Monitoring process mean and variability with one triple EWMA chart

Abstract Control charts are very popular 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 triple exponentially weighted moving average chart (referred as Max-TEWMA chart), that simultaneously detects both upward and downward shifts in the process mean and/or process dispersion. The run length performance and the diagnostic ability of the Max-TEWMA control chart are compared with that of the Max-EWMA, Max-DEWMA and Max-GWMA charts, through Monte-Carlo simulations. The comparisons reveal that the proposed chart is more efficient, than the competing ones, in detecting shifts in the process mean and variability simultaneously. Furthermore, the Max-TEWMA chart provides a satisfactory overall performance for identifying a wide range of shifts in the process mean and variability. Finally, two illustrative examples are presented to explain the application of the Max-TEWMA control chart.


Introduction
An essential quality tool of the Statistical Process Control (SPC) are the control charts.They are efficient in detecting the presence of assignable causes of variation in the production processes.They are usually classified into the location and dispersion charts, in which the first are used to detect shifts in the process mean, while the latter are suitable for detecting shifts in the process variability.Page (1954Page ( , 1963) ) and Roberts (1959) first developed the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts, respectively.These charts are efficient in detecting small and moderate shifts and are also known as memory-type charts, since they consider both current and past information.Zhang and Chen (2005) proposed the double EWMA (DEWMA) chart, Sheu and Lin (2003) and Sheu and Tai (2006) introduced the generally weighted moving average (GWMA) chart, which is an expansion of the EWMA chart, and, recently, Alevizakos, Chatterjee, and Koukouvinos (2021) proposed the Triple EWMA (TEWMA) chart for monitoring the process mean.Additionally, many other authors have proposed memory-type control charts for monitoring process mean or variability, such as Crowder and Hamilton (1992), Steiner (1999), Abbas, Riaz, andDoes (2013a, 2013b), Haq (2013), Haq, Brown, and Moltchanova (2014), Ali and Haq (2018), Tariq et al. (2019), and Chatterjee, Koukouvinos, and Lappa (2021a), to name but a few.
Commonly, two control charts are constructed to monitor the process mean and variability, of which the first is for the process mean, while the other for the process dispersion.Nevertheless, this approach could be time consuming and may increase costs.Moreover, 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 dispersion have been proposed.Domangue and Patch (1991) developed some omnibus EWMA charts for detecting shifts in both location and dispersion, Gan (1995) proposed two EWMA charts for joint monitoring mean and variance, Chen and Cheng (1998) developed a single control chart (Max chart) to simultaneously monitor shifts in both the process mean and standard deviation, and Amin et al. (1999) introduced a joint monitoring of the process mean and variability using a MaxMin EWMA control chart.Furthermore, Xie (1999) presented some EWMA-type charts, such as the Max-EWMA, sum of squares EWMA (SS-EWMA), EWMA-Max, and EWMA semicircle (EWMA-SC) charts, Chen, Cheng, and Xie (2001Xie ( , 2004) ) extended the work of Xie (1999) regarding the Max-EWMA and the EWMA-SC charts, Khoo, Teh, and Wu (2010) proposed a single DEWMA chart (named as Max-DEWMA chart) and Sheu, Huang, and Hsu (2012) extended the single Max-EWMA chart to a single GWMA chart (named as Max-GWMA chart).Other control charts for monitoring mean and/or dispersion concurrently, can be found in the works of Lu and Reynolds (1999), Costa andRahim (2004, 2006), Cheng and Thaga (2006), Teh, Khoo, and Wu (2011), Huang (2014), Huang, Tai, and Lu (2014), Haq, Brown, and Moltchanova (2015), Haq (2017), Haq and Razzaq (2020), Sanusi, Teh, and Khoo (2020) and Chatterjee, Koukouvinos, and Lappa (2021b).
In the current article, motivated by the studies of Xie (1999), Chen, Cheng, and Xie (2001), Khoo, Teh, and Wu (2010) and Alevizakos, Chatterjee, and Koukouvinos (2021), we present a new one Triple EWMA control chart, named as Max-TEWMA chart, for simultaneously monitoring the process mean and variability.A performance comparison study is conducted, using the time-varying control limits, with the Max-EWMA, the Max-DEWMA, and the Max-GWMA charts.The control charts are compared, via Monte-Carlo simulations, using the average run length (ARL), the standard deviation of the run length (SDRL), the Relative Mean Index (RMI) (Han and Tsung 2006) and the diagnostic ability.The comparisons show that the proposed chart is effective in detecting shifts in the process mean and dispersion concurrently.
The rest of this paper is organized as follows.In Section 2, we review some well-known single memory-type control charts, i.e., the Max-EWMA, Max-DEWMA, and Max-GWMA charts.In Section 3, we introduce our proposed Max-TEWMA chart.A performance study is conducted in Section 4 to evaluate the efficiency of the proposed chart and compare its performance with the single charts described in Section 2. In Section 5, two illustrative examples are provided to present the application of the proposed chart and in Section 6 some concluding remarks are given.Finally, some technical details are described in the Appendix section.

Brief review of memorytype control charts
In this section, we briefly present some single EWMA-type control charts for simultaneously monitoring process mean and/or variability.Let X i1 , X i2 , :::, X in i , i ¼ 1, 2, ::: be a sample of n i independent normal, Nðl 0 þ dr 0 , q 2 r 2 0 Þ, random variables, where l 0 and r 0 are the in-control (IC) values of the process mean and standard deviation, respectively, and i is the sample number.The process is considered to be IC if d ¼ 0 and q ¼ 1; otherwise, the process is out-of-control (OOC) and, then d 6 ¼ 0 and/or q 6 ¼ 1: We are interested in identifying a shift in the process mean and/or process variance, from the IC l 0 and r 2 0 values to the OOC l 1 ¼ l 0 þ dr 0 and r 2 1 ¼ q 2 r 2 0 values.

Max-EWMA chart
The Max-EWMA chart combines two EWMA statistics into one single chart, one for monitoring the process mean and the other the variability.In order to design the Max-EWMA chart, the following statistics are defined as and where X i ¼ 1 , i ¼ 1, 2, ::: is the sample variance, U À1 ð:Þ indicates the inverse standard normal distribution function and Hðw; vÞ denotes the v 2 distribution function with v degrees of freedom (Quesenberry 1995).When the process is IC, both U i and V i are independent following a standard normal distribution.The two EWMA statistics, each one for monitoring the mean and the variance, are defined from U i and V i , as follows and where k 2 ð0, 1 is the smoothing parameter and Y 01 ¼ Z 01 ¼ 0 are the starting values, when the process is IC.The Max-EWMA statistic is defined as M i1 ¼ maxfjY i1 j, jZ i1 jg, for i ¼ 1, 2, :::: Since the M i1 statistic is non-negative, the Max-EWMA control chart only requires an upper control limit (UCL), which is given (Xie 1999;Chen, Cheng, and Xie 2001) by where EðM i1 Þ and VarðM i1 Þ are the mean and variance of the M i1 statistic, respectively, when the process is IC, c i1 ¼ k½1Àð1ÀkÞ 2i 2Àk , i ¼ 1, 2, ::: and K 1 > 0 is the control chart multiplier.The process is declared to be OOC when the statistic M i1 exceeds UCL 1 .

Max-DEWMA chart
The Max-DEWMA chart combines two DEWMA statistics into one chart and Khoo, Teh, and Wu (2010) showed that it is more effective compared to the Max-EWMA chart in monitoring shifts in both the mean and variance simultaneously.From the two EWMA statistics given in Eqs. ( 3) and ( 4), two corresponding DEWMA statistics are given as and where k 2 ð0, 1 is the smoothing parameter and Y 02 ¼ Z 02 ¼ 0 are the starting values, when the process is IC.Therefore, the Max-DEWMA statistic is defined as Similarly, because M i2 is non-negative, the Max-DEWMA chart only needs a UCL, which is given (Khoo, Teh, and Wu 2010) by where EðM i2 Þ and VarðM i2 Þ are the mean and variance of M i2 statistic, respectively, when the process is IC, c i2 ¼ k 4 ½1þð1ÀkÞ 2 Àði 2 þ2iþ1Þð1ÀkÞ 2i þð2i 2 þ2iÀ1Þð1ÀkÞ 2iþ2 Ài 2 ð1ÀkÞ 2iþ4 ½1Àð1ÀkÞ 2 3 , i ¼ 1, 2, :::, and K 2 > 0 is a control chart multiplier.The process is considered to be OOC when the statistic M i2 exceeds UCL 2 .

Max-GWMA chart
The Max-GWMA chart combines the Max-EWMA and the GWMA charts into a single control chart.Sheu, Huang, and Hsu (2012) showed that it is more effective compared to the Max-EWMA chart in monitoring shifts in both the mean and variability concurrently.According to the GWMA control chart structure, let M be the number of samples until the first occurrence of an event since its previous occurrence.Therefore, where the probability PðM ¼ 1Þ is the weight of the most recent sample, PðM ¼ 2Þ is the weight of the previous sample, PðM ¼ iÞ is the weight of most out-of-data sample and, consequently, PðM > iÞ is weighted with the target value of the process.The two GWMA statistics, are defined from U i and V i , as follows and where G 0 ¼ H 0 ¼ 0 are the starting values, when the process is IC.For easier computations, Sheu and Lin (2003) considered that PðM ¼ iÞ ¼ PðM > i À 1Þ À PðM > iÞ ¼ q ðiÀ1Þ a À q i a , where PðM > iÞ ¼ q i a , i ¼ 1, 2, :::, q 2 ½0, 1Þ is the design parameter and a > 0 is the adjustment parameter.Thus, the Max-GWMA statistic is defined as Considering that the MG i is non-negative, the Max-GWMA chart also requires only a UCL, which is given (Sheu, Huang, and Hsu 2012) by where EðMG i Þ and VarðMG i Þ are the mean and the variance of the MG i statistic, respectively, Q i ¼ P i j¼1 ½PðM ¼ jÞ 2 , i ¼ 1, 2, :::, when the process is IC and K > 0 is the control chart multiplier.It should be noted that, the EWMA control chart is a special case of the GWMA chart, when a ¼ 1 and q ¼ 1 À k (Roberts 1959).Hence, the Max-GWMA chart reduces to the Max-EWMA chart when a ¼ 1 and q ¼ 1 À k: The process is OOC when the MG i statistic lies above the UCL MG .

The proposed max-TEWMA chart
In the current section, we present a new single memory-type chart, named as maximum triple exponentially weighted moving average control chart (referred as Max-TEWMA chart).From the two EWMA statistics given in Eqs. ( 3) and (4), and the two DEWMA statistics given in Eqs. ( 7) and ( 8), two corresponding TEWMA statistics, can be defined as and where k 2 ð0, 1 is the smoothing parameter and Y 03 ¼ Z 03 ¼ 0 are the starting values of Y i3 and Z i3 statistics, when the process is IC.Hereafter, like Khoo, Teh, and Wu (2010), we assume that the smoothing parameters of the statistics Y ik and Z ik , k ¼ 1, 2, 3 are equal to each other.Hence, the two TEWMA statistics in Eqs. ( 16) and ( 17) are combined into the following Max-TEWMA statistic M i3 ¼ maxfjY i3 j, jZ i3 jg, for i ¼ 1, 2, :::: If the process mean and/or the variance have shifted from their corresponding target values, the M i3 value will be large; otherwise it will be small.Similar to the Max-EWMA and Max-DEWMA statistics, the M i3 statistic is non-negative.The proposed Max-TEWMA control chart takes only a UCL (referred as UCL 3 ), that is given by where EðM i3 Þ and VarðM i3 Þ are the mean and variance of the M i3 statistic, respectively, i ¼ 1, 2, ::: and K 3 > 0 is a control chart multiplier, when the process is IC.The derivation of the mean EðM i3 Þ and the variance VarðM i3 Þ of the M i3 statistic, are provided in Appendix A. A formula that can quicken the computation of the limit UCL 3 for the Max-TEWMA chart, based on desired k and K 3 parameter values, is also obtained in the Appendix A and is given as follows where and h ¼ ð1 À kÞ 2 for i ¼ 1, 2, :::: The construction procedure of the proposed Max-TEWMA chart is equivalent to the single memory-type control charts presented in Section 2. The steps for constructing a Max-TEWMA chart and detecting the potential OOC sample points are briefly described as follows: 1. Estimate the unknown process parameters.If the mean l is unknown, then use l ¼ m as an estimate of l, where X is the grand average and m is the total number of samples.If the standard deviation r is unknown, then use r ¼ R 2. Choose the desired (k, K 3 ) combinations based on the fixed sample size (n), and the IC ARL (ARL 0 ) values.Table 1 presents the (k, K 3 ) combinations when n ¼ 3, 4, 5, and 7, k 2 f0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.40, 0.50, 0.60, 0.70, 0.75, 0.80, 0.90, 1.00g, and ARL 0 % 185, 250 and 370.It should be noted that, the K 3 values are selected via Monte-Carlo simulations using a time-varying UCL (Eq.( 19)). 3. Compute the UCL 3 of the Max-TEWMA chart using Eq. ( 19).
(1) (2) (3) (4) (7) (8) (16) ( 17) and ( 18), respectively, and as the starting values.5. Plot M i3 versus i on the chart and design UCL 3 as the upper control limit.Plot a dot against i when M i3 UCL 3 : On the other hand, when M i3 > UCL 3 , denote the plotted points accordingly with the symbols shown in the Table 2. Particularly, when M i3 > UCL 3 , check both jY i3 j and jZ i3 j against UCL 3 .If only jY i3 j > UCL 3 , draw "mþ" against i when U i > 0 to indicate only an increase in the process mean, whereas draw "mÀ" against i when U i < 0 to indicate that only the process mean has decreased.If only jZ i3 j > UCL 3 , draw "vþ" against i when V i > 0 to indicate that only the process variance has increased, while draw "vÀ" against i when V i < 0 to indicate a decrease only in the process dispersion.If jY i3 j > UCL 3 and jZ i3 j > UCL 3 , draw "þþ" against i when U i > 0 and V i > 0 to suggest that both process mean and variance have increased concurrently.Moreover, draw "þÀ", "Àþ", and "ÀÀ" according to the other three cases U i > 0 and V i < 0, U i < 0 and V i > 0 and, U i < 0 and V i < 0, respectively, with similar clarification.6. Examine and interpret each of the OOC points.

Evaluation performance of the proposed Max-TEWMA chart
The statistical performance of a control chart is typically measured using the ARL and the SDRL.
The ARL is the average number of samples that must be plotted on a control chart until an OOC signal is identified (Montgomery 2013).When the mean and the dispersion of the process are IC, a large ARL 0 value is suggested.However, when the process is OOC, i.e., the mean shifts from l 0 to l 1 ¼ l 0 þ dr 0 (d 6 ¼ 0) and/or the standard deviation shifts from r 0 to r 1 ¼ qr 0 (q 6 ¼ 1) a small OOC ARL (ARL 1 ) value is preferable.Correspondingly, the lower the SDRL value, the better the performance of the chart.Here, the performance efficiency of the proposed control chart is evaluated through the ARL and SDRL measures.
A Monte Carlo simulation algorithm is developed in R language to calculate the run length distribution of the Max-TEWMA control chart with time-varying control limits.The algorithm is run 10000 repetitions to calculate the average and the standard deviation of those 10000 run lengths.For the purpose of studying the performance of the proposed chart, we assume that the underlying process for the IC condition follows the Normal distribution Nðl 0 ¼ 0, r 0 ¼ 1Þ, while the OOC process is Normally distributed with l 1 ¼ l 0 þ dr 0 and r 1 ¼ qr 0 : The examined shifts in the process mean are d 2 f0.00, 0.25, 0.50, 1.00, 1.50, 2.00, 2.50, 3.00g, while the shifts in the process dispersion are q 2 f0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00, 2.50, 3.00g: Note that the shift combination ðd, qÞ ¼ ð0:00, 1:00Þ corresponds to the IC state.The K 3 values are selected, through Monte-Carlo simulations, using the time-varying control limit (UCL 3 ) given in Eq. ( 19), to fix the ARL 0 % 185, 250 and 370 when the sample size n ¼ 5 and k 2 f0.05, 0.10, 0.20, 0.30, 0.50, 0.80, 1.00g (see additionally Table 1).The ARL and SDRL (given in the parenthesis) results of the Max-TEWMA chart using time-varying control limit are presented in Table 3 when ARL 0 % 370 and n ¼ 5.Moreover, Tables A1-A2 in the Supplementary Material present the ARL and SDRL (given in the parenthesis) results of the Max-TEWMA chart using time-varying control limit when the ARL 0 is approximately equal to 185 and 250, and n ¼ 5.
Furthermore, in order to study the effect of the smoothing parameter k and sample size n, on the performance of the Max-TEWMA control chart using time-varying control limit, we randomly choose a shift combination ðd, qÞ ¼ ð0:25, 1:50Þ and draw the scatter-plots of the ARL 1 and SDRL 1 values versus k, for n 2 f3, 4, 5, 7g, and k 2 f0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.40, 0.50, 0.60, 0.70, 0.75, 0.80, 0.90, 1.00g, when ARL 0 % 370: Theses graphs are presented in Figures 1 and 2 for n ¼ 3, 4, 5 and 7, at ARL 0 % 370: From Tables 3 and A1 to A2 in the Supplementary Material, as well as Figures 1 and 2 we observe that, for fixed values of d, q and n, both ARL 1 and SDRL 1 decrease as the ARL 0 decreases and the ARL 1 decreases as k smoothing parameter decreases.Moreover, the performance of the Max-TEWMA chart in detecting small shifts in the process mean and/or variability is better with small k values, hence the sensitivity of the Max-TEWMA chart improves as k is getting smaller.For fixed n and ARL 0 values, the K 3 value increases, with an increase in the k value.It should be noted that, the proposed chart is more efficient in detecting small shifts in the process mean (d 0:50) and small upward shifts in the process dispersion (1:00 q < 1:50), while it is quite effective in identifying moderate and large shifts in the process mean (d > 0:50) and small to large downward shifts in the process dispersion (0:25 q < 1:00).Finally, Figures 1 and 2 indicate that both the ARL 1 and SDRL 1 decrease with an increase in the n value.

Performance comparison of control charts
In this point, we compare the performance of the proposed Max-TEWMA control chart with that of the Max-EWMA, Max-DEWMA and Max-GWMA control charts, that are briefly described in Section 2, in terms of 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 control charts, it is suggested to have a similar ARL 0 value.The smaller ARL 1 value in a certain shift combination ðd, qÞ the better the performance of a control chart.The ARL 0 value of the competing control charts is pre-fixed approximately at 370 and the sample size n is equal to 5. The performance of the Max-EWMA, Max-DEWMA and Max-GWMA charts is compared individually with that of the Max-TEWMA control chart.

Proposed Max-TEWMA chart versus Max-EWMA chart
Comparing the Max-TEWMA and Max-EWMA charts (see Tables 3 and 4), we observe that the first chart has lower ARL 1 values than the latter chart for most of the considered shift combinations ðd, qÞ in the mean and variability.Nevertheless, the competing control chart has better ARL performance than the proposed chart when k ¼ 0:10 at (d ¼ 0:25, q ¼ 1:25) and (d 2 ½0:00, 0:25, q ¼ 1:50), as well as k ¼ 0:20 at (d 0:50, q ¼ 1:50).Additionally, both charts perform similarly, for large shifts in the mean and large downward to small upward shifts in the variability (d > 1:50, 0:25 q 1:25).Considering the SDRL measure, the Max-EWMA chart with k 0:20 produces lower SDRL 1 values compared with the Max-TEWMA chart for small to moderate shifts in the mean and most of the considered shifts in the variability (d < 1:50, 0:50 q 2:00).In case of 0:30 k 0:50, the SDRL performance of the proposed chart improves for small to moderate shifts in the mean and large downward to moderate upward shifts in the dispersion (d < 1:50, 0:50 q 1:75).It should be mentioned that, the Max-TEWMA chart has lower SDRL 1 values for large shifts in the mean and all the considered shifts in the variability (d !1:50, 0:25 q 3:00).Finally, as the k parameter increases, the Max-TEWMA chart outperforms the Max-EWMA chart for both ARL and SDRL measures.

Proposed Max-TEWMA chart versus Max-DEWMA chart
The comparison between the Max-TEWMA and Max-DEWMA charts (see Tables 3 and 5), reveals that the Max-TEWMA chart with k < 0:20 is more effective than the Max-DEWMA chart in detecting moderate to large shifts in the mean and small to large upward and downward shifts in the dispersion (d > 0:50, 0:50 q 3:00), as well as small shifts in the mean and large shifts in the variability (d 0:50, q 2 f0:25, 2:50, 3:00g), whereas for the remaining shifts the opposite happens.As the k value increases, the performance of the proposed chart improves especially for small shifts in the mean and the variability.Particularly, when k ¼ 0:20 the proposed chart has better ARL performance than the competing chart for shift combinations (d 1:00, 0:25 q 1:00), (d 1:00, q > 2:00) and (1:00 < d 2:00, 0:25 q 3:00).In case of k 2 ½0:30, 0:50, the Max-DEWMA chart is less sensitive than the Max-TEWMA chart for shift combinations (d 1:00, 0:25 q 1:50), (d 1:00, q > 2:00) and (1:00 < d 2:00, 0:25 q 3:00) in the mean and the dispersion.Additionally, the proposed chart with k ¼ 0:80 is superior to the competing chart for most of the considered shift combinations (d, q).It should be noted that, both charts present similar performance for large shifts in the mean and downward shifts in the dispersion.(d > 2:00, q < 1:00).Finally, the Max-TEWMA and the Max-DEWMA charts with k ¼ 1:00 perform similarly for moderate to large shifts in the mean and all the examined shifts in the dispersion (d > 1:00, 0:25 q 3:00), while for the remaining shift combinations (d, q) the first chart is better.
Furthermore, the comparison between the Max-TEWMA(k ¼ 0:05) chart (see Table 3) and the run-length results of the Max-GWMA chart presented in Table 6 reveals that the first chart has lower ARL 1 values for small to moderate shifts in the mean and all the considered shifts in the variability (d 1:50, 0:25 q 3:00).Nevertheless, the competing chart has smaller ARL 1 values than the Max-TEWMA(k ¼ 0:05) chart for shift combinations (d 2 f0:00, 0:25g, q ¼ 1:25).It is to be noted that, according to Table 6, the Max-GWMA control chart has lower SDRL 1 values than the proposed chart for shift combinations (d 1:00, 0:50 q 2:50), while the opposite happens for the same amount of shifts in the mean and q 2 f0:25, 3:00g in the dispersion.Considering larger shifts in the mean, i.e., d > 1:50, the Max-TEWMA(k ¼ 0:05) chart has better ARL and SDRL performance for q > 1:00, whereas both charts perform similarly for q 1:00:

Overall performance comparison
Additionally, since we are interested in a wide range of shifts in the process mean and/or variance, it is of interest to measure the overall performance of the competing control charts.Hence, we compute the RMI (Han and Tsung 2006) measure, which is defined as where ARLðd i , q i Þ, i ¼ 1, 2, :::, N, is the ARL 1 value of a chart for a specific shift combination ðd i , q i Þ, ARL Ã ðd i , q i Þ is the smallest ARL 1 value among all the competing control charts for the specific shift combination ðd i , q i Þ and N is the number of shifts considered.The control chart with the smaller RMI value is considered better in its overall performance.Table 7 presents the RMI values using the ARL 1 results presented in Tables 3-5 and A3 to A6 in the Supplementary Material, for all the competing control charts when k 2 f0.05, 0.10, 0.20, 0.30, 0.50, 0.80, 1.00g, q 2 f0.70, 0.80, 0.90, 0.95g, a 2 f0.50, 0.60, 0.70, 0.80, 0.90, 1.10g, ARL 0 % 370 and n ¼ 5 over a wide range of shift combinations (d, q), where 0:00 d 3:00 and 0:25 q 3:00: According to results presented in Table 7, we observe that the proposed chart with k ¼ 0:05 has the best overall performance (RMI < 0.05).Moreover, the Max-TEWMA chart outperforms the Max-EWMA and Max-GWMA control charts over the whole range of shift combinations for all the considered k and ðq, aÞ values.It should be noted that, for k 2 f0:05, 0:10, 0:20g, both the Max-DEWMA and Max-TEWMA control charts perform similarly over the whole range of shift combinations (0:00 d 3:00, 0:25 q 3:00), while the latter is slightly more sensitive.As the k value increases, the overall detection ability of the Max-TEWMA chart enhances versus the Max-DEWMA control chart.Moreover, the Max-DEWMA chart is superior to the Max-EWMA and Max-GWMA charts over the whole process shifts.It is obvious that, the Max-GWMAðq 2 f0:90, 0:95g, a ¼ 0:90) and the Max-EWMA(k 2 f0:05, 0:10gÞ charts, as well as the Max-GWMAðq ¼ 0:80, a ¼ 0:80) and the Max-EWMA(k ¼ 0:20Þ charts perform similarly.Finally, the Max-EWMA, Max-DEWMA and Max-TEWMA charts with k ¼ 1:00 have similar performance over the whole range of shift combinations, with the proposed chart being slightly better.

Diagnostic ability
Moreover, we examine the diagnostic abilities of the Max-TEWMA, Max-EWMA and Max-DEWMA control charts.For each pair of (d, q) values, we simulate 1000 OOC signals for each of the charts.The rules of the Max-TEWMA (Table 2), Max-EWMA and Max-DEWMA charts are used to count the number of OOC points detected by each one of the charts.Setting ARL 0 % 370, and n ¼ 5 we consider the Max-TEWMA, Max-EWMA and Max-DEWMA charts with time-varying control limits and parameter combinations ðk, K 3 Þ ¼ ð0:10, 2:0351Þ, ðk, K 1 Þ ¼ ð0:10, 3:0467Þ, and ðk, K 2 Þ ¼ ð0:10, 2:3262Þ, respectively.The results obtained from the aforementioned study are displayed in Table 8.According to Table 8, it is observed that when the process is IC, the considered charts have same performance in terms of signaling an increase or a decrease in the process mean and dispersion.Furthermore, we observe that the Max-EWMA, Max-DEWMA and Max-TEWMA charts have comparable diagnostic abilities for the OOC cases.

Example 1
In order to illustrate the utilization of the Max-TEWMA chart, a simulated dataset is provided.A dataset is generated having 40 samples of size n ¼ 5, in which the quality characteristics of the process, X ij , i ¼ 1, 2, :::, and j ¼ 1, 2, :::, n, are mutually independent and follow the Normal distribution with mean l 1 ¼ l 0 þ dr 0 and standard deviation r 1 ¼ qr 0 : The first 20 samples are generated from Nðl 0 ¼ 0, r 2 0 ¼ 1Þ, referring to an IC situation, i.e., d ¼ 0 and q ¼ 1.However, a shift of l 1 ¼ l 0 þ 0:25r 0 (d ¼ 0:25) in the mean and a shift of r 1 ¼ 0:75r 0 (q ¼ 0:75) in the standard deviation are added to the remaining 20 samples.Setting ARL 0 % 370, we construct a Max-TEWMA chart with time-varying control limits and parameter combination ðk, K 3 Þ ¼ ð0:10, 2:0351Þ: The generated data, the charting statistic and the upper control limit values of the proposed chart are displayed in Table 9.
Moreover, we construct the Max-EWMA, Max-DEWMA and Max-GWMA charts, with timevarying control limits and parameter combinations ðk, K 1 Þ ¼ ð0:10, 3:0467Þ, ðk, K 2 Þ ¼ ð0:10, 2:3262Þ and ðq, a, KÞ ¼ ð0:90, 0:90, 3:0715Þ, respectively.It should be noted that, the K 1 , K 2 , K and K 3 control chart multipliers are obtained through Monte-Carlo simulations, so that ARL 0 % 370: The values of the charting statistics and the related upper control limits of the aforementioned charts are also presented in Table 9. Figures 3-5 show the Max-EWMA, Max-DEWMA and Max-GWMA control charts, whereas the proposed Max-TEWMA chart is displayed in Figure 6.From these Figures, we observe that the Max-TEWMA chart raises the first OOC signal at the 30th sample, the Max-DEWMA chart at the 36th sample, while the Max-EWMA and Max-GWMA charts fail to detect the shift.In other words, the Max-TEWMA control chart can detect shifts more quickly than the other control charts.

Example 2
In the second example, we demonstrate application of the Max-TEWMA control chart against the Max-EWMA, Max-DEWMA and Max-GWMA charts considering the real data presented in the example of Huang, Tai, and Lu (2014).The fill volume of soft-drink beverage bottles constitutes a significant quality characteristic.The manager of a beverage company in northern Taiwan is interested in understanding the fill volume of their own company's products.Therefore, 25 samples of    sample size n ¼ 5 were taken from a manufacturing process and are presented in Table 10.The estimated values of the process mean and standard deviation are 601.320and 7.093, respectively.

Concluding remarks
The present article develops a new single EWMA-type chart for simultaneously detecting shifts in the process mean and variability.The proposed chart extends the single Max-EWMA and Max-DEWMA charts to a single Triple EWMA chart, referred as Max-TEWMA control chart.The formulae of the mean and the variance of the Max-TEWMA statistic and, consequently, the upper control limit of the chart are calculated and provided in the Appendix.The proposed chart is evaluated in terms of the ARL and SDRL measures.The results reveal that the sensitivity of the proposed chart enhances as k is small and both the ARL 1 and SDRL 1 decrease with an increase in the sample size n.Furthermore, the Max-TEWMA chart is more efficient in detecting small shifts in the process mean and small upward shifts in the process dispersion, while it is quite effective in identifying moderate and large shifts in the process mean and small to moderate downward shifts in the process dispersion.
Moreover, the Max-TEWMA chart is compared with some well-known control charts, such as the Max-EWMA, Max-DEWMA and Max-GWMA charts.The comparison results indicate that the proposed chart is more efficient than the Max-EWMA and Max-GWMA charts.Nevertheless, these competing control charts, with small k, have lower SDRL 1 results for small shifts in the mean and the majority of the examined shifts in the dispersion.Furthermore, the Max-TEWMA chart is more efficient in detecting moderate shifts in the mean and small to large shifts in the dispersion compared with the Max-DEWMA chart for small k.It should be mentioned that, as the smoothing parameter k increases, the performance of the Max-TEWMA chart is better than that of the competing control charts, especially for small shifts in the mean and the variability.Additionally, according to the RMI measure the overall performance comparison reveals that the Max-TEWMA chart is effective over the whole range of shift combinations and constitutes a reliable alternative chart that we encourage quality practitioners to use.Finally, two illustrative examples are provided to explain the implementation of the proposed chart.
Proceeding in a similar way, we get In a similar manner, we get and From Eqs. ( 22) and ( 23), we get Therefore, From Eqs. ( 24) and ( 25), we get The above is obtained because of Y 02 ¼ Y 03 : It is to be noted that where and h ¼ ð1 À kÞ 2 : It is to be noted that c 2 ðiÞ !0 as i !1: From Eqs. (4) ( 8) and ( 17), we get Proceeding as above, it can be shown that It is to be noted that From Eqs. ( 1) and ( 2), it immediately follows that, for i ¼ 1, 2, 3, :::, Lemma 1 If U and V be two independently normally distributed random variables each with zero mean and unit variance and if M ¼ maxðjUj, jVjÞ, then the probability density function (pdf) of M is f ðmÞ ¼ 4/ðmÞð2UðmÞ À 1Þ, m > 0, where /ð:Þ is the pdf of N(0, 1) and Uð:Þ is the distribution function of N(0, 1).

d
2 or r ¼ S c 4 , as an estimate of r, is the average standard deviation.It should be mentioned that, d 2 and c 4 constants depend only on the sample size (Montgomery 2013).

Figure 2 .
Figure 2. SDRL 1 of the Max-TEWMA chart with time-varying control limit when ðd, qÞ ¼ ð0:25, 1:50Þ for various k and n at ARL 0 % 370: Table 8.A comparison of the diagnostic abilities of the Max-EWMA, Max-DEWMA and Max-TEWMA charts.-SIMULATION AND COMPUTATION V R

Table 2 .
Symbols to denote the source and direction of an OOC signal for the Max-TEWMA chart.jZ i3 j > UCL 3

Table 3 .
ARL and SDRL (in the parenthesis)values for the Max-TEWMA chart using time-varying control limit, when

Table 4 .
ARL and SDRL (in the parenthesis)values for the Max-EWMA chart using time-varying control limit, when

Table 5 .
ARL and SDRL (in the parenthesis)values for the Max-DEWMA chart using time-varying control limit, when

Table 6 .
Near optimal ðq, a, KÞ parameter combinations and the corresponding ARL and SDRL (in the parenthesis) values for the Max-GWMA chart using time-varying control limit when

Table 7 .
Overall performance comparison of the competing control charts using RMI measure.

Table 9 .
Data and calculation details of the Max-EWMA, Max-DEWMA, Max-GWMA and Max-TEWMA control charts using simulated data.IN STATISTICS -SIMULATION AND COMPUTATION V R COMMUNICATIONS Lemma 2 If U and V be two independently normally distributed random variables each with zero mean and unit variance and if M ¼ maxðjUj, jVjÞ, then