A Study on Using the Moving Average Chart for Monitoring Time-between-Events

以移動平均管制圖監控事件間隔時間之研究

10.6220/joq.2014.21(6).04

陳榮泰(Jung-Tai Chen)

事件間隔時間 ； 連串法則 ； 移動平均 ； 平均連串長度－不偏 ； 馬可夫鏈 ； time-between-events ； runs rule ； moving average ； ＂ARL＂-unbiased ； Markov chain

品質學報

21卷6期（2014 / 12 / 31）

479 - 500

英文

傳統欲管制製程缺點數常訴諸c chart，然而當現今高產出製程其製程缺點率很低的時候，c chart之管制效果即受到限制；因此，文獻藉由管制缺點發生之間隔時間來達成管制製程缺點的目的，此等管制圖稱之為t chart或cumulative quantity control chart (CQC chart)；最近研究提出以t chart搭配連串法則（簡稱t-runs chart）以提升此等管制圖之管制效果。在判定製程是否在管制狀態時，原始t chart以單一觀測值為依據，而t-runs chart則以兩連續觀測值為依據。本研究提出移動平均管制圖（moving average chart, MA chart）監控缺點之事件間隔時間，為評估MA chart之管制效果，本研究首先針對當事件間隔時間為指數分配時，導出MA chart其平均連串長度（average run length, ARL）之估計方法，研究結果顯示：MA chart在大部分的情況下，MA chart會比t-runs chart有較好的管制效果；另外，在實務操作上，MA chart也較t-runs chart來得簡易。最後本研究也提供將MA chart應用在當事件間隔時間為韋伯分配（Weibull Distribution）或珈瑪分佈（Gamma Distribution）分配時之管制結果，以為實務應用之參考。

Among attributes control charts, the count of nonconformities is often used to monitor the quality of a manufacturing process. When the non-conformity rate in the process is very low, traditional control charts for nonconformities become ineffective. In such a situation, the process control may resort to monitoring the time between successive non-conformities (referred to as the time-between-events, TBE). Control schemes for monitoring TBE can range from simple to complicated, depending on the purpose of the monitoring. Previous studies have proposed a Shewhart-type chart with a runs rule (＂t＂-runs chart) to enhance control performance. This research considers a simple moving average chart (＂MA＂ chart) for the surveillance of TBE. An approximate analytical approach to compute the performance of the ＂MA＂ chart is first derived for exponential TBE data. The ＂MA＂ chart is then compared to the ＂t＂-runs chart in terms of the average run length and the standard deviation of the run length. The comparison results show that the ＂MA＂ chart has the better performance than the ＂t＂-runs chart in most cases. Because the MA chart is simple to interpret and implement, it is a good alternative to the ＂t＂-runs chart. The analytical approach is extended to obtain the design parameters for the ＂MA＂ chart for monitoring Weibull and Gamma TBE. The design parameters for the ＂MA＂ chart are provided for practical implementation.

1. Acosta-Mejia, C. A.,Pignatiello, J. J., Jr.(2000).Monitoring process dispersion without subgrouping.Journal of Quality Technology,32(2),89-102.
2. Chan, L. Y.,Xie, M.,Goh, T. N.(2000).Cumulative quantity control charts for monitoring production processes.International Journal of Production Research,38,397-408.
3. Cheng, C.-S.,Chen, P.-W.(2011).An ARL-unbiased design of time-between-events control charts with runs rules.Journal of Statistical Computation and Simulation,81(7),857-871.
4. Gan, F. F.(1998).Designs of one- and two-sided exponential EWMA charts.Journal of Quality Technology,30(1),55-69.
5. Gan, F. F.(1994).Design of optimal exponential CUSUM control charts.Journal of Quality Technology,26(2),109-124.
6. Huang, X.,Pascual, F.(2011).ARL-unbiased control charts with alarm and warning lines for monitoring Weibull percentiles using the first-order statistic.Journal of Statistical Computation and Simulation,81(11),1677-1696.
7. Jones, L. A.,Champ, C. W.(2002).Phase I control charts for times between events.Quality and Reliability Engineering International,18(6),479-488.
8. Khoo, M. B. C.,Xie, M.(2009).A study of time-between-events control chart for the monitoring of regularly maintained systems.Quality and Reliability Engineering International,25(7),805-819.
9. Liu, J. Y.,Xie, M.,Goh, T. N.(2006).CUSUM chart with transformed exponential data.Communications in Statistics-Theory and Methods,35(10),1829-1843.
10. Liu, J. Y.,Xie, M.,Goh, T. N.,Chan, L. Y.(2007).A study of EWMA chart with transformed exponential data.International Journal of Production Research,45(3),743-763.
11. Liu, J. Y.,Xie, M.,Goh, T. N.,Ranjan, P.(2004).Time-between-events charts for on-line process monitoring system.Proceedings of the IEEE International Engineering Management Conference: Innovation and Entrepreneurship for Sustainable Development
12. Montgomery, D. C.(1997).Introduction to Statistical Quality Control.New York:John Wiley.
13. Pignatiello, J. J., Jr.,Acosta-Mejia, C. A.,Rao, B. V.(1995).The performance of control charts for monitoring process dispersion.Proceedings of the 4th Industrial Engineering Research Conference
14. Sürücü, B.,Sazak, H. S.(2009).Monitoring reliability for a three-parameter Weibull distribution.Reliability Engineering & System Safety,94(2),503-508.
15. Wong, H. B.,Gan, F. F.,Chang, T. C.(2004).Designs of moving average control chart.Journal of Statistical Computation and Simulation,74(1),47-62.
16. Xie, M.,Goh, T. N.,Ranjan, P.(2002).Some effective control chart procedures for reliability monitoring.Reliability Engineering and System Safety,77(2),143-150.
17. Zhang, C. W.,Xie, M.,Goh, T. N.(2005).Economic design of Exponential charts for time between events monitoring.International Journal of Production Research,43(23),5019-5032.
18. Zhang, C. W.,Xie, M.,Liu, J. Y.,Goh, T. N.(2007).A control chart for the Gamma distribution as a model of time between events.International Journal of Production Research,45(23),5649-5666.

1. Chen, Jung-Tai(2018).A CONTROL CHART BASED ON MOVING AVERAGE AND MOVING RATIO FOR MONITORING WEIBULL DATA.品質學報,25(4),211-240.