在這個人工智能時代，科技的日新月異使產品生產技術更加精密複雜，消費者對產品品質要求也日趨嚴苛。製造商為了達到消費者所要求的品質水準，便設法提高產品的壽命以及製程良率，同時嚴格控管製造過程，避免造成不必要的浪費，以降低成本。藉此擴大市場競爭力並達到利潤最大化。
本研究假設產品壽命服從Gompertz分配時，當規格下限L已知，在逐步型I區間設限下，基於使用壽命績效指標之最大概似估計量做為檢定統計量，所建構出的假設檢定程序下，為了達到給定的檢定力水準以及最小的總實驗成本，建構出的最佳抽樣設計。最後，利用一個模擬與一個數值實例來說明這個檢定程序下樣本設計的使用，以檢驗此製程是否有能力。

In this artificial intelligence era, the constantly changing of technology makes production techniques become sophisticated and complicated. Therefore, manufacturers are dedicated to improving the quality of products by increasing the lifetime in order to achieve the quality standards demanded by consumers. Meanwhile, to increase market competitiveness and profits, the process of product manufacturing is strictly controlled to avoid unnecessary waste, and so as to reduce costs.
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. At last, one practical example and one simulation example are given to illustrate the use of this sampling design for the testing procedure to determine whether the process is capable.

目錄
目錄 I
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 3
1.2.1 製程能力指標之發展 3
1.2.2 設限型式 6
1.3 本文架構 8
第二章 壽命績效指標的檢定程序介紹 10
2.1 產品的壽命績效指標C_L 10
2.2 壽命績效指標的假設檢定 14
第三章 可靠度抽樣設計 18
3.1 在固定的觀測區間個數和總觀測時間下，樣本數大小之決定 18
3.2 在固定的總觀測時間下，最佳觀測區間個數與樣本數大小之決定 23
3.3 當總觀測時間為不固定時，最佳觀測區間個數、觀測區間時間以及樣本數之決定 31
第四章 模擬與數值實例分析 40
4.1 數值實例 40
4.2 模擬範例 45
第五章 結論與未來研究 51
5.1 結論 51
5.2 未來研究 52
參考文獻 53


表目錄
表2.1 壽命績效指標C_L值對應之製程良率P_r 13
表4.1 被餵食未飽和飲食的老鼠，未罹患腫瘤天數(單位:200天) 40
附表 1 當形狀參數k=4.47、規格下限L=0.05、總觀測時間T=0.5、型II誤差β=0.25,0.2,0.15、觀測次數m=5(1)8、逐步設限移除率p=0.05(0.025)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)0.95下，所需要的最小樣本數，以及其對應臨界 56
附表 2 當形狀參數k=4.47、規格下限L=0.05、總觀測時間T=0.5、型II誤差β=0.25,0.2,0.15、觀測次數m=5(1)8、逐步設限移除率p=0.05(0.025)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)0.95下，所需要的最小樣本數，以及其對應臨界 57
附表 3 當形狀參數k=4.47、規格下限L=0.05、總觀測時間T=0.5、型II誤差β=0.25,0.2,0.15、觀測次數m=5(1)8、逐步設限移除率p=0.05(0.025)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)0.95下，所需要的最小樣本數，以及其對應臨界 58
附表 4 當形狀參數k=4.47、規格下限L=0.05、總觀測時間T=0.5、觀測次數上界m_0=20、逐步設限移除率p=0.05(0.025)0.1時，在目標值C_0=0.8和實際值C_1=0.825,0.850下，所推薦的最佳觀測區間個數、樣本數，以及其對應之總成本與臨界值 59
附表 5 當形狀參數k=4.47、規格下限L=0.05、總觀測時間T=0.5、觀測次數上界m_0=20、逐步設限移除率p=0.05(0.025)0.1時，在目標值C_0=0.8和實際值C_1=0.875,0.90下，所推薦的最佳觀測區間個數、樣本數，以及其對應之總成本與臨界值 60
附表 6 當形狀參數k=4.47、規格下限L=0.05、觀測次數上界m_0=20、逐步設限移除率p=0.05(0.025)0.1時，在目標值C_0=0.8和實際值C_1=0.825,0.850下，所推薦的最佳觀測區間個數、觀測區間時間、樣本數，以及其對應之總成本與臨界值 61
附表 7 當形狀參數k=4.47、規格下限L=0.05、觀測次數上界m_0=20、逐步設限移除率p=0.05(0.025)0.1時，在目標值C_0=0.8和實際值C_1=0.875,0.90下，所推薦的最佳觀測區間個數、觀測區間時間、樣本數，以及其對應之總成本與臨界值 62


圖目錄
圖1.1 逐步型I區間設限 7
圖3.1.1 當α=0.05、m=5及p=0.05下，不同的檢定力1-β=0.75,0.8,0.85時，所需的最小樣本數n 21
圖3.1.2 當α=0.05、β=0.2及p=0.05下，不同的觀察次數m=5,6,7,8時，所需的最小樣本數n 21
圖3.1.3 當α=0.05、β=0.2及m=5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 22
圖3.1.4 當β=0.2、m=5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 22
圖3.2.1 當β=0.25 ,α=0.01 ,p=0.05 ,c_1=0.825 ,m_0=20時，觀測區間個數與其相對應的總成本曲線 25
圖3.2.2 當β=0.15 ,α=0.1 ,p=0.1 ,c_1=0.9 ,m_0=20時，觀測區間個數與其相對應的總成本曲線 25
圖3.2.3 當α=0.05及p=0.05下，不同的檢定力1-β=0.75,0.8,0.85時，所需的最小觀測區間個數m 27
圖3.2.4 當α=0.05及β=0.2下，不同的逐步設限移除率p=0.05,0.075,0.1時，所需的最小觀測區間個數m 28
圖3.2.5 當β=0.2及p=0.05下，不同的顯著水準α=0.01,0.5,0.1時，所需的最小觀測區間個數m 28
圖3.2.6 當α=0.05及p=0.05下，不同的檢定力1-β=0.75,0.8,0.85時，所達的最小總成本TC* 30
圖3.2.7 當α=0.05及β=0.2下，不同的逐步設限移除率p=0.05,0.075,0.1時，所達的最小總成本TC* 30
圖3.2.8 當β=0.2及p=0.05下，不同的顯著水準α=0.01,0.5,0.1時，所達的最小總成本TC* 31
圖3.3.1 當β=0.25 ,α=0.01 ,p=0.05 ,c_1=0.825 ,m_0=20時，觀測區間個數與其相對應的總成本曲線 33
圖3.3.2 當β=0.15 ,α=0.1 ,p=0.1 ,c_1=0.9 ,m_0=20時，觀測區間個數與其相對應的總成本曲線 33
圖3.3.3 當α=0.01及p=0.05下，不同的檢定力1-β=0.75,0.8,0.85時，所需的最小觀測區間個數m 35
圖3.3.4 當α=0.01及β=0.2下，不同的逐步設限移除率p=0.05,0.075,0.1時，所需的最小觀測區間個數m 36
圖3.3.5 當β=0.2及p=0.05下，不同的顯著水準α=0.01,0.05,0.1時，所需的最小觀測區間個數m 36
圖3.3.6 當α=0.01及p=0.05下，不同的檢定力1-β=0.75,0.8,0.85時，所達的最小總成本TC** 38
圖3.3.7 當α=0.01及β=0.2下，不同的逐步設限移除率p=0.05,0.075,0.1時，所達的最小總成本TC** 38
圖3.3.8 當β=0.2及p=0.05下，不同的顯著水準α=0.01,0.05,0.1時，所達的最小總成本TC** 39
圖4.1 不同k下之p值 42

[1] 	Bai, D. S. and Chung, S.W. (1992). Optimal Design of Partially Acceler-ated Life Tests for the Exponential distribution under Type I Censoring. IEEE Transactions on Reliability, 41, pp. 400-406.
[2] 	Boyles, R. A. (1991). The Taguchi capability index, Journal of Quality Technology, 23(1), pp. 17–26.
[3] 	Chan, L. K., Cheng, S. W. and Spiring, F. A. (1988). A new measure of process capability, Journal of Quality Technology, 20(3), pp. 162-175.
[4] 	Clement, J. A. (1989). Process capability calculations for non-normal distribution, Quality Technology, 22, pp. 95-100.
[5] 	Cohen, A. C. (1963). Progressively censored samples in the life testing, Technometrics, 5, pp. 327-339.
[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] 	Huang, S. R. and Wu, S. J. (2008). Reliability sampling plans under progressive type-I interval censoring using cost functions, IEEE Transactions on Reliability, 57, pp. 445-451.
[8] 	Juran, J. M. (1974). Journal quality control handbook,(3rd ed), McGraw-Hill ,New York.
[9] 	Kane, V. E. (1986). Process capability indices, Journal of Quality Technology, 18, pp. 41-52.
[10] 	Lawless, J. F. (1982). Statistical Methods for Lifetime Data, John Wiley & Sons, New York.
[11] 	Lee, E. T. (1980). Statistical method for survival data analysis, Lifetime Learning Publication, Belmont.
[12] 	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.
[13] 	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.
[14] 	Liu, P. H. and Chen, F. L. (2006). Process capability analysis of non-normal process data using the Burr XII distribution, International Journal of advanced Manufacturing Technology, 27, pp. 975-984.
[15] 	Montgomery, D. C. (1985). Introduction to statistical quality control, John Wiley & Sons, New York.
[16] 	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. 216-233.
[17] 	Pearn, W. L. and Chen, K. S. (1997). Capability indices for non-normal distributions with an application in electrolytic capacitor manufacturing, Microelectronics and Reliability, 37(12), pp.1853-1858.
[18] 	Spurrier, J. D. and Wei, L. J. (1980). A Test of the Parameter of the Exponential Distribution in the Type I Censoring Case, Journal of the American Statistical Association, 75, pp. 405-409.
[19] 	Tong, L. I., Chen, K. T. 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.
[20] 	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.
[21] 	Wu, S. F., and Lin, Y. P. (2016). Computational testing algorithmic procedure of assessment for lifetime performance index of products with one-parameter exponential distribution under progressive type I interval censoring, Mathematics and Computers in Simulation, 120, pp. 79–90.
[22] 	Wu, S. F. and Hsieh, Y. T. (2019). The assessment on the lifetime performance index of products with Gompertz distribution based on the progressive type I interval censored sample, Journal of Computational and Applied Mathematics, 351, pp. 66-76.
[23] 	廖茂峰，《Gompertz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學碩士班，2018。