近年來，製程能力指標已被多數品管工程師廣泛地應用在品質管制方面，以評估製程是否合乎能力水準。然而這些品管工程師通常只是簡單地利用樣本觀察值的資料來計算其製程能力指標的估計值，並且直接以此估計值來判斷製程是否符合能力水準，這樣的處理方式其實是不恰當的。因為此估計值是由點估計所得到的單一數值，能提供給我們的資訊較少，所以並無法評估此估計值的精確度。但是在區間估計中，則以點估計的標準誤來建立區間估計值的範圍，亦即將標準誤的衡量一起納入區間估計中。
    在本文中，為了比較兩個製程表現之差異，我們關心的是兩個製程之製程能力指標Cpmk之比值和差異。在點估計方面，分別比較自然估計法、拔靴法和摺刀法之偏差與均方差的表現；在區間估計方面，我們利用Boyles(1991)法、三個拔靴法和摺刀法來建立兩個製程之製程能力指標Cpmk比值之信賴區間。並利用Chen and Hsu(1995)之漸近常態分配法、三個拔靴法和摺刀法來建立兩個製程之製程能力指標Cpmk差異之信賴區間。我們使用常態分配所產生的樣本分別對不同方法作模擬比較，並以覆蓋率高低來判斷不同方法之優劣。最後，我們給一個實例示範如何使用製程能力指標Cpmk去比較兩個製程能力之優劣。

In recent years, Process Capability Indices (PCIs) have been applied in the quality control by most practitioners, to evaluate the capability of a production process. However, the approach of these practitioners usually simply look at the index value calculated from the given sample and then making a decision on whether the given process is capable or not is intuitively reasonable but not reliable because sampling errors are ignored. Therefore, the confidence interval approach should be provided to decide whether the production process is capable or not.
      For comparing two processes, we consider to estimates the ratio and the difference of two capability indices Cpmk. For point estimation, the bias and the mean square error (MSE) of the given point estimation method are simulated and compared. With respect to the confidence interval estimation, we use the Boyles(1991) method, three bootstrap methods, and the Jackknife method to construct the confidence interval of the ratio of two Cpmk indices. Furthermore, we use the Chen and Hsu’s(1995) method, three bootstrap methods, and the Jackknife method to estimate the difference of two Cpmk indices. The samples generated from normal distribution are used for simulation study and the performance of all methods based the highest average probability are evaluated. At least, one example is given to demonstrate how to compare two processes by using all proposed methods.

目錄
第一章	緒論	1
1.1 研究背景與動機	1
1.2 研究目的	3
1.3 研究範圍	4
1.4 研究架構	4
第二章 相關文獻探討與回顧	6
2.1 常態下之製程能力指標	6
2.1.1 Cp指標	6
2.1.2 Cpk指標	8
2.1.3 Cpm指標	9
2.1.4 Cpmk指標	9
2.2 製程能力指標分析的應用	14
2.3 拔靴模擬法	16
第三章 使用 Cpmk 指標來比較兩個製程之製程能力-點估計	18
3.1 兩個常態製程能力比值Cpmk1/Cpmk2和差異Cpmk1-Cpmk2之點估計	18
3.1.1 拔靴法點估計的建構	20
3.1.2 摺刀法點估計的建構	20
3.2 常態製程能力指標比值Cpmk1/Cpmk2和差異Cpmk1-Cpmk2之不同估計法的偏差及均方差的模擬比較	22
第四章 使用 指標來比較兩個製程之製程能力-區間估計	63
4.1 常態製程能力指標比值θ1=Cpmk1/Cpmk2之區間估計的建構	63
4.1.1 當μ1=μ2=m時，建構五個θ1=Cpmk1/Cpmk2之區間估計的方法	63
4.1.2 當μ1=μ2=m時，常態製程能力指標比值θ1=Cpmk1/Cpmk2之區間估計的模擬比較	67
4.1.3 當μ1≠m或μ2≠m時，建構四個θ1=Cpmk1/Cpmk2之區間估計的方法	74
4.1.4 當μ1≠m或μ2≠m時，常態製程能力指標比值θ1=Cpmk1/Cpmk2之區間估計的模擬比較	74
4.2常態製程能力指標差異θ2=Cpmk1-Cpmk2之區間估計的建構	89
4.2.1 當μ1≠m且μ2≠m時，建構五個θ2=Cpmk1-Cpmk2之區間估計的方法	89
4.2.2 當μ1≠m且μ2≠m時，常態製程能力指標差異θ2=Cpmk1-Cpmk2之區間估計的模擬比較	102
4.2.3 當μ1=m或μ2=m時，建構四個θ2=Cpmk1-Cpmk2之區間估計的方法	115
4.2.4 當μ1=m或μ2=m時，常態製程能力指標差異θ2=Cpmk1-Cpmk2之區間估計的模擬比較	115
第五章  數值實例分析	125
第六章  結論與未來展望	129
6.1結論	129
6.2未來展望	131
參考文獻	132


表目錄
Cpk=1和其他製程能力指標的比較	12
Cpm=1和其他製程能力指標的比較	13
表3.1   20種組合兩個參數θ1=Cpmk1/Cpmk2和θ2=Cpmk1-Cpmk2之真實值	24
表3.2   當T=0.300三種點估計法在θ1=Cpmk1/Cpmk2之模擬偏差和均方差	25
表3.3   當T=0.295三種點估計法在θ1=Cpmk1/Cpmk2之模擬偏差和均方差	29
表3.4   當T=0.305三種點估計法在θ1=Cpmk1/Cpmk2之模擬偏差和均方差	33
表3.5   當T=0.300三種點估計法在θ2=Cpmk1-Cpmk2之模擬偏差和均方差	37
表3.6   當T=0.295三種點估計法在θ2=Cpmk1-Cpmk2之模擬偏差和均方差	41
表3.7   當T=0.305三種點估計法在θ2=Cpmk1-Cpmk2之模擬偏差和均方差	45
表4.1   當T=0.300，μ1=μ2=m，θ1=Cpmk1/Cpmk2之五種方法模擬覆蓋率	68
表4.2   當T=0.295，μ1=μ2=m，θ1=Cpmk1/Cpmk2之五種方法模擬覆蓋率	68
表4.3   當T=0.305，μ1=μ2=m，θ1=Cpmk1/Cpmk2之五種方法模擬覆蓋率	68
表4.1.1 當T=0.300，μ1=μ2=m，五種θ1=Cpmk1/Cpmk2信賴區間方法落在(0.942，0.958)之比例	72
表4.2.1 當T=0.295，μ1=μ2=m，五種θ1=Cpmk1/Cpmk2信賴區間方法落在(0.942，0.958)之比例	72
表4.3.1 當T=0.305，μ1=μ2=m，五種θ1=Cpmk1/Cpmk2信賴區間方法落在(0.942，0.958)之比例	72
表4.4    當T=0.300，μ1≠m或μ2≠m，θ1=Cpmk1/Cpmk2之四種方法模擬覆蓋率	75
表4.5    當T=0.295，μ1≠m或μ2≠m，θ1=Cpmk1/Cpmk2之四種方法模擬覆蓋率	78
表4.6    當T=0.305，μ1≠m或μ2≠m，θ1=Cpmk1/Cpmk2之四種方法模擬覆蓋率	81
表4.4.1  當T=0.300，μ1≠m或μ2≠m，四種θ1=Cpmk1/Cpmk2信賴區間方法落在(0.942，0.958)之比例	87
表4.5.1  當T=0.295，μ1≠m或μ2≠m，四種θ1=Cpmk1/Cpmk2信賴區間方法落在(0.942，0.958)之比例	87
表4.6.1  當T=0.305，μ1≠m或μ2≠m，四種θ1=Cpmk1/Cpmk2信賴區間方法落在(0.942，0.958)之比例	87
表4.7    當T=0.300，μ1≠m且μ2≠m，θ2=Cpmk1-Cpmk2之五種方模擬覆蓋率	104
表4.8    當T=0.295，μ1≠m且μ2≠m，θ2=Cpmk1-Cpmk2之四種方模擬覆蓋率	106
表4.9    當T=0.305，μ1≠m且μ2≠m，θ2=Cpmk1-Cpmk2之四種方模擬覆蓋率	108
表4.7.1  當T=0.300，μ1≠m且μ2≠m，五種θ2=Cpmk1-Cpmk2信賴區間方法落在(0.942，0.958)之比例	113
表4.8.1  當T=0.295，μ1≠m且μ2≠m，五種θ2=Cpmk1-Cpmk2信賴區間方法落在(0.942，0.958)之比例	113
表4.9.1  當T=0.305，μ1≠m且μ2≠m，五種θ2=Cpmk1-Cpmk2信賴區間方法落在(0.942，0.958)之比例	113
表4.10   當T=0.300，μ1=m或μ2=m，θ2=Cpmk1-Cpmk2之四種方模擬覆蓋率	116
表4.11   當T=0.295，μ1=m或μ2=m，θ2=Cpmk1-Cpmk2之四種方模擬覆蓋率	117
表4.12   當T=0.305，μ1=m或μ2=m，θ2=Cpmk1-Cpmk2之四種方模擬覆蓋率	118
表4.10.1  當T=0.300，μ1=m或μ2=m，四種θ2=Cpmk1-Cpmk2信賴區間方法落在(0.942，0.958)之比例	122
表4.11.1  當T=0.295，μ1=m或μ2=m，四種θ2=Cpmk1-Cpmk2信賴區間方法落在(0.942，0.958)之比例	122
表4.12.1  當T=0.305，μ1=m或μ2=m，四種θ2=Cpmk1-Cpmk2信賴區間方法落在(0.942，0.958)之比例	122
表5.1    汽車引擎活塞環內徑資料(Piston rings for an automotive engine) 125
表5.2     兩個單一製程能力指標Cpmk1、Cpmk2之信賴區間	127
表5.3     兩個製程θ1、θ2的95%之信賴區間	128


圖目錄
圖2.1  Cpk=1和其他製程能力指標的比較	12
圖2.2  Cpm=1和其他製程能力指標的比較	13
圖3.1  當T=0.300，n=30,50,70,90時，三種點估計法在θ1=Cpmk1/Cpmk2的模擬偏差圖	49
圖3.2  當T=0.295，n=30,50,70,90時，三種點估計法在θ1=Cpmk1/Cpmk2的模擬偏差圖	50
圖3.3  當T=0.305，n=30,50,70,90時，三種點估計法在θ1=Cpmk1/Cpmk2的模擬偏差圖	51
圖3.4  當T=0.300，n=30,50,70,90時，三種點估計法在θ2=Cpmk1-Cpmk2的模擬偏差圖	52
圖3.5  當T=0.295，n=30,50,70,90時，三種點估計法在θ2=Cpmk1-Cpmk2的模擬偏差圖	53
圖3.6  當T=0.305，n=30,50,70,90時，三種點估計法在θ2=Cpmk1-Cpmk2的模擬偏差圖	54
圖3.7  當T=0.300，n=30,50,70,90時，三種點估計法在θ1=Cpmk1/Cpmk2的模擬均方差圖	55
圖3.8  當T=0.295，n=30,50,70,90時，三種點估計法在θ1=Cpmk1/Cpmk2的模擬均方差圖	56
圖3.9  當T=0.305，n=30,50,70,90時，三種點估計法在θ1=Cpmk1/Cpmk2的模擬均方差圖	57
圖3.10 當T=0.300，n=30,50,70,90時，三種點估計法在θ2=Cpmk1-Cpmk2的模擬均方差圖 58
圖3.11 當T=0.295，n=30,50,70,90時，三種點估計法在θ2=Cpmk1-Cpmk2的模擬均方差圖 59
圖3.12 當T=0.305，n=30,50,70,90時，三種點估計法在θ2=Cpmk1-Cpmk2的模擬均方差圖 60
圖4.1  當T=0.300，μ1=μ2=m，n=30,50,70,90時，θ1=Cpmk1/Cpmk2之四種方法模擬覆蓋率	69
圖4.2  當T=0.295，μ1=μ2=m，n=30,50,70,90時，θ1=Cpmk1/Cpmk2之四種方法模擬覆蓋率	70
圖4.3  當T=0.305，μ1=μ2=m，n=30,50,70,90時，θ1=Cpmk1/Cpmk2之四種方法模擬覆蓋率	71
圖4.4  當T=0.300，μ1≠m或μ2≠m，n=30,50,70,90時，θ1=Cpmk1/Cpmk2之三種方法模擬覆蓋率	84
圖4.5  當T=0.295，μ1≠m或μ2≠m，n=30,50,70,90時，θ1=Cpmk1/Cpmk2之三種方法模擬覆蓋率	85
圖4.6  當T=0.305，μ1≠m或μ2≠m，n=30,50,70,90時，θ1=Cpmk1/Cpmk2之三種方法模擬覆蓋率	86
圖4.7  當T=0.300，μ1≠m且μ2≠m，n=30,50,70,90時，θ2=Cpmk1-Cpmk2之四種方法模擬覆蓋率	110
圖4.8  當T=0.295，μ1≠m且μ2≠m，n=30,50,70,90時，θ2=Cpmk1-Cpmk2之四種方法模擬覆蓋率	111
圖4.9  當T=0.305，μ1≠m且μ2≠m，n=30,50,70,90時，θ2=Cpmk1-Cpmk2之四種方法模擬覆蓋率	112
圖4.10 當T=0.300，μ1=m且μ2=m，n=30,50,70,90時，θ2=Cpmk1-Cpmk2之三種方法模擬覆蓋率	119
圖4.11 當T=0.295，μ1=m且μ2=m，n=30,50,70,90時，θ2=Cpmk1-Cpmk2之三種方法模擬覆蓋率	120
圖4.12 當T=0.305，μ1=m且μ2=m，n=30,50,70,90時，θ2=Cpmk1-Cpmk2之三種方法模擬覆蓋率	121
圖5.1  製程一和製程二的直方圖	126

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