Monitoring process mean and dispersion with one double generally weighted moving average control chart

Abstract

Control charts are widely known quality tools used to detect and control industrial process deviations in Statistical Process Control. In the current paper, we propose a new single memory-type control chart, called the maximum double generally weighted moving average chart (referred as Max-DGWMA), that simultaneously detects shifts in the process mean and/or process dispersion. The run length performance of the proposed Max-DGWMA chart is compared with that of the Max-EWMA, Max-DEWMA, Max-GWMA and SS-DGWMA charts, using time-varying control limits, through Monte–Carlo simulations. The comparisons reveal that the proposed chart is more efficient than the Max-EWMA, Max-DEWMA and Max-GWMA charts, while it is comparable with the SS-DGWMA chart. An automotive industry application is presented in order to implement the Max-DGWMA chart. The goal is to establish statistical control of the manufacturing process of the automobile engine piston rings. The source of the out-of-control signals is interpreted and the efficiency of the proposed chart in detecting shifts faster is evident.

1. Introduction

Control charts are important tools of Statistical Process Control (SPC) and efficient in detecting the presence of special causes of variation in the manufacturing processes. They are classified into Shewhart-type and memory-type control charts. Shewhart-type [20] charts are based on the last observation. Consequently, they are less sensitive to small and moderate shifts in the process parameters. However, the memory-type control charts, such as the cumulative sum (CUSUM) [21] and exponentially weighted moving average (EWMA) [22] charts consider both current and past information. Additionally, concerning the latter case, Zhang and Chen [29] developed the double EWMA (DEWMA) chart and, Sheu and Lin [26] proposed the generally weighted moving average (GWMA) chart, which is an expansion of the EWMA chart. Moreover, Sheu and Hsieh [23] extended the GWMA chart to the double GWMA (DGWMA) chart and they concluded that the DGWMA chart with the time-varying control limits is more effective than the GWMA and DEWMA charts in detecting small shifts in the process location. Additionally, we refer to Chiu and Sheu [8], Lu [18], Karakani et al. [16], Mabude et al. [19] and Alevizakos et al. [1,2], to name but a few.

Due to the rapid progress in the production and manufacturing processes, there are cases where we may not easily recognize whether a shift would arise either in the process mean or dispersion or both. For that reason, control charts for simultaneously monitoring shifts in the process mean and variability have been introduced. For instance, Chen and Cheng [4] developed the Max chart and Xie [28] presented several EWMA-type charts, such as the Max-EWMA, sum of squares EWMA (SS-EWMA), EWMA-Max, and EWMA semicircle (EWMA-SC) charts. Chen et al. [5,6] extended the work of Xie [28] regarding the Max-EWMA and EWMA-SC charts, Khoo et al. [17] proposed the Max-DEWMA chart, Teh et al. [27] introduced the SS-DEWMA chart, Sheu et al. [25] extended the Max-EWMA chart to a Max-GWMA chart Huang [14] extended the SS-EWMA chart to a SS-GWMA chart,, and Huang et al. [15] extended the SS-GWMA chart to a SS-DGWMA chart. Additional research can be found in the works of Costa and Rahim [9,10], Cheng and Thaga [7], Sheu et al. [24], Haq [12], Haq and Razzaq [13] and Chatterjee et al. [3].

This article extends the Max-GWMA chart to a maximum DGWMA chart, namely the Max-DGWMA control chart. A performance comparison study is conducted, using the time-varying control limits, with the Max-EWMA, Max-DEWMA, Max-GWMA and SS-DGWMA charts. The control charts are compared, via Monte–Carlo simulations, using both the average run length (ARL) and the standard deviation of the run length (SDRL) measures. Due to the wide range of the considered shifts in the process mean and dispersion, the relative mean index (RMI) [11] is provided as well. The comparisons indicate that the proposed chart is sensitive in detecting small and moderate shifts in the process mean and dispersion concurrently.

The rest of this paper is organized as follows. In Section 2, we briefly review the Max-GWMA chart. In Section 3, we introduce the Max-DGWMA chart. A performance study is conducted in Section 4 to evaluate the efficiency of the proposed chart and compare its performance with that of the aforementioned competing control charts. In Section 5, a practical application is presented, whereas concluding remarks are provided in Section 6. Finally, some technical details are given in the Appendix section.

2. A brief review of the Max-GWMA control chart

Sheu et al. [25] extended the Max-EWMA chart to a single Max-GWMA chart. They showed that it is more effective compared with the Max-EWMA chart in monitoring shifts in both the process mean and variance simultaneously.

Let $X_{i1},X_{i2},\dots ,X_{i{n}_i}$, $i=1,2,\dots$ be a sample of $n_i$ independent normal, $N({\mu }_0+\delta {\sigma }_0,{\rho }^2{\sigma }_0^2)$, random variables, where ${\mu }_0$ and ${\sigma }_0$ are the in-control (IC) values of the process mean and standard deviation, respectively, and i is the sample number. The process is declared as IC if $\delta =0$ and $\rho =1$; otherwise, the process is out-of-control (OOC) and, then $\delta \ne 0$ and/or $\rho \ne 1$. We are interested in detecting a shift in the process mean and/or process variance, from the IC ${\mu }_0$ and ${\sigma }_0^2$ values to the OOC ${\mu }_1={\mu }_0+\delta {\sigma }_0$ and ${\sigma }_1^2={\rho }^2{\sigma }_0^2$ values.

In order to design the Max-GWMA chart, the following statistics are defined as

$$\begin{array}{rl}{U}_i& =\frac{{\overline{X}}_i-{\mu }_0}{{\sigma }_0/\sqrt{n_i}},\end{array}$$ (1)

$$\begin{array}{rl}\mathrm{and}{V}_i& ={\mathrm{\Phi }}^{-1}\left\{H[\frac{(n_i-1)S_i^2}{{\sigma }_0^2};n_i-1]\right\},\end{array}$$ (2)

where $\overline{X_i}=\frac{1}{n_i}\sum _{j=1}^{n_i}{X}_{ij}$, $i=1,2,\dots$ is the sample mean, $S_i^2=\frac{\sum _{j=1}^{n_i}(X_{ij}-{\overline{X}}_i{)}^2}{n_i-1}$, $i=1,2,\dots$ is the sample variance, ${\mathrm{\Phi }}^{-1}(.)$ represents the inverse standard normal distribution function and $H(w;v)$ denotes the ${\chi }_v^2$ distribution function. When the process is IC, both $U_i$ and $V_i$ are independent following a standard normal distribution.

According to the GWMA control chart scheme, let $N_1$ be the number of samples until the first occurrence of an event since its previous occurrence. Therefore,

$$\sum _{m=1}^{\mathrm{\infty }}P(N_1=m)=P(N_1=1)+P(N_1=2)+\cdots +P(N_1=i)+P(N_1>i)=1,$$ (3)

where the probability $P(N_1=1)$ is the weight of the most recent sample, the probability $P(N_1=2)$ is the weight of the previous sample, the probability $P(N_1=i)$ is the weight of most out-of-data sample and, consequently, the probability $P(N_1>i)$ is weighted with the target value of the process.

In order to construct the Max-GWMA chart for monitoring both the process mean and variability concurrently, two GWMA statistics are defined as follows

$$\begin{array}{rl}{G}_{i1}& =P(N_1=1)U_i+P(N_1=2)U_{i-1}+\cdots +P(N_1=i)U_1+P(N_1>i)G_{01}\end{array}$$ (4)

$$\begin{array}{rl}{G}_{i2}& =P(N_1=1)V_i+P(N_1=2)V_{i-1}+\cdots +P(N_1=i)V_1+P(N_1>i)G_{02}\end{array}$$ (5)

where $i=1,2,\dots$ and, $G_{01}=G_{02}=0$ are the starting values. Because $U_i$ and $V_i$ are independent statistics, the random variables $G_{i1}$ and $G_{i2}$ are also independent. Furthermore, when $\delta =0$, $\rho =1$ and $G_{01}=G_{02}=0$, then both $G_{i1}\sim N(0,Q_{i1})$ and $G_{i2}\sim N(0,Q_{i1})$, where $Q_{i1}=\sum _{j=1}^i[P(N_1=j){]}^2$. For easier computations, Sheu and Lin [26] considered that

$$P(N_1=i)=P(N_1>i-1)-P(N_1>i)=q_1^{(i-1{)}^{\alpha }}-q_1^{i^{\alpha }},$$ (6)

where $P(N_1>i)=q_1^{i^{\alpha }}$, $i=1,2,\dots$, $q_1\in [0,1)$ is the design parameter and $\alpha >0$ is the adjustment parameter. Both the design and the adjustment parameters are determined by the practitioner.

Therefore, the Max-GWMA statistic is defined as

$$M{G}_i=max\{|G_{i1}|,|G_{i2}|\},\mathrm{for}i=1,2,\dots .$$ (7)

Because the $M{G}_i$ statistic is non-negative, the Max-GWMA chart needs only an upper time-varying control limit $(UC{L}_{MG})$ [25], which is given by

$$UC{L}_{MG}=(1.12838+0.60281L_1)\sqrt{Q_{i1}},$$ (8)

where $i=1,2,\dots$ and $L_1>0$ is the control chart multiplier when the process is IC. The process is declared to be OOC when the $M{G}_i$ statistic exceeds the $UC{L}_{MG}$. It is to be noted that the EWMA chart is a special case of the GWMA chart, when $\alpha =1$ and $q_1=1-\lambda$ [22]. Similarly, the Max-GWMA chart reduces to the Max-EWMA chart, when $\alpha =1$ and $q_1=1-\lambda$, as well.

3. The proposed Max-DGWMA chart

Taking into consideration the GWMA chart scheme described above, let $N_2$ be the number of samples until the first occurrence of an event since its previous occurrence, among the sequence of independent samples. From the two GWMA statistics $G_{i1}$ and $G_{i2}$ given in Equations (4) and (5), respectively, two corresponding DGWMA statistics can be computed as follows:

$$\begin{array}{rl}{G}_{i3}& =P(N_2=1)G_{i1}+P(N_2=2)G_{(i-1)1}+\cdots +P(N_2=i)G_{11}+P(N_2>i)G_{03}\end{array}$$ (9)

$$\begin{array}{rl}{G}_{i4}& =P(N_2=1)G_{i2}+P(N_2=2)G_{(i-1)2}+\cdots +P(N_2=i)G_{12}+P(N_2>i)G_{04}\end{array}$$ (10)

where $i=1,2,\dots$ and $G_{03}=G_{04}=0$ are the starting values. It is to be noted that the statistics $G_{i1}$ and $G_{i2}$ have weight sequence $\{P(N_1=i)\}$, while the statistics $G_{i3}$ and $G_{i4}$ have weight sequence $\{P(N_2=i)\}$. For easier computations, we consider

$$P(N_2=i)=q_2^{(i-1{)}^{\beta }}-q_2^{i^{\beta }},$$ (11)

where $P(N_2>i)=q_2^{i^{\beta }}$, $i=1,2,\dots$, $q_2\in [0,1)$ is the design parameter and $\beta >0$ is the adjustment parameter. Both the design and the adjustment parameters are also determined by the practitioner. Moreover, as shown in the Appendix, the $G_{i3}$ and $G_{i4}$ statistics, can be rewritten, respectively, as

$$\begin{array}{rl}{G}_{i3}& =\sum _{j=1}^i{w}_j{U}_j,\end{array}$$ (12)

$$\begin{array}{rl}\mathrm{and}{G}_{i4}& =\sum _{j=1}^i{w}_j{V}_j,\end{array}$$ (13)

where $i=1,2,\dots$ and $w_j=\sum _{j_1=j}^i(q_2^{(i-j_1{)}^{\beta }}-q_2^{(i-j_1+1{)}^{\beta }})(q_1^{(j_1-j{)}^{\alpha }}-q_1^{(j_1-j+1{)}^{\alpha }}).$ Similarly, $G_{i3}$ and $G_{i4}$ are independent, since $G_{i1}$ and $G_{i2}$ are independent. Moreover, when the process is IC and $G_{03}=G_{04}=0$, then $G_{i3}\sim N(0,Q_{i2})$ and $G_{i4}\sim N(0,Q_{i2})$, where $Q_{i2}=\sum _{j_1=j}^i{w}_j^2$. The Max-DGWMA statistic is defined as

$$MD{G}_i=max\{|G_{i3}|,|G_{i4}|\},\mathrm{for}i=1,2,\dots .$$ (14)

If the process mean and/or the variance have shifted from their respective target values, the $MD{G}_i$ statistic will be large; otherwise, it will be small. Because the $MD{G}_i$ statistic is non-negative, the proposed Max-DGWMA chart requires only an upper time-varying control limit $(UC{L}_{MDG})$, which is given by

$$\begin{array}{rl}UC{L}_{MDG}& =E(MD{G}_i)+L_2\sqrt{Var(MD{G}_i)}\\ & =(1.12838+0.60281L_2)\sqrt{Q_{i2}}\end{array}$$ (15)

where $i=1,2,\dots$, $E(MD{G}_i)$ and $Var(MD{G}_i)$ are the mean and the variance of the $MD{G}_i$ statistic, respectively, when the process is IC, and $L_2>0$ is the control chart multiplier. The derivation of the $UC{L}_{MDG}$ is provided also in the Appendix. The process is declared to be OOC when the $MD{G}_i$ statistic exceeds the $UC{L}_{MDG}$.

Moreover, hereafter, we take the simple approach to set the design parameter ( $q_1=q_2=q$) and the adjustment parameter ( $\alpha =\beta$) the same for $G_{i1}$, $G_{i2}$, $G_{i3}$ and $G_{i4}$ in Equations (4), (5), (9) and (10), respectively. It should be noted that, the DEWMA chart is a special case of the DGWMA chart, when $\alpha =1$ and $q=1-\lambda$. Consequently, the Max-DGWMA chart reduces to the Max-DEWMA chart when $\alpha =1$ and $q=1-\lambda$ as well.

The design procedure of the proposed Max-DGWMA chart is equivalent to that of the Max-GWMA chart given in Sheu et al. [25]. The main steps involved in constructing the Max-DGWMA chart are briefly described as:

1. In case of unknown parameters, mean and standard deviation should be estimated first. If the mean is unknown, then use $\overline{\overline{X}}=\sum _{i=1}^m\frac{{\overline{X}}_i}{m}$ as its estimate, where $\overline{\overline{X}}$ is the grand average and m is the total number of the samples. If the standard deviation is unknown, then use $\frac{\overline{S}}{c_4}$ as its estimate, where $\overline{S}=\frac{\sum _{i=1}^m{S}_i}{m}$ is the average standard deviation and the $c_4$ constant depends only on the sample size (n) [20].

2. Choose the desired ( $q,\alpha ,L_2$) combinations based on the fixed sample size (n), and the IC ARL ( $AR{L}_0$) values. Particularly, Table 1 presents the ( $q,\alpha ,L_2$) combinations when $n=$ 5, $q\in \{$0.50, 0.60, 0.70, 0.80, 0.90, 0.95 $\}$, $\alpha \in \{0.05,0.10,\dots ,1.90,2.00,2.50,3.00\}$ and $AR{L}_0\approx$ 185, 250 and 370. Note that the $L_2$ values in Table 1 are selected via Monte–Carlo simulations using the time-varying upper control limit given in Equation (15) for various $(q,\alpha )$ combinations, when $AR{L}_0\approx$ 185, 250 and 370 and n = 5.

3. Compute the time-varying $UC{L}_{MDG}$ of the Max-DGWMA chart using Equation (15) for $i=1,\dots ,m$. Note that the computation of the $Q_{i2}$ for $i=1,\dots ,m$, is required, before the calculation of the $UC{L}_{MDG}$.

4. Compute the $U_i$ and $V_i$ statistics for $i=1,\dots ,m$, using Equations (1) and (2), respectively.

5. Compute the $G_{i1}$, $G_{i2}$, $G_{i3}$ and $G_{i4}$ statistics, for $i=1,\dots ,m$, using Equations (4), (5), (9) and (10), respectively, and $G_{01}=G_{02}=G_{03}=G_{04}=0$ as the starting values.

6. Compute the $MD{G}_i$ statistic, for $i=1,\dots ,m$, using Equation (14).

7. Plot the $MD{G}_i$ statistic versus i (where $i=1,\dots ,m$) on the chart and design $UC{L}_{MDG}$ as the upper control limit. Draw a dot against i when $MD{G}_i\le UC{L}_{MDG}$. On the other hand, when $MD{G}_i>UC{L}_{MDG}$, denote the plotted points accordingly with the symbols presented in Table 2. Particularly, when $MD{G}_i>UC{L}_{MDG}$, check both $|G_{i3}|$ and $|G_{i4}|$ against $UC{L}_{MDG}$.

   1. If only $|G_{i3}|>UC{L}_{MDG}$, then draw “ $m+$” against i when $U_i>0$ to indicate that only a rise in the process mean has occurred, whereas draw “ $m-$” versus i when $U_i<0$ to point out that only the process mean has decreased.
   2. If only $|G_{i4}|>UC{L}_{MDG}$, then draw “ $v+$” against i when $V_i>0$ to show that only the process variability has increased, while draw “ $v-$” against i when $V_i<0$ to point out a decrease only in the process dispersion.
   3. If $|G_{i3}|>UC{L}_{MDG}$ and $|G_{i4}|>UC{L}_{MDG}$, then draw: “++” versus i when $U_i>0$ and $V_i>0$ to suggest that both process mean and dispersion have concurrently rised; “+−” versus i when $U_i>0$ and $V_i<0$ to indicate an increase in the process mean and a drop in the process dispersion simultaneously; “− +” versus i when $U_i<0$ and $V_i>0$ to show a decrease in the process mean and a rise in the process variability concurrently, and; “−−” versus i when $U_i<0$ and $V_i<0$ to suggest a simultaneous reduction of the process mean and variability.

8. Examine and interpret the causes of all the OOC points.

Table 1.

$(q,\alpha ,L_2)$ parameter combinations for the Max-DGWMA control chart using time-varying control limits with sample size $n=$5 at $AR{L}_0\approx$ 185, 250, 370.

$AR{L}_0$	185	250	370	185	250	370	185	250	370	185	250	370	185	250	370	185	250	370
q	0.50			0.60			0.70			0.80			0.90			0.95
α	$L_2$
0.05	3.094	3.238	3.437	3.094	3.238	3.436	3.092	3.235	3.435	3.091	3.237	3.436	3.091	3.237	3.437	3.091	3.237	3.437
0.10	3.089	3.242	3.434	3.086	3.243	3.434	3.084	3.243	3.432	3.084	3.241	3.433	3.085	3.238	3.432	3.086	3.237	3.430
0.20	3.082	3.235	3.427	3.074	3.230	3.421	3.062	3.221	3.410	3.039	3.200	3.353	3.005	3.161	3.353	2.983	3.135	3.327
0.30	3.058	3.215	3.412	3.026	3.190	3.386	2.957	3.127	3.328	2.846	3.001	3.211	2.634	2.776	2.954	2.457	2.571	2.727
0.40	3.028	3.190	3.391	2.965	3.125	3.335	2.827	2.995	3.213	2.562	2.738	2.962	2.114	2.249	2.428	1.729	1.819	1.949
0.50	3.000	3.166	3.372	2.905	3.074	3.285	2.708	2.890	3.116	2.372	2.553	2.785	1.790	1.940	2.145	1.334	1.442	1.587
0.60	2.977	3.138	3.353	2.860	3.027	3.243	2.645	2.825	3.048	2.281	2.465	2.706	1.667	1.830	2.052	1.192	1.317	1.498
0.70	2.956	3.123	3.329	2.828	2.999	3.210	2.618	2.795	3.012	2.254	2.439	2.681	1.663	1.841	2.070	1.180	1.330	1.527
0.80	2.933	3.109	3.305	2.808	2.981	3.191	2.605	2.781	2.996	2.270	2.458	2.695	1.721	1.901	2.145	1.241	1.407	1.629
0.90	2.926	3.095	3.295	2.800	2.969	3.182	2.605	2.781	3.000	2.299	2.493	2.720	1.799	1.996	2.233	1.342	1.524	1.760
1.00	2.922	3.090	3.285	2.802	2.965	3.183	2.619	2.799	3.016	2.348	2.528	2.7666	1.894	2.086	2.3262	1.464	1.655	1.898
1.10	2.917	3.084	3.285	2.808	2.975	3.187	2.641	2.810	3.041	2.402	2.590	2.813	1.993	2.180	2.423	1.594	1.787	2.0295
1.20	2.921	3.085	3.289	2.820	2.986	3.198	2.672	2.843	3.066	2.460	2.638	2.861	2.085	2.275	2.511	1.715	1.912	2.155
1.30	2.925	3.093	3.296	2.835	2.999	3.210	2.073	2.873	3.091	2.510	2.690	2.911	2.172	2.358	2.596	1.835	2.023	2.277
1.40	2.937	3.097	3.303	2.857	3.020	3.226	2.740	2.901	3.123	2.565	2.740	2.963	2.260	2.443	2.678	1.948	2.140	2.376
1.50	2.947	3.106	3.306	2.875	3.038	3.244	2.768	2.933	3.147	2.611	2.793	3.008	2.340	2.527	2.755	2.054	2.238	2.477
1.60	2.958	3.120	3.318	2.895	3.055	3.260	2.797	2.964	3.175	2.661	2.835	3.050	2.410	2.592	2.820	2.137	2.331	2.568
1.70	2.964	3.127	3.327	2.908	3.070	3.275	2.825	2.993	3.201	2.706	2.871	3.091	2.473	2.651	2.878	2.227	2.415	2.653
1.80	2.976	3.135	3.336	2.925	3.082	3.288	2.850	3.018	3.224	2.744	2.912	3.124	2.532	2.712	2.936	2.310	2.498	2.726
1.90	2.985	3.144	3.343	2.938	3.100	3.303	2.875	3.042	3.245	2.780	2.947	3.154	2.585	2.763	2.985	2.380	2.560	2.795
2.00	2.992	3.151	3.348	2.954	3.112	3.315	2.898	3.060	3.264	2.813	2.976	3.185	2.632	2.815	3.031	2.440	2.620	2.854
2.50	3.013	3.172	3.367	3.000	3.157	3.356	2.975	3.131	3.336	2.923	3.090	3.290	2.818	2.985	3.191	2.690	2.856	3.075
3.00	3.020	3.180	3.375	3.020	3.178	3.373	3.024	3.180	3.373	3.000	3.160	3.359	2.925	3.088	3.292	2.842	3.009	3.215

Table 2.

Symbols to denote the source and direction of an OOC signal for the Max-DGWMA chart.

			$|G_{i4}|>UC{L}_{MDG}$
		$|G_{i4}|<UC{L}_{MDG}$	$V_i>0$	$V_i<0$
$|G_{i3}|<UC{L}_{MDG}$			$v+$	$v-$
$|G_{i3}|>UC{L}_{MDG}$	$U_i>0$	$m+$	++	+−
	$U_i<0$	$m-$	− +	−−

4. Performance study

4.1. Evaluation performance of the proposed Max-DGWMA chart

The statistical performance of a control chart is commonly measured using the ARL and the SDRL. The ARL is the average number of samples that must be plotted on a chart until an OOC signal is raised [20]. When the process mean and process variability are IC, a large value of $AR{L}_0$ is suggested to avoid false alarms. Nevertheless, when the process is OOC, namely the mean shifts from ${\mu }_0$ to ${\mu }_1={\mu }_0+\delta {\sigma }_0$ ( $\delta \ne 0$) and/or the standard deviation shifts from ${\sigma }_0$ to ${\sigma }_1=\rho {\sigma }_0$ ( $\rho \ne 1$) a small OOC ARL ( $AR{L}_1$) value is preferable so as to rapidly detect the shift combination. Furthermore, the SDRL measures the stability of the run length distribution, i.e. the smaller the SDRL, the better will be the ARL performance for a given control chart. A control chart with the smaller $AR{L}_1$ and OOC SDRL ( $SDR{L}_1$) compared with its counterparts, having the same $AR{L}_0$ value, is considered to be more efficient in detecting a shift in the process mean and variability [13,25]. In the present article, the Max-DGWMA control chart is evaluated through the ARL and SDRL performance measures.

A Monte–Carlo simulation algorithm is developed in R statistical programming language to calculate the run length distribution of the initial-state Max-DGWMA control chart with time-varying control limits. The algorithm is run 10,000 iterations to calculate the average and the standard deviation of those 10,000 run lengths. In order to study the performance of the Max-DGWMA control chart, we consider that the underlying process for the IC condition follows the Normal distribution with mean ${\mu }_0=0$ and standard deviation ${\sigma }_0=1$, while the OOC process is Normally distributed with mean ${\mu }_1={\mu }_0+\delta {\sigma }_0$ and standard deviation ${\sigma }_1=\rho {\sigma }_0$. The considered shifts in the process mean are $\delta \in \{$0.00, 0.10, 0.25, 0.50, 1.00, 1.50, 2.00, 2.50, 3.00 $\}$ and the shifts in the process dispersion are $\rho \in \{$0.25, 0.50, 0.75, 0.95, 1.00, 1.05, 1.25, 1.50, 1.75, 2.00, 2.50, 3.00 $\}$, where the shift combination $(\delta ,\rho )=(0.00,1.00)$ corresponds to the IC state. The control chart multiplier $L_2$ is chosen, via Monte–Carlo simulations, using the time-varying upper control limit (Equation (15)), to set the $AR{L}_0\approx$ 370, when the sample size n = 5, $q\in \{$0.70, 0.80, 0.90, 0.95 $\}$, and $\alpha \in \{$0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10 $\}$. Tables A1 to A4 in the Supplementary Material present the ARL and SDRL (given in the parenthesis) values of the Max-DGWMA chart with a time-varying control limit for various $(q,\alpha )$ combinations, n = 5 and $AR{L}_0\approx$ 370. The smallest $AR{L}_1$ value for each ( $\delta ,\rho$) shift combination and each $q\in \{$0.70, 0.80, 0.90, 0.95 $\}$ is indicated with bold print in these Tables.

From Tables A1 to A4 in the Supplementary Material, we observe that for a fixed value of α (q), the performance of the proposed chart improves as the value of q (α) increases (decreases) for both ARL and SDRL measures. For example, when $(\delta ,\rho )=(0.25,1.25)$, $\alpha =0.50$ and $q=$ 0.70, 0.80, 0.90 and 0.95, the $AR{L}_1(SDR{L}_1)$ values are 10.84(8.14), 9.41(7.45), 6.88(6.16) and 4.96(4.87), respectively. Additionally, when $(\delta ,\rho )=(0.10,0.95)$, q = 0.95 and $\alpha =$ 0.50, 0.60, 0.70, 0.80, 0.90, 1.00 and 1.10, the corresponding $AR{L}_1(SDR{L}_1)$ values are 35.39(41.17), 40.89(46.31), 47.90(50.25), 55.56(53.64), 63.11(57.46), 71.20(64.10) and 80.61(72.89). It should be mentioned that the Max-DGWMA chart with $q\in \{$0.70, 0.80, 0.90, 0.95 $\}$ and $\alpha \in [0.50,0.90]$ is more sensitive for small to moderate shifts and especially for $0.00\le \delta \le 1.00$ and $0.50\le \rho \le 1.50$. Furthermore, it is more efficient in detecting small upward shifts in the process variability regardless the amount of the shift in the process mean. Tables A1–A4 in the Supplementary Material, reveal that the Max-DGWMA chart with q = 0.95 has the smallest $AR{L}_1$ and $SDR{L}_1$ values. Particularly, the bold printed values of the Max-DGWMA(q = 0.95) in Table A4 in the Supplementary Material correspond to the smallest $AR{L}_1$ values for each shift in the process mean and dispersion. It is to be noted that, the parameter combination [25] that leads to the smallest $AR{L}_1$ value for each $(\delta ,\rho )$ shift combination is usually considered as the optimal parameter combination. For example, according to Table A4, the optimal parameter combination of the optimal Max-DGWMA chart at $(\delta ,\rho )=(0.10,1.50)$ is $(q,\alpha ,L_2)=(0.95,0.50,1.587)$. Table 4 also contains the optimal $(q,\alpha ,L_2)$ parameter combinations for the optimal Max-DGWMA chart, considering various $(\delta ,\rho )$ shift combinations at $AR{L}_0\approx 370$ and n = 5.

Table 4.

$(q,\alpha ,L_2)$ and $(q,\alpha ,L)$ combinations and the corresponding ARL results for the optimal Max-DGWMA and optimal SS-DGWMA charts, for various $(\delta ,\rho )$ shift combinations at $AR{L}_0\approx 370$ and n = 5.

		δ
ρ		0.00	0.10	0.25	0.50	1.00	1.50	2.00	2.50	3.00
0.25	Max-DGWMA	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)
		1.13	1.13	1.13	1.13	1.03	1.00	1.00	1.00	1.00
	SS-DGWMA	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		1.22	1.21	1.18	1.08	1.00	1.00	1.00	1.00	1.00
0.50	Max-DGWMA	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)
		2.47	2.47	2.46	2.22	1.27	1.00	1.00	1.00	1.00
	SS-DGWMA	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		2.67	2.62	2.42	1.88	1.15	1.00	1.00	1.00	1.00
0.75	Max-DGWMA	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)
		8.56	8.27	6.57	3.42	1.42	1.04	1.00	1.00	1.00
	SS-DGWMA	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		8.83	8.16	5.84	3.12	1.40	1.04	1.00	1.00	1.00
0.95	Max-DGWMA	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.50,1.587)
		87.94	35.39	10.57	3.85	1.52	1.08	1.00	1.00	1.00
	SS-DGWMA	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		87.72	34.17	10.34	3.81	1.52	1.08	1.01	1.00	1.00
1.00	Max-DGWMA		(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)
			38.48	10.53	3.85	1.53	1.09	1.01	1.00	1.00
	SS-DGWMA		(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
			38.32	10.34	3.82	1.54	1.10	1.01	1.00	1.00
1.05	Max-DGWMA	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.50,1.587)
		65.65	29.11	9.79	3.79	1.55	1.10	1.01	1.00	1.00
	SS-DGWMA	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		64.31	28.12	9.49	3.73	1.54	1.10	1.01	1.00	1.00
1.25	Max-DGWMA	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)
		6.60	6.25	4.96	3.11	1.55	1.14	1.02	1.00	1.00
	SS-DGWMA	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		6.37	5.99	4.69	2.95	1.52	1.13	1.02	1.00	1.00
1.50	Max-DGWMA	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.50,1.587)
		2.72	2.68	2.53	2.15	1.46	1.15	1.04	1.01	1.00
	SS-DGWMA	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.70,1.485)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		2.61	2.57	2.42	2.05	1.33	1.14	1.03	1.01	1.00
1.75	Max-DGWMA	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)
		1.79	1.79	1.75	1.64	1.34	1.14	1.05	1.01	1.00
	SS-DGWMA	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		1.73	1.72	1.69	1.58	1.30	1.13	1.04	1.01	1.00
2.00	Max-DGWMA	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.60,1.498)
		1.44	1.43	1.42	1.37	1.23	1.12	1.05	1.02	1.00
	SS-DGWMA	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		1.40	1.40	1.38	1.34	1.21	1.11	1.04	1.01	1.00
2.50	Max-DGWMA	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.60,1.498)
		1.16	1.17	1.16	1.15	1.11	1.07	1.04	1.02	1.01
	SS-DGWMA	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		1.15	1.15	1.15	1.14	1.10	1.07	1.03	1.02	1.01
3.00	Max-DGWMA	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.60,1.498)	(0.95,0.50,1.587)
		1.08	1.08	1.08	1.07	1.06	1.04	1.03	1.02	1.01
	SS-DGWMA	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.60,1.432)	(0.95,0.50,1.551)	(0.95,0.50,1.551)	(0.95,0.50,1.551)
		1.07	1.07	1.07	1.06	1.05	1.04	1.02	1.01	1.01

In addition, we study the effect of the α parameter and the sample size (n), on the performance of the Max-DGWMA chart with the time-varying control limits. The q parameter is fixed at 0.95, since for that case the proposed chart has smaller $AR{L}_1$ values. We randomly select a shift combination $(\delta ,\rho )=(0.10,1.25)$ and design the scatter plots of the ARL and SDRL measures against α, for $n\in \{$3, 5, 7 $\}$, q = 0.95 and $\alpha \in \{0.05,0.10,\dots ,1.90,2.00,2.50,3.00\}$, when $AR{L}_0\approx 370$. These scatter plots are presented in Figure 1. From Table A4 in the Supplementary Material and Figure 1, it is obvious that the Max-DGWMA chart with q = 0.95 and $\alpha \in \{0.50,0.60\}$ has the best overall run length performance and, both $AR{L}_1$ and $SDR{L}_1$ are affected from the sample size n, i.e. they decrease as n increments.

Figure 1.

$AR{L}_1$ and $SDR{L}_1$ values for the Max-DGWMA chart with q = 0.95 and shift combination $(\delta ,\rho )=(0.10,1.25)$ for various α and n at $AR{L}_0\approx 370$.

4.2. Performance comparison of control charts

In this section, we compare the performance of the proposed Max-DGWMA chart with that of the Max-EWMA, Max-DEWMA, Max-GWMA and SS-DGWMA charts for monitoring the process mean and/or dispersion, using the ARL, SDRL and RMI measures. Note that the time-varying control limits are taken into consideration for all the competing charts. In order to examine the performance of the considered control charts, it is preferable to have a similar $AR{L}_0$ value. The $AR{L}_0$ value of the competing control charts is pre-fixed approximately at 370 and the sample size n is equal to 5. The considered charts are briefly presented and compared individually with the Max-DGWMA chart.

The run length performance of the Max-EWMA and Max-DEWMA charts with $\lambda \in \{0.05,0.10,0.20,0.30\}$ is presented in Tables A5–A8 and A1–A4 in the Supplementary material, respectively, due to $q=1-\lambda$ and $\alpha =1.00$. Tables A5–A12 in the Supplementary material show the ARL and SDRL (in the parenthesis) values of the Max-GWMA and SS-DGWMA charts with $q\in \{$0.70, 0.80, 0.90, 0.95 $\}$, and $\alpha \in \{$0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10 $\}$. Furthermore, the control chart multipliers of the competing charts, are selected via Monte–Carlo simulations, using the corresponding time-varying control limits, when $AR{L}_0\approx$ 370, n = 5, and the aforementioned q, α and λ values. According to Tables A1–A12 in the Supplementary material, the smaller ARL and SDRL values correspond to $\lambda =0.05$ for the Max-EWMA and Max-DEWMA charts and q = 0.95 for the Max-GWMA and SS-DGWMA charts. Therefore, for comparison purposes, Table 3 presents the ARL and SDRL (in the parenthesis) values of the Max-GWMA, Max-DGWMA and SS-DGWMA charts with q = 0.95, and $\alpha \in \{$0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10 $\}$, and the Max-EWMA and Max-DEWMA charts with $\lambda =0.05$ (i.e. the columns that correspond to Max-GWMA(q = 0.95, $\alpha =1.00$) and Max-DGWMA(q = 0.95, $\alpha =1.00$)) for representative small to large shifts in the process mean and dispersion, such as $\delta \in \{$0.00, 0.10, 0.50, 1.00, 1.50, 2.00 $\}$ and $\rho \in \{$0.25, 0.50, 0.95, 1.00, 1.05, 1.25, 1.50 $\}$.

• Proposed Max-DGWMA chart versus Max-EWMA chart

Table 3.

ARL and SDRL (in the parenthesis) values for the Max-GWMA, Max-DGWMA and SS-DGWMA charts with q = 0.95 and $\alpha =0.50,\dots ,1.10$ at n = 5 and $AR{L}_0\approx 370$.

		Max-GWMA(q = 0.95)							Max-DGWMA(q = 0.95)							SS-DGWMA(q = 0.95)
		$\alpha =$0.50	0.60	0.70	0.80	0.90	1.00 ${}^{\mathrm{a}}$	1.10	$\alpha =$0.50	0.60	0.70	0.80	0.90	1.00 ${}^{\mathrm{b}}$	1.10	$\alpha =$0.50	0.60	0.70	0.80	0.90	1.00	1.10
δ	ρ	$L_1$=3.230	3.0795	2.943	2.850	2.792	2.770	2.759	$L_2=$1.587	1.498	1.527	1.629	1.760	1.898	2.030	$L=$1.551	1.432	1.485	1.641	1.838	2.055	2.275
0.00	0.25	2.13	1.95	1.84	1.77	1.73	1.71	1.71	1.16	1.13	1.14	1.17	1.22	1.28	1.34	1.26	1.22	1.24	1.29	1.35	1.43	1.52
		(0.72)	(0.63)	(0.58)	(0.55)	(0.54)	(0.54)	(0.54)	(0.37)	(0.34)	(0.35)	(0.38)	(0.42)	(0.46)	(0.50)	(0.44)	(0.42)	(0.43)	(0.46)	(0.49)	(0.53)	(0.67)
	0.50	5.78	4.99	4.50	4.22	4.07	4.02	4.03	2.55	2.47	2.54	2.72	2.97	3.28	3.60	2.76	2.67	2.76	2.99	3.28	3.62	4.02
		(2.43)	(2.08)	(1.87)	(1.75)	(1.70)	(1.67)	(1.67)	(1.23)	(1.21)	(1.27)	(1.38)	(1.51)	(1.67)	(1.82)	(1.30)	(1.28)	(1.35)	(1.47)	(1.63)	(1.79)	(1.95)
	0.95	196.19	182.09	180.66	191.38	207.10	229.28	252.04	87.94	104.00	120.55	139.53	159.95	180.60	202.25	87.72	103.65	120.02	139.26	157.98	179.27	201.25
		(162.80)	(158.57)	(164.26)	(180.91)	(199.78)	(225.33)	(251.59)	(123.94)	(134.26)	(140.82)	(152.16)	(169.50)	(186.22)	(205.92)	(124.31)	(133.62)	(141.32)	(152.51)	(166.73)	(184.73)	(203.29)
	1.00	370.37	370.10	370.88	370.41	370.10	370.29	370.80	370.02	370.36	370.22	370.35	370.76	370.32	369.81	370.05	370.88	370.58	370.44	370.53	370.92	370.80
		(401.27)	(407.91)	(405.85)	(395.03)	(388.62)	(384.83)	(384.43)	(768.45)	(639.22)	(547.60)	(479.65)	(447.30)	(415.93)	(401.43)	(783.92)	(637.00)	(546.54)	(488.60)	(445.35)	(417.95)	(401.90)
	1.05	135.18	129.25	130.64	137.13	146.75	157.41	166.12	65.65	81.59	97.28	113.12	128.55	144.76	157.13	64.31	80.11	94.81	110.93	125.70	141.30	153.46
		(122.79)	(123.37)	(128.33)	(137.52)	(149.46)	(160.74)	(168.73)	(105.31)	(120.88)	(130.61)	(138.25)	(147.56)	(161.09)	(169.10)	(106.79)	(121.26)	(128.60)	(135.64)	(145.27)	(157.80)	(164.93)
	1.25	15.60	14.25	13.59	13.46	13.74	14.21	14.68	6.60	7.20	8.32	9.77	11.39	12.92	14.27	6.37	6.98	8.10	9.55	11.18	12.74	14.10
		(11.88)	(11.23)	(11.02)	(11.13)	(11.43)	(11.81)	(12.32)	(7.14)	(8.21)	(9.41)	(10.60)	(11.61)	(12.29)	(12.84)	(7.06)	(8.09)	(9.33)	(10.49)	(11.63)	(12.37)	(12.84)
	1.50	5.37	4.97	4.73	4.66	4.68	4.79	4.93	2.72	2.77	3.00	3.38	3.82	4.31	4.78	2.61	2.68	2.91	3.27	3.71	4.18	4.68
		(3.86)	(3.63)	(3.51)	(3.53)	(3.61)	(3.74)	(3.86)	(2.29)	(2.52)	(2.83)	(3.28)	(3.69)	(4.08)	(4.41)	(2.23)	(2.44)	(2.80)	(3.22)	(3.65)	(4.03)	(4.39)
0.10	0.25	2.13	1.95	1.84	1.77	1.73	1.71	1.71	1.16	1.13	1.14	1.17	1.22	1.28	1.34	1.25	1.21	1.23	1.28	1.34	1.42	1.51
		(0.72)	(0.63)	(0.58)	(0.55)	(0.54)	(0.54)	(0.54)	(0.37)	(0.34)	(0.35)	(0.38)	(0.42)	(0.46)	(0.50)	(0.43)	(0.41)	(0.42)	(0.45)	(0.49)	(0.52)	(0.56)
	0.50	5.78	4.99	4.50	4.22	4.07	4.02	4.03	2.55	2.47	2.54	2.72	2.97	3.28	3.60	2.71	2.62	2.71	2.93	3.22	3.55	3.94
		(2.43)	(2.08)	(1.87)	(1.75)	(1.70)	(1.67)	(1.67)	(1.23)	(1.21)	(1.27)	(1.38)	(1.51)	(1.67)	(1.82)	(1.27)	(1.26)	(1.32)	(1.44)	(1.59)	(1.75)	(1.91)
	0.95	92.07	81.79	78.26	79.72	85.26	94.41	106.25	35.39	40.89	47.90	55.56	63.11	71.20	80.61	34.17	38.73	45.33	52.65	59.55	66.97	76.11
		(66.70)	(61.55)	(61.43)	(65.05)	(73.25)	(84.32)	(99.39)	(41.17)	(46.31)	(50.25)	(53.64)	(57.46)	(64.10)	(72.89)	(39.54)	(43.87)	(47.47)	(50.67)	(54.32)	(60.42)	(69.01)
	1.00	93.83	85.53	82.79	85.49	91.09	100.52	111.60	38.48	45.27	53.46	62.33	71.02	80.54	89.99	38.32	44.62	53.33	62.28	70.75	79.84	90.45
		(73.03)	(69.02)	(69.70)	(74.44)	(82.41)	(94.67)	(107.44)	(49.20)	(56.41)	(61.61)	(66.05)	(70.97)	(78.51)	(86.20)	(50.06)	(56.35)	(61.63)	(65.91)	(70.66)	(77.70)	(87.32)
	1.05	70.64	64.55	63.09	64.85	68.71	74.69	81.04	29.11	34.55	41.46	48.91	55.82	62.68	69.38	28.12	32.85	39.66	46.69	53.34	59.38	65.80
		(57.45)	(54.47)	(54.80)	(57.63)	(62.51)	(70.50)	(78.13)	(38.63)	(45.26)	(49.84)	(53.30)	(56.81)	(61.83)	(67.22)	(37.74)	(42.82)	(47.63)	(50.94)	(54.75)	(58.40)	(63.72)
	1.25	14.71	13.51	12.86	12.77	12.99	13.47	13.90	6.25	6.79	7.79	9.20	10.76	12.20	13.49	5.99	6.52	7.54	8.91	10.35	11.72	13.02
		(11.09)	(10.48)	(10.31)	(10.47)	(10.74)	(11.13)	(11.48)	(6.66)	(7.67)	(8.75)	(9.92)	(10.96)	(11.59)	(12.13)	(6.50)	(7.47)	(8.61)	(9.77)	(10.65)	(11.32)	(11.83)
	1.50	5.28	4.90	4.68	4.60	4.62	4.72	4.84	2.68	2.73	2.96	3.32	3.75	4.23	4.71	2.57	2.63	2.86	3.21	3.63	4.10	4.58
		(3.79)	(3.56)	(3.45)	(3.47)	(3.56)	(3.67)	(3.79)	(2.25)	(2.45)	(2.77)	(3.19)	(3.60)	(3.99)	(4.33)	(2.19)	(2.41)	(2.74)	(3.12)	(3.54)	(3.94)	(4.30)
0.50	0.25	2.13	1.95	1.84	1.77	1.73	1.71	1.71	1.16	1.13	1.14	1.17	1.22	1.28	1.34	1.10	1.08	1.09	1.12	1.17	1.23	1.29
		(0.72)	(0.63)	(1.68)	(0.55)	(0.54)	(0.54)	(0.54)	(0.37)	(0.34)	(0.35)	(0.38)	(0.42)	(0.46)	(0.50)	(0.31)	(0.27)	(0.29)	(0.33)	(0.37)	(0.42)	(0.46)
	0.50	5.61	4.81	4.32	4.02	3.87	3.83	3.83	2.31	2.22	2.28	2.45	2.69	2.98	3.29	1.96	1.88	1.93	2.05	2.21	2.42	2.64
		(2.25)	(1.89)	(1.68)	(1.56)	(1.50)	(1.48)	(1.48)	(1.02)	(1.00)	(1.04)	(1.13)	(1.26)	(1.41)	(1.56)	(0.84)	(0.81)	(0.84)	(0.90)	(0.98)	(1.09)	(1.20)
	0.95	8.75	7.69	7.07	6.77	6.65	6.69	6.77	3.85	3.88	4.15	4.61	5.19	5.81	6.41	3.81	3.83	4.15	4.65	5.25	5.90	6.54
		(4.86)	(4.32)	(4.03)	(3.92)	(3.90)	(3.94)	(4.00)	(2.72)	(2.88)	(3.15)	(3.48)	(3.83)	(4.15)	(4.41)	(2.75)	(2.88)	(3.19)	(3.54)	(3.92)	(4.26)	(4.53)
	1.00	8.52	7.53	6.95	6.69	6.60	6.64	6.74	3.85	3.88	4.18	4.63	5.22	5.84	6.45	3.82	3.86	4.17	4.68	5.27	5.93	6.61
		(4.97)	(4.44)	(4.16)	(4.05)	(4.05)	(4.10)	(4.17)	(2.84)	(3.00)	(3.30)	(3.63)	(4.02)	(4.35)	(4.62)	(2.88)	(3.05)	(3.35)	(3.74)	(4.11)	(4.47)	(4.77)
	1.05	8.21	7.34	6.83	6.57	6.50	6.57	6.68	3.79	3.82	4.12	4.59	5.17	5.80	6.42	3.73	3.77	4.09	4.59	5.19	5.84	6.49
		(5.00)	(4.52)	(4.27)	(4.17)	(4.16)	(4.25)	(4.33)	(2.90)	(3.06)	(3.38)	(3.74)	(4.13)	(4.50)	(4.79)	(2.90)	(3.08)	(3.40)	(3.79)	(4.22)	(4.58)	(4.89)
	1.25	6.36	5.82	5.50	5.36	5.34	5.45	5.57	3.11	3.14	3.39	3.81	4.33	4.87	5.41	2.95	3.00	3.23	3.61	4.06	4.53	5.01
		(4.26)	(3.96)	(3.82)	(3.80)	(3.85)	(3.95)	(4.04)	(2.53)	(2.71)	(3.03)	(3.49)	(3.91)	(4.30)	(4.65)	(2.38)	(2.55)	(2.86)	(3.25)	(3.62)	(3.95)	(4.25)
	1.50	3.96	3.68	3.51	3.44	3.44	3.50	3.58	2.15	2.16	2.29	2.51	2.80	3.11	3.43	2.05	2.06	2.18	2.37	2.61	2.89	3.18
		(2.70)	(2.53)	(2.45)	(2.46)	(2.50)	(2.58)	(2.67)	(1.60)	(1.70)	(1.91)	(2.20)	(2.52)	(2.81)	(3.08)	(1.51)	(1.59)	(1.79)	(2.05)	(2.30)	(2.58)	(2.83)
1.00	0.25	2.04	1.86	1.76	1.71	1.68	1.67	1.66	1.04	1.03	1.03	1.06	1.09	1.15	1.23	1.00	1.00	1.00	1.00	1.00	1.01	1.01
		(0.63)	(0.53)	(0.48)	(0.47)	(0.47)	(0.48)	(0.48)	(0.20)	(0.16)	(0.18)	(0.23)	(0.29)	(0.36)	(0.42)	(0.02)	(0.02)	(0.02)	(0.03)	(0.06)	(0.08)	(0.11)
	0.50	2.73	2.43	2.23	2.12	2.06	2.03	2.03	1.31	1.27	1.28	1.33	1.41	1.50	1.60	1.18	1.15	1.17	1.20	1.25	1.31	1.38
		(0.91)	(0.79)	(0.70)	(0.66)	(0.63)	(0.62)	(0.63)	(0.48)	(0.46)	(0.47)	(0.50)	(0.53)	(0.58)	(0.64)	(0.39)	(0.36)	(0.38)	(0.41)	(0.45)	(0.49)	(0.54)
	0.95	2.73	2.52	2.36	2.27	2.22	2.21	2.21	1.55	1.52	1.54	1.61	1.70	1.82	1.94	1.56	1.52	1.55	1.63	1.73	1.85	1.99
		(1.40)	(1.26)	(1.17)	(1.12)	(1.09)	(1.09)	(1.10)	(0.76)	(0.75)	(0.78)	(0.85)	(0.94)	(1.05)	(1.17)	(0.77)	(0.76)	(0.80)	(0.57)	(0.97)	(1.09)	(1.20)
	1.00	2.73	2.52	2.37	2.28	2.23	2.23	2.23	1.57	1.53	1.56	1.63	1.73	1.84	1.97	1.57	1.54	1.57	1.64	1.75	1.87	2.01
		(1.44)	(1.29)	(1.21)	(1.16)	(1.14)	(1.14)	(1.15)	(0.79)	(0.78)	(0.82)	(0.89)	(0.99)	(1.10)	(1.22)	(0.80)	(0.79)	(0.84)	(0.91)	(1.02)	(1.14)	(1.26)
	1.05	2.72	2.51	2.37	2.28	2.25	2.24	2.25	1.58	1.55	1.58	1.65	1.75	1.87	1.99	1.57	1.54	1.58	1.66	1.76	1.88	2.03
		(1.48)	(1.34)	(1.25)	(1.20)	(1.18)	(1.18)	(1.20)	(0.82)	(0.81)	(0.85)	(0.93)	(1.03)	(1.16)	(1.28)	(0.82)	(0.81)	(0.86)	(0.95)	(1.06)	(1.18)	(1.31)
	1.25	2.61	2.44	2.33	2.26	2.22	2.23	2.24	1.58	1.55	1.59	1.67	1.77	1.90	2.03	1.54	1.52	1.45	1.63	1.73	1.85	1.98
		(1.53)	(1.41)	(1.34)	(1.30)	(1.29)	(1.30)	(1.34)	(0.88)	(0.88)	(0.94)	(1.04)	(1.17)	(1.30)	(1.44)	(0.85)	(0.86)	(0.83)	(1.02)	(1.13)	(1.26)	(1.40)
	1.50	2.33	2.20	2.11	2.06	2.04	2.05	2.07	1.48	1.46	1.50	1.56	1.66	1.78	1.89	1.43	1.42	1.33	1.51	1.58	1.68	1.79
		(1.43)	(1.33)	(1.27)	(1.25)	(1.25)	(1.27)	(1.31)	(0.81)	(0.82)	(0.89)	(0.99)	(1.13)	(1.28)	(1.43)	(0.75)	(0.77)	(0.69)	(0.92)	(1.04)	(1.17)	(1.30)
1.50	0.25	1.11	1.05	1.02	1.01	1.01	1.01	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
		(0.31)	(0.23)	(0.15)	(0.11)	(0.10)	(0.09)	(0.09)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.01)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)
	0.50	1.29	1.23	1.18	1.15	1.13	1.13	1.13	1.00	1.00	1.00	1.00	1.01	1.01	1.02	1.00	1.00	1.00	1.00	1.01	1.01	1.01
		(0.46)	(0.42)	(0.38)	(0.36)	(0.34)	(0.34)	(0.33)	(0.07)	(0.06)	(0.06)	(0.08)	(0.09)	(0.11)	(0.14)	(0.05)	(0.04)	(0.05)	(0.06)	(0.07)	(0.09)	(0.10)
	0.95	1.45	1.39	1.35	1.32	1.30	1.30	1.30	1.09	1.08	1.08	1.09	1.11	1.13	1.16	1.09	1.08	1.09	1.10	1.12	1.15	1.17
		(0.62)	(0.58)	(0.54)	(0.52)	(0.51)	(0.50)	(0.50)	(0.29)	(0.28)	(0.29)	(0.31)	(0.34)	(0.37)	(0.41)	(0.30)	(0.29)	(0.29)	(0.32)	(0.35)	(0.39)	(0.43)
	1.00	1.46	1.41	1.36	1.33	1.32	1.31	1.31	1.10	1.09	1.09	1.11	1.13	1.15	1.18	1.10	1.10	1.10	1.12	1.13	1.16	1.19
		(0.64)	(0.59)	(0.56)	(0.54)	(0.52)	(0.52)	(0.52)	(0.32)	(0.30)	(0.31)	(0.33)	(0.36)	(0.40)	(0.44)	(0.32)	(0.30)	(0.31)	(0.34)	(0.37)	(0.41)	(0.46)
	1.05	1.47	1.42	1.38	1.35	1.33	1.33	1.32	1.11	1.10	1.11	1.12	1.14	1.17	1.19	1.11	1.10	1.11	1.12	1.15	1.17	1.20
		(0.66)	(0.61)	(0.57)	(0.55)	(0.54)	(0.54)	(0.54)	(0.33)	(0.32)	(0.33)	(0.35)	(0.39)	(0.43)	(0.46)	(0.34)	(0.32)	(0.33)	(0.36)	(0.39)	(0.43)	(0.47)
	1.25	1.52	1.46	1.42	1.39	1.38	1.37	1.37	1.15	1.14	1.14	1.16	1.18	1.21	1.24	1.14	1.13	1.14	1.15	1.18	1.20	1.24
		(0.73)	(0.67)	(0.63)	(0.61)	(0.60)	(0.60)	(0.60)	(0.39)	(0.38)	(0.39)	(0.42)	(0.45)	(0.51)	(0.55)	(0.38)	(0.37)	(0.38)	(0.41)	(0.45)	(0.49)	(0.55)
	1.50	1.51	1.46	1.42	1.40	1.39	1.38	1.39	1.17	1.15	1.16	1.18	1.21	1.24	1.28	1.15	1.14	1.15	1.16	1.18	1.21	1.25
		(0.76)	(0.71)	(0.68)	(0.66)	(0.65)	(0.65)	(0.66)	(0.42)	(0.41)	(0.43)	(0.48)	(0.15)	(0.57)	(0.63)	(0.40)	(0.39)	(0.41)	(0.44)	(0.48)	(0.53)	(0.59)
2.00	0.25	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
		(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)
	0.50	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
		(0.05)	(0.04)	(0.03)	(0.02)	(0.02)	(0.02)	(0.02)	(0.00)	(0.00)	(0.00)	(0.00)	(0.01)	(0.01)	(0.01)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.01)
	0.95	1.07	1.06	1.05	1.04	1.04	1.04	1.04	1.01	1.00	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01
		(0.26)	(0.24)	(0.22)	(0.21)	(0.20)	(0.19)	(0.19)	(0.08)	(0.07)	(0.07)	(0.08)	(0.10)	(0.11)	(0.11)	(0.08)	(0.08)	(0.08)	(0.09)	(0.10)	(0.11)	(0.12)
	1.00	1.08	1.07	1.06	1.05	1.05	1.05	1.05	1.01	1.01	1.01	1.01	1.01	1.01	1.02	1.01	1.01	1.01	1.01	1.01	1.02	1.02
		(0.28)	(0.26)	(0.24)	(0.22)	(0.22)	(0.21)	(0.21)	(0.09)	(0.09)	(0.09)	(0.10)	(0.11)	(0.12)	(0.13)	(0.09)	(0.09)	(0.09)	(0.10)	(0.11)	(0.12)	(0.14)
	1.05	1.09	1.08	1.07	1.06	1.06	1.06	1.05	1.01	1.01	1.01	1.01	1.02	1.02	1.02	1.01	1.01	1.01	1.01	1.02	1.02	1.02
		(0.29)	(0.28)	(0.26)	(0.24)	(0.23)	(0.23)	(0.23)	(0.11)	(0.10)	(0.11)	(0.11)	(0.21)	(0.14)	(0.15)	(0.11)	(0.10)	(0.10)	(0.11)	(0.13)	(0.14)	(0.16)
	1.25	1.14	1.12	1.10	1.09	1.09	1.09	1.09	1.02	1.02	1.02	1.03	1.03	1.04	1.04	1.02	1.02	1.02	1.03	1.03	1.04	1.04
		(0.37)	(0.34)	(0.32)	(0.30)	(0.29)	(0.29)	(0.29)	(0.16)	(0.15)	(0.15)	(0.16)	(0.18)	(0.20)	(0.22)	(0.15)	(0.15)	(0.15)	(0.16)	(0.18)	(0.20)	(0.21)
	1.50	1.18	1.16	1.14	1.13	1.13	1.12	1.12	1.04	1.04	1.04	1.05	1.05	1.06	1.01	1.04	1.03	1.04	1.04	1.05	1.05	1.06
		(0.42)	(0.39)	(0.38)	(0.36)	(0.36)	(0.36)	(0.35)	(0.21)	(0.20)	(0.20)	(0.22)	(0.24)	(0.26)	(0.28)	(0.19)	(0.18)	(0.19)	(0.20)	(0.22)	(0.24)	(0.27)

${}^{\mathrm{a}}$ Max-EWMA( $\lambda =0.05$), ${}^{\mathrm{b}}$ Max-DEWMA( $\lambda =0.05$) due to $q=1-\lambda ,\alpha =1.00$

The two EWMA statistics, each one for monitoring the mean and the variance, are defined as follows

$$\begin{array}{rl}{Y}_{i1}& =(1-\lambda )Y_{(i-1)1}+\lambda U_i,\end{array}$$ (16)

$$\begin{array}{rl}{Z}_{i1}& =(1-\lambda )Z_{(i-1)1}+\lambda V_i,\end{array}$$ (17)

where $i=1,2,\dots$, $\lambda \in (0,1]$ is the smoothing parameter and $Y_{01}=Z_{01}=0$ are the starting values. The Max-EWMA statistic is defined as

$$M{E}_i=max\{|Y_{i1}|,|Z_{i1}|\},\mathrm{for}i=1,2,\dots .$$ (18)

The upper control limit ( $UC{L}_{ME}$) of the Max-EWMA [6,28] chart is given by

$$UC{L}_{ME}=(1.12838+0.60281K_1)\sqrt{c_1(i)}$$ (19)

where $c_1(i)=\frac{\lambda [1-(1-\lambda {)}^{2i}]}{2-\lambda }$, $i=1,2,\dots$ and $K_1>0$ is the control chart multiplier, when the process is IC. The process is declared to be OOC when the statistic $M{E}_i$ exceeds $UC{L}_{ME}$.

From Tables 3 and, A1–A4 and A5–A8 in the Supplementary Material we observe that the Max-DGWMA chart outperforms the Max-EWMA chart for most of the considered shift combination scenarios. Particularly, the Max-DGWMA chart has lower $AR{L}_1$ and $SDR{L}_1$ values for monitoring small and moderate shifts in the process mean ( $\delta \le 2.00$) and all the considered shifts in the process variability ( $0.25\le \rho \le 3.00$). Nevertheless, for fixed λ and q values, there are shift combinations ( $\delta \le 0.50$, $\rho \ge 1.25$) in which the Max-EWMA chart has lower $SDR{L}_1$ values than the Max-DGWMA(q, $\alpha \in [1.00,1.10]$). For example, for $(\delta ,\rho )=(0.10,1.25)$ the Max-DGWMA(q = 0.95, $\alpha =1.10$) chart has $SDR{L}_1=12.13$, while the Max-EWMA( $\lambda =0.05$) chart has $SDR{L}_1=11.13$. For larger shifts of size $\delta >2.00$ in the process mean and $0.25\le \rho \le 1.25$ in the process variability both charts have similar performance, while for $\rho >1.25$ the Max-DGWMA chart is more efficient.

• Proposed Max-DGWMA chart versus Max-DEWMA chart

From the two EWMA statistics of $Y_{i1}$ and $Z_{i1}$ given in Equations (16) and (17), respectively, two corresponding DEWMA statistics are given as follows

$$\begin{array}{rl}{Y}_{i2}& =(1-\lambda )Y_{(i-1)2}+\lambda Y_{i1},\end{array}$$ (20)

$$\begin{array}{rl}{Z}_{i2}& =(1-\lambda )Z_{(i-1)2}+\lambda Z_{i1},\end{array}$$ (21)

where $i=1,2,\dots$, $\lambda \in (0,1]$ is the smoothing parameter and $Y_{02}=Z_{02}=0$ are the starting values. Hereafter, as Khoo et al. [17], we set the smoothing parameters the same, for the statistics $Y_{ik}$ and $Z_{ik}$, k = 1, 2. Therefore, the Max-DEWMA statistic is defined as

$$MD{E}_i=\max \{|Y_{i2}|,|Z_{i2}|\},\mathrm{for}i=1,2,\dots .$$ (22)

The upper control limit ( $UC{L}_{MDE}$) of the Max-DEWMA [17] chart is given as

$$UC{L}_{MDE}=(1.12838+0.60281K_2)\sqrt{c_2(i)}$$ (23)

where $c_2(i)=\frac{{\lambda }^4[1+(1-\lambda {)}^2-(i^2+2i+1)(1-\lambda {)}^{2i}+(2i^2+2i-1)(1-\lambda {)}^{2i+2}-i^2(1-\lambda {)}^{2i+4}]}{[1-(1-\lambda {)}^2{]}^3}$, $i=1,2,\dots$, and $K_2>0$ is a control chart multiplier, when the process is IC. The process is declared to be OOC when the statistic $MD{E}_i$ exceeds $UC{L}_{MDE}$.

Tables 3 and A1–A4 in the Supplementary Material point out that the Max-DGWMA chart with $\alpha \le 0.90$ is superior to the Max-DEWMA chart (i.e. Max-DGWMA $(q=1-\lambda ,\alpha =1.00)$), since it has lower $AR{L}_1$ and $SDR{L}_1$ values for monitoring small and moderate shifts in the process mean ( $\delta \le 2.00$) and all the examined shifts in the process dispersion ( $0.25\le \rho \le 3.00$). For example, the $AR{L}_1$ values of the Max-DGWMA( $q=0.95,\alpha =0.50$) and Max-DEWMA( $\lambda =0.05$) charts for $(\delta ,\rho )=(0.10,0.95)$ are 35.39 and 71.20, respectively. However, the Max-DGWMA( $q,\alpha =1.10$) chart is less efficient than the Max-DEWMA chart for the same amounts of shifts in process mean and variability (i.e. ( $\delta \le 2.00$, $0.25\le \rho \le 3.00$)). Finally, for large shifts in the process mean ( $\delta >2.00$) and regardless the amount of the shift in the process variability, both charts have similar performance.

• Proposed Max-DGWMA chart versus Max-GWMA chart

From Tables 3 and, A1–A4 and A5–A8 in the Supplementary Material we observe that the Max-DGWMA chart seems to outperform the Max-GWMA chart for most of the shifts in the process mean and/or variability for both measures. Specifically, the Max-DGWMA chart has smaller $AR{L}_1$ and $SDR{L}_1$ for monitoring small and moderate shifts in the process mean ( $\delta \le 2.00$) and all the upward and downward shifts in the process dispersion ( $0.25\le \rho \le 3.00$). However, for fixed q and $\alpha \in [1.00,1.10]$, the Max-GWMA chart has smaller $SDR{L}_1$ values than the Max-DGWMA chart for small shifts in the process mean and moderate to large increasing shifts in the process variability ( $\delta \le 0.50$, $\rho \ge 1.25$). Eventually, the proposed and the Max-GWMA charts perform similarly for large shifts in the process mean ( $\delta >2.00$) and for downward to small upward shifts in the process variability ( $0.25\le \rho \le 1.25$), whereas for $\delta >2.00$ and $\rho >1.25$ the Max-DGWMA chart is more sensitive.

• Proposed Max-DGWMA chart versus SS-DGWMA chart

The two DGWMA statistics are given by Equations (9) and (10). The SS-DGWMA statistic is defined as

$$SSD{G}_i=G_{i3}^2+G_{i4}^2,i=1,2,\dots .$$ (24)

The upper control limit ( $UC{L}_{SSDG}$) [15] of the SS-DGWMA chart, is defined as

$$UC{L}_{SSDG}=2(1+L)\sum _{j=1}^i{w}_j^2,$$ (25)

where L>0 is the control chart multiplier, when the process is IC. The process is considered to be OOC when the $SSD{G}_i$ exceeds $UC{L}_{SSDG}$. It should be noted that the SS-DGWMA chart reduces to the SS-DEWMA [27] chart when $\alpha =1.00$ and $q=1-\lambda$.

Tables 3 and, A1–A4 and A9–A12 in the Supplementary Material reveal that, the competing charts have comparable performance. The Max-DGWMA chart is more sensitive for small shifts in the mean and moderate to large downward shifts in the variability ( $(\delta =0.00,\rho <0.95)$ and $(0.10\le \delta \le 0.25,0.25\le \rho <0.75)$), and for moderate to large shifts in the mean and small shifts in the variability $(0.50\le \delta \le 1.50,0.95\le \rho \le 1.05)$. However, the SS-DGWMA chart performs better for small shifts in the mean and small downward to moderate upward shifts in the variability, i.e. $(\delta \le 0.25,0.95<\rho \le 1.75)$ and $(0.10\le \delta \le 0.25,0.75\le \rho \le 0.95)$. In addition, it is more effective when $(0.50\le \delta \le 1.00,0.25\le \rho \le 0.75)$. It must be mentioned that the Max-DGWMA and SS-DGWMA charts perform almost similarly, with the latter to be slightly better, when $0.00\le \delta <2.00$ and $\rho \ge 2.00$. Finally, both charts have similar efficiency in detecting large shifts in the mean and small to large upward and downward shifts in the variability, i.e. $(\delta \ge 1.50,\rho \le 0.95)$ and $(\delta \ge 2.00,\rho \ge 1.00)$.

Furthermore, the bold printed values in Tables A4, A8, and A12 in the Supplementary Material, and 3 correspond to the smallest $AR{L}_1$ values for each shift combination of the Max-DGWMA, Max-GWMA and SS-DGWMA charts, while the smallest $AR{L}_1$ results of the Max-EWMA and Max-DEWMA charts correspond to $\lambda =0.05$ (see Max-GWMA $(q=0.95,\alpha =1.00)$ and Max-DGWMA $(q=0.95,\alpha =1.00)$, respectively, in the aforementioned Tables). According to the smallest $AR{L}_1$ values of these Tables, and especially from Table 3, the Max-DGWMA chart outperforms the Max-EWMA, Max-DEWMA and Max-GWMA charts for most of the $(\delta ,\rho )$ cases. Furthermore, the proposed chart seems to be more sensitive for small and moderate shifts in the mean and variability ( $\delta \le 1.50$, $0.50\le \rho \le 1.50$). It is to be noted that the optimal parameter combinations of the Max-EWMA, Max-DEWMA and Max-GWMA charts can easily be obtained, from these Tables, for each shift combination. For instance, the $AR{L}_1$ values at $(\delta ,\rho )=(0.50,0.95)$, of the optimal Max-EWMA, optimal Max-DEWMA and optimal Max-GWMA charts are 6.69, 5.81 and 6.65, respectively, when ( $\lambda =0.05,K_1=2.770$), ( $\lambda =0.05,K_2=1.898$) and ( $q=0.95,\alpha =0.90,L_1=2.792$) (see Table 3).

Taking into consideration, the previous conclusions, the Max-DGWMA and SS-DGWMA charts show comparable performance. In order to extend the comparisons between these charts, Table 4 presents the $(q,\alpha ,L_2)$ and $(q,\alpha ,L)$ combinations and the corresponding ARL results for the optimal Max-DGWMA and optimal SS-DGWMA charts with time-varying control limits when $AR{L}_0\approx 370$ and n = 5. Comparing this Table, it is obvious that the Max-DGWMA chart is more sensitive for small shifts in the mean and moderate to large downward shifts in the variability ( $\delta \le 0.25,0.25\le \rho \le 0.75$). Whereas the SS-DGWMA chart is more efficient for small shifts in the mean and moderate downward to moderate upward in the dispersion ( $\delta \le 0.50,0.75<\rho \le 1.75$). Furthermore, both charts perform similarly for the majority of the shift combinations, at ( $\delta \le 1.00,\rho \ge 2.00$) and ( $\delta >1.00,0.25\le \rho \le 3.00$).

Since we are interested in a wide range of shifts in the process mean and/or variance, it is helpful to measure the overall performance of the competing charts as well. Therefore, we use the RMI [11] measure, which is given by

$$RMI=\frac{1}{N}\sum _{i=1}^N\frac{ARL({\delta }_i,{\rho }_i)-AR{L}^{\ast }({\delta }_i,{\rho }_i)}{AR{L}^{\ast }({\delta }_i,{\rho }_i)}$$ (26)

where N is the number of shifts considered, $ARL({\delta }_i,{\rho }_i)$, $i=1,2,\dots ,N$ is the $AR{L}_1$ value of a chart for a specific shift combination $({\delta }_i,{\rho }_i)$ and $AR{L}^{\ast }({\delta }_i,{\rho }_i)$ is the smallest $AR{L}_1$ value among all the competing charts for the specific shift combination $({\delta }_i,{\rho }_i)$. According to this index, a control chart with a smaller RMI value is considered better in its overall performance. Table 5 presents the RMI values using the $AR{L}_1$ values presented in Tables A1–A12 in the Supplementary Material, for all the competing charts when $AR{L}_0\approx 370$, n = 5, $\lambda \in \{$0.05, 0.10, 0.20, 0.30 $\}$, $q\in \{$0.70, 0.80, 0.90, 0.95 $\}$ and $\alpha \in \{$0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10 $\}$, over a wide range of shift combinations ( $\delta ,\rho$), where $0.00\le \delta \le 3.00$, and $0.25\le \rho \le 3.00$. Concerning this performance measure, Table 5 reveals that the Max-DGWMA and SS-DGWMA charts with q = 0.95 and $\alpha \in [0.50,0.60]$ have the best overall performance ( $RMI\le 0.05$). It is to be noted that the Max-DGWMA and SS-DGWMA charts show almost similar overall performance, with the latter to be slightly better. For example, the RMI value of the Max-DGWMA( $q=0.95,\alpha =0.50$) chart is equal to 0.03, while the RMI values for the Max-EWMA( $\lambda =0.05$), Max-DEWMA( $\lambda =0.05$), Max-GWMA( $q=0.95,\alpha \in [0.80,0.90]$) and SS-GWMA( $q=0.95,\alpha =0.50$) charts are 0.42, 0.26, 0.40 and 0.02, respectively. In addition, the Max-DGWMA and SS-DGWMA charts are superior to the Max-EWMA and Max-GWMA charts, for all the considered λ and $(q,\alpha )$ values, over the whole range of the shift combinations. Finally, both the proposed and SS-DGWMA charts with $q\in [0.70,0.95]$ and $\alpha \in [0.50,0.90]$ have better overall performance than the Max-DEWMA chart, whereas for $\alpha =1.10$ the opposite happens as q decreases.

Table 5.

Overall performance comparison of the competing control charts using RMI measure.

	RMI		RMI		RMI		RMI
Max-EWMA( $\lambda =$0.05)	0.42	Max-EWMA( $\lambda =$0.10)	0.54	Max-EWMA( $\lambda =$0.20)	0.74	Max-EWMA( $\lambda =$0.30)	0.94
Max-DEWMA( $\lambda =$0.05)	0.26	Max-DEWMA( $\lambda =$0.10)	0.39	Max-DEWMA( $\lambda =$0.20)	0.54	Max-DEWMA( $\lambda =$0.30)	0.66
Max-GWMA( $q=0.95,\alpha =0.50$)	0.59	Max-GWMA( $q=0.90,\alpha =0.50$)	0.66	Max-GWMA( $q=0.80,\alpha =0.50$)	0.78	Max-GWMA( $q=0.70,\alpha =0.50$)	0.97
Max-GWMA( $q=0.95,\alpha =0.60$)	0.49	Max-GWMA( $q=0.90,\alpha =0.60$)	0.58	Max-GWMA( $q=0.80,\alpha =0.60$)	0.73	Max-GWMA( $q=0.70,\alpha =0.60$)	0.88
Max-GWMA( $q=0.95,\alpha =0.70$)	0.43	Max-GWMA( $q=0.90,\alpha =0.70$)	0.54	Max-GWMA( $q=0.80,\alpha =0.70$)	0.69	Max-GWMA( $q=0.70,\alpha =0.70$)	0.87
Max-GWMA( $q=0.95,\alpha =0.80$)	0.40	Max-GWMA( $q=0.90,\alpha =0.80$)	0.52	Max-GWMA( $q=0.80,\alpha =0.80$)	0.70	Max-GWMA( $q=0.70,\alpha =0.80$)	0.99
Max-GWMA( $q=0.95,\alpha =0.90$)	0.40	Max-GWMA( $q=0.90,\alpha =0.90$)	0.53	Max-GWMA( $q=0.80,\alpha =0.90$)	0.71	Max-GWMA( $q=0.70,\alpha =0.90$)	0.91
Max-GWMA( $q=0.95,\alpha =1.10$)	0.44	Max-GWMA( $q=0.90,\alpha =1.10$)	0.57	Max-GWMA( $q=0.80,\alpha =1.10$)	0.77	Max-GWMA( $q=0.70,\alpha =1.10$)	0.98
SS-DGWMA( $q=0.95,\alpha =0.50$)	0.02	SS-DGWMA( $q=0.90,\alpha =0.50$)	0.15	SS-DGWMA( $q=0.80,\alpha =0.50$)	0.34	SS-DGWMA( $q=0.70,\alpha =0.50$)	0.48
SS-DGWMA( $q=0.95,\alpha =0.60$)	0.03	SS-DGWMA( $q=0.90,\alpha =0.60$)	0.16	SS-DGWMA( $q=0.80,\alpha =0.60$)	0.35	SS-DGWMA( $q=0.70,\alpha =0.60$)	0.47
SS-DGWMA( $q=0.95,\alpha =0.70$)	0.05	SS-DGWMA( $q=0.90,\alpha =0.70$)	0.20	SS-DGWMA( $q=0.80,\alpha =0.70$)	0.37	SS-DGWMA( $q=0.70,\alpha =0.70$)	0.49
SS-DGWMA( $q=0.95,\alpha =0.80$)	0.12	SS-DGWMA(( $q=0.90,\alpha =0.80$)	0.24	SS-DGWMA( $q=0.80,\alpha =0.80$)	0.41	SS-DGWMA( $q=0.70,\alpha =0.80$)	0.52
SS-DGWMA( $q=0.95,\alpha =0.90$)	0.18	SS-DGWMA( $q=0.90,\alpha =0.90$)	0.30	SS-DGWMA( $q=0.80,\alpha =0.90$)	0.45	SS-DGWMA( $q=0.70,\alpha =0.90$)	0.57
SS-DGWMA( $q=0.95,\alpha =1.00$)	0.24	SS-DGWMA( $q=0.90,\alpha =1.00$)	0.36	SS-DGWMA( $q=0.80,\alpha =1.00$)	0.50	SS-DGWMA( $q=0.70,\alpha =1.00$)	0.62
SS-DGWMA( $q=0.95,\alpha =1.10$)	0.31	SS-DGWMA( $q=0.90,\alpha =1.10$)	0.42	SS-DGWMA( $q=0.80,\alpha =1.10$)	0.55	SS-DGWMA( $q=0.70,\alpha =1.10$)	0.67
Max-DGWMA( $q=0.95,\alpha =0.50$)	0.03	Max-DGWMA( $q=0.90,\alpha =0.50$)	0.18	Max-DGWMA( $q=0.80,\alpha =0.50$)	0.38	Max-DGWMA( $q=0.70,\alpha =0.50$)	0.52
Max-DGWMA( $q=0.95,\alpha =0.60$)	0.04	Max-DGWMA( $q=0.90,\alpha =0.60$)	0.18	Max-DGWMA( $q=0.80,\alpha =0.60$)	0.38	Max-DGWMA( $q=0.70,\alpha =0.60$)	0.51
Max-DGWMA( $q=0.95,\alpha =0.70$)	0.08	Max-DGWMA( $q=0.90,\alpha =0.70$)	0.22	Max-DGWMA( $q=0.80,\alpha =0.70$)	0.40	Max-DGWMA( $q=0.70,\alpha =0.70$)	0.53
Max-DGWMA( $q=0.95,\alpha =0.80$)	0.13	Max-DGWMA(( $q=0.90,\alpha =0.80$)	0.27	Max-DGWMA( $q=0.80,\alpha =0.80$)	0.44	Max-DGWMA( $q=0.70,\alpha =0.80$)	0.56
Max-DGWMA( $q=0.95,\alpha =0.90$)	0.20	Max-DGWMA( $q=0.90,\alpha =0.90$)	0.33	Max-DGWMA( $q=0.80,\alpha =0.90$)	0.49	Max-DGWMA( $q=0.70,\alpha =0.90$)	0.61
Max-DGWMA( $q=0.95,\alpha =1.10$)	0.33	Max-DGWMA( $q=0.90,\alpha =1.10$)	0.45	Max-DGWMA( $q=0.80,\alpha =1.10$)	0.60	Max-DGWMA( $q=0.70,\alpha =1.10$)	0.73

5. Illustrative example

In the SPC literature, it is usual to consider either a real or a simulated dataset to illustrate the application of the proposed control chart. Here the real dataset is taken from Montgomery [20], so as to demonstrate the practical importance and detection abilities of the proposed chart against the Max-DEWMA chart. The goal is to establish statistical control for a manufacturing process, where the insider diameter measurements (in mm) of an automobile engine piston ring is regarded as the quality characteristic. The dataset consists of 40 samples, each of sample size n = 5. The first 25 samples (see Table 6.3 of [20]) represent the Phase I observations. The estimates of the process mean and standard deviation are 74.001mm and 0.01mm, respectively. The last 15 samples (see Table 6E.8 of [20]) for the piston ring process were taken after the initial control charts were established (Phase II). Setting $AR{L}_0\approx 370$, we construct the Max-DEWMA and Max-DGWMA charts with a time-varying upper control limit and parameter combinations $(\lambda ,K_2)=(0.10,2.3262)$, and $(q,\alpha ,L_2)=(0.90,0.50,2.145)$, respectively. The calculated values of the charting statistics are provided in Table 6, whereas the control charts are displayed in Figures 2 and 3. It is obvious that the control charts do not raise OOC signals using the first 25 samples, which implies that the process remains in the IC state. However, for the remaining samples, both charts raise an OOC signal to indicate that the process is OOC. It can be seen that the Max-DGWMA chart triggers the first OOC signal at the 37th sample, while the Max-DEWMA chart at the 39th sample. According to Table 2, the OOC samples 39-40 for the Max-DEWMA chart and 37-40 for the Max-DGWMA chart are related with an increase in the mean, and they are labelled with $‘‘m{+}^{″}$ in Figures 2 and 3. Therefore, the proposed chart is more sensitive than the Max-DEWMA chart.

Table 6.

Calculation details of the Max-DEWMA and Max-DGWMA control charts using Piston Ring Diameter Data.

Sample,	Max-DEWMA		Max-DGWMA		Sample,	Max-DEWMA		Max-DGWMA
i	$MD{E}_i$	$UC{L}_{MDE}$	$MD{G}_i$	$UC{L}_{MDG}$	i	$MD{E}_i$	$UC{L}_{MDE}$	$MD{G}_i$	$UC{L}_{MDG}$
1	0.020	0.025	0.020	0.024	21	0.114	0.375	0.016	0.067
2	0.035	0.052	0.016	0.031	22	0.118	0.379	0.021	0.068
3	0.062	0.081	0.031	0.035	23	0.112	0.384	0.013	0.069
4	0.087	0.109	0.033	0.039	24	0.107	0.387	0.015	0.070
5	0.112	0.137	0.036	0.042	25	0.084	0.391	0.004	0.071
6	0.118	0.164	0.026	0.045	26	0.045	0.394	0.021	0.072
7	0.119	0.189	0.021	0.047	27	0.010	0.396	0.021	0.072
8	0.110	0.212	0.023	0.049	28	0.012	0.398	0.013	0.073
9	0.107	0.234	0.019	0.051	29	0.024	0.400	0.009	0.074
10	0.097	0.254	0.011	0.053	30	0.027	0.402	0.006	0.075
11	0.072	0.272	0.019	0.055	31	0.032	0.403	0.009	0.075
12	0.052	0.288	0.032	0.056	32	0.034	0.404	0.017	0.076
13	0.037	0.302	0.025	0.058	33	0.027	0.405	0.009	0.077
14	0.048	0.316	0.031	0.059	34	0.054	0.406	0.031	0.077
15	0.061	0.327	0.017	0.061	35	0.101	0.407	0.053	0.078
16	0.076	0.338	0.027	0.062	36	0.144	0.407	0.055	0.078
17	0.093	0.347	0.026	0.063	37	0.212	0.408	0.087	0.079
18	0.097	0.355	0.021	0.064	38	0.306	0.408	0.123	0.079
19	0.107	0.363	0.022	0.065	39	0.429	0.409	0.165	0.080
20	0.115	0.369	0.025	0.066	40	0.551	0.409	0.181	0.081

The signals for OOC are in bold print.

Figure 2.

Max-DEWMA chart for piston-ring data.

Figure 3.

Max-DGWMA chart for piston-ring data.

6. Concluding remarks

The present article introduces a new single control chart for simultaneously detecting shifts in the process mean and variability. The proposed chart extends the single Max-GWMA chart to a single Max-Double GWMA, named as Max-DGWMA control chart. The formulae of the mean and the variance of the Max-DGWMA statistic and, consequently, the upper control limit of the chart are calculated. The proposed chart is evaluated in terms of the ARL and SDRL measures. The results reveal that for a fixed α (q) value, the performance of the Max-DGWMA chart enhances as q (α) increases (decreases), it is more sensitive for small to moderate shifts ( $0.00\le \delta \le 1.00$, $0.50\le \rho \le 1.50$), while it has better overall run length performance when q = 0.95 and $\alpha \in [0.50,0.60]$. Furthermore, it is more efficient in detecting small upward shifts in the process variability regardless the amount of the shift in the process mean, and its run length performance is affected by the sample size.

Moreover, the Max-DGWMA chart is compared with the existing counterparts, such as the Max-EWMA, Max-DEWMA, Max-GWMA and SS-DGWMA charts, through the ARL and SDRL measures. The comparison results indicate that the proposed chart outperforms the Max-EWMA, Max-DEWMA and Max-GWMA charts in detecting small to moderate shifts in the mean and all the considered shifts in the dispersion ( $\delta \le 2.00$, $0.25\le \rho \le 3.00$), whereas they perform similarly for large shifts in the mean. However, for $\alpha \in [1.00,1.10]$, both Max-EWMA and Max-GWMA charts have lower $SDR{L}_1$ values when ( $\delta \le 0.50,\rho \ge 1.25$), while for $\alpha =1.10$ the Max-DEWMA chart is better than the proposed chart. Additionally, the Max-DGWMA and SS-DGWMA charts are comparable. In particular, they have similar performance for $(\delta \ge 1.50,0.25\le \rho \le \mathrm{3.0.0})$ and $(0\le \delta \le 3.00,\rho \ge 2.00)$. However, the first chart is more sensitive for moderate shifts in the mean and small shifts in the variability, as well as small shifts in the mean and moderate to large downward shifts in the variability, whereas the latter is better for small shifts in the mean and small downward to moderate upward shifts in the dispersion. Furthermore, according to the RMI measure, the Max-DGWMA chart is superior to the Max-EWMA, Max-DEWMA and Max-GWMA charts, whereas it is slightly less sensitive than the SS-DGWMA chart for detecting a wide range of shifts in the mean and dispersion ( $0.00\le \delta \le 3.00$, $0.25\le \rho \le 3.00$). Finally, an illustrative example is provided to explain the implementation of the proposed chart. Consequently, we recommend practitioners using the Max-DGWMA chart as a tool for detecting small and moderate shifts in the process mean and variability concurrently.

Appendix. Derivation of the Max-DGWMA chart.

Through these transformations, we consider that $U_i\sim N(0,1)$, $V_i\sim N(0,1)$, and both $U_i$ and $V_i$ are independently distributed when the process is IC ( $\delta =0$, $\rho =1$).

From Equation (4) and the assumption $G_{01}=0$, we get

$$\begin{array}{rl}{G}_{i1}& =\sum _{j=1}^{i}P(N_1=j)U_{i-j+1}\\ & =\sum _{j=1}^i(q_1^{(j-1{)}^{\alpha }}-q_1^{j^{\alpha }})U_{i-j+1}\\ & =\sum _{j=1}^i(q_1^{(i-j{)}^{\alpha }}-q_1^{(i-j+1{)}^{\alpha }})U_j.\end{array}$$ (A1)

Similarly, using the assumption $G_{03}=0$, we get from Equation (9)

$$G_{i3}=\sum _{j=1}^i(q_2^{(i-j{)}^{\beta }}-q_2^{(i-j+1{)}^{\beta }})G_{j1}.$$ (A2)

From Equations (A1) and (A2), we get

$$\begin{array}{rl}{G}_{i3}& =\sum _{j_1=1}^i\sum _{j_2=1}^{j_1}(q_2^{(i-j_1{)}^{\beta }}-q_2^{(i-j_1+1{)}^{\beta }})(q_1^{(j_1-j_2{)}^{\alpha }}-q_1^{(j_1-j_2+1{)}^{\alpha }})U_{j_2}\\ & =\sum _{j_2=1}^i\left(\sum _{j_1=j_2}^i(q_2^{(i-j_1{)}^{\beta }}-q_2^{(i-j_1+1{)}^{\beta }})(q_1^{(j_1-j_2{)}^{\alpha }}-q_1^{(j_1-j_2+1{)}^{\alpha }})U_{j_2}\right)\\ & =\sum _{j=1}^i{w}_j{U}_j,\text{say},\end{array}$$ (A3)

where $w_j=\sum _{j_1=j}^i(q_2^{(i-j_1{)}^{\beta }}-q_2^{(i-j_1+1{)}^{\beta }})(q_1^{(j_1-j{)}^{\alpha }}-q_1^{(j_1-j+1{)}^{\alpha }})$. It is easy to note that $G_{i3}\sim N(0,Q_{i2})$, where $Q_{i2}=\sum _{j_1=j}^i{w}_j^2$. Similarly, it can be shown that

$$G_{i4}=\sum _{j=1}^i{w}_j{V}_j\text{and}{G}_{i4}\sim N(0,Q_{i2}).$$ (A4)

From Equation (14), we get

$$MD{G}_i=max\{|G_{i3}|,|G_{i4}|\},i=1,2,\dots$$

The following lemma will be helpful in developing the chart.

Lemma A.1

If U and V be two independently normally distributed random variables each with zero mean and unit variance and if $M=max(|U|,|V|)$, then the probability density function (pdf) of M is

$$f(m)=4\varphi (m)(2\mathrm{\Phi }(m)-1),m>0,$$

where $\varphi (.)$ is the pdf of $N(0,1)$ and $\mathrm{\Phi }(.)$ is the distribution function of $N(0,1)$.

Lemma A.2

If U and V be two independently normally distributed random variables each with zero mean and unit variance and if $M=max(|U|,|V|)$, then

$$E(M)=\frac{2}{\sqrt{\pi }}\mathrm{and}Var(M)=1-\frac{2}{\pi }.$$

Proof.

By definition $E(M)={\int }_0^{\mathrm{\infty }}mf(m)dm=4{\int }_o^{\mathrm{\infty }}m\varphi (m)(2\mathrm{\Phi }(m)-1)dm.$ Now we know that ${\varphi }^{\mathrm{\prime }}(m)=\frac{\mathrm{d}\varphi (m)}{\mathrm{d}m}=-m\varphi (m).$ Therefore,

$$\begin{array}{rl}E(M)& =4{\int }_0^{\mathrm{\infty }}m\varphi (m)(2\mathrm{\Phi }(m)-1)dm=-4{\int }_0^{\mathrm{\infty }}{\varphi }^{\mathrm{\prime }}(m)(2\mathrm{\Phi }(m)-1)dm\\ & ={[-4(2\mathrm{\Phi }(m)-1)\varphi (m)]}_0^{\mathrm{\infty }}+8{\int }_0^{\mathrm{\infty }}(\varphi (m){)}^{2}dm\\ & =\frac{8}{2\pi }{\int }_0^{\mathrm{\infty }}{e}^{{}^{-m^2}}dm=\frac{2}{\sqrt{\pi }}.\end{array}$$

Also

$$\begin{array}{rl}E(M^2)& =4{\int }_0^{\mathrm{\infty }}{m}^2\varphi (m)(2\mathrm{\Phi }(m)-1)dm=-4{\int }_0^{\mathrm{\infty }}m{\varphi }^{\mathrm{\prime }}(m)(2\mathrm{\Phi }(m)-1)dm\\ & ={[-4m(2\mathrm{\Phi }(m)-1)\varphi (m)]}_0^{\mathrm{\infty }}+4{\int }_0^{\mathrm{\infty }}[(2\mathrm{\Phi }(m)-1)+2m\varphi (m)]\varphi (m)dm\\ & =4{\int }_0^{\mathrm{\infty }}(2\mathrm{\Phi }(m)-1)d(\mathrm{\Phi }(m))+\frac{4}{\pi }{\int }_0^{\mathrm{\infty }}m{e}^{{}^{-m^2}}dm=1+\frac{2}{\pi }.\end{array}$$

Therefore, $Var(M)=1-\frac{2}{\pi }$.

This implies

$$E(MD{G}_i)=\frac{2\sqrt{Q_{i2}}}{\sqrt{\pi }}\mathrm{and}Var(MD{G}_i)=(1-\frac{2}{\pi })Q_{i2}.$$ (A5)

Therefore, the upper control limit $(UC{L}_{MDG})$ of the Max-DGWMA chart, is given by

$$\begin{array}{rl}UC{L}_{MDG}& =E(MD{G}_i)+L_2\sqrt{Var(MD{G}_i)}\\ & =(1.12838+0.60281L_2)\sqrt{Q_{i2}}\end{array}$$ (A6)

Disclosure statement

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