Simulation modeling with memory-type control charts for monitoring the process variability

: Memory-type control charts, renowned for their effectiveness in identifying small deviations in the process variance, are commonly used to monitor the process variability. In this article, we introduce a new tool, the Quadruple Exponentially Weighted Moving Average (QEWMA) chart, which is designed for the specific purpose of monitoring changes in the process variability. We refer to this chart as the S 2 -QEWMA chart. The performance of the S 2 -QEWMA chart is assessed through an extensive series of Monte-Carlo simulations, carefully considering the run-length distribution. Comparing it with other well-known memory-type charts, it becomes evident that the S 2 -QEWMA chart excels in its ability to effectively detect small shifts in the process dispersion. To illustrate the practical application of this chart, we provide an example


Introduction
In Statistical Process Control (SPC), two types of variations exist within a production process: common causes and assignable causes.A production process is considered to be in statistical control (IC) when it is influenced solely by common causes of variation.Conversely, when assignable causes of variation stem from external sources, they result in a process going out of statistical control (OOC).Control charts play a pivotal role in SPC by detecting assignable causes of variation that can impact process parameters, specifically the mean or variance of the process.These charts can be categorized into two types: location charts, which are effective at identifying deviations in the process mean, and dispersion charts, which are well-suited for detecting variations in process dispersion.
Shewhart-type control charts primarily rely on the most recent observations, making them effective at detecting large shifts in process parameters.Conversely, memory-type control charts, such as the cumulative sum (CUSUM) chart [1,2] and the exponentially weighted moving average (EWMA) chart [3], take into account both current and past data, rendering them more sensitive in identifying small to moderate shifts.Additionally, innovations have been introduced in this field.Shamma and Shamma [4] and Zhang and Chen [5] developed the Double EWMA (DEWMA) chart, while Sheu and Lin [6] extended the EWMA chart to create the generally weighted moving average (GWMA) chart.Haq [7,8] proposed the Hybrid EWMA (HEWMA) chart, and Alevizakos, Chatterjee, and Koukouvinos [9,10] introduced the Triple EWMA (TEWMA) and Quadruple EWMA (QEWMA) charts for monitoring the process mean.
In this current article, inspired by the research of Castagliola [11] and Alevizakos, Chatterjee, and Koukouvinos [10], we introduce a novel control chart for monitoring process dispersion.This chart is based on a 3-parameter logarithmic transformation to S 2 and is referred to as the S 2 -QEWMA chart.To assess its effectiveness, we conduct a comparative study with other control charts, including the S 2 -EWMA, S 2 -CUSUM, CS-EWMA, S 2 -HEWMA, S 2 -TEWMA, S 2 -GWMA, and S 2 -DGWMA charts, using asymptotic control limits.To evaluate these control charts, we employ several Monte-Carlo simulations and consider well-established performance measures such as the average runlength (ARL) and the standard deviation of the run-length (SDRL).These comparisons reveal the efficiency of the proposed memory-type chart in detecting minor shifts in process variability.
The remainder of this article is organized as follows: In Section 2, we will develop the S 2 -QEWMA chart.In Section 3, we will examine the performance of this newly developed chart.Following that, in Section 4, we will compare its performance with the previously mentioned memory-type dispersion charts using measures like the ARL and the SDRL.Section 5 will provide an illustrative example to explain how to implement the S 2 -QEWMA chart.Concluding remarks will be summarized in Section 6, and additional technical details of the S 2 -QEWMA chart can be found in the Appendix.

The proposed 𝐒 𝟐 -QEWMA control chart
Consider a sample (or a subgroup) X k1 , . . ., X kn , of n(> 1) independent normal distributed N(µ 0 , τσ 0 ) random variables, where µ 0 and σ 0 are assumed to be the IC process mean and standard deviation, respectively.Here, k represents the sample number, with k taking on values of 1, 2, . ... If τ = 1, then the process is considered to be IC, whereas the process is declared as OOC when τ ≠ 1.Our objective is to promptly detect a shift in the process dispersion, from the IC σ 0 2 value to the OOC σ 1 2 = (τσ 0 ) 2 , where τ ≠ 1, while ensuring that the process mean remains at its IC value (µ 0 ).
The proposed S 2 -QEWMA chart considers a 3-parameter (a, b, c) logarithmic transformation applied to S 2 [40,41], thus where is the sample variance and is the sample mean.According to Castagliola [11], the proper selection of the a, b, and c, implies that the statistic T k ≈ N(µ T (n), σ Τ 2 (n)).Table 1 provides the values of A(n), B(n), C(n), µ T (n) and σ T (n) for sample size n ∈ {3, 4, . . ., 15}, that originally presented in Table I of Castagliola [11].The plotting statistic Q k of the S 2 -QEWMA control chart for monitoring the process variability is given by where λ is the smoothing parameter with 0 < λ ≤ 1, and are the starting values.The Q 0 values are also provided in Table 1 for n ∈ {3, 4, . . ., 15}.
The mean of the statistic Q k is given by The variance of the statistic Q k is defined as where V(d, k) is given by (5) and d = (1 − λ) 2 .For large values of k (k → ∞), the asymptotic variance of the statistic Q k becomes where The derivation of the mean and the variance of the statistic Q k is provided in the Appendix in detail.
Consequently, the control limits of the S 2 -QEWMA chart are given by with L >0 being the control chart multiplier.For simplicity purposes, the asymptotic control limits are used for the construction of the S 2 -QEWMA control chart, by computing the variance of the statistic Q k given in Eq. ( 6).The S 2 -QEWMA chart is designed by plotting the statistic Q k versus the subgroup number k.The process is considered to be OOC when otherwise, it is said to be IC.

Performance evaluation of the 𝐒 𝟐 -QEWMA chart
In the current section, we examine the efficiency of the S 2 -QEWMA chart.Traditionally, the statistical performance of a control chart is measured using the ARL, the SDRL and the percentile points.Particularly, the ARL is described as the average number of statistics that must be plotted on a chart until an OOC signal is raised.When the process variability is IC (τ = σ 1 σ 0 ⁄ = 1), a large value of IC ARL (ARL 0 ) is suggested to avoid false alarms.Nevertheless, when the process is OOC, i.e. τ ≠ 1, a small OOC ARL (ARL 1 ) value is preferable so as to detect the shift in the process variability quickly.Here, both the ARL and SDRL measures are used to examine the performance of the S 2 -QEWMA control chart.
The run-length distribution of the proposed S 2 -QEWMA chart is evaluated via a Monte-Carlo simulation algorithm using the R statistical software.The algorithm is run 10000 iterations, so as to calculate the mean and the standard deviation of the 10000 run-lengths.We assume that the process for the IC state is normally distributed with µ 0 = 0 and σ 1 = τσ 0 (τ = 1.00), whereas the OOC process follows the Normal distribution with mean µ 1 = 0 and standard deviation σ 1 = τσ 0 (τ ≠ 1.00).Furthermore, the statistical design of the S 2 -QEWMA chart requires the finding of the (λ, L) design parameter combinations, in order to achieve a pre-fixed ARL 0 value for a specified value of the sample size n.Consequently, the L values are obtained via Monte-Carlo simulations considering the asymptotic control limits of the S 2 -QEWMA chart given in Eq. ( 8), by calculating the asymptotic variance of statistic Q k given in Eq. ( 6), for various λ and n values when ARL 0 ≈200, 370 and 500.
• Generally, the efficiency of the S 2 -QEWMA chart improves, as the sample size n increases.
• The S 2 -QEWMA chart is more sensitive in detecting the upward shifts than the downward shifts in the variability.For instance, when ARL 0 ≈370 and n = 5, the ARL 1 values of the S 2 -QEWMA control chart with λ = 0.10, 0.20 and 0.30, at τ = 0.90, are 78.08,82.17 and 101.20, respectively, whereas the corresponding ARL 1 values at τ = 1.10 are 48.85,63.86, and 71.19.Nevertheless, the results show that the proposed chart with λ = 0.50 is less efficient in detecting large upward shifts than large downward shifts in the variability.

Performance comparisons
Here, we compare the performance of the S 2 -QEWMA chart with that of some recently developed memory-type control charts in the literature, such as the S 2 -GWMA, S 2 -EWMA, S 2 -CUSUM, CS-EWMA, S 2 -HEWMA, S 2 -TEWMA and S 2 -DGWMA charts.We use run length measures, like the ARL and the SDRL.For a pre-fixed ARL 0 value, the control chart with the smaller ARL 1 value can detect the shift faster compared with the other competing control charts.Consequently, in order to have fair comparisons, we take into consideration these control charts assuming two-sided asymptotic control limits, ARL 0 ≈370 as well as n = 5.Tables A3 to A8 in the Supplementary Material present the ARL and SDRL (in the parenthesis) results of these charts, for the same τ values (0.50 ≤ τ ≤ 2.00) as in Section 3. Note that the design parameters of the S 2 -GWMA, S 2 -EWMA, S 2 -CUSUM, CS-EWMA, S 2 -HEWMA, S 2 -TEWMA and S 2 -DGWMA charts are obtained through Monte-Carlo simulations such that ARL 0 ≈370 and n = 5.The considered control charts are briefly described, and compared individually with the proposed S 2 -QEWMA chart.

According to Tables
The plotting statistics of the S 2 -CUSUM chart are given by where the statistic T k is given by Eq. (1), K(≥ 0) is the reference value and C 0 − = C 0 + = 0 are the starting values.The C k − and C k + statistics are plotted against the decision interval H.The process is declared as OOC, if either of the two statistics is plotted above H(≥ 0).Table A4 in the Supplementary Material presents the ARL and SDRL (in the parenthesis) results of the S 2 -CUSUM chart for various (K, H) combinations when ARL 0 ≈370 and n = 5.
Tables A2 and A4 in the Supplementary Material indicate that the S 2 -QEWMA chart is more efficient than the S 2 -CUSUM chart in identifying large downward to small upward shifts in the variability.For example, the S 2 -QEWMA chart with λ ∈ {0.15, 0.20, 0.25} is more effective than the S 2 -CUSUM chart with K = 0.50 at 0.80 ≤ τ ≤ 1.20.Furthermore, the S 2 -QEWMA chart with λ = 0.25 is better than the S 2 -CUSUM chart with K = 1.00 at 0.60 ≤ τ ≤ 1.20, and the S 2 -QEWMA chart with λ = 0.40 is more efficient than the S 2 -CUSUM chart with K = 1.25 at 0.50 ≤ τ < 1.00 as well as 1.10 ≤ τ ≤ 1.30.Nevertheless, the opposite is observed for the ARL measure in case of moderate to large upward shifts.For instance, the S 2 -CUSUM chart with K = 0.50 has lower ARL 1 values than the S 2 -QEWMA chart with λ ∈ {0.15, 0.20, 0.25} at τ > 1.20.Finally, the SDRL performance of the S 2 -QEWMA chart is better than that of the S 2 -CUSUM chart for most of the examined cases.

• 𝐒 𝟐 -QEWMA chart versus CS-EWMA chart
The charting statistics of the CS-EWMA chart are defined as where  A5 in the Supplementary Material for various (λ, K CS , H CS ) combinations when ARL 0 ≈370 and n = 5.

• 𝐒 𝟐 -QEWMA chart versus 𝐒 𝟐 -HEWMA chart
The plotting statistic Y k of the S 2 -HEWMA chart is given through the following system of equations where T k is given by Eq. ( 1), λ 1 , λ 2 ∈ (0, 1] are the smoothing parameters, and Y 0 = Z 0 = Q 0 are the starting values.Given λ 1 ≠ λ 2 , the asymptotic control limits of the S 2 -HEWMA chart are given by where L(> 0) is the control chart multiplier.When λ 1 = λ 2 , the asymptotic control limits of the S 2 -HEWMA chart are given by The S 2 -HEWMA control chart is designed by plotting the statistic Y k versus the subgroup number k.The process is considered to be IC, if LCL < Y k < UCL.Table A6 in the Supplementary Material provides the ARL and the SDRL (in the parenthesis) values of the S 2 -HEWMA chart for various (λ 1 , λ 2 , L) combinations when ARL 0 ≈370 and n = 5.A2 and A6 in the Supplementary Material indicates that the proposed chart is better than the S 2 -HEWMA chart in detecting small deviations in the process dispersion.Particularly, as the λ value increases, the ARL performance of the S 2 -QEWMA chart improves in detecting moderate to large downward shifts compared with the competing chart.Additionally, the proposed chart has better SDRL performance than that of the S 2 -HEWMA chart for most of the examined cases and τ values.For example, the S 2 -QEWMA (λ = 0.15) chart has lower ARL 1 values at 0.90 ≤ τ ≤ 1.10, as well as lower SDRL 1 values at 0.50 ≤ τ ≤ 2.00, in comparison with the S 2 -HEWMA (λ 1 = 0.15, λ 2 ∈ {0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.50}) chart.Furthermore, comparing the S 2 -QEWMA (λ = 0.35) and S 2 −HEWMA (λ 1 = 0.35, λ 2 ∈ {0.35, 0.40, 0.50}) charts, we observe that the first chart has better ARL performance at 0.70 ≤ τ < 1.00 and τ = 1.10, as well as better SDRL performance at 0.50 ≤ τ < 1.00 and 1.10 ≤ τ ≤ 2.00 than the latter chart.

The comparison of the results between Tables
•   -QEWMA chart versus   -TEWMA chart The plotting statistic W k of the S 2 -TEWMA chart is given through the following system of equations where T k is given by Eq. ( 1), λ ∈ (0, 1] is the smoothing constant, and W 0 = Y 0 = Z 0 = Q 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 k versus the sample number k and the process raises an OOC signal, when W k ≤ LCL or W k ≥ UCL.The ARL and SDRL (in the parenthesis) values of the S 2 -TEWMA chart are presented in Table A7 in the Supplementary Material for various (λ, L) combinations when ARL 0 ≈370 and n = 5.
•   -QEWMA chart versus   -DGWMA chart The plotting statistic of the S 2 -DGWMA chart is defined through the following system of equations where the statistic T k is given by Eq. ( 1), q ∈ [0, 1) is the design parameter, α > 0 is the adjustment parameter and DG 0 = G 0 = Q 0 are the starting values.The asymptotic control limits of the S 2 -DGWMA chart are given by where The S 2 -DGWMA chart is designed by plotting the statistic DG k versus the sample number k.The process is declared as IC, when LCL < DG k < UCL; otherwise, it is considered to be OOC.It is important to note 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 .Table A8 in the Supplementary Material presents the ARL and SDRL (in the parenthesis) values of the S2-DGWMA chart for various (q, α, L) combinations when ARL 0 ≈370 and n = 5.It is to be noted that the ARL and SDRL results of the S 2 -DGWMA (q = 1 − λ, α = 1) chart are presented in Table A6 for λ = λ 1 = λ 2 .
The comparison of the results between Tables A2 and A8 in the Supplementary Material shows that the S 2 -QEWMA chart is better than the S 2 -DGWMA chart in detecting small shifts in the process dispersion.Furthermore, as the λ value increments, the ARL performance of the S 2 -QEWMA chart is better in detecting moderate to large downward shifts in comparison with the competing chart.Additionally, the newly developed chart has better SDRL performance than that of the S 2 -DGWMA chart for most of the examined scenarios and τ values.For instance, the S 2 -QEWMA (λ = 0.25) chart has lower ARL 1 values at 0.80 ≤ τ ≤ 1.20 in comparison with the S 2 -DGWMA (q = 0.75, α = 1.20) chart.Furthermore, the S 2 -QEWMA (λ = 0.30) chart has better SDRL 1 results at τ ≤ 1.00 and τ ≥ 1.40 in comparison with the S 2 -DGWMA (q = 0.70, α = 0.80) chart.

Conclusions
In this article, we introduce a novel control chart called the S 2 -QEWMA chart, which utilizes a three-parameter logarithmic transformation to the sample variance, serving as an EWMA-type chart for monitoring the process dispersion.We conduct numerous Monte-Carlo simulations to determine the design parameters for the S 2 -QEWMA chart.Our evaluation study reveals that this chart exhibits increased sensitivity as the sample size grows.Additionally, we recommend using small  values for detecting small deviations in the process variability, while larger λ values are more suitable for identifying moderate to large upward shifts.Furthermore, we perform a comparative analysis of the newly proposed S 2 -QEWMA chart against several established memory-type control charts designed for monitoring the process variability, including the S 2 -GWMA, S 2 -EWMA, S 2 -CUSUM, CS-EWMA, S 2 -HEWMA, S 2 -TEWMA, and S 2 -DGWMA

B. Derivation of an explicit form of 𝐐 𝐤
Eq. ( 2) can be rewritten as From Eq. ( 20), we get after algebraic simplification the following C. Derivation of (  ) From Eq. ( 21), we get Again, after simplification, we get From above, it follows that (  ) =   ().
2, ... , λ ∈ (0, 1] is the smoothing parameter, M 0 − = M 0 + = 0 are the starting values, and K ′ = K CS √ λ 2−λ is the reference value with K CS ≥ 0. The statistics M k − and M k + are plotted against the decision interval Η ′ = H CS √ λ 2−λ , while the process raises an OOC signal if either M k − or M k + is plotted over the Η ′ .The ARL and the SDRL (in the parenthesis) values for the CS-EWMA chart are displayed in Table

Table 3 .
Data and Calculation Details