隨著科技的日新月異，產品的製程逐漸變得繁複且精密。消費者在購買產品時，對於產品品質的要求也變得更為嚴苛。為了達到消費者所需求的品質，生產者在製造產品時，會對製程的要求更加嚴格，如何節省成本及避免造成不必要的浪費，以追求更多的利潤，已是當今生產者所必須重視的問題。
    本研究假設產品的壽命服從Burr XII分配時，在逐步型I區間設限下，計算出壽命績效指標C_L之最大概似估計量，並使用此估計量作為檢定統計量進行假設檢定程序，本研究在逐步型I區間設限下決定最佳實驗設計以達到最小的總實驗成本，並以表格形式列出相關值供實際使用。最後，我們用兩個數值實例說明如何使用本研究所提出的最佳實驗設計的檢定程序。

It is a very important topic these days to assessing the lifetime performance of products in manufacturing or service industries. Lifetime performance indices CL is used to measure the larger-the-better type quality characteristics to evaluate the process performance for the improvement of quality and productivity. The lifetimes of products are assumed to have Burr XII distribution. Based on the hypothesis testing procedure using the maximum likelihood estimator as testing statistic, the sampling design is determined and the related values are tabulated for practical use to reach the given power level or minimize the total experimental cost under progressive type I interval censoring. Some numerical studies are also given to demonstrate the proposed sampling design. Finally, two practical examples are given to illustrate the use of this testing algorithmic procedure to determine whether the process is capable.

目錄 I
表目錄 III
圖目錄 VII
第一章 緒論 1
1.1 研究動機與目的 1
1.2 文獻探討 4
1.2.1 製程能力指標之發展 4
1.2.2 設限型式 7
1.3 本文架構 9
第二章 壽命績效指標與其估計 10
2.1 產品的壽命績效指標C_L	13
2.2 壽命績效指標的估計量 17
第三章 可靠度實驗設計 21
3.1 固定實驗終止時間T和觀察區間數m後求得所需樣本數大小n 21
3.2 固定實驗終止時間T下求得所需樣本數大小n及觀察區間數m 26
3.3 不固定實驗終止時間T下求得所需觀察區間數m、區間時間t和樣本數大小n 34
第四章 模擬與數值實例分析 43
4.1 模擬範例 43
4.2 數值實例 48
第五章  結論與未來研究 54
5.1 結論 54
5.2 未來研究 54
參考文獻	56

表2.1 壽命績效指標C_L值對應之製程良率P_r 15 
表4.1 36個電器產品的失效時間(單位:小時/1000) 48
附表1 當形狀參數c=1.37、規格下限L=0.02、總觀測時間T=1.0、檢定力1-β=0.75,0.80,0.85、觀測次數m=5,6,7,8及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.01、目標值c_0=0.8和實際值c_1=0.825(0.025)0.9下，所需要的最小樣本數n以及臨界值C_L^0 58
附表2 當形狀參數c=1.37、規格下限L=0.02、總觀測時間T=1.0、檢定力1-β=0.75,0.80,0.85、觀測次數m=5,6,7,8及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.01、目標值c_0=0.8和實際值c_1=0.925,0.95,0.96,0.975下，所需要的最小樣本數n以及臨界值C_L^0 60
附表3 當形狀參數c=1.37、規格下限L=0.02、總觀測時間T=1.0、檢定力1-β=0.75,0.80,0.85、觀測次數m=5,6,7,8及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.05、目標值c_0=0.8和實際值c_1=0.825(0.025)0.9下，所需要的最小樣本數n以及臨界值C_L^0 62
附表4 當形狀參數c=1.37、規格下限L=0.02、總觀測時間T=1.0、檢定力1-β=0.75,0.80,0.85、觀測次數m=5,6,7,8及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.05、目標值c_0=0.8和實際值c_1=0.925,0.95,0.96,0.975下，所需要的最小樣本數n以及臨界值C_L^0 64
附表5 當形狀參數c=1.37、規格下限L=0.02、總觀測時間T=1.0、檢定力1-β=0.75,0.80,0.85、觀測次數m=5,6,7,8及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.1、目標值c_0=0.8和實際值c_1=0.825(0.025)0.9下，所需要的最小樣本數n以及臨界值C_L^0 66
附表6 當形狀參數c=1.37、規格下限L=0.02、總觀測時間T=1.0、檢定力1-β=0.75,0.80,0.85、觀測次數m=5,6,7,8及逐步設限移除率p=0.05,0.075,0.1時，在顯著水準α=0.1、目標值c_0=0.8和實際值c_1=0.925,0.95,0.96,0.975下，所需要的最小樣本數n以及臨界值C_L^0 68
附表7 當形狀參數c=1.37、規格下限L=0.02、總觀測時間T=1.0、檢定力1-β=0.75,0.80,0.85、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^*、最低總成本TC^*以及臨界值C_L^0 70
附表8 當形狀參數c=1.37、規格下限L=0.02、總觀測時間T=1.0、檢定力1-β=0.75,0.80,0.85、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=0.875,0.9下，最佳的觀測次數m^*、最小樣本數n^*、最低總成本TC^*以及臨界值C_L^0 71
附表9 當形狀參數c=1.37、規格下限L=0.02、總觀測時間T=1.0、檢定力1-β=0.75,0.80,0.85、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=0.925,0.95下，最佳的觀測次數m^*、最小樣本數n^*、最低總成本TC^*以及臨界值C_L^0 72
附表10 當形狀參數c=1.37、規格下限L=0.02、檢定力1-β=0.75,0.80,0.85、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=0.825,0.850下，最佳的觀測次數m^*、最小樣本數n^*、區間時間t^*、最低總成本TC^**以及臨界值C_L^0 73
附表11 當形狀參數c=1.37、規格下限L=0.02、檢定力1-β=0.75,0.80,0.85、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^*、區間時間t^*、最低總成本TC^**以及臨界值C_L^0 74
附表12 當形狀參數c=1.37、規格下限L=0.02、檢定力1-β=0.75,0.80,0.85、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=0.925,0.95下，最佳的觀測次數m^*、最小樣本數n^*、區間時間t^*、最低總成本TC^**以及臨界值C_L^0 75

圖1.1 逐步型I區間設限圖 8
圖2.1 尺度參數k=1,形狀參數c=0.5,1,2,3,5,10時的p.d.f. 11
圖2.2 尺度參數k=2,形狀參數c=0.5,1,2,3,5,10時的p.d.f. 11
圖2.3 尺度參數k=1,形狀參數c=0.5,1,2,3,5,10時的失效率函數 12
圖2.4 尺度參數k=2,形狀參數c=0.5,1,2,3,5,10時的失效率函數 12
圖3.1.1 當1-β=0.85、m=8及p=0.05下，不同的顯著水準α=0.01,0.05,0.1時所需的最小樣本數n 24
圖3.1.2 當α=0.01、1-β=0.85及p=0.1下，不同的觀察次數m=5,6,7,8時所需的最小樣本數n 24
圖3.1.3 當α=0.01、1-β=0.85及m=8下，不同的逐步設限移除率p=0.05,0.075,0.1時所需的最小樣本數n 25
圖3.1.4 當α=0.01、m=8及p=0.1下，不同的檢定力1-β=0.85,0.8,0.75時所需的最小樣本數n 25
圖3.2(a) m=2(1)m_0的總成本曲線 28
圖3.2(b) m=2(1)m_0的總成本曲線 28
圖3.2.1 當1-β=0.85及p=0.1下，不同的顯著水準α=0.01,0.05,0.1時的最低總成本TC^*	30
圖3.2.2 當α=0.01及1-β=0.85下，不同的逐步設限移除率p=0.05,0.075,0.1時的最低總成本TC^* 31
圖3.2.3 當α=0.01、p=0.1下，不同的檢定力1-β=0.85,0.8,0.75時的最低總成本TC^*	31
圖3.2.4 當1-β=0.85及p=0.05下，不同的顯著水準α=0.01,0.05,0.1時所需觀察區間數m 33
圖3.2.5 當α=0.05及1-β=0.85下，不同的逐步設限移除率p=0.05,0.075,0.1時所需觀察區間數m 33
圖3.2.6 當α=0.05、p=0.05下，不同的檢定力1-β=0.85,0.8,0.75時所需觀察區間數m 34
圖3.3(a) m=2)(1)_m_0的總成本曲線 36
圖3.3(b) m=1(1)m_0的總成本曲線 36
圖3.3.1 當1-β=0.85及p=0.1下，不同的顯著水準α=0.01,0.05,0.1時的最低總成本TC^(**) 38
圖3.3.2 當α=0.01及1-β=0.85下，不同的逐步設限移除率p=0.05,0.075,0.1時的最低總成本TC^(**) 39
圖3.3.3 當α=0.01、p=0.1下，不同的檢定力1-β=0.85,0.8,0.75時的最低總成本TC^(**) 39
圖3.3.4 當1-β=0.85及p=0.05下，不同的顯著水準α=0.01,0.05,0.1時所需觀察區間數m 41
圖3.3.5 當α=0.01及1-β=0.85下，不同的逐步設限移除率p=0.05,0.075,0.1時所需觀察區間數m 41
圖3.3.6 當α=0.01、p=0.05下，不同的檢定力1-β=0.85,0.8,0.75時所需觀察區間數m 42
圖4.1 不同c下之p值 50

[1]Boyles, R.A. (1991), The Taguchi capability index, Journal of Quality Technology, 23(1), pp. 17-26
[2]Burr, I.W. (1942). Cumulative frequency function, The Annals of Mathematical Statistics, 13, pp. 215–232.
[3]Chan, L. K., Cheng, S. W. and Spiring, F. A.(1988). A new measure of process capability:C_pm. Journal of Quality Technology, 20(3), pp.162-175.
[4]Cohen, A. C. (1963), Progressively censored samples in life testing, Technometrics, 5(3), pp. 327–339.
[5]Cohen, A. C. (1991), Truncated and censored sample, Marcel Dekker, New York.
[6]Gill, M.H. and Gastwirth J.L. (1978), A sacle-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]Hong, C. W., Wu, J. W., and Cheng, C. H. (2008), Computational procedure businesses based on confidence interval, Applied Soft Computing , 8(1), pp. 698-705
[8]Huang, S. R. and Wu, S. J. (2008), Reliability sampling plans under peogressive type-I interval censoring using cost functions, IEEE Transactions on Reliability, 57, pp.445-451.
[9]Juran,J. M. (1974), Journal Quality Control Handbook, 3rd Edition, McGraw-Hill,New York.
[10]Kane, V. E. (1986), Process capability indices, Journal of Quality Technology, 18, pp. 41–52.
[11]Lawless, J. F. (1982), Statistical Methods for Lifetime Data, John Wiley and Sons, New York.
[12]Lawless, J. F. (2003), Statistical Models and Methods for Lifetime Data, (2nded), John Wiley and Sons, New York.
[13]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.
[14]Lee, W. C., Wu, J. W. and Hong, C. W. (2009), Assessing the lifetime performance index of products from progressively type II right censored data using Burr XII model, Mathematics and Computers in  Simulation, 79(7), pp. 2167–2179.
[15]Lee, W. C., Wu, J. W. and Lei, C. L. (2010), Evaluating the lifetime performance index for the exponential lifetime products, Applied Mathematical Modelling, 34(5), pp. 1217–1224.
[16]Montgomery, D. C. (1985), Introduction to statistical quality control, John Wiley and Sons, New York.
[17]Pearn, W. L.,Kotz, S. and Johnson, N. L.(1992),Distributional and inferential properties of process capability indices, Journal of Quality Technology, 24(4), pp. 216–231.
[18]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.
[19]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.(a)
[20]Wu, J. W., Lee, H. M. and Lei, C. L. (2007), Computational testing algorithmic procedure of assessment for lifetime performance index of products with two-parameter exponential distribution, Applied Mathematics and Computation, 190, pp. 116–125.(b)
[21]Wu, J. W., Lee, W. C., Hong, C. W. and Yeh, S. Y. (2013) , Implementing lifetime performance index of Burr XII products with progressively type II right censored sample, International Journal of Innovative Computing, Information and Control, 10(2), pp. 671–693.
[22]Wu, S. F. , Tzu-Chin Chen, Wen-Jui Chang, Chia-Wei Chang and Chen Lin(2018), A hypothesis testing procedure for the evaluation on the lifetime performance index of products with Burr XII distribution under progressive type I interval censoring, Communications in Statistics: Simulation and Computation, 47(9), 2670-2683.
[23]康哲維, 《Burr XII 分配產品的壽命績效指標在逐步型I區間設限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碩士班, 2018.