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。

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.

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

Bibliography
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