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系統識別號 U0002-1206200812412800
中文論文名稱 監控變異數成分之模擬Shewhart管制圖
英文論文名稱 The Simulated Shewhart Control Chart for Monitoring the Variance Components
校院名稱 淡江大學
系所名稱(中) 統計學系碩士班
系所名稱(英) Department of Statistics
學年度 96
學期 2
出版年 97
研究生中文姓名 謝亦瑋
研究生英文姓名 Yi-Wei Hsieh
學號 695650258
學位類別 碩士
語文別 英文
口試日期 2008-05-17
論文頁數 55頁
口試委員 指導教授-蔡宗儒
委員-吳柏林
委員-廖敏治
委員-蘇懿
中文關鍵字 管制圖  變異數成分  隨機效應  平均連串長度  品質管制 
英文關鍵字 Control chart  Average run length  Variance components  Random effects  Quality control 
學科別分類 學科別自然科學統計
中文摘要 在許多工業製程中, 製程的總變異可被分解為變異數成分(variance component) 之組合, 而變異數成分即為由某特定原因所造成之變異。倘若能針對變異數成分個別進行監控以取代對總變異的監控方式, 當製程出現失控訊息時, 便可縮小尋找可歸屬原因的範圍。本論文討論當製程使用單因子隨機效應模型下, 如何對其變異數成分進行監控。文獻上已有學者對此問題進行研究, Chang 和Gan [1] 提出近似迴歸法(approximate regression method) 來解決此一問題, 但此方法受限於某些參數組合的限制, 限縮了此一方法的使用範圍, 因此本論文提出以數值的方法來建立Shewhart 管制圖以監控變異數成分。模擬的結果顯示本文所提出的Shewhart 管制圖之績效優於近似迴歸法所建構的Shewhart 管制圖。文中並將本論文建立的管制圖運用在真實的工業資料中。
英文摘要 In many manufacturing processes, the overall process variation can be decomposed into relevant components of variation. If the associated special causes of respective
variance components can be identified, it is more effective and appropriate to monitor these components with separate control chart instead of monitoring the overall
variance with single control chart. This thesis develops Shewhart control charts for monitoring the variance components under the random effects model with single-
factor design based on a numerical method. A numerical study is conducted for the comparison of performances based on the proposed method with the other existing
methods in the literature. The numerical results indicate that the proposed method is more sensitive to detect the changes of variance components in process. Moreover,
the proposed method is illustrated with real manufacturing data.
論文目次 1 Introduction 1
2 Literature Review 6
2.1 The Random Effects Model with Single-factor Design . . . . . . . . . . . . 6
2.2 Estimation of Variance Components . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Shewhart Control Charts based on Variance Components . . . . . . . . . . 10
3 The Proposed Method 13
3.1 The Simulated One-sided Shewhart ˆσ2
L Control Chart . . . . . . . . . . . . 13
3.2 The Simulated Two-sided Shewhart ˆσ2
L Control Chart . . . . . . . . . . . . 16
4 Numerical Study 19
4.1 The Determination of Parameters for the Proposed Control Chart . . . . . 19
4.2 The Measurement of Performance . . . . . . . . . . . . . . . . . . . . . . . 21
5 Examples 45
6 Conclusions 52
Bibliography 54

List of Tables
4.1 The numerical ARLs for r = 15, n = 3, and m = 300. . . . . . . . . . . . . 21
4.2 The ARLs and the squared MSRLs for α = 0.0020, r = 5, n = 2, and σǫ = 1. 29
4.3 The ARLs and the squared MSRLs for α = 0.0020, r = 4, n = 5, and σǫ = 1. 30
4.4 The ARLs and the squared MSRLs for α = 0.0020, r = 5, n = 4, and σǫ = 1. 31
4.5 The ARLs and the squared MSRLs for α = 0.0020, r = 10, n = 2, and
σǫ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6 The ARLs and the squared MSRLs for α = 0.0020, r = 3, n = 10, and
σǫ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.7 The ARLs and the squared MSRLs for α = 0.0020, r = 6, n = 5, and σǫ = 1. 34
4.8 The ARLs and the squared MSRLs for α = 0.0020, r = 10, n = 3, and
σǫ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.9 The ARLs and the squared MSRLs for α = 0.0020, r = 3, n = 15, and
σǫ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.10 The ARLs and the squared MSRLs for α = 0.0020, r = 5, n = 9, and σǫ = 1. 37
4.11 The ARLs and the squared MSRLs for α = 0.0020, r = 9, n = 5, and σǫ = 1. 38
4.12 The ARLs and the squared MSRLs for α = 0.0020, r = 2, n = 5, and σǫ = 1. 39
4.13 The ARLs and the squared MSRLs for α = 0.0020, r = 2, n = 10, and
σǫ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.14 The ARLs and the squared MSRLs for α = 0.0020, r = 5, n = 6, and σǫ = 1. 41
4.15 The ARLs and the squared MSRLs for α = 0.0020, r = 15, n = 2, and
σǫ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.16 The ARLs and the squared MSRLs for α = 0.0020, r = 15, n = 3, and
σǫ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

List of Figures
4.1 The flow chart for the simulation procedure . . . . . . . . . . . . . . . . . 28
4.2 The scatter plot for in-control ARLs of the proposed control chart under
some selected parameter combinations of (r, n, m). . . . . . . . . . . . . . 44
5.1 (a) Two-sided Shewhart ˆσ2
ǫ control chart with α = 0.0050. (b) One-sided
Shewhart ˆσ2
L control chart with α = 0.0010. . . . . . . . . . . . . . . . . . 50
5.2 (a) Two-sided Shewhart ˆσ2
ǫ control chart with α = 0.0050. (b) One-sided
Shewhart ˆσ2
L control chart with α = 0.0010. . . . . . . . . . . . . . . . . . 51
參考文獻 Bibliography
[1] Chang, T. C. and Gan, F. F. (2004). Shewhart charts for monitoring the variance
components. Journal of Quality Technology, 36(3): 293-308.
[2] Kim, K. S. and Yum, B. J. (1999). Control charts for random and fixed components
of variation in the case of fixed wafer locations and measurement positions. IEEE
Transactions on Semiconductor Manufacturing, 12(2): 214-228.
[3] Montgomery, D. C. (2005). Design and Analysis of Experiments. 6th ed., John Wiley
and Sons, New York, NY.
[4] Roes, K. C. and Does, R. J. M. (1995). Shewhart-type charts in nonstandard situa-
tions. Technometrics, 37(1): 15-24.
[5] Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance
components. Biometrics Bulletin, 2: 110-114.
[6] Woodall, W. H. and Thomas, E. V. (1991). Statistical process control with several
components of common cause variability. IIE Transactions, 27(6): 757-764, Dec.
1995.
[7] Yashchin, E. (1994). Monitoring variance components. Technometrics, 36(4): 379-
393.
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