Evaluating Sensitivity of Double EWMA Chart for ARL Under Trend SAR(1) Model and Applications
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The goal of this study is to offer the precise average run length (ARL) on the Double Exponentially Weighted Moving Average (double EWMA) control chart for the data underlying the first-order seasonal autoregressive (SAR(1)L) with trend model. A comparison was made between the explicit formula and the computed ARL obtained using the numerical integral equation (NIE) approach, employing four quadrature methods: the midpoint, Simpson’s, trapezoidal, and Boole’s rules. The comparison was based on accuracy percentage (%Acc) and computation time (in seconds). The results showed that there was not much variation in accuracy between the ARL results of the explicit ARL and ARL via the NIE method. The findings indicate that the explicit ARL and NIE approaches produce very consistent accuracy values; however, the explicit formula is significantly more rapid (instantaneous compared to 1.5–26 seconds). The advantage of the double EWMA chart compared to the extended EWMA chart in identifying process changes is demonstrated, encompassing evaluations under both one-sided and two-sided setups with varied LCL values. The results are additionally corroborated by sensitivity measures (AEQL, PCI, RMI) and checked with actual durian export data, guaranteeing that the conclusions are firmly established in both simulated and empirical evidence.
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[1] Freitas, L. L. G., Kalbusch, A., Henning, E., & Walter, O. M. F. C. (2021). Using statistical control charts to monitor building water consumption: A case study on the replacement of toilets. Water (Switzerland), 13(18), 2474. doi:10.3390/w13182474.
[2] Kovarik, M., & Klimek, P. (2012). The Usage of Time Series Control Charts for Financial Process Analysis. Journal of Competitiveness, 4(3), 29–45. doi:10.7441/joc.2012.03.03.
[3] Khan, N., Aslam, M., Jeyadurga, P., & Balamurali, S. (2021). Monitoring of production of blood components by attribute control chart under indeterminacy. Scientific Reports, 11(1), 922. doi:10.1038/s41598-020-79851-5.
[4] Raza, S. M. M., Sial, M. H., Hassan, N. ul, Mekiso, G. T., Tashkandy, Y. A., Bakr, M. E., & Kumar, A. (2024). Use of improved memory type control charts for monitoring cancer patients recovery time censored data. Scientific Reports, 14(1), 5604. doi:10.1038/s41598-024-55731-0.
[5] Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technometrics, 1(3), 239–250. doi:10.1080/00401706.1959.10489860.
[6] Page, E. S. (1954). Continuous Inspection Schemes. Biometrika, 41(1–2), 100–115. doi:10.1093/biomet/41.1-2.100.
[7] Naveed, M., Azam, M., Khan, N., & Aslam, M. (2018). Design of a Control Chart Using Extended EWMA Statistic. Technologies, 6(4), 108–122. doi:10.3390/technologies6040108.
[8] Khan, N., Aslam, M., & Jun, C. (2016). Design of a Control Chart Using a Modified EWMA Statistic. Quality and Reliability Engineering International, 33(5), 1095–1104. doi:10.1002/qre.2102.
[9] Shamma, S. E., & Shamma, A. K. (1992). Development and Evaluation of Control Charts Using Double Exponentially Weighted Moving Averages. International Journal of Quality & Reliability Management, 9(6), 18–25. doi:10.1108/02656719210018570.
[10] Mahmoud, M. A., & Woodall, W. H. (2010). An Evaluation of the Double Exponentially Weighted Moving Average Control Chart. Communications in Statistics - Simulation and Computation, 39(5), 933–949. doi:10.1080/03610911003663907.
[11] Ibazizen, M., & Fellag, H. (2003). Bayesian estimation of an AR(1) process with exponential white noise. Statistics, 37(5), 365–372. doi:10.1080/0233188031000078042.
[12] Brook, D., & Evans, D. A. (1972). An approach to the probability distribution of CUSUM run length. Biometrika, 59(3), 539–549. doi:10.1093/biomet/59.3.539.
[13] Champ, C. W., & Rigdon, S. E. (1991). A comparison of the Markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts. Communications in Statistics - Simulation and Computation, 20(1), 191–204. doi:10.1080/03610919108812948.
[14] Peerajit, W. (2023). Developing average run length for monitoring changes in the mean on the presence of long memory under seasonal fractionally integrated MAX model. Mathematics and Statistics, 11(1), 34–50. doi:10.13189/ms.2023.110105.
[15] Supharakonsakun, Y. (2021). Statistical design for monitoring process mean of a modified EWMA control chart based on autocorrelated data. Walailak Journal of Science and Technology, 18(12), 19813. doi:10.48048/wjst.2021.19813.
[16] Ng, P. S., Goh, S. Y., Saha, S., & Yeong, W. C. (2021, April). An enhanced double EWMA chart for monitoring the process mean shifts. International Conference on Robotics, Vision, Signal Processing and Power Applications, Springer Nature, Singapore. doi:10.1007/978-981-99-9005-4_48.
[17] Capizzi, G., & Masarotto, G. (2003). An adaptive exponentially weighted moving average control chart. Technometrics, 45(3), 199-207. doi:10.1198/004017003000000023.
[18] Karoon, K., Areepong, Y., & Sukparungsee, S. (2023). Modification of ARL for detecting changes on the double EWMA chart in time series data with the autoregressive model. Connection Science, 35(1), 2219040. doi:10.1080/09540091.2023.2219040.
[19] Horng Shiau, J. J., & Ya-Chen, H. (2005). Robustness of the EWMA control chart to non-normality for autocorrelated processes. Quality Technology & Quantitative Management, 2(2), 125-146. doi:10.1080/16843703.2005.11673089.
[20] Bualuang, D., & Peerajit, W. (2022). Performance of the CUSUM Control Chart Using Approximation to ARL for Long-Memory Fractionally Integrated Autoregressive Process with Exogenous Variable. Applied Science and Engineering Progress, 16(2), 5917. doi:10.14416/j.asep.2022.05.003.
[21] Karoon, K., & Areepong, Y. (2024). Enhancing the Performance of an Adjusted MEWMA Control Chart Running on Trend and Quadratic Trend Autoregressive Models. Lobachevskii Journal of Mathematics, 45(4), 1601-1617. doi:10.1134/S1995080224601577.
[22] Muangngam, T., Areepong, Y., & Sukparungsee, S. (2024). Performance Evaluation of Extended EWMA Chart for AR Model with Exogenous Variables. HighTech and Innovation Journal, 5(4), 901–917. doi:10.28991/HIJ-2024-05-04-03.
[23] Sunthornwat, R., Sukparungsee, S., & Areepong, Y. (2024). The Development and Evaluation of Homogenously Weighted Moving Average Control Chart based on an Autoregressive Process. HighTech and Innovation Journal, 5(1), 16–35. doi:10.28991/HIJ-2024-05-01-02.
[24] Karoon, K., & Areepong, Y. (2025). The Efficiency of the New Extended EWMA Control Chart for Detecting Changes Under an Autoregressive Model and Its Application. Symmetry, 17(1), 104. doi:10.3390/sym17010104.
[25] Peerajit, W. (2025). Optimizing the EWMA Control Chart to Detect Changes in the Mean of a Long-Memory Seasonal Fractionally Integrated Moving Average and an Exogenous Variable Process with Exponential White Noise and its Application to Electrical Output Data. WSEAS Transactions on Systems and Control, 20, 25–41. doi:10.37394/23203.2025.20.4.
[26] Polyeam, D., & Phanyaem, S. (2025). Enhancing Average Run Length Efficiency of the Exponentially Weighted Moving Average Control Chart under the SAR(1)L Model with Quadratic Trend. Malaysian Journal of Fundamental and Applied Sciences, 21(3), 2174–2193. doi:10.11113/mjfas.v21n3.4290.
[27] Polyeam, D., & Phanyaem, S. (2025). Application of EWMA Control Chart for Analyzing Changes in SAR(P)L Model with Quadratic Trend. Mathematics and Statistics, 13(3), 163–174. doi:10.13189/ms.2025.130305.
[28] Almousa, M. (2020). Adomian Decomposition Method with Modified Bernstein Polynomials for Solving Nonlinear Fredholm and Volterra Integral Equations. Mathematics and Statistics, 8(3), 278–285. doi:10.13189/ms.2020.080305.
[29] Sofonea, M., Han, W., & Shillor, M. (2005). Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman and Hall/CRC, Boca Raton, United States. doi:10.1201/9781420034837.
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