在可靠度領域中，對於監控元件或系統的失效間隔時間(time between failures)一直是感興趣的。失效間隔時間可以使用Weibull分配進行建模，因為Weibull分配可以更靈活的描述不同形狀的失效時間分配。在失效間隔時間為Weibull分配和單次測量(individual measurements)的情況下，分別使用EWMA-Tukey control chart (EWMA-TCC)、CUSUM-TCC、t-chart與normal-transformed EWMA-TCC (NEWMA-TCC)這四種管制圖(control chart)進行監控與比較。本研究的目的為在不同的Weibull形狀參數 (shape parameter) 以及尺度參數 (scale parameter) 中，監控失效間隔時間的平均偏移。其中三個測量值，像是平均連串長度 (average run length)、串連長度標準差(standard deviation of run length)和中位數串連長度 (median run length)，使用〖10〗^5次迭代的蒙地卡羅模擬(Monte-Carlo simulation) 進行計算。相對平均指標(relative mean index)的測量也被用來比較不同管制圖的整體表現。以兩個闡述性範例說明四個管制圖應用於監控失效間隔時間的平均偏移。結果發現當EWMA-TCC的λ=0.25時，在失效間隔時間為Weibull分配且形狀參數固定的情況下檢測平均偏移的表現優於其他管制圖。EWMA-TCC還能夠根據相對平均指標快速檢測出尺度參數與形狀參數的變化，這種情況在業界非常的普遍。另一方面，在監控形狀參數的改變上，t-chart是最有效的管制圖。

In the reliability area, monitoring the time between failures (TBF) of the component or system is often of interest. The TBF can be modeled by using Weibull distribution because it is more flexible to describe the failure time distributions with many different shapes. Four different control charts including EWMA-Tukey control chart (EWMA-TCC), CUSUM-TCC, t-chart, and normal-transformed EWMA-TCC (NEWMA-TCC) are compared to monitor the Weibull-distributed TBF with individual measurements. The objective of this research is to monitor the TBF mean shifts under the different values of the Weibull scale and shape parameters. Three measures such as average run length (ARL), standard deviation of run length (SDRL), and median run length (MDRL) are computed using Monte-Carlo simulation with 105 iterations. Relative mean index (RMI) is also used as a measure for comparing the overall performance of different control charts. Two illustrative examples are provided to show the application of four control charts for monitoring the TBF mean shifts. Based on the results, EWMA-TCC with λ=0.25 performed better than the other control charts for detecting the mean shift of the Weibull-distributed TBF with fixed shape parameter. EWMA-TCC also able to detect the change of the scale and the shape parameter quickly in terms of RMI. These parameter changes are very common in the industry. On the other hand, t-chart is the most effective control chart for monitoring the shape parameter change.

1. Shafae MS, Dickinson RM, Woodall WH, Camelio JA. Cumulative sum control charts for monitoring Weibull‐distributed time between events. Quality and Reliability Engineering International 2015; 31(5):839-849. doi:10.1002/qre.1643
2. Batson RG, Jeong Y, Fonseca DJ, Ray PS. Control charts for monitoring field failure data. Quality and Reliability Engineering International 2006; 22(7):733-755. doi:10.1002/qre.725
3. Pascual F, Nguyen D. Moving range charts for monitoring the Weibull shape parameter with single observation samples. Quality and Reliability Engineering International 2011; 27(7):905-919. doi:10.1002/qre.1180
4. Pascual F. Individual and moving ratio charts for Weibull processes. Stochastic Orders in Reliability and Risk 2013, Springer:331-350. doi:10.1007/978-1-4614-6892-9_17
5. Chen JT. A Shewhart-type control scheme to monitor Weibull data without subgrouping. Quality and Reliability Engineering International 2014; 30(8):1197-1214. doi:10.1002/qre.1542
6. Akhundjanov SB, Pascual F. Moving range EWMA control charts for monitoring the Weibull shape parameter. Journal of Statistical Computation and Simulation 2015; 85(9):1864-1882. doi:10.1080/00949655.2014.907574.
7. Wang FK. MaxEWMA control chart for a Weibull process with individual measurements. Quality and Reliability Engineering International 2017; 33(2):369-379. doi:10.1002/qre.2013.
8. Gan F. Design of optimal exponential CUSUM control charts. Journal of Quality Technology 1994; 26(2): 109-124.
9. Gan F. Designs of one-and two-sided exponential EWMA charts. Journal of Quality Technology 1998; 30(1): 55.
10. Chan LY, Xie M, Goh TN. Cumulative quantity control charts for monitoring production processes. International Journal of Production Research 2000; 38(2): 397-408.
11. Xie M, Goh TN, Ranjan P. Some effective control chart procedures for reliability monitoring. Reliability Engineering & System Safety 2002; 77(2):143-150. doi:10.1016/S0951-8320(02)00041-8.
12. Khoo MBC, Xie M. A study of time-between-events control chart for the monitoring of regularly maintained systems. Quality and Reliability Engineering International 2009; 25(7):805-819. doi:10.1002/qre.977
13. Alemi F. Tukey’s control charts. Quality Management Health Care 2004; 13(4):216-221.
14. Sukparungsee S. Robustness of Tukey’s control chart in detecting a changes of parameter of skew distributions. International Journal of Applied Physics and Mathematics 2012; 2(5):379-382.
15. Sukparungsee S. Asymmetric Tukey’s control chart robust to skew and non-skew process observation. International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 2013; 7(8):1275-1278.
16. Khaliq QUA, Riaz M, Alemi F. Performance of Tukey’s and individual/moving range control charts. Quality and Reliability Engineering International 2015; 31(6):1063-1077. doi:10.1002/qre.1664.
17. Khaliq QUA, Riaz M. Robust Tukey-CUSUM control chart for process monitoring. Quality and Reliability Engineering International 2015; 32(3):933-948. doi:10.1002/qre.1804.
18. Khaliq QUA, Riaz M, Ahmad S. On designing a new Tukey-EWMA control chart for process monitoring. The International Journal of Advanced Manufacturing Technology 2016; 82:1-23. doi:10.1007/s00170-015-7289-6.
19. Aslam M. A mixed EWMA–CUSUM control chart for Weibull‐distributed quality characteristics. Quality and Reliability Engineering International 2016; 32(8):2987-2994. doi:10.1002/qre.1982
20. Montgomery DC. Statistical quality control. New York: Wiley; 2009.
21. Dickinson RM, Roberts DAO, Driscoll AR, Woodall WH, Vining GG. CUSUM charts for monitoring the characteristic life of censored Weibull lifetimes. Journal of Quality Technology 2014; 46(4):340-358.
22. Polunchenko AS, Sokolov G, Tartakovsky AG. Optimal design and analysis of the exponentially weighted moving average chart for exponential data. Journal of Applied Statistics 2013; 15(2):55-82. arXiv:1307.7126v3
23. Faraz A, Saniga EM, Heuchenne C. Shewhart control charts for monitoring reliability with Weibull lifetimes. Quality and Reliability Engineering International 2015; 31(8):1565-74. doi:10.1002/qre.1692
24. Jiao JR, Helo PT. Optimization design of a CUSUM control chart based on Taguchi's loss function. The International Journal of Advanced Manufacturing Technology 2008; 35(11):1234-1243. doi:10.1007/s00170-006-0803-0
25. Han D, Tsung F. A reference-free cuscore chart for dynamic mean change detection and a unified framework for charting performance comparison. Journal of the American Statistical Association 2006; 101(473):368-86. doi:10.1198/016214505000000556
26. Zhang CW, Ye Z, Xie M. Monitoring the shape parameter of the Weibull renewal process. IISE Transactions 2017; 0(0):1-14. doi:10.1080/24725854.2016.1278315