指數加權移動平均管制法在幾何卜瓦松製程中之品管設計與使用

A Study on Detecting Small Shifts in Quality Levels with the Geometric Poisson Process Using EWMA Control Schemes

陳慶文(Ching-Wen Chen)；吳一聲(Yi-Sheng Wu)；蘇國瑋(Kuo-Wei Su)

幾何卜瓦松 ； 指數加權移動平均 ； 平均連串長度 ； 馬可夫鏈 ； geometric Poisson ； exponentially weighted moving average EWMA ； average run length ARL ； Markov chain

品質學報

13卷1期（2006 / 03 / 01）

85 - 97

繁體中文

指數加權移動平均(EWMA)管制法比傳統的3個標準差管制圖更能及早發現製程微量的變化，但有效的EWMA管制法需配合著能正確地描述品質特性的機率分配。在製程管制當中，缺點數(defects)之分配常被假設為卜瓦松分配(Poisson distribution)。然而當代的生產製程通常是非常的複雜，且加工步驟又多，以卜瓦松分配為基礎的製程管制常會誤估了製程的平均缺點數與製程變異。通常這種情形應以複合卜瓦松過程(compound Poisson process)為基礎的製程管制方法才能提供適宜的製程品管。本研究以Brook and Evans的馬可夫鏈法來計算在幾何卜瓦松製程下，某特定規格之EWMA管制圖所需之平均連串長度(ARL)，並建構出幾何卜瓦松EWMA管制圖(geometric Poisson EWMA control schemes)，以供製程偵測微小變動及增進製程品質之用。

It is well known that the exponentially weighted moving average (EWMA) control schemes can better detect small shift of process mean in a short period of time than the conventional 3-sigma control charts do. However, an effective EWMA control scheme has to be with correct underling distribution in describing the characteristic of process quality. In process control, the common situation is to assume that the Poisson distribution holds for count data, such as the conventional c-chart. In contemporary modern production environment, the production process is usually more complex than others. The conventional Poisson-based control schemes usually under- or overestimates the mean and variance of defects in the process. In fact the control schemes based on compound Poisson distribution are more appropriately used in controlling the defects. Applying the Markov chain approach by Brook and Evans to calculate the average run length (ARL), we take one step further, in this study, to develop the EWMA control scheme for the geometric Poisson production process to detect a small sustained shift in the mean. Using the EWMA control schemes we can detect possible shift at the early beginning of the process change, and improve the quality level of the production process.

1. Albin, S. L.,Friedman, D. J.(1989).The impact of clustered defect distributions in ic fabrication.Management Sciences,35(9),1066-1078.
2. Borror, C. M.,Champ, C. W.,Rigdon, S. E.(1998).Poisson EWMA control charts.Journal of Quality Technology,30(4),352-361.
3. Brook, D.,Evans, D. A.(1972).An approach to the probability distribution of CUSUM run length.Biometrika,59(3),539-549.
4. 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.Communication in Statistics-Computation and Simulation,20,191-204.
5. Chen, C. W.,Randolph, P. H.,Liou, T. S.(2005).Using CUSUM control schemes for monitoring quality levels in compound Poisson production environment: the geometric Poisson process.Quality Engineering,17(2),207-217.
6. Crowder, S. V.(1989).Design of exponentially weighted moving average schemes.Journal of Quality Technology,21,155-162.
7. Friedman, D. J.,Albin, S. L.(1991).Clustered defects in IC fabrication: impact on process control charts.IEEE Transactions on Semiconductor Manufacturing,4(1),36-42.
8. Fu, J. C.,Spiringa, F. A.,Xie, H.(2002).On the average run lengths of quality control schemes using a Markov chain approach.Statistics & Probability Letters,56,369-380.
9. Gan, F. F.(1990).Monitoring Poisson observations using modified exponentially weighted moving average control charts.Communications in Statistics-Simulation and Computation,19,103-124.
10. Gardiner, J. S.(1987).Seattle, Washington,University of Washington.
11. Hunter, J. S.(1986).The exponentially weighted moving average.Journal of Quality Technology,18(4),203-219.
12. Jackson, E. J.(1972).All count distribution are not alike.Journal of Quality Technology,4(2),86-92.
13. Johnson, N. L.,Kotz, S.(1969).Discrete Distribution.Boston, MA:Houghton Mifflin Company.
14. Lucas, J. M.,Crosier, R. B.(1990).Exponentially weighted moving average control schemes: properties and enhancements.Technometrics,32,1-12.
15. Montgomery, D. C.(1996).Introduction to Statistical Quality Control.New York, NY:John Wiley & Sons, Inc..
16. Page, E. S.(1954).Continuous inspection schemes.Biometrika,41,100-115.
17. Parzen, E.(1962).Stochastic Processes.San Francisco, CA:Holden-Day. Inc..
18. Randolph, H. P.,Sahinoglu, M.(1995).A stopping rule for a compound Poisson random variable.Applied Stochastic Models and Data Analysis,11,135-143.
19. Roberts, S. W.(1959).Control charts based on geometric moving average.Technometrics,1,234-250.
20. Ross, S. M.(1993).Introduction to Probability Models.San Diego, CA:Academic Press, Inc..
21. Ross, S. M.(1996).Stochastic Process.New York, NY:John Wiley & Sons.
22. Sheaffer, R. L.,Leavenworth, R. S.(1976).The negative binomial model for counts in units of varying size.Journal of Quality Technology,8(3),158-163.
23. Sherbrooke, C. C.(1966).Memorandum RM-4831-PR.Santa Monica, CA:The Rand Corporation.
24. Shewhart, W. A.(1931).Economic Control of Quality of Manufactured Product.New York, NY:Van Nostrand.
25. Stapper, C. H.(1985).The Effects of Wafer to Wafer Defect Density Variations on Integrated Circuit Defect and Fault Distribution.IBM Journal of Research Development,29(1),87-97.
26. 陳慶文(2001)。實驗貝氏分析在幾何卜瓦松製程品質管制上之應用。品質學報，8(1)，107-131。