品質（quality）不僅是消費者評斷產品的重要依據，對生產者而言也是提升競爭力的方法，因此有許多企業將學界所提出的品質管理方法應用於實務生產當中，其中一種量化工具為製程良率（process yield），為合格品佔總產品的比率，也是用來衡量製程能力（process capability）的基準之一。但製程良率僅將產品分成通過與否，無法進一步區分合格品的品質，為解決此問題，有學者提出了客戶導向的品質良率指標（quality-yield index）Yq，以良率為基礎進一步考量消費者的損失，能夠有效衡量製程實際的品質水準。
過去Yq指標相關的研究包含保守之近似估計方法與無母數的複式抽樣法等，皆侷限於常態製程，實際上製程多半為非常態性，故本研究將Yq指標推廣至非常態製程的伽瑪分配與韋伯分配當中，並以複式抽樣法與貝氏方法結合不同馬可夫鏈蒙地卡羅法（Markov Chain Monte Carlo, MCMC）之技巧，建構出品質良率指標Yq的信賴區間，並進行模擬評估與比較。結果顯示，以貝氏方法結合MCMC的技巧所建構的區間估計，在涵蓋率的表現最為優異，可惜其執行時間較長，建議在時間受限的情況之下，若樣本數夠大（n>=100）可選擇性地採用複式抽樣法做為決策依據，待時間充裕時重新使用MCMC技巧做為驗證。已知伽瑪與韋伯分配可收斂為指數分配，故本研究亦針對此情況進行模擬分析，證實收斂後結果相符，操作時可擇一使用。最後，本研究建立一圖形化使用者操作界面，並搭配伽瑪與韋伯分配之實際案例，可協助決策者快速上手與應用至其他案例當中，有效提升實務價值。

Quality is not only important for customers to evaluate a product but also for producers to enhance their competitiveness. Therefore, there’re a growing of quality management techniques were applied in actual processing. A quantitative measurement called process yield is widely used in manufacturing industry for measuring process performance. However, process yield only considers whether the product quality characteristic meeting the specification limits and it can’t classify the quality of the product. A customer-orientated index was proposed to solve the problem, named quality-yield index ( Yq). Yq is based on process yield and further reflecting to customers loss, which could provide an more appropriate measure of actual quality levels.
Researches related to Yq are limited in normal distribution, approximate distribution (AD) approach, bootstrap method and others are included. Due to the nonnormality in most of actual process, this study would focus on Gamma and Weibull distributions, try to establish an interval estimation for Yq by using Bootstrap and Bayesian method with different Markov Chain Monte Carlo (MCMC) techniques. After simulation and analysis, the result showed using MCMC techniques is the best to build interval estimation in our case. Considering the executing time, we suggested using bootstrap method at first under the time restriction then check the result by MCMC later. It’s known that Gamma and Weibull distribution would reduce to exponential distribution and we found that the result is consistent. The both methods could be used to exponential distribution. Moreover, a graphical user interface is developed then using two application cases to demonstrate the procedure and help decision maker to learn and use it.

致謝 i
摘要 ii
Abstract iii
目錄 iv
圖目錄 vi
表目錄 viii
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 研究架構 3
第二章 文獻回顧 6
2.1 製程能力指標 6
2.1.1 良率指標 6
2.1.2 損失指標 9
2.1.3 製程良率 10
2.1.4 品質良率指標 12
2.2 品質良率指標之區間估計方法 15
2.2.1 近似估計方法 16
2.2.2 複式抽樣法 17
2.2.3 廣義信賴區間法 20
2.2.4 貝氏方法 22
2.3 馬可夫鏈蒙地卡羅法 24
2.3.1 Metropolis-Hastings 演算法 25
2.3.2 Gibbs 抽樣法 28
2.3.3 適應性拒絕抽樣法 30
2.3.4 適應性拒絕Metropolis抽樣法 32
2.4 伽瑪分配 35
2.5 韋伯分配 37
第三章 基於伽瑪分配下區間估計方法之分析與探討 40
3.1 區間估計方法說明 40
3.1.1 複式信賴區間 40
3.1.2 以複式抽樣法建立信賴區間 43
3.1.3 貝氏區間估計 44
3.1.4 以馬可夫鏈蒙地卡羅法建構可信區間 47
3.2 數值模擬分析之結果與比較 48
3.2.1 參數設定與執行步驟 48
3.2.2 涵蓋率與平均寬度 53
3.2.3 ARMS初始值範圍敏感度分析 55
3.2.4 複式抽樣法與馬可夫鏈蒙地卡羅法模擬結果 59
第四章 基於韋伯分配下區間估計方法之分析與探討 66
4.1 區間估計方法說明 66
4.1.1 複式信賴區間 66
4.1.2 以複式抽樣法建立信賴區間 67
4.1.3 貝氏區間估計 68
4.1.4 以馬可夫鏈蒙地卡羅法建構可信區間 71
4.2 數值模擬分析之結果與比較 72
4.2.1 參數設定與執行步驟 72
4.2.2 ARMS初始值範圍設定敏感度分析 77
4.2.3 複式抽樣法與馬可夫鏈蒙地卡羅法模擬結果 81
4.2.4 收斂至指數分配之比較 88
4.3 配適度檢定 91
第五章 圖形化使用者介面與案例分析 94
5.1 圖形化使用者介面介紹 94
5.2 案例分析 105
5.2.1 後段封裝製程銲球尺寸 105
5.2.2 表面安裝零件高度 111
第六章 結論與未來研究方向 116
6.1 結論 116
6.2 未來研究方向 118
參考文獻 119
附錄A 123
附錄B 129

一、 中文文獻
1. 林孟慈 (2020)。以品質良率指標為基礎探討供應商評選方法之研究，國立清華大學工業工程與工程管理學系碩士論文，未出版，新竹市。
2. 廖律瑋 (2009)。考慮韋伯製程變異數發生變動下之製程能力調整，國立交通大學工業工程與管理學系碩士論文，未出版，新竹市。
3. 鄭惇勻 (2019)。壽命績效指標於伽瑪與韋伯分配下區間估計方法比較研究，國立清華大學工業工程與工程管理學系碩士論文，未出版，新竹市。
二、 英文文獻
1. Anderson, T. W., & Darling, D. A. (1952). Asymptotic theory of certain" goodness of fit" criteria based on stochastic processes. The annals of mathematical statistics, 193-212.
2. Balamurali, S., & Kalyanasundaram, M. (2002). Bootstrap lower confidence limits for the process capability indices Cp, Cpk and Cpm. International Journal of Quality & Reliability Management, 19, 1088-1097.
3. Bernardo, J. M., & Smith, A. F. M. (1993). Bayesian theory: John Wiley & Sons.
4. Box, G. E., & Tiao, G. C. (2011). Bayesian inference in statistical analysis (Vol. 40): John Wiley & Sons.
5. Boyles, R. A. (1991). The Taguchi Capability Index. Journal of Quality Technology, 23(1), 17-26.
6. Chan, L. K., Cheng, S. W., & Spiring, F. A. (1988). A new measure of process capability: Cpm. Journal of Quality Technology, 20(3), 162-175.
7. Cheng, S. W., & Spiring, F. A. (1989). Assessing Process Capability: A Bayesian Approach. IIE Transactions, 21(1), 97-98.
8. Cowles, M. K., & Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association, 91(434), 883-904.
9. Efron, B. (1979). Bootstrap methods: another look at the jackknife. The Annals of Statistics, 7(1), 1-26.
10. Efron, B. (1981). Nonparametric estimates of standard error: the jackknife, the bootstrap and other methods. Biometrika, 68(3), 589-599.
11. Efron, B. (1982). The jackknife, the bootstrap and other resampling plans: SIAM.
12. Efron, B. (1987). Better bootstrap confidence intervals. Journal of the American Statistical Association, 82(397), 171-185.
13. Efron, B., & Gong, G. (1983). A leisurely look at the bootstrap, the jackknife, and cross-validation. The American Statistician, 37(1), 36-48.
14. Efron, B., & Tibshirani, R. (1986). Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy. Statist. Sci., 1(1).
15. Franklin, L. A., & Wasserman, G. S. (1992). Bootstrap lower confidence limits for capability indices. Journal of Quality Technology, 24(4), 196-210.
16. Gelfand, A. E., & Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85(410), 398-409.
17. Geman, S., & Geman, D. (1984). Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6(6), 721-741.
18. Geyer, C. J. (1992). Practical Markov chain Monte Carlo. Statistical science, 473-483.
19. Gilks, W. R. (1992). Derivative-free adaptive rejection sampling for Gibbs sampling. Bayesian statistics, 4(2), 641-649.
20. Gilks, W. R., Best, N. G., & Tan, K. (1995). Adaptive rejection Metropolis sampling within Gibbs sampling. Journal of the Royal Statistical Society: Series C (Applied Statistics), 44(4), 455-472.
21. Gilks, W. R., & Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Journal of the Royal Statistical Society: Series C (Applied Statistics), 41(2), 337-348.
22. Gunter, B. H. (1989). The use and abuse of Cpk. Quality Progress, 22(1), 72-73.
23. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1), 97-109.
24. Hsiang, T. C., & Taguchi, G. (1985). A tutorial on quality control and assurance-the Taguchi methods. ASA Annual Meeting, Las Vegas, Nevada, U.S.A.
25. Hsu, Y.-C., Pearn, W. L., & Wu, P.-C. (2008). Capability adjustment for gamma processes with mean shift consideration in implementing Six Sigma program. European Journal of Operational Research, 191(2), 517-529.
26. Jeffreys, H. (1998). The theory of probability: OUP Oxford.
27. Johnson, T. (1992). The relationship of Cpm to squared error Loss. Journal of Quality Technology, 24(4), 211-215.
28. Juran, J. (1974). Juran’s Quality Control Handbook: McGraw-Hill.
29. Kane, V. E. (1986). Process capability indices. Journal of Quality Technology, 18(1), 41-52.
30. Kolmogorov, A. (1933). Sulla determinazione empirica di una lgge di distribuzione. Inst. Ital. Attuari, Giorn., 4, 83-91.
31. Liao, M. Y. (2016). Markov chain Monte Carlo in Bayesian models for testing gamma and lognormal S-type process qualities. International Journal of Production Research, 54(24), 7491-7503.
32. Liao, M. Y. (2017). Efficient technique for assessing actual non‐normal quality loss: Markov chain Monte Carlo. Quality and Reliability Engineering International, 33(5), 945-957.
33. Liao, M. Y., & Wu, C. W. (2018). Supplier selection based on normal process yield: the Bayesian inference. Neural Computing and Applications, 32(8), 4121-4133.
34. Mathew, T., Sebastian, G., & Kurian, K. (2007). Generalized confidence intervals for process capability indices. Quality and Reliability Engineering International, 23(4), 471-481.
35. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6), 1087-1092.
36. Ng, K. K., & Tsui, K. L. (1992). Expressing variability and yield with a focus on the customer. Quality Engineering, 5(2), 255-267.
37. Pearn, W. L., Chang, Y. C., & Wu, C. W. (2004). Quality-yield measure for production processes with very low fraction defective. International Journal of Production Research, 42(23), 4909-4925.
38. Pearn, W. L., Chang, Y. C., & Wu, C. W. (2005). Bootstrap approach for estimating process quality yield with application to light emitting diodes. The International Journal of Advanced Manufacturing Technology, 25(5-6), 560-570.
39. Pearn, W. L., Kotz, S., & Johnson, N. L. (1992). Distributional and inferential properties of process capability indices. Journal of Quality Technology, 24(4), 216-231.
40. Pearn, W. L., Lin, G. H., & Chen, K. S. (1998). Distributional and inferential properties of the process accuracy and process precision indices. Communications in Statistics - Theory and Methods, 27(4), 985-1000.
41. Pearn, W. L., Wu, C. C., & Wu, C. H. (2015). Estimating process capability index Cpk: classical approach versus Bayesian approach. Journal of Statistical Computation and Simulation, 85(10), 2007-2021.
42. Pearn, W. L., & Wu, C. W. (2005). Process capability assessment for index Cpk based on bayesian approach. Metrika, 61(2), 221-234.
43. Pearson, K. (1900). X. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 50(302), 157-175.
44. Robert, C., & Casella, G. (2013). Monte Carlo statistical methods: Springer Science & Business Media.
45. Singh, S. K., Singh, U., & Kumar, D. (2013). Bayes estimators of the reliability function and parameter of inverted exponential distribution using informative and non-informative priors. Journal of Statistical Computation and Simulation, 83(12), 2258-2269.
46. Son, Y. S., & Oh, M. (2006). Bayesian estimation of the two-parameter Gamma distribution. Communications in Statistics-Simulation and Computation, 35(2), 285-293.
47. Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association, 69(347), 730-737.
48. Tierney, L. (1994). Markov chains for exploring posterior distributions. The Annals of Statistics, 1701-1728.
49. Weerahandi, S. (1995). Generalized Confidence Intervals. In S. Weerahandi (Ed.), Exact Statistical Methods for Data Analysis (pp. 143-168). New York, NY: Springer New York.
50. Wu, C. W., & Huang, P. H. (2010). Generalized Confidence Intervals for Comparing the Capability of Two Processes. Communications in Statistics - Theory and Methods, 39(13), 2351-2364.
51. Wu, C. W., & Pearn, W. L. (2005). Capability testing based on Cpm with multiple samples. Quality and Reliability Engineering International, 21(1), 29-42.
52. Yang, R., & Berger, J. O. (1996). A catalog of noninformative priors: Institute of Statistics and Decision Sciences, Duke University.