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

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


Introduction
There are two types of variation that are present in a production process, namely, the common causes and assignable causes of variation. The process is declared as in-control (IC) when it operates with the common causes of variation. Nevertheless, when assignable causes of variation arise from external sources, they lead to an out-of-control (OOC) process [34]. The control charts constitute an essential part of the Statistical Process Control (SPC) in detecting assignable causes of variation that may affect either the mean or variance of the process. They are also classified into location and dispersion charts, where the first are used to identify shifts in the process mean, while the latter are suitable for detecting shifts in the process variance.
The Shewhart-type charts, likeX, R and S charts, are efficient in detecting large shifts due to their memoryless property [34]. However, the memory-type charts are more sensitive in detecting small and moderate shifts because they take into consideration both the current and past information. The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts are the first memory-type charts, which were developed by Page [35,36] and Roberts [38], respectively. Shamma and Shamma [39] developed the Double EWMA (DEWMA) chart, Haq [21,22] proposed the Hybrid EWMA (HEWMA) chart, and Alevizakos et al. [4] introduced the Triple EWMA (TEWMA) chart. Sheu and Lin [41] proposed the generally weighted moving average (GWMA) chart as an extension of the EWMA chart, while Sheu and Hsieh [40] extended the GWMA chart to a Double GWMA (DGWMA) chart. Particularly, the DGWMA charting scheme is a weighted moving average of a weighted moving average, that is, the smoothing process is performed twice. More information about the DGWMA scheme can be found in Chiu and Sheu [19], Tai et al. [43], Kang and Baik [28], Huang et al. [25], Chiu and Lu [18], Lu [31], Alevizakos et al. [7], Karakani et al. [29], Alevizakos et al. [6], Mabude et al. [32] and Chatterjee et al. [16].
In many industrial applications, it is important to monitor the presence of shifts in the process dispersion rather than the process mean. Therefore, many researchers have developed memory-type control charts for monitoring the process variability. For instance, Castagliola [11] and Castagliola et al. [12] used a three-parameter logarithmic transformation to sample variance (S 2 ) in order to construct the S 2 -EWMA and S 2 -CUSUM charts, respectively. Furthermore, Castagliola et al. [14] introduced a Variable Sampling Interval (VSI) version of the S 2 -EWMA [11] chart, referred to as VSI S 2 -EWMA chart. Taking into account the aforementioned logarithmic transformation, Abbas, Riaz and Does [1] developed the CS-EWMA chart, Tariq et al. [44] introduced a HEWMA chart for monitoring the process variability (named as HEWMTn chart, but hereafter, for simplicity purposes, it is denoted by S 2 -HEWMA), Chatterjee et al. [15] proposed the S 2 -TEWMA chart for monitoring the process dispersion and Alevizakos et al. [5] developed the S 2 -GWMA chart as an expansion of the S 2 -EWMA chart. Other works about the control charting technique for monitoring the process variability are those of Castagliola et al. [13], Huwang et al. [26], Sheu and Lu [42], Abbasi et al. [3], Ali and Haq [8,9], Haq [23,24], Riaz et al. [37], Mahadik et al. [33], Abbasi et al. [2], Arshad et al. [10], Li and Mukherjee [30], and Zaman [45,46], to name a few.
To the best of our knowledge, as well as the research of Mabude et al. [32], most of the works on the DGWMA scheme are related with monitoring the process location or jointly monitoring the process location and variability. Therefore, motivated by Sheu and Hsieh [40], and Castagliola [11], the current article extends the work of Alevizakos et al. [5] to a DGWMA chart based on a three-parameter logarithmic transformation to S 2 , named as S 2 -DGWMA chart for monitoring the process variability. The proposed chart is compared with several dispersion control charts such as the S 2 -EWMA, CS-EWMA, S 2 -HEWMA, S 2 -TEWMA, S 2 -GWMA and VSI S 2 -EWMA charts. The remainder of this article is structured as follows. In Section 2, we present the proposed S 2 -DGWMA chart. A simulation study is conducted in Section 3 to evaluate its efficiency through the run-length distribution, while its performance is compared with those of the previously mentioned charts in Section 4. A real data example is presented in Section 5 and some concluding remarks are summarized in Section 6.

The proposed S 2 -DGWMA control chart
Assume that X ij , i = 1, 2, . . ., j = 1, 2, . . . , n, is the jth observation in the ith random sample of size n ( > 1) and X ij iid ∼ N(μ 0 , τ σ 0 ), where μ 0 and σ 0 are the corresponding IC values Table 1. Values of A(n), B(n), C(n), μ T (n), σ T (n) and DG 0 for n = 3, 4, . . . , 15. of the process mean and standard deviation. The process is considered to be IC, if τ = 1; otherwise, the process is declared as OOC and consequently, τ = 1. Here, we are interested in monitoring the shifts in the process variance from an IC value σ 2 0 to an OOC σ 2 1 = (τ σ 0 ) 2 , where τ = 1, considering that the process mean μ 0 remains stable. The sample variance S 2 i is given by . . is the sample mean. In order to monitor the process variability, the three-parameter logarithmic transformation applied to S 2 i [27], is utilized, i.e.
where a = A(n) − 2B(n) ln(σ 0 ), b = B(n), c = C(n)σ 2 0 and, the functions A(n), B(n) and C(n) depend only on n. Castagliola [11] first showed that, for a fixed n, if the constants a, b and c are suitably selected, then the statistic T i is approximately a normal random variable with mean μ T (n) and standard deviation σ T (n). Table 1 shows the values of A(n), B(n), C(n), μ T (n) and σ T (n) for n = 3, 4, . . . , 15, that were originally presented in Table I of Castagliola [11].
The plotting statistic of the proposed S 2 -DGWMA chart is defined through the following system of equations: where T i is given by Equation (2), q ∈ [0, 1) is the design parameter, α > 0 is the adjustment parameter and DG 0 = G 0 = A(n) + B(n) ln(1 + C(n)) are the starting values. The values of the DG 0 are provided in the last column of Table 1. Moreover, the DG i statistic follows approximately the normal distribution with mean E(DG i ) = μ T (n) and variance The time-varying control limits of the S 2 -DGWMA chart are given by where L is a positive control chart multiplier, when the process is IC. For large values of i, the asymptotic control limits of the S 2 -DGWMA chart are given by where W = lim i→∞ W i . Hereafter, for simplicity, we use the asymptotic control limits, given by Equation (5), in order to develop the S 2 -DGWMA chart. The proposed chart is designed by plotting the statistic DG i versus the sample number i. The process is declared as IC, when LCL < DG i < UCL; otherwise, it is considered to be OOC. It should be mentioned that the S 2 -DGWMA chart reduces to the S 2 -HEWMA chart when q = 1 − λ, α = 1 and λ = λ 1 = λ 2 , where 0 < λ 1 , λ 2 ≤ 1 are the smoothing constants of the S 2 -HEWMA chart.

Performance evaluation of the proposed chart
The average run-length (ARL) and the standard deviation of the run-length (SDRL) are most commonly used to measure the performance of a control chart. Particularly, the ARL is the average number of the charting statistics that must be drawn on a control chart until an OOC signal is triggered [34]. When the process dispersion is IC (τ = 1.00), a large value of ARL 0 is preferred. Nevertheless, when the process variability is OOC, that is, the standard deviation shifts from σ 0 to σ 1 = τ σ 0 (with τ = 1.00), a small OOC ARL (ARL 1 ) value is preferred. Here, both the ARL and SDRL measures are utilized in order to evaluate the performance of the proposed chart. Furthermore, we calculate the performance of the control chart over a range of shifts, through the expected ARL (EARL) which is defined as where ARL(τ ) is the ARL 0 if τ = 1 or the ARL 1 value corresponding to a chart specific shift (τ = 1) and f τ (τ ) is the probability density function of the magnitude of the process shift when τ ∈ [τ min , τ max ].
A Monte-Carlo simulation algorithm is performed in R statistical software to calculate the run-length distribution of the two-sided S 2 -DGWMA chart with asymptotic control limits (given by Equation (5)). The algorithm is run 10,000 iterations to calculate the average and the standard deviation of those 10,000 run-lengths. The steps of the simulation algorithm are briefly described as follows: (1) For a fixed n value, generate 10,000 random subgroups that follow the N(0, τ σ 0 ) distribution with σ 0 = 1. (2) For various q and a values, obtain the L value, such that the ARL 0 is approximately equal to the pre-fixed values. (3) Compute the S 2 i , T i , G i and DG i statistics for each subgroup (i.e. i = 1, 2, . . . , 10,000) using Equations (1), (2), and (3). (4) Calculate the asymptotic control limits given by Equation (5). (5) For the purpose of computing the run-length, compare each DG i statistic with the control limits given by Equation (5) for i = 1, 2, . . . , 10,000. If the process is OOC, stop the simulations and record the run-length value. (6) Repeat Steps (1) through (5) for 10,000 times and compute the ARL and the SDRL.
Particularly, Table 2 presents the L values of the S 2 -DGWMA control chart for various combinations of the design parameters (q, α) and sample size n = 3, 5, 7 and 9, when ARL 0 ≈ 370. Tables 3 and A1 in the Supplementary Material present the ARL, SDRL (in the parenthesis) and EARL results of the S 2 -DGWMA chart using asymptotic control limits with q ∈ {0.50, 0.60, 0.70, 0.80, 0.90, 0.95}, and α ∈ {0.70, 0.80, 0.90, 1.00, 1.20, 1.50}, when ARL 0 ≈ 370 as well as n = 5 and 9, respectively. Furthermore, Table A2 in the Supplementary Material presents the ARL, SDRL (in the parenthesis) and EARL results, along with the L values of the S 2 -DGWMA chart with the same (q, α) combinations using asymptotic control limits, when ARL 0 ≈ 200 and n = 5. The smallest ARL 1 value for each τ and n values is indicated with bold print in the aforementioned Tables, as well.

Performance comparisons of control charts
In the current section, we compare the performance of the proposed chart with that of the S 2 -GWMA, S 2 -EWMA, CS-EWMA, S 2 -TEWMA and VSI S 2 -EWMA charts for monitoring the process variability. For fair comparisons, it is recommended to have a similar pre-fixed ARL 0 value. The control chart with the lowest ARL 1 value in a certain shift τ in the process variability, can detect faster than the other control charts. Therefore, we consider two-sided asymptotic control limits for all the competing control charts, the ARL 0 is set approximately equal to 370 and the sample size n is set at 5. Furthermore, we consider the cases that ARL 0 ≈ 370 and n = 9, as well as ARL The plotting statistic of the S 2 -GWMA chart is given by where q ∈ [0, 1) is the design parameter, α > 0 is the adjustment parameter and G 0 = DG 0 is the starting value. The asymptotic control limits of the S 2 -GWMA chart are given by where L(> 0) is the control chart multiplier, Q = lim i→∞ Q i and Q i = i j=1 (q (j−1) α − q j α ) 2 , i = 1, 2, . . .. It should be noted that, the S 2 -GWMA chart reduces to the S 2 -EWMA chart when q = 1 − λ and α = 1.00, where λ ∈ (0, 1] is the smoothing constant of the latter chart. The S 2 -GWMA chart is constructed by plotting the statistic G i versus the sample number i. The process is declared as OOC, when G i ≤ LCL or G i ≥ UCL. According to Tables 3 and A1, the S 2 -DGWMA chart is more efficient than the S 2 -GWMA chart, as well as the S 2 -EWMA chart, for small shifts in the process variability. As q decreases, it becomes better for downward shifts and small to moderate upward shifts. See, for example, (i) the  Tables 3 and A1). The performance of the S 2 -DGWMA chart enhances compared with the S 2 -GWMA chart for small to moderate upward shifts as the sample size n increases and the parameter q decreases. In particular, (i) the S 2 -DGWMA(q = 0.80, α = 1.00, L = 2.515) chart is better than the S 2 -GWMA(q = 0.80, α = 1.00, L = 2.8215) (i.e.  Tables A1 and B1 in the Supplementary Material). In case the ARL 0 decreases, the proposed chart is slightly less efficient than the S 2 -GWMA chart, in detecting upward shifts in the dispersion with a decrease in the parameter q. For example, the S 2 -DGWMA(q = 0.60, α = 0.70, L = 2.560) chart is better than the S 2 -GWMA(q = 0.60, α = 0.70, L = 2.631) chart at 0.50 ≤ τ < 1.00 and 1.20 ≤ τ ≤ 1.30, and the S 2 -DGWMA(q = 0.50, α = 0.80, L = 2.620) chart is more efficient than the S 2 -GWMA(q = 0.50, α = 0.80, L = 2.632) chart at 0.50 ≤ τ ≤ 1.00 and 1.20 ≤ τ ≤ 1.30 (see Tables A2 and B2 in the Supplementary Material). It should be noted that, the EARL of the proposed chart is better than that of the S 2 -GWMA chart for most of the examined cases.

• S 2 -DGWMA chart versus CS-EWMA chart
The charting statistics of the CS-EWMA chart are given by where

• S 2 -DGWMA chart versus S 2 -TEWMA chart
The plotting statistic W i of the S 2 -TEWMA chart is given through the following system of equations: where λ ∈ (0, 1] is the smoothing constant, and W 0 = Y 0 = Z 0 = DG 0 are the starting values. The asymptotic control limits of the S 2 -TEWMA chart are given by where L(> 0) is the control chart multiplier. The S 2 -TEWMA chart is constructed by plotting the statistic W i versus the sample number i and the process raises an OOC signal, when W i ≤ LCL or W i ≥ UCL. Tables 3 and A3 indicate that the S 2 -DGWMA chart is more efficient for upward shifts. In addition, its performance improves for downward shifts as q(λ) increases (decreases). For instance, the  Tables 3 and A1 -A3). According to Table 4, the S 2 -DGWMA chart is the most efficient in detecting small to moderate upward, and large downward shifts. Furthermore, the CS-EWMA is better for small to large downward shifts, whereas the S 2 -GWMA chart is the most effective for large upward shifts. Table 5 reveals that, as the sample size n increases, the S 2 -DGWMA chart is the most sensitive for small to moderate upward shifts, the CS-EWMA chart is the most efficient for small to moderate downward shifts, while the S 2 -GWMA chart is the most efficient for large upward and downward shifts. Table 6 shows that as the ARL 0 decreases, the S 2 -DGWMA chart is the most efficient for upward shifts, whereas the CS-EWMA chart is the most effective for downward shifts. It should be noted that the S 2 -DGWMA and S 2 -TEWMA charts perform almost similarly for downward shifts. Finally, the proposed chart is more efficient than the S 2 -EWMA and the S 2 -HEWMA charts for most of the considered τ values, while it is better than the S 2 -GWMA for small downward to moderate upward shifts in the variability. Table 4. ARL values and the corresponding near optimal design combinations of the control charts at n = 5 and ARL 0 ≈ 370.  Table 5. ARL values and the corresponding near optimal design combinations of the control charts at n = 9 and ARL 0 ≈ 370.  Table 6. ARL values and the corresponding near optimal design combinations of the control charts at n = 5 and ARL 0 ≈ 200. Furthermore, we compare the performance of the proposed chart with that of the VSI S 2 -EWMA chart, that is, briefly described as follows. The sampling interval, and particularly, the time between two successive samples T i and T i+1 , depends on the current value of the Z i = λT i + (1 − λ)Z i−1 , λ ∈ (0, 1] statistic. A longer sampling interval h L is utilized when the Z i statistic lies in the region R L = [LWL, UWL], defined as  [14] present the minimal ATS results of the VSI S 2 -EWMA chart when n = 5 and 9, that they were obtained through a Markov Chain approach. Comparing Tables 4 and 5  above, with Tables 2 and 3 of Castagliola et al. [14], we observe that the proposed chart is more efficient than the VSI S 2 -EWMA chart for small shifts, and vice versa for the rest range of shifts.

Illustrative example
In the current section, we demonstrate the application of the S 2 -DGWMA control chart against the S 2 -EWMA, S 2 -HEWMA, S 2 -TEWMA and S 2 -GWMA control charts considering the real data given in DeVor et al. [20]. The aforementioned dataset is widely used by numerous scholars, such as Chen et al. [17] and Tariq et al. [44]. These data are measurements of the inside diameter of cylinder bores in an engine block. Table 7  The charting statistics of these charts are presented in Table 7, as well. Figures 1-4 plot the S 2 -EWMA, S 2 -HEWMA, S 2 -TEWMA, S 2 -GWMA control charts, whereas the proposed S 2 -DGWMA chart is displayed in Figure 5. We observe that the S 2 -DGWMA chart triggers an OOC signal at the 1, 10 and 11 samples, the S 2 -GWMA chart at the 6th sample, while the remaining charts fail to detect any shift.

Concluding remarks
In the present article, we develop a new memory-type control chart for monitoring both upward and downward shifts in the process variability. The proposed chart extends the S 2 -GWMA chart to the S 2 -DGWMA chart by applying a three-parameter logarithmic transformation to the S 2 on the DGWMA control charting scheme. The proposed chart is evaluated through the ARL and SDRL measures, using asymptotic control limits. The results indicate that for a fixed value of the α (q) parameter, the ARL performance of the S 2 -DGWMA chart improves for small to moderate downward and small to large   Moreover, the S 2 -DGWMA chart is compared with several well-known memory-type control charts for monitoring the process variability. The results indicate that the S 2 -DGWMA chart is more effective in detecting small shifts in the process variability, and particularly, more efficient in identifying upward shifts. Specifically, the proposed chart with α < 1.00 is more efficient than the S 2 -HEWMA chart for detecting moderate downward to large upward shifts, while the performance of the S 2 -DGWMA chart with α > 1.00 improves in detecting downward shifts against the S 2 -HEWMA chart as q rises. It is more efficient than the S 2 -EWMA and S 2 -GWMA charts for downward to moderate upward shifts. The S 2 -DGWMA chart is better than the CS-EWMA chart for large downward and all the considered upward shifts, while it is better than the S 2 -TEWMA chart for  upward shifts. It should be noted that, the VSI S 2 -EWMA chart is less sensitive than the S 2 -DGWMA chart for small shifts, whereas the opposite is observed for the remaining considered shifts in the variability. Furthermore, an illustrative example is displayed to explain the application of the proposed chart. Consequently, our findings indicate that the proposed chart is a reliable alternate control chart that quality practitioners should utilize for monitoring the process variability. For future research, it would be interesting to investigate the VSI version of the S 2 -DGWMA chart.