近年來，科技的創新與發展下，經濟實惠的產品相繼問世，除了產品的功能及價格之外，產品品質亦是消費者相當重視的一環。為了把關產品品質，生產者採用各種統計品質管制的方法進行線上即時監控產品製程，以降低製程變異，進而評估製程的產出績效。而製程能力指標可以量化的方式衡量製程能力，並說明一個製程符合規格之能力，其中最常被使用的基本型製程能力指標為Cpk。但此基本型指標僅適用於符合常態分配的製程資料，若製程資料為非常態分配，則將會錯估其製程能力。實務上，雖許多資料服從常態分配，但也有一些資料並不符合常態分配，例如：壽命資料、月降雨量、可靠度資料等。過去已有許多學者提出各式非常態型的製程能力指標，期能更適切地反映實際製程之能力。但此類非常態型製程能力指標與實際產出品質水準並無對應的關係，再加上此類指標在不同分配下的表現不盡相同，故即使求算出指標值後，仍無法直接對應推論出製程能力的優劣。因此，本研究首先針對伽瑪分配，在不同參數下，分析及探討幾個常見的非常態型指標(包括Cjkp、Cs、Cpk(WV)、Cpk(WSD))的估計偏誤，並選出Cpk(WV)做為後續的研究對象。其次，利用曲線擬合(curve fitting)方法找出常態型指標Cpk及非常態型指標Cpk(WV)的修正函數，使修正後的指標能更精確地反映製程能力。再者，為了讓使用者能更方便的應用此方法，本研究將上述之演算流程發展一套完整的平台及操作介面，使用者只需輸入製程規格及資料，即可得修正後的指標估計值。最後，引用案例的實際數據進行操作，以說明本研究的修正法在實務中的應用。

In the recent year, more and more goods are both excellent in quality and reasonable in price. People put more attention on the quality of products. To maintain the quality and lower the variation, manufacturers tend to use lots of statistical quality control method to achieve real-time monitoring of the manufacturing processes. Process capability index (PCI) is commonly adopted for measuring the quality of products, which is used to measure whether the process is in-control. Production department can also trace and improve the process performance by analyzing PCIs. The most widely used PCIs is Cpk, which is critical to the normality assumption. If a process violates this assumption, Cpk tends to misestimate the capability. Even if most of processes are under normal assumption in practice, there are still some processes with non-normal distributions. For example, life time data, monthly rainfall data, reliability data. In order to assess process capability more accuracy, some studies proposed PCIs for non-normal processes. However, there are no corresponding relationship between non-normal PCIs and quality condition. Moreover, these PCIs perform differently under various non-normal distributions. Therefore, this study focus on Cpk(WV) which perform best under gamma distributions among Cjkp, Cs, Cpk(WV) and Cpk(WSD). Secondly, correction functions of both Cpk and Cpk(WV) are obtained by curve fitting method and thus revising the PCIs. The results show that the revised PCIs give more accuracy estimation. An example is presented to illustrate the results. Besides, a graphic user interface is also constructed to help user modify their PCIs by this method.

摘要 iii
Abstract iiiii
致謝 iii
目錄 iv
圖目錄 vi
表目錄 ix
第一章 緒論 1
1.1 研究背景與動機 1
1.2 研究目的 2
1.3 研究架構 3
第二章 文獻回顧與探討 5
2.1 製程能力指標 5
2.1.1 常態型製程能力指標 5
2.1.2 非常態型製程能力指標 11
2.2 曲線擬合法 16
2.3 伽瑪分配及最大概似估計法 23
第三章 伽瑪製程之曲線擬合修正法 27
3.1 各種非常態製程能力指標在伽瑪分配下相對偏誤之分析與比較 27
3.2 製程能力指標真值估計伽瑪分配資料之結果 35
3.3 曲線擬合法應用於伽瑪分配之製程能力指標 36
3.3.1 模型之參數設定 37
3.3.2 模型建立之流程 38
3.3.3 利用修正模型修正製程能力指標估計流程 41
第四章 曲線擬合修正法之結果驗證與分析 45
4.1 曲線擬合修正法之結果驗證 45
4.1.1 曲線擬合修正法應用於常態指標之結果驗證 45
4.1.2 曲線擬合修正法應用於非常態指標之結果驗證 50
4.2 曲線擬合修正法之敏感度分析 53
4.2.1 形狀參數對修正結果之影響 53
4.2.2 不同樣本參數範圍對於修正結果之影響 56
4.2.3 良率與指標轉換式上下界之選擇 61
第五章 圖形化使用者介面與案例分析 65
5.1 圖形化使用者介面操作及說明 65
5.2 案例分析 73
第六章 結論及未來展望 77
6.1 結論 77
6.2 未來展望 78
參考文獻 79
附錄 A 83
附錄 B 89
附錄 C 93

一、中文參考文獻
1. 李顯宏. (2004). MATLAB 7. x 程式開發與應用技巧: 文魁資訊.
2. 蕭文龍. (2007). 多變量分析最佳入門實用書: SPSS+ LISREALΔΚ: 碁峰資訊.
二、英文參考文獻
1. Akima, H. (1969). A Method of Smooth Curve Fitting: Institute for Telecommunication Sciences.
2. Akima, H. (1970). A new method of interpolation and smooth curve fitting based on local procedures. Journal of the ACM (JACM), 17(4), 589-602.
3. Aldowaisan, T., Nourelfath, M., & Hassan, J. (2015). Six sigma performance for non-normal processes. European Journal of Operational Research, 247(3), 968-977.
4. Arlinghaus, S. (1994). Practical Handbook of Curve Fitting: CRC press.
5. Bai, D. S., & Choi, I. (1995). X and R control charts for skewed populations. Journal of Quality Technology, 27(2), 120-131.
6. Barger, G. L., & Thom, H. C. (1949). Evaluation of drought hazard. Agronomy Journal.
7. Box, G. E., & Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society. Series B (Methodological), 211-252.
8. Boyles, R. A. (1991). The Taguchi capability index. Journal of Quality Technology, 23, 17-26.
9. Chang, Y. S., & Bai, D. S. (2001). Control charts for positively‐skewed populations with weighted standard deviations. Quality and Reliability Engineering International, 17(5), 397-406.
10. Chang, Y. S., Choi, I. S., & Bai, D. S. (2002). Process capability indices for skewed populations. Quality and Reliability Engineering International, 18(5), 383-393.
11. Choi, S. C., & Wette, R. (1969). Maximum likelihood estimation of the parameters of the gamma distribution and their bias. Technometrics, 11(4), 683-690.
12. Choobineh, F., & Ballard, J. (1987). Control-limits of QC charts for skewed distributions using weighted-variance. IEEE Transactions on Reliability, 36(4), 473-477.
13. Choobineh, F., & Branting, D. (1986). A simple approximation for semivariance. European Journal of Operational Research, 27(3), 364-370.
14. Clements, J. (1989). Process capability indices for non-normal calculations. Quality Progress, 22, 49-55.
15. Cowden, D. J. (1957). Statistical Methods in Quality Control: Prentice-Hall: Princeton.
16. Forsythe, G. E., Moler, C. B., & Malcolm, M. A. (1977). Computer Methods for Mathematical Computations.
17. Hahn, G. J. S., & Samuel, S. (1967). Statistical Models in Engineering: John Wiley & Sons (New York).
18. 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.
19. Husak, G. J., Michaelsen, J., & Funk, C. (2007). Use of the gamma distribution to represent monthly rainfall in Africa for drought monitoring applications. International Journal of Climatology, 27(7), 935-944.
20. Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika, 36(1/2), 149-176.
21. Johnson, N. L., Kotz, S., & Pearn, W. L. (1994). Flexible process capability indices. Pakistan Journal of Statistics, 10A, 23-31.
22. Juran, J. M. (1974). Quality Control Handbook.
23. Kane, V. E. (1986). Process capability indices. Journal of Quality Technology, 18(1), 41-52.
24. Lesh, F. H. (1959). Multi-dimensional least-squares polynomial curve fitting. Communications of the ACM, 2(9), 29-30.
25. Liao, M. Y., Pearn, W. L., & Liu, Y. L. (2015). Assessing the actual Gamma process quality – a curve-fitting approach for modifying the non-normal flexible index. International Journal of Production Research, 53(15), 4720-4734.
26. Montgomery, D. C. (2009). Introduction to Statistical Quality Control: John Wiley & Sons (New York).
27. Pearn, W. L., & Chen, K. (1997). A practical implementation of the process capability index Cpk. Quality Engineering, 9(4), 721-737.
28. Pearn, W. L., & Chen, K. (1999). Making decisions in assessing process capability index C-pk. Quality and Reliability Engineering International, 15(4), 321-326.
29. Pearn, W. L., Kotz, S., & Johnson, N. L. (1992). Distributional and inferential properties of process capability indexes. Journal of Quality Technology, 24(4), 216-231.
30. Piao, C., & Zhi-Sheng, Y. (2018). A systematic look at the gamma process capability indices. European Journal of Operational Research, 265(2), 589-597.
31. Pyzdek, T. (1992). Process capability analysis using personal computers. Quality Engineering, 4(3), 419-440.
32. Reinsch, C. H. (1967). Smoothing by spline functions. Numerische Mathematik, 10(3), 177-183.
33. Schoenberg, I. J. (1964). On interpolation by spline functions and its minimal properties. In On Approximation Theory/Über Approximationstheorie (pp. 109-129): Springer.
34. Somerville, S. E., & Montgomery, D. C. (1996). Process capability indices and non-normal distributions. Quality Engineering, 9(2), 305-316.
35. Van Noortwijk, J. (2009). A survey of the application of gamma processes in maintenance. Reliability Engineering & System Safety, 94(1), 2-21.
36. Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (1993). Probability and Statistics for Engineers and Scientists (Vol. 5): Macmillan New York.
37. Weisberg, S. (2005). Applied Linear Regression (Vol. 528): John Wiley & Sons (New York).
38. Wright, P. A. (1995). A process capability index sensitive to skewness. Journal of Statistical Computation and Simulation, 52(3), 195-203.
39. Wu, H. H., Swain, J. J., Farrington, P. A., & Messimer, S. L. (1999). A weighted variance capability index for general non‐normal processes. Quality and Reliability Engineering International, 15(5), 397-402.