隨著科學技術日新月異的變化，許多電子產品對我們的日常生活越來越重要，降低產品成本也成為重要的研究方向。在實驗設計中製程能力指標(process capability indices, PCIs) 可用於測量產品質量，並提供有用的信息來評估製程的性能。
    當產品的壽命假設服從Weibull分配時，在逐步型I區間設限下，計算出壽命績效指標C下標L之最大概似估計量並以其為檢定統計量發展出檢定程序。我們還考慮檢測次數m是否固定或終止時間T是否固定之下，給定顯著水準和檢定力下，決定最佳實驗的抽樣設計以達到最低實驗總成本。最後，藉由一個數值例子和一個模擬例子說明如何使用本研究所提出的最佳實驗設計之檢定程序以及評估方法。

With the ever-changing nature of technology and science, many electronic products are becoming more demanding for our daily life. Furthermore, for the customers, they will choose the products with longer life expectancy and lower failure rate. In addition, the manufacturer will tend to lower the cost of the product but the quality will hold or even better. For measuring the quality of products, Process capability indices (PCI) have provided very useful information to evaluate the performance and the capability of the process. As for the lifetime of products, we use C_L to assess the performance of the lifetime with Weibull distribution.
Our research is focusing on the evaluation of lifetime performance of products with Weibull distribution. The maximum likelihood estimator is used to estimate the lifetime performance index based on the progressive type I interval censored sample and it’s used to develop a hypothesis testing procedure.  In the condition of either the number of inspections m is fixed or not or the termination time T is fixed or not, we proposed the algorithm to achieve the optimal experimental design to attain the minimum total experimental cost. Finally, we give one simulation example and one practical example to illustrate the use of the proposed experimental design to conduct the testing algorithmic procedure to determine whether the process is capable.

目錄
表目錄	III
圖目錄	V
第一章	緒論	1
1-1 研究動機與目的	1
1-2 文獻探討	2
1-2-1 製程能力指標之發展	2
1-2-2 設限形式	3
1-3 本文架構	5
第二章 Weibull分配下壽命績效指標與其估計	6
2-1 產品壽命績效指標 	7
2-2 壽命績效指標之估計量	10
第三章 可靠度抽樣設計	14
3-1 在固定m以及T之下決定樣本的大小	14
3.2 在固定T之下尋找最佳的m和n	19
3.3 在實驗終止時間T不固定之下計算最佳的m,t和n	28
第四章 模擬與數值實例分析	37
4-1 模擬範例	37
4-2 數值範例	41
第五章 結論與未來研究	46
5-1 結論	46
5-2 未來研究	47
參考文獻	48
附錄	50
表目錄
附表1 當形狀參數δ=1.97、規格下限L=0.3，總觀測時間T=3.0，型二誤差β=0.15,0.20,0.25及逐步移除率p=0.05,0.075,0.1時，實際值c_1=0.825(0.025)0.95在目標值c_0=0.8和顯著水準α=0.01下，最小的樣本數..……………...…….50
附表2 當形狀參數δ=1.97、規格下限L=0.3，總觀測時間T=3.0，型二誤差β=0.15,0.20,0.25及逐步移除率p=0.05,0.075,0.1時，實際值c_1=0.825(0.025)0.95在目標值 c_0=0.8和顯著水準α=0.05下，最小的樣本數	52
附表3 當形狀參數δ=1.97、規格下限L=0.3、總觀測時間T=3.0，型二誤差β=0.15,0.20,0.25及逐步移除率p=0.05,0.075,0.1時，實際值c_1=0.825(0.025)0.95在目標值c_0=0.8和顯著水準α=0.1下，最小的樣本數	54
附表4. 當形狀參數δ=1.97、規格下限L=0.3、總觀測時間T=3.0、型二誤差β=0.25,0.20,0.15、及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.01,0.05,0.1、目標值c_0=0.8和實際值c_1=0.825,0.85下，最佳的觀測次數m^*、最小樣本數n^*、總成本TC以及臨界值C_L^0………………………………………………….56
附表5. 當形狀參數δ=1.97、規格下限L=0.3、總觀測時間T=3.0、型二誤差β=0.25,0.20,0.15、及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.01,0.05,0.1、目標值c_0=0.8和實際值c_1=0.875,0.90下，最佳的觀測次數m^*、最小樣本數n^*、總成本TC以及臨界值C_L^0	57
附表6. 當形狀參數δ=1.97、規格下限L=0.3、型二誤差β=0.25,0.20,0.15、及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.01,0.05,0.1、目標值c_0=0.8和實際值c_1=0.825,0.85下，最佳的觀測次數m^*、最小樣本數n^*、總成本TC以及臨界值C_L^0	58
附表7. 當形狀參數δ=1.97、規格下限L=0.3、型二誤差β=0.25,0.20,0.15、及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.01,0.05,0.1、目標值c_0=0.8和實際值c_1=0.875,0.90下，最佳的觀測次數m^*、最小樣本數n^*、總成本TC以及臨界值C_L^0	59
圖目錄
圖1.2.1 逐步型I 區間設限 ....................................................................... 5
圖2.1.1 Weibull分配 lambda=1 或lambda=2，不同delta時的機率密度函數圖 ................ 6
圖2.1.2 Weibull分配 lambda=1 或lambda=2，不同delta時的失效率函數圖 .................... 7
圖3.1.1 當alpha = 0.05, delta = 1.97 及p = 0.05下，型二誤差1 − beta =
0.75, 0.80, 0.85時的最小樣本數n .................................................... 16
圖3.1.2 當alpha = 0.01, delta = 1.97 及p = 0.05下，檢測次數m = 5(1)8時的最小樣本數n ................................................................................. 17
圖3.1.3 當alpha = 0.05, delta = 1.97, 及1 − beta = 0.85下，逐步移除率p =
0.05,0.075, 0.1時的最小樣本數n .................................................... 17
圖3.1.4 當p = 0.05, delta = 1.97, 及1 − beta = 0.85, m = 5下，顯著水準
alpha = 0.01,0.05,0.1時的最小樣本數n ............................................... 18
圖3.2(a): m = 2(1)m0的總成本曲線 ................................................... 21
圖3.2(b): m = 2(1)m0的總成本曲線 ................................................... 22
圖3.2.1 當alpha = 0.05、delta = 1.97、p = 0.05 ，不同檢定力 1 − beta =
0.75, 0.80,0.85 時的最小檢定次數m ............................................. 24
圖3.2.2 當alpha = 0.05、delta = 1.97、1 − beta = 0.75 ，不同的逐步設限移除率 p = 0.05,0.075,0.1 時的最小檢定次數m ............................ 24
圖3.2.3 當 p=0.05、 delta=1.97、 1−beta=0.75 ，不同的顯著水準
alpha=0.01,0.05,0.1 時的最小檢定次數 m 25
圖3.2.4 當 alpha=0.05、 delta=1.97、 p=0.05 ，不同檢定力 1−beta=0.75,0.80,0.85 時的最小總成本 TC 26
圖3.2.5當 alpha=0.05、 delta=1.97、 1−beta=0.75 ，不同的逐步設限移
除率 p=0.05,0.075,0.1 時的最小總成本 TC 27
圖3.2.6 當 p=0.05、 delta=1.97、 1−beta=0.75 ，不同的顯著水準
alpha=0.01,0.05,0.1 時的最小總成本 TC 27
圖3.3(a) m=2(1)m0的總成本曲線 30
圖3.3(b) m=1(1)m0的總成本曲線 30
圖3.3.1當 alpha=0.05、 delta=1.97、 p=0.05 ，不同檢定力 1−beta=0.75,0.80,0.85 時的最小檢定次數 m 33
圖3.3.2 當 alpha=0.05、 delta=1.97、 1−beta=0.75 ，不同的逐步設限移
除率 p=0.05,0.075,0.1 時的最小檢定次數 m 33
圖3.3.3 當 p=0.05、 delta=1.97、 1−beta=0.75 ，不同的顯著水準
alpha=0.01,0.05,0.1 時的最小檢定次數 m 34
圖3.3.4 當 alpha=0.05、 delta=1.97、 p=0.05 ，不同檢定力 1−beta=0.75,0.80,0.85 時的最小總成本 TC 35
圖3.3.5當 alpha=0.05、 delta=1.97、 1−beta=0.75 ，不同的逐步設限移
除率 p=0.05,0.075,0.1 時的最小總成本 TC 36
圖3.3.6 當 p=0.05、 delta=1.97、 1−beta=0.75 ，不同的顯 著水準
alpha=0.01,0.05,0.1 時的最小總成本 TC 36
圖4.2.1不同形狀參數 delta下之 p值 42

[1] Boyles, R. A. (1991), The Taguchi capability index, Journal of Quality Technology, 23(1), pp. 17–26.
[2] Caroni, C. (2002a), Modeling the reliability of ball bearings. Journal of Statistics Education, 10(3), pp. 1-8.
[3] Caroni, C. (2002b), The correct ”ball bearings” data. Lifetime Data Analysis, 8(4), pp.395-399.
[4] Chan, L. K., Cheng, S. W. and Spiring, F. A. (1988), A new measure of process capability: Cpm, Journal of Quality Technology, 20(3), pp. 162-175.
[5] Cohen, A. C. (1963), Progressively Censored Samples in Life Testing, Technometrics, 5(3), pp. 327–339.
[6] Gill, M. H. and Gastwirth, J. L. (1978), A scale-free goodness-of-fit Test for the Exponential Distribution Based on the Gini Statistic, Journal of the Royal Statistical Society, Series B(Methodological), 40, pp. 350–357.
[7] Huang, S. R. and Wu, S. T. (2008), Reliability Sampling Plans Under Progressive Type-I Interval Censoring Using Cost Functions, IEEE Transactions on Reliability, 57(3), pp.445-451.
[8] Juran, J. M. (1974), Journal Quality Control Handbook, 3rd Edition, McGraw-Hill, New York.
[9] Kane, V. E. (1986), Process capability indices, Journal of Quality Technology, 18, pp. 41–52.
[10] Lawless, J. F. (2003), Statistical Models and Methods for lifetime Data. (Second Edition). John Wiley and Sons, New York.
[11] Lee, H. M., Wu, J. W., and Lei, C. L. (2013), Assessing the lifetime performance index of exponential products with step-stress accelerated life-testing data, IEEE Transactions on Reliability, 62(1), pp. 296–304.
[12] Montgomery, D. C. (1985), Introduction to statistical quality control, John Wiley and Sons, New York.
[13] Tong, L. I., Chen, K. S. and Chen, H. T. (2002), Statistical testing for assessing the performance of lifetime index of electronic components with exponential distribution, International Journal of Quality Reliability Management, 19(7), pp. 812–824.
[14] Weibull, W. (1951), A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), pp.293-297
[15] Wu, J. W., Lee, W. C. and Hou, H. C. (2007), Assessing the performance for the products with Rayleigh lifetime, Journal of Quantitative Management, 4, pp.147-160.
[16] Wu, S. F. and Lin, M. J. (2017), Computational testing algorithmic procedure of assessment for lifetime performance index of products with Weibull distribution under progressive type I interval censoring , Journal of Computational and Applied Mathematics , 311, pp. 364-374.
[17] 吳奇翰 《 Weibull分配產品的壽命績效指標在逐步型Ι區間下
之檢定程序的檢定力分析 》， 碩士論文， 淡江大學 統計學系 應
用統計 學碩士班 2018.