266	0.9502	0.237
5	−0.8969	2.3647	0.5979	0.00748	0.9670	0.211
6	−0.9940	2.5941	0.5801	0.00485	0.9765	0.193
7	−1.0827	2.8042	0.5678	0.00335	0.9825	0.178
8	−1.1647	2.9992	0.5588	0.00243	0.9864	0.167
9	−1.2413	3.1820	0.5519	0.00182	0.9892	0.157
10	−1.3135	3.3548	0.5465	0.00141	0.9912	0.149
11	−1.3820	3.5189	0.5421	0.00112	0.9927	0.142
12	−1.4473	3.6757	0.5384	0.00090	0.9938	0.136
13	−1.5097	3.8260	0.5354	0.00074	0.9947	0.131
14	−1.5697	3.9705	0.5327	0.00062	0.9955	0.126
15	−1.6275	4.1100	0.5305	0.00052	0.9960	0.122

The plotting statistic of the proposed $S^2$-DGWMA chart is defined through the following system of equations:

$$\{\begin{array}{l}{\displaystyle G_i=\sum _{j=1}^i(q^{{(j-1)}^{\alpha }}-q^{j^{\alpha }})T_{i-j+1}+q^{i^{\alpha }}{G}_0,}\\ {\displaystyle D{G}_i=\sum _{j=1}^i(q^{{(j-1)}^{\alpha }}-q^{j^{\alpha }})G_{i-j+1}+q^{i^{\alpha }}D{G}_0,}\end{array}i=1,2,\dots ,$$ (3)

where $T_i$ is given by Equation (2), $q\in [0,1)$ is the design parameter, $\alpha >0$ is the adjustment parameter and $D{G}_0=G_0=A\left(n\right)+B\left(n\right)\ln (1+C(n\left)\right)$ are the starting values. The values of the $D{G}_0$ are provided in the last column of Table 1. Moreover, the $D{G}_i$ statistic follows approximately the normal distribution with mean $E\left(D{G}_i\right)={\mu }_T\left(n\right)$ and variance $Var\left(D{G}_i\right)=W_i{\sigma }_T^2\left(n\right)$, where

$$W_i=\sum _{j=1}^i{\left(\sum _{u=j}^i\right(q^{{(i-u)}^{\alpha }}-q^{{(i-u+1)}^{\alpha }}\left)\right(q^{{(u-j)}^{\alpha }}-q^{{(u-j+1)}^{\alpha }}\left)\right)}^2.$$

The time-varying control limits of the $S^2$-DGWMA chart are given by

$$\begin{array}{rl}LC{L}_i& ={\mu }_T\left(n\right)-L{\sigma }_T\left(n\right)\sqrt{W_i},\\ C{L}_i& ={\mu }_T\left(n\right),\\ UC{L}_i& ={\mu }_T\left(n\right)+L{\sigma }_T\left(n\right)\sqrt{W_i},\end{array}$$ (4)

where L is a positive control chart multiplier, when the process is IC. For large values of i, the asymptotic control limits of the $S^2$-DGWMA chart are given by

$$\begin{array}{rl}LCL& ={\mu }_T\left(n\right)-L{\sigma }_T\left(n\right)\sqrt{W},\\ CL& ={\mu }_T\left(n\right),\\ UCL& ={\mu }_T\left(n\right)+L{\sigma }_T\left(n\right)\sqrt{W},\end{array}$$ (5)

where $W=\underset{i\to \mathrm{\infty }}{lim}{W}_i$. Hereafter, for simplicity, we use the asymptotic control limits, given by Equation (5), in order to develop the $S^2$-DGWMA chart. The proposed chart is designed by plotting the statistic $D{G}_i$ versus the sample number i. The process is declared as IC, when $LCL<D{G}_i<UCL$; otherwise, it is considered to be OOC. It should be mentioned that the $S^2$-DGWMA chart reduces to the $S^2$-HEWMA chart when $q=1-\lambda$, $\alpha =1$ and $\lambda ={\lambda }_1={\lambda }_2$, where $0<{\lambda }_1,{\lambda }_2\le 1$ are the smoothing constants of the $S^2$-HEWMA chart.

3. Performance evaluation of the proposed chart

The average run-length (ARL) and the standard deviation of the run-length (SDRL) are most commonly used to measure the performance of a control chart. Particularly, the ARL is the average number of the charting statistics that must be drawn on a control chart until an OOC signal is triggered [34]. When the process dispersion is IC ( $\tau =1.00$), a large value of ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ is preferred. Nevertheless, when the process variability is OOC, that is, the standard deviation shifts from ${\sigma }_0$ to ${\sigma }_1=\tau {\sigma }_0$ (with $\tau \ne 1.00$), a small OOC ARL ( ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$) value is preferred. Here, both the ARL and SDRL measures are utilized in order to evaluate the performance of the proposed chart. Furthermore, we calculate the performance of the control chart over a range of shifts, through the expected ARL (EARL) which is defined as

$$\mathrm{E}\mathrm{A}\mathrm{R}\mathrm{L}={\int }_{\tau ={\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}}^{{\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}}\mathrm{A}\mathrm{R}\mathrm{L}\left(\tau \right)f_{\tau }\left(\tau \right)\mathrm{d}\tau ,$$

where ARL(τ) is the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ if $\tau =1$ or the ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ value corresponding to a chart specific shift ( $\tau \ne 1$) and $f_{\tau }\left(\tau \right)$ is the probability density function of the magnitude of the process shift when $\tau \in [{\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}]$.

A Monte-Carlo simulation algorithm is performed in R statistical software to calculate the run-length distribution of the two-sided $S^2$-DGWMA chart with asymptotic control limits (given by Equation (5)). The algorithm is run 10,000 iterations to calculate the average and the standard deviation of those 10,000 run-lengths. The steps of the simulation algorithm are briefly described as follows:

1. For a fixed n value, generate 10,000 random subgroups that follow the $N(0,\tau {\sigma }_0)$ distribution with ${\sigma }_0=1$.

2. For various q and a values, obtain the L value, such that the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ is approximately equal to the pre-fixed values.

3. Compute the $S_i^2$, $T_i$, $G_i$ and $D{G}_i$ statistics for each subgroup (i.e. $i=1,2,\dots ,\mathrm{10,000}$) using Equations (1), (2), and (3).

4. Calculate the asymptotic control limits given by Equation (5).

5. For the purpose of computing the run-length, compare each $D{G}_i$ statistic with the control limits given by Equation (5) for $i=1,2,\dots ,\mathrm{10,000}$. If the process is OOC, stop the simulations and record the run-length value.

6. Repeat Steps (1) through (5) for 10,000 times and compute the ARL and the SDRL.

It should be mentioned that, when the process is IC, then $\tau =1.00$ while $\tau \ne 1.00$ for an OOC process. In addition, the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ is fixed approximately equal to 200 and 370, while the examined shifts in the process variability are $\tau =\frac{{\sigma }_1}{{\sigma }_0}\in \{0.50,0.60,0.70,0.80,0.90,0.95,1.05,1.10,1.20,1.30,1.40,1.50,1.60,1.70,1.80,1.90,2.00\}$, with $\tau <1.00$ referring to the downward shifts, and $\tau >1.00$ corresponding to the upward shifts. The control chart multiplier L of the $S^2$-DGWMA chart with $q\in \{0.50,0.60,0.70,0.80,0.90,0.95\}$ and $\alpha \in \{0.70,0.80,0.90,1.00,1.20,1.50\}$ is obtained, via the above Monte-Carlo simulation algorithm, using the asymptotic control limits given by Equation (5), to set the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 200$ and 370.

Particularly, Table 2 presents the L values of the $S^2$-DGWMA control chart for various combinations of the design parameters $(q,\alpha )$ and sample size n = 3, 5, 7 and 9, when ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$. Tables 3 and A1 in the Supplementary Material present the ARL, SDRL (in the parenthesis) and EARL results of the $S^2$-DGWMA chart using asymptotic control limits with $q\in \{0.50,0.60,0.70,0.80,0.90,0.95\}$, and $\alpha \in \{0.70,0.80,0.90,1.00,1.20,1.50\}$, when ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$ as well as n = 5 and 9, respectively. Furthermore, Table A2 in the Supplementary Material presents the ARL, SDRL (in the parenthesis) and EARL results, along with the L values of the $S^2$-DGWMA chart with the same $(q,\alpha )$ combinations using asymptotic control limits, when ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 200$ and n = 5. The smallest ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ value for each τ and n values is indicated with bold print in the aforementioned Tables, as well.

Table 2.

$(q,\alpha ,L)$ parameter combinations for the $S^2$-DGWMA control chart using asymptotic control limits with sample size n = 3, 5, 7 and 9 at ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$.

q								q
n	α	0.50	0.60	0.70	0.80	0.90	0.95	n	α	0.50	0.60	0.70	0.80	0.90	0.95
3	0.70	2.8640	2.7860	2.6700	2.5000	2.9121	5.0274	7	0.70	2.8340	2.7848	2.6710	2.4510	2.2007	2.2374
	0.80	2.8590	2.7760	2.6600	2.4770	2.4155	2.72375		0.80	2.8330	2.7790	2.6645	2.4650	2.1050	2.4255
	0.90	2.8530	2.7720	2.6620	2.4840	2.2290	2.9217		0.90	2.8290	2.7731	2.6650	2.4860	2.1480	1.9885
	1.00	2.8540	2.7700	2.6680	2.5080	2.2350	2.4045		1.00	2.8260	2.7735	2.6742	2.5150	2.2130	1.9328
	1.20	2.8590	2.7760	2.6900	2.5670	2.3480	2.1230		1.20	2.8270	2.7770	2.7005	2.5760	2.3492	2.0942
	1.50	2.8770	2.8090	2.7420	2.6550	2.4980	2.3240		1.50	2.8280	2.7950	2.7452	2.6650	2.5100	2.3300
5	0.70	2.8200	2.7780	2.6650	2.4590	2.3870	3.8403	9	0.70	2.8480	2.7930	2.6700	2.4380	2.1140	2.8627
	0.80	2.8160	2.7710	2.6630	2.4580	2.1650	2.8520		0.80	2.8420	2.7845	2.6660	2.4525	2.0805	2.1678
	0.90	2.8150	2.7640	2.6670	2.4850	2.1630	2.2677		0.90	2.8370	2.7820	2.6725	2.4810	2.1370	1.8850
	1.00	2.8110	2.7610	2.6770	2.5170	2.2170	2.0030		1.00	2.8370	2.7810	2.6810	2.5150	2.2070	1.9020
	1.20	2.8090	2.7640	2.6930	2.5790	2.3490	2.1040		1.20	2.8370	2.7855	2.7080	2.5870	2.3450	2.0915
	1.50	2.8100	2.7780	2.7310	2.6570	2.5090	2.3290		1.50	2.8395	2.8040	2.7540	2.6740	2.5110	2.3240

Table 3.

ARL, SDRL (in the parenthesis) and EARL values of the $S^2$-DGWMA control chart using asymptotic control limits at n = 5 and ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$.

	q = 0.95						0.90
τ	$\alpha =0.70$	0.80	0.90	1.00	1.20	1.50	0.70	0.80	0.90	1.00	1.20	1.50
0.50	29.96	24.37	25.43	20.07	15.12	11.53	20.64	15.99	13.56	11.99	10.01	8.32
	(27.57)	(16.45)	(2.72)	(1.07)	(0.89)	(0.77)	(1.48)	(1.16)	(1.01)	(0.92)	(0.82)	(0.78)
0.60	24.08	20.72	28.41	23.12	17.20	13.06	25.00	19.14	16.03	14.07	11.66	9.83
	(31.96)	(20.80)	(6.71)	(1.82)	(1.51)	(1.38)	(3.24)	(2.02)	(1.76)	(1.61)	(1.50)	(1.74)
0.70	20.70	18.21	31.68	28.17	20.77	16.18	31.92	24.62	20.40	17.85	15.07	14.14
	(36.78)	(24.88)	(13.15)	(3.35)	(2.94)	(3.54)	(7.69)	(3.87)	(3.46)	(3.34)	(3.72)	(5.90)
0.80	19.71	17.91	37.36	38.62	29.35	27.23	44.33	36.72	30.53	27.44	25.99	31.51
	(46.63)	(32.07)	(23.84)	(8.16)	(8.77)	(13.97)	(19.10)	(9.59)	(9.22)	(9.98)	(13.35)	(22.83)
0.90	27.65	24.94	59.32	77.52	74.55	93.06	78.81	80.21	74.40	75.34	89.12	126.97
	(92.38)	(63.05)	(62.88)	(42.22)	(50.47)	(78.44)	(64.22)	(44.19)	(45.45)	(52.87)	(75.31)	(116.85)
0.95	78.79	57.73	125.44	172.47	186.36	238.66	155.79	178.10	179.50	189.46	230.49	296.82
	(374.86)	(205.40)	(192.56)	(152.19)	(165.59)	(227.07)	(193.08)	(151.86)	(152.44)	(169.17)	(217.29)	(287.51)
1.00	368.21	368.41	370.43	370.43	370.94	369.94	369.77	370.80	369.79	370.09	370.38	370.10
	(1872.21)	(1623.40)	(838.05)	(453.42)	(369.55)	(361.31)	(741.61)	(458.29)	(385.64)	(372.60)	(364.89)	(370.19)
1.05	16.28	18.72	47.97	93.66	139.20	167.49	45.83	89.39	118.07	136.49	161.33	176.64
	(147.74)	(106.57)	(122.35)	(125.85)	(139.39)	(163.35)	(107.45)	(121.03)	(129.02)	(137.57)	(159.92)	(171.68)
1.10	2.50	3.81	12.38	31.65	54.66	70.35	12.77	30.11	44.41	53.28	66.22	78.29
	(15.41)	(18.18)	(31.08)	(40.45)	(50.75)	(63.67)	(27.68)	(38.37)	(46.22)	(50.96)	(61.77)	(73.70)
1.20	1.26	1.53	3.72	10.94	20.33	24.66	4.18	10.38	15.39	18.71	22.36	25.89
	(2.70)	(3.79)	(7.44)	(10.64)	(13.98)	(17.44)	(6.50)	(10.11)	(12.22)	(14.00)	(16.85)	(21.31)
1.30	1.10	1.23	2.22	6.78	12.91	14.81	2.58	6.28	9.37	11.23	12.96	14.14
	(0.86)	(1.58)	(3.31)	(5.01)	(6.24)	(7.55)	(3.08)	(4.87)	(5.77)	(6.42)	(7.41)	(9.13)
1.40	1.06	1.12	1.71	5.16	10.10	11.31	1.99	4.66	6.99	8.38	9.57	10.03
	(0.57)	(0.86)	(2.03)	(3.08)	(3.71)	(4.32)	(1.95)	(3.08)	(3.54)	(3.82)	(4.30)	(5.10)
1.50	1.04	1.08	1.45	4.30	8.63	9.58	1.67	3.79	5.75	6.91	7.87	8.06
	(0.42)	(0.60)	(1.35)	(2.16)	(2.65)	(2.87)	(1.38)	(2.19)	(2.55)	(2.73)	(2.91)	(3.26)
1.60	1.03	1.05	1.31	3.76	7.70	8.57	1.48	3.25	4.98	6.02	6.88	7.00
	(0.31)	(0.44)	(1.01)	(1.67)	(2.04)	(2.16)	(1.05)	(1.69)	(1.98)	(2.09)	(2.20)	(2.36)
1.70	1.02	1.04	1.22	3.39	7.06	7.88	1.37	2.87	4.43	5.40	6.20	6.31
	(0.24)	(0.35)	(0.79)	(1.36)	(1.68)	(1.72)	(0.86)	(1.38)	(1.60)	(1.69)	(1.76)	(1.84)
1.80	1.02	1.03	1.17	3.12	6.57	7.38	1.30	2.61	4.04	4.94	5.72	5.83
	(0.20)	(0.28)	(0.64)	(1.16)	(1.43)	(1.46)	(0.72)	(1.18)	(1.36)	(1.43)	(1.47)	(1.50)
1.90	1.01	1.02	1.13	2.91	6.19	6.99	1.24	2.41	3.75	4.59	5.35	5.46
	(0.16)	(0.22)	(0.53)	(1.02)	(1.26)	(1.26)	(0.62)	(1.04)	(1.18)	(1.25)	(1.27)	(1.27)
2.00	1.01	1.02	1.11	2.75	5.88	6.67	1.19	2.24	3.50	4.31	5.04	5.17
	(0.13)	(0.20)	(0.46)	(0.91)	(1.11)	(1.13)	(0.54)	(0.95)	(1.06)	(1.11)	(1.12)	(1.11)
EARL	35.19	33.47	45.16	53.19	58.48	64.01	48.10	51.79	53.90	56.06	60.93	68.67
	q = 0.80						0.70
τ	$\alpha =0.70$	0.80	0.90	1.00	1.20	1.50	0.70	0.80	0.90	1.00	1.20	1.50
0.50	10.21	9.06	8.29	7.71	6.91	6.25	7.63	7.02	6.58	6.25	5.83	5.70
	(1.17)	(1.04)	(0.95)	(0.90)	(0.88)	(1.05)	(1.20)	(1.10)	(1.06)	(1.05)	(1.13)	(1.67)
0.60	12.85	11.25	10.23	9.49	8.66	8.52	10.01	9.18	8.64	8.33	8.23	9.32
	(2.14)	(1.94)	(1.85)	(1.84)	(2.09)	(3.22)	(2.34)	(2.28)	(2.32)	(2.52)	(3.28)	(5.23)
0.70	17.85	15.59	14.30	13.62	13.59	16.18	14.98	14.02	13.75	13.99	15.69	21.44
	(4.38)	(4.17)	(4.34)	(4.85)	(6.56)	(10.75)	(5.25)	(5.58)	(6.35)	(7.40)	(10.24)	(16.99)
0.80	29.72	26.86	26.27	27.04	32.40	46.79	28.58	29.06	31.28	35.08	45.45	69.98
	(11.22)	(11.97)	(13.84)	(16.37)	(24.22)	(40.65)	(15.28)	(17.96)	(22.17)	(27.53)	(39.65)	(65.55)
0.90	80.77	81.47	89.32	101.29	135.17	194.50	98.17	110.49	126.85	148.08	191.69	278.84
	(53.60)	(61.28)	(73.75)	(88.79)	(125.48)	(187.89)	(80.82)	(97.76)	(116.41)	(140.29)	(183.33)	(276.86)
0.95	196.69	208.81	233.41	258.42	397.38	381.92	254.25	279.18	306.36	330.86	379.15	458.64
	(168.36)	(188.51)	(220.32)	(248.17)	(296.95)	(377.97)	(240.79)	(267.24)	(299.22)	(324.12)	(372.54)	(457.90)
1.00	370.66	369.69	370.52	370.70	370.27	370.25	370.02	370.19	370.49	370.40	370.31	370.12
	(385.07)	(372.21)	(370.68)	(370.74)	(370.87)	(370.15)	(372.02)	(370.07)	(370.50)	(367.06)	(367.95)	(370.58)
1.05	119.40	137.51	153.14	163.25	174.62	177.78	142.26	154.94	163.82	171.00	174.31	177.76
	(129.66)	(140.30)	(154.20)	(162.34)	(173.02)	(173.85)	(145.94)	(154.52)	(161.22)	(170.56)	(172.00)	(175.21)
1.10	44.61	53.84	61.31	67.38	76.78	83.06	56.45	63.70	69.78	75.14	81.35	84.46
	(46.45)	(52.23)	(58.16)	(63.79)	(73.21)	(79.22)	(54.70)	(61.04)	(67.09)	(72.35)	(78.14)	(82.81)
	q = 0.80						0.70
τ	$\alpha =0.70$	0.80	0.90	1.00	1.20	1.50	0.70	0.80	0.90	1.00	1.20	1.50
1.20	15.19	18.07	20.25	21.98	24.62	27.24	18.60	20.72	22.30	23.86	25.97	28.46
	(12.68)	(14.63)	(16.43)	(18.11)	(21.32)	(24.07)	(15.73)	(17.66)	(19.32)	(21.13)	(23.45)	(26.41)
1.30	8.78	10.25	11.27	11.95	12.97	13.94	10.18	11.09	11.78	12.28	13.10	14.06
	(6.11)	(6.80)	(7.41)	(7.96)	(9.22)	(10.67)	(7.25)	(7.92)	(8.69)	(9.21)	(10.31)	(11.44)
1.40	6.26	7.28	7.93	8.33	8.78	9.17	6.99	7.51	7.89	8.17	8.51	9.01
	(3.77)	(4.12)	(4.39)	(4.60)	(5.16)	(5.85)	(4.39)	(4.67)	(4.96)	(5.27)	(5.77)	(6.62)
1.50	4.94	5.77	6.25	6.54	6.80	6.98	5.38	5.77	6.01	6.16	6.35	6.58
	(2.73)	(2.93)	(3.02)	(3.13)	(3.36)	(3.79)	(3.10)	(3.23)	(3.34)	(3.46)	(3.75)	(4.13)
1.60	4.16	4.84	5.26	5.52	5.72	5.80	4.44	4.75	4.94	5.06	5.19	5.30
	(2.12)	(2.24)	(2.30)	(2.36)	(2.47)	(2.71)	(2.39)	(2.45)	(2.50)	(2.58)	(2.72)	(2.94)
1.70	3.60	4.21	4.61	4.85	5.04	5.07	3.80	4.08	4.25	4.36	4.47	4.54
	(1.71)	(1.81)	(1.86)	(1.88)	(1.94)	(2.08)	(1.91)	(1.95)	(1.98)	(2.03)	(2.12)	(2.67)
1.80	3.20	3.77	4.14	4.36	4.54	4.57	3.33	3.60	3.76	3.87	3.95	4.00
	(1.43)	(1.51)	(1.54)	(1.55)	(1.57)	(1.64)	(1.58)	(1.61)	(1.62)	(1.64)	(1.67)	(1.74)
1.90	2.91	3.42	3.79	4.00	4.18	4.20	3.00	3.25	3.39	3.49	3.58	3.63
	(1.25)	(1.30)	(1.32)	(1.31)	(1.29)	(1.32)	(1.36)	(1.36)	(1.36)	(1.37)	(1.38)	(1.43)
2.00	2.68	3.17	3.51	3.73	3.92	3.95	2.74	2.98	3.13	3.22	3.30	3.35
	(1.11)	(1.15)	(1.17)	(1.15)	(1.13)	(1.14)	(1.20)	(1.20)	(1.20)	(1.20)	(1.19)	(1.20)
EARL	53.74	55.70	58.70	61.74	71.52	78.52	58.79	62.08	65.69	69.63	77.10	90.99
	q = 0.60						0.50
τ	$\alpha =0.70$	0.80	0.90	1.00	1.20	1.50	0.70	0.80	0.90	1.00	1.20	1.50
0.50	6.61	6.19	5.88	5.69	5.57	5.91	6.33	6.03	5.88	5.81	6.01	6.97
	(1.38)	(1.33)	(1.34)	(1.39)	(1.73)	(2.56)	(1.74)	(1.76)	(1.88)	(2.07)	(2.69)	(4.12)
0.60	9.19	8.70	8.48	8.49	9.16	11.57	9.64	9.51	9.72	10.18	11.90	16.11
	(2.86)	(3.00)	(3.27)	(3.73)	(5.09)	(8.03)	(4.04)	(4.48)	(5.18)	(6.01)	(8.36)	(13.02)
0.70	15.25	15.18	15.69	16.87	20.78	30.96	18.65	19.58	21.61	24.48	32.29	48.40
	(7.27)	(8.24)	(9.58)	(11.50)	(16.33)	(27.11)	(11.68)	(13.62)	(16.34)	(19.90)	(28.48)	(45.06)
0.80	35.24	38.74	43.78	50.24	67.37	105.91	53.45	60.64	70.15	81.18	110.21	165.99
	(24.53)	(30.09)	(37.03)	(44.21)	(61.85)	(101.43)	(45.02)	(53.23)	(64.20)	(76.11)	(106.48)	(163.68)
0.90	145.69	166.33	188.72	212.82	270.54	384.85	248.47	269.98	300.58	330.56	406.89	548.22
	(135.74)	(159.33)	(181.54)	(205.44)	(266.96)	(378.85)	(242.22)	(263.48)	(295.83)	(330.95)	(401.51)	(540.64)
0.95	350.60	371.15	386.53	407.41	459.69	538.02	480.97	486.85	504.05	512.64	553.45	630.50
	(341.42)	(363.93)	(383.53)	(412.42)	(466.69)	(536.35)	(476.82)	(482.85)	(501.53)	(505.43)	(545.70)	(625.44)
1.00	369.77	369.76	370.96	370.40	369.69	369.91	370.39	370.33	369.78	370.87	369.93	369.98
	(362.83)	(368.90)	(368.64)	(366.71)	(370.18)	(363.44)	(368.66)	(370.85)	(367.86)	(371.15)	(365.22)	(367.99)
1.05	150.66	159.79	164.23	168.24	173.15	176.47	149.66	157.59	162.85	166.54	170.57	171.70
	(149.48)	(158.80)	(163.02)	(165.09)	(171.41)	(175.42)	(147.48)	(156.85)	(161.92)	(165.29)	(166.85)	(172.03)
1.10	63.25	69.18	73.84	76.98	82.04	85.88	66.59	70.88	74.99	77.67	82.83	85.91
	(61.40)	(67.31)	(71.83)	(75.06)	(80.55)	(84.45)	(64.59)	(69.23)	(73.04)	(76.58)	(82.32)	(84.79)
1.20	20.62	22.28	23.58	24.82	27.12	29.29	21.87	23.36	24.72	25.88	27.75	29.67
	(18.01)	(19.78)	(21.27)	(22.61)	(25.20)	(27.90)	(19.71)	(21.43)	(23.04)	(24.20)	(26.24)	(28.24)
1.30	10.83	11.56	12.08	12.57	13.45	14.31	11.23	11.87	12.42	12.91	13.66	14.51
	(8.29)	(9.06)	(9.59)	(10.23)	(11.23)	(12.19)	(9.11)	(9.87)	(10.47)	(11.03)	(11.79)	(12.86)
1.40	7.17	7.52	7.80	8.02	8.49	8.94	7.25	7.52	7.80	8.06	8.46	8.87
	(4.85)	(5.12)	(5.45)	(5.71)	(6.39)	(7.05)	(5.35)	(5.60)	(5.89)	(6.26)	(6.79)	(7.28)
1.50	5.41	5.63	5.79	5.92	6.12	6.35	5.34	5.52	5.68	5.78	5.99	6.23
	(3.35)	(3.47)	(3.61)	(3.74)	(4.02)	(4.40)	(3.60)	(3.71)	(3.86)	(3.97)	(4.24)	(4.56)
1.60	4.38	4.56	4.67	4.75	4.88	5.04	4.24	4.39	4.50	4.59	4.73	4.88
	(2.56)	(2.61)	(2.67)	(2.74)	(2.92)	(3.17)	(2.69)	(2.76)	(2.84)	(2.92)	(3.11)	(3.33)
1.70	3.69	3.87	3.97	4.04	4.13	4.21	3.54	3.66	3.76	3.82	3.92	4.01
	(2.03)	(2.07)	(2.11)	(2.16)	(2.25)	(2.40)	(2.14)	(2.17)	(2.22)	(2.25)	(2.35)	(2.49)
1.80	3.21	3.36	3.46	3.52	3.59	3.65	3.04	3.15	3.23	3.29	3.38	3.45
	(1.67)	(1.66)	(1.68)	(1.70)	(1.75)	(1.89)	(1.74)	(1.74)	(1.76)	(1.78)	(1.84)	(1.93)
1.90	2.85	3.01	3.10	3.16	3.23	3.27	2.69	2.79	2.87	2.93	3.01	3.06
	(1.42)	(1.41)	(1.40)	(1.40)	(1.45)	(1.50)	(1.48)	(1.48)	(1.46)	(1.47)	(1.50)	(1.55)
2.00	2.59	2.74	2.84	2.90	2.96	3.00	2.43	2.52	2.60	2.66	2.74	2.80
	(1.26)	(1.24)	(1.22)	(1.21)	(1.22)	(1.26)	(1.29)	(1.29)	(1.28)	(1.28)	(1.27)	(1.29)
EARL	67.72	71.59	75.36	79.43	89.14	107.26	83.30	86.99	92.04	96.88	102.32	131.97

According to Tables 3, A1 and A2, the results reveal that:

• For a fixed value of α (q), the performance of the proposed chart improves for small to moderate downward and small to large upward shifts in the process variability, as the value of q (α) increases (decreases). For example, when $\tau =1.10$, n = 5, $\alpha =0.90$ and q = 0.50, 0.60, 0.70, 0.80, 0.90 and 0.95, the ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ ( ${\mathrm{S}\mathrm{D}\mathrm{R}\mathrm{L}}_1$) values are 74.99 (73.04), 73.84 (71.83), 69.78 (67.09), 61.31 (58.16), 44.41 (46.22) and 12.38 (31.08), respectively (see Table 3). Additionally, if $\tau =0.80$, n = 5, $\alpha =1.50$ and q = 0.50, 0.60, 0.70, 0.80, 0.90 and 0.95, then the ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ ( ${\mathrm{S}\mathrm{D}\mathrm{R}\mathrm{L}}_1$) values are 78.60 (74.88), 57.09 (53.29), 42.03 (37.86), 31.21 (25.81), 23.83 (15.49) and 23.21 (10.56), respectively (see Table A2 in the Supplementary Material). When $\tau =1.10$, n = 5, q = 0.80 and $\alpha =0.70$, 0.80, 0.90, 1.00, 1.20 and 1.50, the corresponding ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ ( ${\mathrm{S}\mathrm{D}\mathrm{R}\mathrm{L}}_1$) values are 44.61 (46.45), 53.84 (52.23), 61.31 (58.16), 67.38 (63.79), 76.78 (73.21) and 83.06 (79.22)(see Table 3).

• As n increases, the sensitivity of the proposed chart improves for most of the considered $(q,\alpha )$ combinations and τ shifts. For example, the ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ values of the $S^2$-DGWMA chart with (q = 0.90, $\alpha =0.90$) at $\tau =1.05$, are 118.07 and 81.27, when n = 5 and 9, respectively (see Tables 3 and A1 in the Supplementary Material). However, the opposite is observed, e.g. for the $S^2$-DGWMA chart with (q = 0.90, $\alpha =0.80$) at $1.30\le \tau \le 2.00$, and the $S^2$-DGWMA chart with (q = 0.95, $\alpha =1.00$) at $1.20\le \tau \le 2.00$.

• Both ${\mathrm{S}\mathrm{D}\mathrm{R}\mathrm{L}}_0$ and ${\mathrm{S}\mathrm{D}\mathrm{R}\mathrm{L}}_1$ decrease with an increase in the sample size.

• The $S^2$-DGWMA chart with (q = 0.95, $\alpha =0.70$) is the most efficient chart in detecting small to large upward shifts in the process dispersion.

• The $S^2$-DGWMA chart with (q = 0.95, $\alpha \in [0.70,0.80]$) shows the best performance for small to moderate downward shifts.

• The $S^2$-DGWMA chart with ( $q\in [0.50,0.80],\alpha \in [1.20,1.50]$) are preferable for moderate to large downward shifts.

• It is more efficient in detecting small to moderate upward shifts. Particularly, the proposed chart is more sensitive in identifying small upward shifts than small downward shifts. For example, the ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ values of the $S^2$-DGWMA (q = 0.90, $\alpha =0.90$, L = 2.163) chart are 74.40 and 44.41 when n = 5 at $\tau =0.90$ and 1.10, respectively (see Table 3). Additionally, the ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ values of the $S^2$-DGWMA (q = 0.95, $\alpha =1.20$, L = 1.828) chart are 124.08 and 88.77 when n = 5 at $\tau =0.95$ and 1.05, respectively (see Table A2 in the Supplementary Material).

• It is ARL-biased for small downward shifts ( $0.95\le \tau <1.00$) and n = 5, i.e. when (i) $(q=0.50,\alpha \in [0.70,1.50\left]\right)$ at ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 200,370$, (ii) $(q=0.60,\alpha \in [0.80,1.50\left]\right)$ at ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$ , (iii) $(q\in [0.70,0.80],\alpha \in [1.20,1.50\left]\right)$ at ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$, ( iv) $(q=0.60,\alpha \in [1.00,1.50\left]\right)$ at ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 200$ and v) $(q=0.70,\alpha =1.50)$ at ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 200$.

• For fixed τ and n values, the ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ decreases as the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ decreases. Additionally, the ${\mathrm{S}\mathrm{D}\mathrm{R}\mathrm{L}}_1$ decreases with a decrease in the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ value, for most of the considered τ and $(q,\alpha )$ cases.

• As mentioned in Section 2, the $S^2$-HEWMA chart is a special case of the proposed chart. Therefore, its charting statistic and the corresponding control limits are given by Equations (3) and (5), respectively, using $q=1-\lambda$, $\alpha =1.00$ and ${\lambda }_1={\lambda }_2=\lambda$. The performance of the competing chart is also presented in Table 3, as well as Tables A1 and A2 in the Supplementary Material when $\lambda \in \{0.05,0.10,0.20,0.30,0.40,0.50\}$, $q=1-\lambda$, and $\alpha =1.00$. We observe that, the $S^2$-DGWMA chart with $\alpha <1.00$ is more efficient than the $S^2$-HEWMA chart for detecting moderate downward to large upward shifts. For example, the $S^2$-DGWMA( $q=0.80,\alpha =0.90,L=2.485$) chart is better than the $S^2$-HEWMA( ${\lambda }_1=0.20,{\lambda }_2=0.20,L=2.517$) (i.e. $S^2$-DGWMA( $q=0.80,\alpha =1.00,L=2.517$)) chart at $0.70<\tau \le 2.00$ (see Table 3). Additionally, as the parameter q rises, the performance of the $S^2$-DGWMA chart with $\alpha >1.00$ improves in detecting downward shifts, compared with the competing chart. For example, the $S^2$-DGWMA( $q=0.90,\alpha =1.20,L=2.349$) chart is more efficient than the $S^2$-HEWMA( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=2.217$) chart at $0.50\le \tau \le 0.80$ (see Table 3). It should be noted that, similar conclusions are extracted, either the sample size n or the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ changes. For instance, the $S^2$-DGWMA( $q=0.80,\alpha =0.80,L=2.4525$) chart is more effective than the $S^2$-HEWMA( ${\lambda }_1=0.20,{\lambda }_2=0.20,L=2.515$) chart at $0.80<\tau \le 2.00$ (see Table A1 in the Supplementary Material). Furthermore, the $S^2$-HEWMA( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=1.955$) chart is less sensitive than the $S^2$-DGWMA( $q=0.90,\alpha =1.50,L=2.260$) chart at $0.50\le \tau \le 0.80$ (see Table A2 in the Supplementary Material). Finally, the $S^2$-DGWMA chart with $\alpha <1.00$ has lower EARL results than the $S^2$-HEWMA chart (see Tables 3 and A1–A2 in the Supplementary Material).

4. Performance comparisons of control charts

In the current section, we compare the performance of the proposed chart with that of the $S^2$-GWMA, $S^2$-EWMA, CS-EWMA, $S^2$-TEWMA and VSI $S^2$-EWMA charts for monitoring the process variability. For fair comparisons, it is recommended to have a similar pre-fixed ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ value. The control chart with the lowest ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ value in a certain shift τ in the process variability, can detect faster than the other control charts. Therefore, we consider two-sided asymptotic control limits for all the competing control charts, the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ is set approximately equal to 370 and the sample size n is set at 5. Furthermore, we consider the cases that ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$ and n = 9, as well as ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 200$ and n = 5, i.e. whether the sample size or the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ changes. The competing control charts are briefly described, and compared individually with the $S^2$-DGWMA chart. Note that the control chart multipliers L of the $S^2$-GWMA, $S^2$-EWMA and $S^2$-TEWMA charts, as well as the parameter $H_{\mathrm{C}\mathrm{S}}$ of the CS-EWMA chart, are selected through Monte-Carlo simulations similar to Section 3. Tables A1–A3 in the Appendix present the ARL and EARL results of the $S^2$-GWMA, $S^2$-EWMA, CS-EWMA and $S^2$-TEWMA charts when ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$ and n = 5, while Tables B1–B2, C1–C2 and D1–D2 in the Supplementary Material present the ARL and EARL results of these charts, when ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$ and n = 9 as well as ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 200$ and n = 5. The bold printed fonts in these Tables indicate the smallest ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ values for each τ. The ${\mathrm{S}\mathrm{D}\mathrm{R}\mathrm{L}}_0$ values are also provided.

• $S^2$-DGWMA chart versus $S^2$-GWMA chart

The plotting statistic of the $S^2$-GWMA chart is given by

$$G_i=\sum _{j=1}^i(q^{{(j-1)}^{\alpha }}-q^{j^{\alpha }})T_{i-j+1}+q^{i^{\alpha }}{G}_0,i=1,2,\dots$$ (6)

where $q\in [0,1)$ is the design parameter, $\alpha >0$ is the adjustment parameter and $G_0=D{G}_0$ is the starting value. The asymptotic control limits of the $S^2$-GWMA chart are given by

$$\begin{array}{rl}LCL& ={\mu }_T\left(n\right)-L{\sigma }_T\left(n\right)\sqrt{Q},\\ CL& ={\mu }_T\left(n\right),\\ UCL& ={\mu }_T\left(n\right)+L{\sigma }_T\left(n\right)\sqrt{Q},\end{array}$$ (7)

where $L(>0)$ is the control chart multiplier, $Q=\underset{i\to \mathrm{\infty }}{lim}{Q}_i$ and $Q_i=\sum _{j=1}^i(q^{{(j-1)}^{\alpha }}-q^{j^{\alpha }}{)}^2$, $i=1,2,\dots$. It should be noted that, the $S^2$-GWMA chart reduces to the $S^2$-EWMA chart when $q=1-\lambda$ and $\alpha =1.00$, where $\lambda \in (0,1]$ is the smoothing constant of the latter chart. The $S^2$-GWMA chart is constructed by plotting the statistic $G_i$ versus the sample number i. The process is declared as OOC, when $G_i\le LCL$ or $G_i\ge UCL$.

According to Tables 3 and A1, the $S^2$-DGWMA chart is more efficient than the $S^2$-GWMA chart, as well as the $S^2$-EWMA chart, for small shifts in the process variability. As q decreases, it becomes better for downward shifts and small to moderate upward shifts. See, for example, (i) the $S^2$-DGWMA( $q=0.90,\alpha =1.00,L=2.217$) chart versus the $S^2$-GWMA( $q=0.90,\alpha =1.00,L=2.686$) (i.e. $S^2$-EWMA( $\lambda =0.10,L=2.686$)) chart at $0.80\le \tau \le 1.20$, (ii) the $S^2$-DGWMA( $q=0.80,\alpha =0.80,L=2.458$) chart versus the $S^2$-GWMA( $q=0.80,\alpha =0.80,L=2.810$) chart at $0.70\le \tau <1.00$ and $1.05<\tau \le 1.20$, (iii) the $S^2$-DGWMA( $q=0.70,\alpha =1.00,L=2.677$) chart versus the $S^2$-GWMA( $q=0.70,\alpha =1.00,L=2.824$) (i.e. $S^2$-EWMA( $\lambda =0.30,L=2.824$)) chart at $\tau \le 1.00$, (iv) the $S^2$-DGWMA( $q=0.60,\alpha =1.20,L=2.764$) chart versus the $S^2$-GWMA( $q=0.60,\alpha =1.20,L=2.833$) chart at $\tau \le 1.00$ and (v) the $S^2$-DGWMA( $q=0.50,\alpha =0.80,L=2.816$) chart versus the $S^2$-GWMA( $q=0.50,\alpha =0.80,L=2.820$) chart at $\tau <1.00$ and $1.10\le \tau \le 1.40$ (see Tables 3 and A1). The performance of the $S^2$-DGWMA chart enhances compared with the $S^2$-GWMA chart for small to moderate upward shifts as the sample size n increases and the parameter q decreases. In particular, (i) the $S^2$-DGWMA( $q=0.80,\alpha =1.00,L=2.515$) chart is better than the $S^2$-GWMA( $q=0.80,\alpha =1.00,L=2.8215$) (i.e. $S^2$-EWMA( $\lambda =0.20,L=2.8215$)) chart at $0.80\le \tau \le 1.20$, (ii) the $S^2$-DGWMA( $q=0.70,\alpha =0.80,L=2.666$) chart is more sensitive than the $S^2$-GWMA( $q=0.70,\alpha =0.80,L=2.8605$) chart at $0.70\le \tau \le 1.20$ and (iii) the $S^2$-DGWMA( $q=0.60,\alpha =1.20,L=2.7855$) chart is more effective than the $S^2$-GWMA( $q=0.60,\alpha =1.20,L=2.868$) chart at $0.60\le \tau <1.00$ and $1.00<\tau \le 1.30$ (see Tables A1 and B1 in the Supplementary Material). In case the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ decreases, the proposed chart is slightly less efficient than the $S^2$-GWMA chart, in detecting upward shifts in the dispersion with a decrease in the parameter q. For example, the $S^2$-DGWMA( $q=0.60,\alpha =0.70,L=2.560$) chart is better than the $S^2$-GWMA( $q=0.60,\alpha =0.70,L=2.631$) chart at $0.50\le \tau <1.00$ and $1.20\le \tau \le 1.30$, and the $S^2$-DGWMA( $q=0.50,\alpha =0.80,L=2.620$) chart is more efficient than the $S^2$-GWMA( $q=0.50,\alpha =0.80,L=2.632$) chart at $0.50\le \tau \le 1.00$ and $1.20\le \tau \le 1.30$ (see Tables A2 and B2 in the Supplementary Material). It should be noted that, the EARL of the proposed chart is better than that of the $S^2$-GWMA chart for most of the examined cases.

• $S^2$-DGWMA chart versus CS-EWMA chart

The charting statistics of the CS-EWMA chart are given by

$$\begin{array}{rl}{M}_i^{-}& =max[0,M_{i-1}^{-}+{\mu }_T(n)-Z_i-K^{\prime }],\\ M_i^{+}& =max[0,M_{i-1}^{+}+Z_i-{\mu }_T(n)-K^{\prime }],\end{array}i=1,2,\dots ,$$ (8)

where $Z_i=\lambda T_i+(1-\lambda )Z_{i-1}$, $\lambda \in (0,1]$, $M_0^{-}=M_0^{+}=0$ are the starting values and $K^{\prime }=K_{\mathrm{C}\mathrm{S}}\sqrt{\frac{\lambda }{2-\lambda }}$ ( $K_{\mathrm{C}\mathrm{S}}\ge 0$) is the reference value. The $M_i^{-}$ and $M_i^{+}$ statistics are plotted against the decision interval $H^{\prime }=H_{\mathrm{C}\mathrm{S}}\sqrt{\frac{\lambda }{2-\lambda }}$, while the process raises an OOC signal if either $M_i^{-}$ or $M_i^{+}$ is plotted above $H^{\prime }$. Comparing Tables 3 and A2, we observe that the proposed chart is more effective than the CS-EWMA chart for moderate to large downward and small to large upward shifts in the variability. For example, the $S^2$-DGWMA( $q=0.95,\alpha =1.20,L=2.104)$ chart is better than the CS-EWMA( $\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=17.10)$ chart at $0.50\le \tau \le 0.80$, and $1.10<\tau \le 2.00$, while the corresponding EARL values are 58.48 and 58.98 (see Tables 3 and A2). Finally, similar results are observed, if the sample size increases or the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ decreases. For example, the $S^2$-DGWMA( $q=0.95,\alpha =1.20,L=2.0915)$ chart is more effective compared with the CS-EWMA( $\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=17.85)$ chart at $0.50<\tau \le 0.90$, and $1.20\le \tau \le 2.00$ (see Tables A1 and C1 in the Supplementary Material). Additionally, the $S^2$-DGWMA( $q=0.95,\alpha =1.50,L=2.070)$ chart is better than the CS-EWMA( $\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=0.50,H_{\mathrm{C}\mathrm{S}}=29.40)$ chart at $0.50\le \tau \le 0.80$ and $1.20\le \tau \le 2.00$, with the corresponding EARL values equal to 43.47 and 46.92 (see Tables A2 and C2 in the Supplementary Material).

• $S^2$-DGWMA chart versus $S^2$-TEWMA chart

The plotting statistic $W_i$ of the $S^2$-TEWMA chart is given through the following system of equations:

$$\{\begin{array}{ll}& Z_i=\lambda T_i+(1-\lambda )Z_{i-1},\\ & Y_i=\lambda Z_i+(1-\lambda )Y_{i-1},\\ & W_i=\lambda Y_i+(1-\lambda )W_{i-1},\end{array}i=1,2,\dots ,$$ (9)

where $\lambda \in (0,1]$ is the smoothing constant, and $W_0=Y_0=Z_0=D{G}_0$ are the starting values. The asymptotic control limits of the $S^2$-TEWMA chart are given by

$$\begin{array}{rl}LCL& ={\mu }_T\left(n\right)-L{\sigma }_T\left(n\right)\sqrt{[\frac{6(1-\lambda {)}^6\lambda }{(2-\lambda {)}^5}+\frac{12(1-\lambda {)}^4{\lambda }^2}{(2-\lambda {)}^4}+\frac{7(1-\lambda {)}^2{\lambda }^3}{(2-\lambda {)}^3}+\frac{{\lambda }^4}{(2-\lambda {)}^2}]},\\ CL& ={\mu }_T\left(n\right),\\ UCL& ={\mu }_T\left(n\right)+L{\sigma }_T\left(n\right)\sqrt{[\frac{6(1-\lambda {)}^6\lambda }{(2-\lambda {)}^5}+\frac{12(1-\lambda {)}^4{\lambda }^2}{(2-\lambda {)}^4}+\frac{7(1-\lambda {)}^2{\lambda }^3}{(2-\lambda {)}^3}+\frac{{\lambda }^4}{(2-\lambda {)}^2}]},\end{array}$$ (10)

where $L(>0)$ is the control chart multiplier. The $S^2$-TEWMA chart is constructed by plotting the statistic $W_i$ versus the sample number i and the process raises an OOC signal, when $W_i\le LCL$ or $W_i\ge UCL$. Tables 3 and A3 indicate that the $S^2$-DGWMA chart is more efficient for upward shifts. In addition, its performance improves for downward shifts as $q\left(\lambda \right)$ increases (decreases). For instance, the $S^2$-DGWMA( $q=0.70,\alpha =0.80,L=2.663)$ chart is better than the $S^2$-TEWMA( $\lambda =0.30,L=2.5156$) chart for shifts of size $0.50\le \tau <0.70$, $\tau =0.90$ and $1.00\le \tau \le 2.00$, while the corresponding EARL results are 62.08 and 64.99. Furthermore, the $S^2$-DGWMA( $q=0.90,\alpha =0.90,L=2.163)$ chart is more efficient than the $S^2$-TEWMA( $\lambda =0.10,L=2.020$) chart at $0.50\le \tau \le 0.90$ and $1.00\le \tau \le 2.00$, as well as the corresponding EARL results are 53.90 and 58.81 (see Tables 3 and A3). It should be noted that as the sample size increases, the performance of the $S^2$-DGWMA chart slightly improves in detecting downward shifts in comparison with the $S^2$-TEWMA chart. In particular, the $S^2$-DGWMA( $q=0.70,\alpha =0.80,L=2.666)$ chart has lower ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ values than the $S^2$-TEWMA( $\lambda =0.30,L=2.5152$) chart at $0.50\le \tau <0.80$, $0.90\le \tau <1.00$ and $1.00<\tau \le 2.00$, while the corresponding EARL values are 45.16 and 47.45 (see Tables A1 and D1 in the Supplementary Material). Finally, similar conclusions are observed, if the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ changes, e.g. the $S^2$-DGWMA( $q=0.70,\alpha =0.80,L=2.430)$ chart is more sensitive than the $S^2$-TEWMA( $\lambda =0.30,L=2.275$) chart at $0.50\le \tau \le 0.60$, $\tau =0.90$ and $1.00<\tau \le 2.00$ (see Tables A2 and D2 in the Supplementary Material).

Tables 4 to 6 present the ‘near optimal’ combinations of the design parameters for the aforementioned charts, as well as the corresponding ${\mathrm{A}\mathrm{R}\mathrm{L}}_1$ values for all the considered τ shifts when (i) ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$ and n = 5, (ii) ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$ and n = 9 as well as (iii) ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 200$ and n = 5. It should be mentioned that we do not consider some parameter combinations due to the large ${\mathrm{S}\mathrm{D}\mathrm{R}\mathrm{L}}_0$, e.g. the $S^2$-DGWMA $(q=0.95,\alpha =0.70,L=3.8403)$, $S^2$-GWMA $(q=0.95,\alpha =0.70,L=2.843)$, $S^2$-HEWMA $({\lambda }_1=0.05,{\lambda }_2=0.05,L=2.003)$ and $S^2$-TEWMA $(\lambda =0.05,L=2.14537)$ charts (see Tables 3 and A1 – A3). According to Table 4, the $S^2$-DGWMA chart is the most efficient in detecting small to moderate upward, and large downward shifts. Furthermore, the CS-EWMA is better for small to large downward shifts, whereas the $S^2$-GWMA chart is the most effective for large upward shifts. Table 5 reveals that, as the sample size n increases, the $S^2$-DGWMA chart is the most sensitive for small to moderate upward shifts, the CS-EWMA chart is the most efficient for small to moderate downward shifts, while the $S^2$-GWMA chart is the most efficient for large upward and downward shifts. Table 6 shows that as the ${\mathrm{A}\mathrm{R}\mathrm{L}}_0$ decreases, the $S^2$-DGWMA chart is the most efficient for upward shifts, whereas the CS-EWMA chart is the most effective for downward shifts. It should be noted that the $S^2$-DGWMA and $S^2$-TEWMA charts perform almost similarly for downward shifts. Finally, the proposed chart is more efficient than the $S^2$-EWMA and the $S^2$-HEWMA charts for most of the considered τ values, while it is better than the $S^2$-GWMA for small downward to moderate upward shifts in the variability.

Table 4.

ARL values and the corresponding near optimal design combinations of the control charts at n = 5 and ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$.

τ	$S^2$-DGWMA	$S^2$-EWMA	$S^2$-HEWMA	CS-EWMA	$S^2$-TEWMA	$S^2$-GWMA
0.50	$(q=0.60,\alpha =1.20,L=2.764)$	( $\lambda =0.20,L=2.800)$	( ${\lambda }_1=0.40,{\lambda }_2=0.40,L=2.761)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.50,L=2.731)$	$(q=0.80,\alpha =1.50,L=2.787)$
	5.57	6.26	5.69	5.79	5.65	5.68
0.60	$(q=0.70,\alpha =1.20,L=2.693)$	( $\lambda =0.20,L=2.800)$	( ${\lambda }_1=0.30,{\lambda }_2=0.30,L=2.677)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.40,L=2.6415)$	$(q=0.90,\alpha =1.50,L=2.699)$
	8.23	9.02	8.33	8.02	8.27	8.17
0.70	$(q=0.80,\alpha =1.20,L=2.579)$	( $\lambda =0.10,L=2.686)$	( ${\lambda }_1=0.20,{\lambda }_2=0.20,L=2.517)$	$(\lambda =0.30,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=6.61)$	$(\lambda =0.30,L=2.5156)$	$(q=0.95,\alpha =1.50,L=2.576)$
	13.59	14.81	13.62	13.07	13.50	13.32
0.80	$(q=0.90,\alpha =1.20,L=2.349)$	( $\lambda =0.10,L=2.686)$	( ${\lambda }_1=0.20,{\lambda }_2=0.20,L=2.517)$	$(\lambda =0.40,K_{\mathrm{C}\mathrm{S}}=0.50,H_{\mathrm{C}\mathrm{S}}=11.72)$	$(\lambda =0.20,L=2.332)$	$(q=0.95,\alpha =1.20,L=2.513)$
	25.99	28.68	27.04	24.55	26.16	26.27
0.90	$(q=0.90,\alpha =0.90,L=2.163)$	( $\lambda =0.05,L=2.513)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=2.217)$	$(\lambda =0.20,K_{\mathrm{C}\mathrm{S}}=0.50,H_{\mathrm{C}\mathrm{S}}=19.91)$	$(\lambda =0.10,L=2.020)$	$(q=0.95,\alpha =1.00,L=2.513)$
	74.40	81.06	75.34	68.31	74.41	81.06
0.95	$(q=0.95,\alpha =1.00,L=2.003)$	( $\lambda =0.05,L=2.513)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=2.217)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=0.10,H_{\mathrm{C}\mathrm{S}}=25.82)$	$(\lambda =0.10,L=2.020)$	$(q=0.95,\alpha =0.90,L=2.549)$
	172.47	201.13	189.46	153.55	179.00	200.45
1.05	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.05,L=2.513)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=2.217)$	$(\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=17.10)$	$(\lambda =0.10,L=2.020)$	$(q=0.95,\alpha =0.90,L=2.549)$
	89.39	120.60	136.49	120.27	128.47	98.08
1.10	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.05,L=2.513)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=2.217)$	$(\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=17.10)$	$(\lambda =0.10,L=2.020)$	$(q=0.95,\alpha =0.90,L=2.549)$
	30.11	45.31	53.28	48.75	50.70	35.79
1.20	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.05,L=2.513)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=2.217)$	$(\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=17.10)$	$(\lambda =0.10,L=2.020)$	$(q=0.95,\alpha =0.90,L=2.549)$
	10.38	15.25	18.71	21.31	19.50	12.24
1.30	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.05,L=2.513)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=2.217)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.40,L=2.6415)$	$(q=0.95,\alpha =0.90,L=2.549)$
	6.28	8.73	11.23	12.92	12.80	7.14
1.40	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.05,L=2.513)$	( ${\lambda }_1=0.40,{\lambda }_2=0.40,L=2.761)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.50,L=2.731)$	$(q=0.95,\alpha =0.90,L=2.549)$
	4.66	6.17	8.02	8.47	8.35	5.08
1.50	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.05,L=2.513)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.811)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.50,L=2.731)$	$(q=0.95,\alpha =0.90,L=2.549)$
	3.79	4.84	5.78	6.36	6.15	4.01
1.60	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.05,L=2.513)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.811)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.50,L=2.731)$	$(q=0.95,\alpha =0.90,L=2.549)$
	3.25	4.04	4.59	5.19	4.97	3.31
1.70	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.50,L=2.824)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.811)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.50,L=2.731)$	$(q=0.95,\alpha =0.90,L=2.549)$
	2.87	3.45	3.82	4.43	4.24	2.84
1.80	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.50,L=2.824)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.811)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.50,L=2.731)$	$(q=0.95,\alpha =0.90,L=2.549)$
	2.61	2.91	3.29	3.89	3.71	2.51
1.90	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.50,L=2.824)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.811)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.50,L=2.731)$	$(q=0.95,\alpha =0.90,L=2.549)$
	2.41	2.54	2.93	3.51	3.35	2.27
2.00	$(q=0.90,\alpha =0.80,L=2.165)$	( $\lambda =0.50,L=2.824)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.811)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.35)$	$(\lambda =0.50,L=2.731)$	$(q=0.95,\alpha =0.90,L=2.549)$
	2.24	2.26	2.66	3.21	3.08	2.08

Table 6.

ARL values and the corresponding near optimal design combinations of the control charts at n = 5 and ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 200$.

τ	$S^2$-DGWMA	$S^2$-EWMA	$S^2$-HEWMA	CS-EWMA	$S^2$-TEWMA	$S^2$-GWMA
0.50	$(q=0.60,\alpha =1.20,L=2.557)$	( $\lambda =0.30,L=2.634)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.6105)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=3.578)$	( $\lambda =0.50,L=2.516)$	$(q=0.80,\alpha =1.50,L=2.585)$
	4.97	5.45	5.05	5.17	5.12	4.95
0.60	$(q=0.70,\alpha =1.20,L=2.473)$	( $\lambda =0.20,L=2.592)$	( ${\lambda }_1=0.40,{\lambda }_2=0.40,L=2.5486)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=3.578)$	( $\lambda =0.50,L=2.516)$	$(q=0.90,\alpha =1.50,L=2.478)$
	7.19	7.85	7.29	7.11	7.20	7.18
0.70	$(q=0.80,\alpha =1.20,L=2.337)$	( $\lambda =0.10,L=2.452)$	( ${\lambda }_1=0.30,{\lambda }_2=0.30,L=2.4437)$	$(\lambda =0.30,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=5.17)$	( $\lambda =0.40,L=2.416)$	$(q=0.95,\alpha =1.50,L=2.334)$
	11.58	13.02	11.70	11.40	11.76	11.53
0.80	$(q=0.80,\alpha =1.00,L=2.270)$	( $\lambda =0.10,L=2.452)$	( ${\lambda }_1=0.20,{\lambda }_2=0.20,L=2.27)$	$(\lambda =0.40,K_{\mathrm{C}\mathrm{S}}=0.50,H_{\mathrm{C}\mathrm{S}}=9.67)$	( $\lambda =0.20,L=2.07)$	$(q=0.95,\alpha =1.20,L=2.266)$
	22.11	23.82	22.11	20.99	22.41	22.44
0.90	$(q=0.95,\alpha =1.20,L=1.828)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=1.955)$	$(\lambda =0.20,K_{\mathrm{C}\mathrm{S}}=0.50,H_{\mathrm{C}\mathrm{S}}=15.47)$	( $\lambda =0.10,L=1.756)$	$(q=0.95,\alpha =1.20,L=2.266)$
	58.38	63.63	58.45	54.51	60.64	63.00
0.95	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=1.955)$	$(\lambda =0.40,K_{\mathrm{C}\mathrm{S}}=0.10,H_{\mathrm{C}\mathrm{S}}=23.50)$	( $\lambda =0.10,L=1.756)$	$(q=0.95,\alpha =1.00,L=2.266)$
	123.83	133.44	125.98	111.22	123.35	133.44
1.05	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=1.955)$	$(\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=10.62)$	( $\lambda =0.10,L=1.756)$	$(q=0.90,\alpha =0.70,L=2.566)$
	55.69	77.13	88.95	81.27	83.01	62.10
1.10	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=1.955)$	$(\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=10.62)$	( $\lambda =0.10,L=1.756)$	$(q=0.90,\alpha =0.70,L=2.566)$
	19.51	32.37	38.79	35.97	35.77	25.29
1.20	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=1.955)$	$(\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=10.62)$	( $\lambda =0.10,L=1.756)$	$(q=0.95,\alpha =0.90,L=2.321)$
	6.83	11.82	14.75	16.19	14.61	9.33
1.30	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=1.955)$	$(\lambda =0.40,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.24)$	( $\lambda =0.10,L=1.756)$	$(q=0.95,\alpha =0.90,L=2.321)$
	4.05	6.99	9.19	11.01	9.79	5.63
1.40	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=1.955)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=3.578)$	( $\lambda =0.50,L=2.516)$	$(q=0.95,\alpha =0.90,L=2.321)$
	3.01	5.03	6.92	7.41	7.29	4.07
1.50	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.6105)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=3.578)$	( $\lambda =0.50,L=2.516)$	$(q=0.95,\alpha =0.90,L=2.321)$
	2.43	4.00	5.17	5.64	5.53	3.22
1.60	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.6105)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=3.578)$	( $\lambda =0.50,L=2.516)$	$(q=0.95,\alpha =0.90,L=2.321)$
	2.08	3.32	4.14	4.63	4.52	2.69
1.70	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.6105)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=3.578)$	( $\lambda =0.50,L=2.516)$	$(q=0.90,\alpha =0.70,L=2.566)$
	1.85	2.87	3.47	3.96	3.88	2.33
1.80	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.6105)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=3.578)$	( $\lambda =0.50,L=2.516)$	$(q=0.90,\alpha =0.70,L=2.566)$
	1.68	2.55	3.02	3.49	3.42	2.06
1.90	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.05,L=2.266)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.6105)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=3.578)$	( $\lambda =0.50,L=2.516)$	$(q=0.90,\alpha =0.70,L=2.566)$
	1.56	2.32	2.70	3.16	3.11	1.87
2.00	$(q=0.90,\alpha =0.80,L=1.985)$	( $\lambda =0.50,L=2.638)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.6105)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=3.578)$	( $\lambda =0.50,L=2.516)$	$(q=0.90,\alpha =0.70,L=2.566)$
	1.47	2.09	2.46	2.91	2.88	1.72

Table 5.

ARL values and the corresponding near optimal design combinations of the control charts at n = 9 and ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$.

τ	$S^2$-DGWMA	$S^2$-EWMA	$S^2$-HEWMA	CS-EWMA	$S^2$-TEWMA	$S^2$-GWMA
0.50	$(q=0.50,\alpha =1.50,L=2.8395)$	( $\lambda =0.40,L=2.8645)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.837)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.60,\alpha =1.50,L=2.870)$
	3.15	3.21	3.29	3.81	3.80	2.97
0.60	$(q=0.50,\alpha =1.20,L=2.837)$	( $\lambda =0.30,L=2.854)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.837)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.80,\alpha =1.50,L=2.8137)$
	4.44	4.70	4.47	4.86	4.72	4.42
0.70	$(q=0.70,\alpha =1.20,L=2.708)$	( $\lambda =0.20,L=2.8215)$	( ${\lambda }_1=0.40,{\lambda }_2=0.40,L=2.781)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.90,\alpha =1.50,L=2.7118)$
	7.31	7.72	7.35	7.16	7.33	7.27
0.80	$(q=0.80,\alpha =1.00,L=2.515)$	( $\lambda =0.10,L=2.690)$	( ${\lambda }_1=0.20,{\lambda }_2=0.20,L=2.515)$	$(\lambda =0.30,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=7.004)$	( $\lambda =0.30,L=2.5152)$	$(q=0.95,\alpha =1.50,L=2.5762)$
	14.40	15.22	14.40	14.02	14.49	14.34
0.90	$(q=0.90,\alpha =1.00,L=2.207)$	( $\lambda =0.05,L=2.494)$	( ${\lambda }_1=0.10,{\lambda }_2=0.10,L=2.207)$	$(\lambda =0.30,K_{\mathrm{C}\mathrm{S}}=0.50,H_{\mathrm{C}\mathrm{S}}=15.46)$	( $\lambda =0.20,L=2.327)$	$(q=0.95,\alpha =1.00,L=2.494)$
	41.90	44.26	41.90	40.20	44.50	44.26
0.95	$(q=0.95,\alpha =1.00,L=1.902)$	( $\lambda =0.05,L=2.494)$	( ${\lambda }_1=0.05,{\lambda }_2=0.05,L=1.902)$	$(\lambda =0.10,K_{\mathrm{C}\mathrm{S}}=0.50,H_{\mathrm{C}\mathrm{S}}=31.78)$	( $\lambda =0.10,L=1.996)$	$(q=0.95,\alpha =0.90,L=2.5115)$
	109.11	119.52	109.11	101.19	110.96	119.46
1.05	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.05,L=2.494)$	( ${\lambda }_1=0.05,{\lambda }_2=0.05,L=1.902)$	$(\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=17.85)$	( $\lambda =0.10,L=1.996)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	42.26	83.29	69.23	82.64	88.21	56.41
1.10	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.05,L=2.494)$	( ${\lambda }_1=0.05,{\lambda }_2=0.05,L=1.902)$	$(\lambda =0.05,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=17.85)$	( $\lambda =0.10,L=1.996)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	14.85	28.79	25.36	33.52	33.52	20.06
1.20	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.05,L=2.494)$	( ${\lambda }_1=0.05,{\lambda }_2=0.05,L=1.902)$	$(\lambda =0.30,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=7.004)$	( $\lambda =0.30,L=2.5152)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	6.20	10.76	11.66	14.03	13.91	7.88
1.30	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.05,L=2.494)$	( ${\lambda }_1=0.40,{\lambda }_2=0.40,L=2.781)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	4.12	6.56	7.43	7.76	7.64	4.84
1.40	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.30,L=2.854)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.837)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	3.18	4.66	4.88	5.39	5.19	3.52
1.50	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.40,L=2.8645)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.837)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	2.66	3.43	3.68	4.26	4.06	2.80
1.60	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.50,L=2.8599)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.837)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	2.31	2.69	3.02	3.59	3.43	2.35
1.70	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.50,L=2.8599)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.837)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	2.06	2.24	2.61	3.15	3.01	2.03
1.80	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.50,L=2.8599)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.837)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	1.87	1.93	2.33	2.84	2.74	1.81
1.90	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.50,L=2.8599)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.837)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	1.72	1.72	2.11	2.61	2.54	1.64
2.00	$(q=0.90,\alpha =0.70,L=2.114)$	( $\lambda =0.50,L=2.8599)$	( ${\lambda }_1=0.50,{\lambda }_2=0.50,L=2.837)$	$(\lambda =0.50,K_{\mathrm{C}\mathrm{S}}=1.00,H_{\mathrm{C}\mathrm{S}}=4.657)$	( $\lambda =0.50,L=2.749)$	$(q=0.95,\alpha =0.80,L=2.5719)$
	1.60	1.57	1.95	2.44	2.39	1.51

• $S^2$-DGWMA chart versus VSI $S^2$-EWMA chart

Furthermore, we compare the performance of the proposed chart with that of the VSI $S^2$-EWMA chart, that is, briefly described as follows. The sampling interval, and particularly, the time between two successive samples $T_i$ and $T_{i+1}$, depends on the current value of the $Z_i=\lambda T_i+(1-\lambda )Z_{i-1}$, $\lambda \in (0,1]$ statistic. A longer sampling interval $h_L$ is utilized when the $Z_i$ statistic lies in the region $R_L=[LWL,UWL]$, defined as

$$\begin{array}{rl}LWL& ={\mu }_T\left(n\right)-W{\sigma }_T\left(n\right)\sqrt{\frac{\lambda }{2-\lambda }},\\ UWL& ={\mu }_T\left(n\right)+W{\sigma }_T\left(n\right)\sqrt{\frac{\lambda }{2-\lambda }},\end{array}$$ (11)

where W is the warning limit coefficient that establishes the proportion of times that the $Z_i$ statistic lies within the long and short sampling regions. Similarly, the short sampling interval $h_S$ is utilized when the control statistic lies within the region $R_S=[LCL,LWL]\cup [UWL,UCL]$. The process is declared as OOC, if the statistic $Z_i$ lies outside the range $[LCL,UCL]$. Castagliola et al. [14] used the Average Time to Signal (ATS), that is, the expected value of the time from the start of the process to the time when the chart signals, as a performance measure. For a Fixed Sampling Interval (FSI) model, the ATS is defined as: ${\mathrm{A}\mathrm{T}\mathrm{S}}^{\mathrm{F}\mathrm{S}\mathrm{I}}=h_0\times {\mathrm{A}\mathrm{R}\mathrm{L}}^{\mathrm{F}\mathrm{S}\mathrm{I}}$. In the case of a VSI model, the ATS is given by: ${\mathrm{A}\mathrm{T}\mathrm{S}}^{\mathrm{V}\mathrm{S}\mathrm{I}}=E\left(h\right)\times {\mathrm{A}\mathrm{R}\mathrm{L}}^{\mathrm{V}\mathrm{S}\mathrm{I}},$ where $E\left(h\right)$ is the expected value of the sampling interval. For fair comparisons between a VSI chart and a FSI chart, we assume that $h_S=h_L=h_0=1$. Therefore, the IC average sampling interval for a VSI chart should be $E_0\left(h\right)=1$. Moreover, if ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370,$ then ${\mathrm{A}\mathrm{T}\mathrm{S}}_0\approx 370$ for both the FSI and VSI charts. Table 2 and 3 of Castagliola et al. [14] present the minimal ATS results of the VSI $S^2$-EWMA chart when n = 5 and 9, that they were obtained through a Markov Chain approach. Comparing Tables 4 and 5 above, with Tables 2 and 3 of Castagliola et al. [14], we observe that the proposed chart is more efficient than the VSI $S^2$-EWMA chart for small shifts, and vice versa for the rest range of shifts.

5. Illustrative example

In the current section, we demonstrate the application of the $S^2$-DGWMA control chart against the $S^2$-EWMA, $S^2$-HEWMA, $S^2$-TEWMA and $S^2$-GWMA control charts considering the real data given in DeVor et al. [20]. The aforementioned dataset is widely used by numerous scholars, such as Chen et al. [17] and Tariq et al. [44]. These data are measurements of the inside diameter of cylinder bores in an engine block. Table 7 shows the data for the first 35 samples, with sample size n = 5. It should be noted that each observation in this table is recorded in the last three digits of its actual measurement. For instance, if the actual measurements are 3.5205 and 3.5202, then the corresponding observations will be 205 and 202. The estimated values of the process mean and standard deviation are 200.251 and 3.306, respectively.

Table 7.

Data for the inside diameter of cylinder bores and charting statistics.

i	$X_1$	$X_2$	$X_3$	$X_4$	$X_5$	$S_i^2$	$T_i$	$S^2$-EWMA	$S^2$-HEWMA	$S^2$-TEWMA	$S^2$-GWMA	$S^2$-DGWMA
1	205	202	204	207	205	3.300	−1.146	0.143	0.208	0.211	0.143	0.208
2	202	196	201	198	202	7.200	−0.357	0.118	0.203	0.210	0.142	0.204
3	201	202	199	197	196	6.500	−0.480	0.088	0.197	0.210	0.126	0.200
4	205	203	196	201	197	14.800	0.685	0.118	0.193	0.209	0.170	0.199
5	199	196	201	200	195	6.700	−0.444	0.090	0.188	0.208	0.136	0.196
6	203	198	192	217	196	93.700	4.343	0.303	0.194	0.207	0.359	0.204
7	202	202	198	203	202	3.800	−1.029	0.236	0.196	0.207	0.220	0.205
8	197	196	196	200	204	11.800	0.326	0.241	0.198	0.206	0.233	0.206
9	199	200	204	196	202	9.200	−0.035	0.227	0.200	0.206	0.216	0.207
10	202	196	204	195	197	15.700	0.782	0.255	0.202	0.206	0.247	0.209
11	205	204	202	208	205	4.700	−0.832	0.200	0.202	0.206	0.183	0.207
12	200	201	199	200	201	0.700	−1.873	0.097	0.197	0.205	0.096	0.202
13	205	196	201	197	198	13.300	0.512	0.117	0.193	0.205	0.155	0.199
14	202	199	200	198	200	2.200	−1.427	0.040	0.185	0.204	0.080	0.193
15	200	200	201	205	201	4.300	−0.918	−0.008	0.176	0.202	0.061	0.187
16	201	187	209	202	200	63.700	3.502	0.168	0.175	0.201	0.261	0.191
17	202	202	204	198	203	5.200	−0.729	0.123	0.173	0.199	0.162	0.189
18	201	198	204	201	201	4.500	−0.874	0.073	0.168	0.198	0.119	0.186
19	207	206	194	197	201	31.500	2.052	0.172	0.168	0.196	0.234	0.188
20	200	204	198	199	199	5.500	−0.669	0.130	0.166	0.195	0.160	0.187
21	203	200	204	199	200	4.700	−0.832	0.082	0.162	0.193	0.121	0.183
22	196	203	197	201	194	13.700	0.560	0.106	0.159	0.192	0.162	0.182
23	197	199	203	200	196	7.500	−0.306	0.085	0.155	0.190	0.136	0.180
24	201	197	196	199	207	19.000	1.110	0.136	0.154	0.188	0.194	0.181
25	204	196	201	199	197	10.300	0.125	0.136	0.153	0.186	0.177	0.181
26	206	206	199	200	203	10.700	0.180	0.138	0.153	0.185	0.176	0.180
27	204	203	199	199	197	8.800	−0.096	0.126	0.151	0.183	0.162	0.179
28	199	201	201	194	200	8.500	−0.143	0.113	0.149	0.181	0.151	0.178
29	201	196	197	204	200	10.300	0.125	0.113	0.148	0.180	0.157	0.177
30	203	206	201	196	201	13.300	0.512	0.133	0.147	0.178	0.177	0.177
31	203	197	199	197	201	6.800	−0.427	0.105	0.145	0.176	0.142	0.175
32	197	194	199	200	199	5.700	−0.630	0.069	0.141	0.174	0.114	0.172
33	200	201	200	197	200	2.300	−1.400	−0.005	0.134	0.172	0.054	0.166
34	199	199	201	201	201	1.200	−1.714	−0.090	0.123	0.170	−0.003	0.158
35	200	204	197	197	199	8.300	−0.174	−0.094	0.112	0.167	0.030	0.151

Assuming ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$, we construct the $S^2$-EWMA, $S^2$-HEWMA, $S^2$-TEWMA, $S^2$-GWMA and $S^2$-DGWMA control charts with asymptotic control limits and design parameters $(\lambda ,L)=(0.05,2.513)$, $({\lambda }_1,{\lambda }_2,L)=(0.05,0.05,2.003)$, $(\lambda ,L)=(0.05,2.14537)$, $(q,\alpha ,L)=(0.95,0.70,2.843)$ and $(q,\alpha ,L)=(0.95,0.70,3.8403)$, respectively. It should be noted that, the control chart multipliers are obtained through Monte-Carlo simulations, so that ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$ and n = 5.

The charting statistics of these charts are presented in Table 7, as well. Figures 1– 4 plot the $S^2$-EWMA, $S^2$-HEWMA, $S^2$-TEWMA, $S^2$-GWMA control charts, whereas the proposed $S^2$-DGWMA chart is displayed in Figure 5. We observe that the $S^2$-DGWMA chart triggers an OOC signal at the 1, 10 and 11 samples, the $S^2$-GWMA chart at the 6th sample, while the remaining charts fail to detect any shift.

Figure 2.

The $S^2$-HEWMA control chart for the cylinder diameter data.

Figure 3.

The $S^2$-TEWMA control chart for the cylinder diameter data.

Figure 1.

The $S^2$-EWMA control chart for the cylinder diameter data.

Figure 4.

The $S^2$-GWMA control chart for the cylinder diameter data.

Figure 5.

The $S^2$-DGWMA control chart for the cylinder diameter data.

6. Concluding remarks

In the present article, we develop a new memory-type control chart for monitoring both upward and downward shifts in the process variability. The proposed chart extends the $S^2$-GWMA chart to the $S^2$-DGWMA chart by applying a three-parameter logarithmic transformation to the $S^2$ on the DGWMA control charting scheme. The proposed chart is evaluated through the ARL and SDRL measures, using asymptotic control limits. The results indicate that for a fixed value of the α (q) parameter, the ARL performance of the $S^2$-DGWMA chart improves for small to moderate downward and small to large upward shifts in the process dispersion, as the value of q (α) increases (decreases). Generally, as the value of the sample size increases, the sensitivity of the proposed chart enhances. Furthermore, the $S^2$-DGWMA chart with (q = 0.95, $\alpha =0.70$) is efficient in detecting small to large upward shifts in the process variability, the $S^2$-DGWMA chart with (q = 0.95, $\alpha \in [0.70,0.80]$) is effective for small to moderate downward shifts, while the $S^2$-DGWMA chart with ( $q\in [0.50,0.80]$, $\alpha \in [1.20,1.50])$ is recommended for moderate to large downward shifts.

Moreover, the $S^2$-DGWMA chart is compared with several well-known memory-type control charts for monitoring the process variability. The results indicate that the $S^2$-DGWMA chart is more effective in detecting small shifts in the process variability, and particularly, more efficient in identifying upward shifts. Specifically, the proposed chart with $\alpha <1.00$ is more efficient than the $S^2$-HEWMA chart for detecting moderate downward to large upward shifts, while the performance of the $S^2$-DGWMA chart with $\alpha >1.00$ improves in detecting downward shifts against the $S^2$-HEWMA chart as q rises. It is more efficient than the $S^2$-EWMA and $S^2$-GWMA charts for downward to moderate upward shifts. The $S^2$-DGWMA chart is better than the CS-EWMA chart for large downward and all the considered upward shifts, while it is better than the $S^2$-TEWMA chart for upward shifts. It should be noted that, the VSI $S^2$-EWMA chart is less sensitive than the $S^2$-DGWMA chart for small shifts, whereas the opposite is observed for the remaining considered shifts in the variability. Furthermore, an illustrative example is displayed to explain the application of the proposed chart. Consequently, our findings indicate that the proposed chart is a reliable alternate control chart that quality practitioners should utilize for monitoring the process variability. For future research, it would be interesting to investigate the VSI version of the $S^2$-DGWMA chart.

Appendix.

Table A1.

ARL and EARL values of the $S^2$-GWMA control chart using asymptotic control limits at n = 5 and ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$.

	q = 0.95						0.90
	$\alpha =0.70$	0.80	0.90	1.00	1.20	1.50	0.70	0.80	0.90	1.00	1.20	1.50
τ	L = 2.843	2.641	2.549	2.513	2.513	2.576	2.767	2.721	2.695	2.686	2.683	2.699
0.50	17.15	13.43	11.35	10.02	8.38	7.03	10.58	9.13	8.17	7.51	6.61	5.86
0.60	22.97	17.62	14.63	12.74	10.46	8.79	14.41	12.25	10.86	9.88	8.70	8.17
0.70	33.77	25.21	20.60	17.84	14.65	13.32	21.70	18.22	16.14	14.81	13.79	15.25
0.80	57.21	42.18	34.23	29.70	26.27	29.89	39.01	32.78	29.70	28.68	31.21	43.43
0.90	128.55	98.48	85.16	81.06	86.81	122.16	105.11	95.36	95.37	102.47	126.43	183.94
0.95	251.83	212.83	200.45	201.13	226.88	290.13	255.41	244.36	249.36	264.98	304.82	370.00
1.00	369.33	370.15	370.03	370.69	370.53	369.53	370.98	370.69	370.13	370.24	370.01	370.06
	(666.40)	(491.30)	(415.79)	(382.43)	(372.17)	(368.53)	(423.42)	(391.22)	(375.63)	(371.24)	(368.35)	(368.46)
1.05	44.20	72.32	98.08	120.60	147.93	172.21	93.13	113.64	130.41	144.95	162.25	173.34
1.10	15.72	24.70	35.79	45.31	58.74	73.80	34.30	43.49	51.38	57.77	68.51	79.99
1.20	6.30	8.93	12.24	15.25	19.51	23.62	12.18	14.87	17.09	19.00	22.03	25.50
1.30	4.01	5.41	7.14	8.73	10.82	12.49	7.06	8.33	9.42	10.30	11.63	12.87
1.40	2.96	3.91	5.08	6.17	7.57	8.48	4.96	5.78	6.47	7.02	7.76	8.38
1.50	2.40	3.09	4.01	4.84	5.93	6.55	3.81	4.44	4.97	5.40	5.90	6.24
1.60	2.05	2.57	3.31	4.04	4.95	5.50	3.13	3.63	4.08	4.44	4.83	5.10
1.70	1.81	2.23	2.84	3.47	4.30	4.80	2.65	3.07	3.46	3.79	4.16	4.37
1.80	1.65	1.99	2.51	3.07	3.83	4.31	2.32	2.68	3.03	3.31	3.66	3.86
1.90	1.53	1.81	2.27	2.77	3.47	3.95	2.08	2.39	2.70	2.97	3.30	3.48
2.00	1.44	1.68	2.08	2.54	3.20	3.67	1.91	2.17	2.46	2.72	3.04	3.24
EARL	57.53	53.12	52.70	53.92	57.61	65.71	56.44	56.16	57.45	59.76	65.31	75.59
	q = 0.80						0.70
	$\alpha =0.70$	0.80	0.90	1.00	1.20	1.50	0.70	0.80	0.90	1.00	1.20	1.50
τ	L = 2.815	2.810	2.805	2.800	2.790	2.787	2.819	2.824	2.825	2.824	2.825	2.823
0.50	8.02	7.20	6.66	6.26	5.76	5.68	7.59	6.95	6.53	6.27	6.14	6.90
0.60	11.46	10.27	9.50	9.02	8.79	10.18	11.70	10.87	10.47	10.44	11.47	15.60
0.70	18.59	16.94	16.24	16.21	17.83	25.19	21.62	21.07	21.75	23.43	29.77	46.36
0.80	39.22	37.93	39.48	42.86	54.23	84.63	59.63	63.03	69.43	78.84	103.45	162.67
0.90	152.24	159.89	173.13	191.83	234.03	325.56	309.26	318.81	331.22	351.18	410.52	547.55
0.95	393.46	390.11	394.51	407.17	432.78	499.81	610.02	576.45	556.01	560.35	578.32	636.50
1.00	370.93	369.42	370.07	370.13	370.80	370.22	370.55	370.06	370.43	370.83	370.05	370.97
	(379.35)	(368.81)	(367.04)	(366.36)	(370.32)	(370.00)	(373.73)	(373.35)	(374.46)	(372.75)	(369.57)	(369.43)
1.05	122.84	136.02	145.61	154.57	163.53	171.59	132.76	141.03	148.12	153.10	163.39	169.55
1.10	50.94	57.27	62.29	67.50	74.66	81.86	57.91	62.63	66.73	70.78	77.14	83.38
1.20	17.22	18.91	20.33	21.72	24.37	27.37	19.52	21.01	22.24	23.49	25.91	28.65
1.30	9.30	10.02	10.68	11.30	12.30	13.54	10.24	10.81	11.35	11.83	12.78	14.06
1.40	6.23	6.66	7.01	7.33	7.81	8.44	6.69	6.98	7.22	7.47	7.98	8.56
1.50	4.67	4.99	5.24	5.45	5.73	6.06	4.89	5.09	5.28	5.44	5.71	6.01
1.60	3.74	4.00	4.22	4.39	4.59	4.81	3.88	4.02	4.16	4.28	4.48	4.72
1.70	3.13	3.34	3.52	3.68	3.88	4.02	3.21	3.34	3.45	3.55	3.70	3.86
1.80	2.69	2.88	3.05	3.19	3.36	3.47	2.75	2.86	2.95	3.04	3.17	3.31
1.90	2.38	2.55	2.70	2.83	2.99	3.11	2.42	2.52	2.60	2.68	2.80	2.93
2.00	2.15	2.30	2.44	2.55	2.73	2.85	2.18	2.26	2.34	2.41	2.53	2.65
EARL	67.99	69.34	71.69	75.00	82.07	97.28	92.13	92.86	94.68	98.35	108.37	131.05
	q = 0.60						0.50
	$\alpha =0.70$	0.80	0.90	1.00	1.20	1.50	0.70	0.80	0.90	1.00	1.20	1.50
τ	L = 2.820	2.825	2.828	2.831	2.833	2.833	2.819	2.820	2.822	2.824	2.827	2.835
0.50	8.37	7.81	7.51	7.47	7.94	10.12	11.31	10.96	11.03	11.56	13.66	20.34
0.60	14.67	14.26	14.54	15.45	18.73	28.34	26.73	27.79	29.96	33.46	42.92	69.61
0.70	34.87	36.89	40.48	45.41	59.69	93.99	97.35	104.48	114.60	127.46	161.12	253.17
0.80	135.40	146.34	159.36	177.42	223.29	332.67	516.75	506.50	517.21	545.51	629.46	898.16
0.90	747.04	705.81	696.21	710.79	761.47	825.19	2052.45	1791.33	1630.09	1547.88	1488.31	1621.76
0.95	893.25	832.74	785.13	765.36	740.63	754.80	1054.77	991.54	952.51	920.53	873.35	872.58
1.00	370.47	370.07	370.57	370.52	370.67	370.79	369.59	368.52	370.37	370.10	368.68	370.39
	(372.45)	(369.29)	(373.66)	(370.69)	(370.90)	(374.10)	(372.40)	(369.61)	(373.18)	(371.83)	(370.31)	(366.81)
1.05	138.98	144.25	149.65	154.07	160.17	166.61	146.59	148.39	150.91	154.23	157.94	164.40
1.10	64.33	67.64	71.46	73.72	79.08	84.00	70.87	72.96	75.00	77.23	80.76	84.94
1.20	21.89	22.92	23.97	25.16	26.94	29.06	24.21	25.04	25.78	26.64	27.91	29.94
1.30	11.12	11.54	12.00	12.44	13.32	14.42	12.09	12.38	12.69	13.05	13.73	14.73
1.40	7.08	7.30	7.49	7.70	8.13	8.68	7.54	7.65	7.82	7.99	8.35	8.85
1.50	5.07	5.20	5.37	5.50	5.73	6.06	5.30	5.40	5.49	5.59	5.81	6.13
1.60	3.97	4.07	4.16	4.26	4.43	4.65	4.09	4.14	4.20	4.27	4.42	4.63
1.70	3.26	3.33	3.40	3.48	3.60	3.77	3.33	3.37	3.40	3.45	3.55	3.69
1.80	2.77	2.84	2.89	2.95	3.05	3.18	2.80	2.83	2.87	2.91	2.98	3.09
1.90	2.43	2.49	2.54	2.60	2.68	2.79	2.45	2.48	2.51	2.54	2.60	2.70
2.00	2.18	2.23	2.27	2.32	2.40	2.51	2.19	2.21	2.23	2.26	2.31	2.39
EARL	148.27	144.69	144.67	148.14	158.88	181.35	307.96	284.54	272.87	270.05	277.13	328.66

${}^{\ast }$SDRL ${}_0$ values in the parenthesis.

Table A2.

ARL and EARL values of the CS-EWMA control chart using asymptotic control limits at n = 5 and ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$.

	$\lambda =0.05$			0.10			0.20			0.30			0.40			0.50
	$K_{\mathrm{C}\mathrm{S}}=0.10$	0.50	1.00	0.10	0.50	1.00	0.10	0.50	1.00	0.10	0.50	1.00	0.10	0.50	1.00	0.10	0.50	1.00
τ	$H_{\mathrm{C}\mathrm{S}}=94$	43.4	17.1	69.2	30.58	12.81	48.1	19.91	8.74	37.52	14.77	6.61	30.67	11.72	5.28	25.82	9.62	4.35
0.50	28.67	20.76	15.64	22.01	14.98	11.26	17.45	10.90	8.23	15.47	9.07	6.94	14.25	8.02	6.22	13.40	7.32	5.79
0.60	33.09	24.14	18.48	25.71	17.59	13.46	20.81	13.06	10.13	18.73	11.09	8.82	17.43	10.01	8.22	16.51	9.31	8.02
0.70	40.29	29.70	23.35	31.93	22.08	17.46	26.62	17.05	13.96	24.42	15.00	13.07	23.00	13.97	13.22	21.98	13.39	14.10
0.80	54.20	40.83	33.62	44.52	31.68	27.00	38.85	26.47	25.22	36.51	24.86	27.25	34.89	24.55	31.88	33.69	24.88	38.72
0.90	94.67	77.59	73.74	83.70	68.84	73.30	77.82	68.31	89.84	75.30	72.80	111.50	73.42	79.65	142.50	72.05	88.40	180.87
0.95	173.43	161.60	171.05	161.90	160.08	186.07	156.51	173.30	234.46	154.92	191.27	280.86	154.08	212.51	326.15	153.55	236.82	376.60
1.00	370.67	370.64	369.49	370.23	370.92	370.66	369.25	369.87	370.33	370.04	370.42	370.21	370.58	370.57	369.59	369.76	370.51	370.91
	(273.46)	(348.29)	(371.74)	(275.23)	(346.77)	(360.79)	(284.39)	(350.83)	(360.39)	(293.61)	(355.98)	(363.30)	(304.43)	(355.84)	(363.17)	(309.70)	(358.66)	(366.99)
1.05	139.09	121.31	120.27	147.57	135.55	143.51	149.81	144.66	161.96	149.64	150.11	168.46	148.33	156.14	170.87	147.37	159.89	170.44
1.10	74.29	55.03	48.75	76.43	59.50	58.22	76.19	62.19	67.08	74.87	63.88	72.70	73.39	66.02	75.53	71.94	67.77	77.02
1.20	43.08	28.94	21.31	40.63	27.38	22.41	38.44	25.62	23.14	37.07	24.74	23.72	35.85	24.35	25.56	34.79	24.17	25.00
1.30	33.05	21.76	15.00	29.56	19.10	14.33	26.73	16.60	13.43	25.27	15.31	13.06	24.16	14.58	12.97	23.27	14.11	12.92
1.40	27.83	18.27	12.26	24.08	15.42	11.10	21.04	12.78	9.78	19.56	11.45	9.11	18.52	10.67	8.71	17.73	10.11	8.47
1.50	24.56	16.13	10.68	20.79	13.28	9.36	17.71	10.69	7.91	16.24	9.38	7.13	15.23	8.59	6.67	14.50	8.03	6.36
1.60	22.29	14.65	9.62	18.56	11.85	8.25	15.53	9.35	6.79	14.08	8.07	6.01	13.13	7.30	5.52	12.42	6.76	5.19
1.70	20.59	13.55	8.84	16.95	10.83	7.47	13.97	8.42	6.03	12.56	7.19	5.26	11.62	6.44	4.77	10.95	5.92	4.43
1.80	19.26	12.69	8.24	15.72	10.05	6.89	12.81	7.72	5.49	11.41	6.54	4.73	10.51	5.82	4.24	9.87	5.31	3.89
1.90	18.20	12.00	7.77	14.73	9.43	6.43	11.90	7.18	5.06	10.53	6.04	4.32	9.65	5.35	3.85	9.02	4.85	3.51
2.00	17.31	11.43	7.39	13.94	8.93	6.06	11.17	6.75	4.73	9.83	5.65	4.01	8.96	4.97	3.55	8.35	4.48	3.21
EARL	81.27	66.29	58.98	74.65	61.64	58.35	69.79	58.90	60.85	67.51	58.61	64.41	65.86	59.43	69.09	64.54	60.75	74.66

${}^{\ast }$SDRL ${}_0$ values in the parenthesis.

Table A3.

ARL and EARL values of the $S^2$-TEWMA control chart using asymptotic control limits at n = 5 and ${\mathrm{A}\mathrm{R}\mathrm{L}}_0\approx 370$.

	$\lambda =0.05$	0.10	0.20	0.30	0.40	0.50
τ	L = 2.14537	2.0200	2.3320	2.5156	2.6415	2.7310
0.50	33.68	18.09	10.38	7.63	6.28	5.65
0.60	35.37	20.37	12.05	9.28	8.27	8.41
0.70	36.45	24.20	15.44	13.50	14.15	17.00
0.80	38.71	32.69	26.16	28.94	37.06	51.14
0.90	56.69	74.41	87.55	114.90	155.88	214.61
0.95	124.91	179.00	226.24	277.58	334.77	400.90
1.00	369.17	370.41	370.57	370.21	370.50	370.37
	(788.14)	(378.90)	(361.60)	(367.79)	(369.08)	(370.67)
1.05	59.99	128.47	159.59	171.05	174.21	172.89
1.10	16.65	50.70	65.37	73.09	78.06	80.44
1.20	4.88	19.50	22.28	23.83	24.96	25.99
1.30	2.79	12.95	13.05	12.82	12.80	13.02
1.40	1.99	10.40	9.73	8.93	8.54	8.35
1.50	1.61	9.04	8.05	7.05	6.49	6.15
1.60	1.41	8.16	7.06	6.00	5.37	4.97
1.70	1.29	7.53	6.40	5.33	4.67	4.24
1.80	1.21	7.06	5.91	4.84	4.17	3.71
1.90	1.16	6.69	5.54	4.48	3.82	3.35
2.00	1.12	6.40	5.24	4.20	3.54	3.08
EARL	47.55	58.81	60.76	64.99	71.35	80.19

${}^{\ast }$SDRL ${}_0$ values in the parenthesis.

Disclosure statement

No potential conflict of interest was reported by the author(s).