在生產過程中，製程常會受到許多因素干擾而產生變異，進而對品質特性產生影響，降低品質水準。當製程因干擾而產生變異時，統計製程管制(Statistical Process Control，簡稱SPC)能迅速地偵測出造成製程變異的可歸屬原因，以利品質管制人員採取修正措施並降低不良品的產生數量 。
本論文延伸Cassady 和 Nachlas (2006) 所提出之三個分類水準Shewhart 管制圖到三個分類水準的指數加權移動平均(Exponentially Weighted Moving Average，EWMA)管制圖及Shewhart-EWMA 管制圖 。在本研究中，管制圖的管制界限是在給定管制狀態下的平均連串長度，使用馬可夫鏈方法計算得到
的 。基本上，本論文所提出的管制圖可以改善三個分類水準Shewhart 管制圖在對製程小偏移的偵測速度。

The thesis extends the three-level Shewhart control chart proposed by Cassady and Nachlas [8] to exponentially weighted moving average and Shewhart-EWMA control charts for monitoring the quality of three-level (conforming,
marginal, nonconforming) products. The control limits of the proposed control chart are established based on the zero-state average run lengths using Markov chain approximation. Basically, the proposed control charts improve the performance of the three-level Shewhart control chart signi¯cantly and are
able to detect small shifts in a process more quickly.

Contents

1 Introduction...........................................1
2 The Three-level Shewhart control chart.................4
3 The Proposed Control Charts............................8
3.1 The Three-level EWMA Control Chart...................8 
3.2 The Three-level Shewhart-EWMA Control Chart.........10 
4 Performance Measures..................................12
4.1 The Average Run Length..............................12
4.2 The Calculation of ARL for the Three-level EWMA 
    Control Chart.......................................13
4.3 The Calculation of ARL for the Three-level Shewhart-    EWMA Control Chart..................................16
5 Numerical study.......................................20
6 Conclusions...........................................44

List of Tables

5.1 The out-of-control ARLs under delta1_1=1.50 and delta_2=1.14. . . . . . . . . . . . 22
5.2 The out-of-control ARLs under delta_1=2.00 and delta_2=1.39. . . . . . . . . . . . 23
5.3 The out-of-control ARLs under delta_1=2.50 and delta_2=1.60. . . . . . . . . . . . . 24
5.4 The out-of-control ARLs under delta_1=3.00 and delta_2=1.78. . . . . . . . . . . . . 25
5.5 The out-of-control ARLs under delta_1=3.50 and delta_2=1.94. . . . . . . . . . . . . 26
5.6 The out-of-control ARLs under de1ta_1=4.00 and delta_2=2.09. . . . . . . . . . . . . 27
5.7 The out-of-control ARLs under delta_1=2.00 and delta_2=1.27. . . . . . . . . . . . . 28
5.8 The out-of-control ARLs under delta_1=2.50 and delta_2=1.49. . . . . . . . . . . . . 29
5.9 The out-of-control ARLs under delta_1=3.00 and delta_2=1.68. . . . . . . . . . . . . 30
5.10 The out-of-control ARLs under delta_1=3.50 and delta_2=1.85. . . . . . . . . . . . . 31
5.11 The out-of-control ARLs under delta_1=4.00 and delta_2=2.00. . . . . . . . . . . . . 32
5.12 The out-of-control ARLs under delta_1=4.50 and delta_2=2.14. . . . . . . . . . . . . 33
5.13 The out-of-control ARLs under delta_1=3.00 and delta_2=1.47. . . . . . . . . . . . . 34
5.14 The out-of-control ARLs under delta_1=3.50 and delta_2=1.66. . . . . . . . . . . . . 35
5.15 The out-of-control ARLs under delta_1=4.00 and delta_2=1.82. . . . . . . . . . . . . 36
5.16 The out-of-control ARLs under delta_1=4.50 and delta_2=1.97. . . . . . . . . . . . . 37
5.17 The out-of-control ARLs under delta_1=5.00 and delta_2=2.11. . . . . . . . . . . . . 38
5.18 The out-of-control ARLs under delta_1=5.50 and delta_2=2.23. . . . . . . . . . . . . 39
5.19 The out-of-control ARLs under lambda=0.1 and N=101. . . . . . . . . . . . . . 40
5.20 The out-of-control ARLs under lambda=0.15 and N=101. . . . . . . . . . . . . 41
5.21 The out-of-control ARLs under lambda=0.2 and N=101. . . . . . . . . . . . . . 42
5.22 The out-of-control ARLs under lambda=0.3 and N=101. . . . . . . . . . . . . . 43

List of Figures

4.1 In-control partition of interval. . . . . . . . . . . . . . . . . . . . . . . . . . 13

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