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 製程能力指標Cpm可以充分反應製程損失與製程良率，因此是工業界常用的指標之一。若以其指標的點估計值評估製程品質優劣，會因為抽樣誤差的產生而導致對製程能力指標的誤判；而在區間估計裡考慮了點估計的標準誤，並可藉由信賴水準下限判斷製程的優劣；但因為其估計的機率密度分配函數相當的複雜，因此其信賴區間的推導也相對的困難。本文將指標Cpm化為δ＝（μ－T）/ d與γ＝ σ / d的函數，並以δ與γ所構成的聯合信賴區間行解區域及Cpm為目標函數，運用數學規劃方法求出指標Cpm的信賴區間。再以蒙地卡羅模擬法評估其涵蓋率來驗證本方法的正確性。如此不但省去統計方法的煩瑣過程，且求解為最佳解，沒有誤差的問題，在觀念與實際應用上也更為容易了解，以便幫助其製程能力的判讀，應用上也更為有效率。
 The process capability index (Cpm), which can respond to both process loss and process yield, is one of the indices frequently used in industries. If values of individual index points are used to evaluate the process quality, errors in sampling will lead to erroneous results. Using the interval estimation method, the standard deviation for point estimation is considered thus avoiding the errors. Additionally, the process quality can also be determined based on the confidence lower bound. However, because the probability density function of Cpm is quite complex, the confidence intervals for Cpm is difficult to derive.In this article, Cpm is transferred to δ and γ functions, δ＝(μ－T) / d and γ＝σ / d; the confidence sets for δ and γ as well as Cpm are considered as the objective functions. A mathematical programming model is derived to calculate confidence intervals for Cpm, whose coverage percentage is estimated using the Monte Carlo simulation. This method, which can leave out complex statistics procedures to obtain optimized and error-free answers, is easy to understand and implement for determining the capacity of differentiate processes.
 中文摘要 …………………………………………………………………… i英文摘要 …………………………………………………………………… ii誌謝 …………………………………………………………………… iii目錄 …………………………………………………………………… iv表目錄 …………………………………………………………………… v圖目錄 …………………………………………………………………… vi符號說明 …………………………………………………………………… vii第一章 緒論……………………………………………………………… 11.1 研究背景………………………………………………………… 11.2 研究動機與目標………………………………………………… 41.3 研究架構………………………………………………………… 5第二章 文獻回顧………………………………………………………… 62.1 單邊規格與雙邊規格製程能力指標…………………………… 62.2 製程能力指標之分配與統計性質……………………………… 82.3 製程能力指標之信賴區間……………………………………… 9第三章 應用數學規劃模式求解指標信賴區間………………………… 103.1 建立指標Cpm信賴區間之數學規劃模式………………………. 103.2 數學規劃模式之求解…………………………………………… 133.2.1 推導指標Cpm信賴區間下限…………………………………… 133.2.2 推導指標Cpm信賴區間下限.......................... 15第四章 蒙地卡羅模擬法………………………………………………… 19第五章 結論……………………………………………………………… 22參考文獻 …………………………………………………………………… 23附錄 蒙地卡羅模擬法程式碼………………………………………… 25
 【1】 Bissell, A.F., 1990, “How reliable is your capability index?”, Appl. Statist., Vol. 39, No. 3, pp. 331-340.【2】 Boyles, R.A., 1991, “The Taguchi capability index”, Journal of Quality Technology, Vol. 23, pp. 17-26.【3】 Chan, L.K., Cheng, S.W. and Spring, F.A., 1988, “A new measure of process capability, Cpm”, Journal of Quality Technology, Vol. 20, pp. 160-175.【4】 Chan, L.K., Xiong, Z. and Zhang, D., 1990, “On the Asymptotic distribution of some process capability indices”, Communications in Statistics-Theory and Methods, Vol. 19, pp. 11-18.【5】 Chou, Y.M. and Owen, D.B., 1989, “On the distribution of the estimated process capability indices”, Communication in Statistics-Theory and Methods, Vol. 18, pp. 4549-4560【6】 Chou, Y.; Owen, D. B.; and Borrego, S. A., 1990, “Lower Confidence Limits on Process Capability Indices”, Journal of Quality Technology, Vol. 22, pp. 223-229.【7】 Chen, J. P. and Chen, K. S., 2004, “Comparing the Capability of Two Process Using Cpm”, Journal of Quality Technology, Vol. 36(3), pp. 329-335【8】 Franklin, L. A. and Wasserman, G.S., 1992, “Bootstrap Confidence Limits for Capability Indices”, Journal of Quality Technology, Vol. 24(4), pp. 196-210.【9】 Hammersley, J. M. and Handscomb, D. C., 1964, Monte Carlo Methods, Chapman and Hall, London.【10】 Juran, J.M. 1974, Jurans Quality Control Handbook, 3rd edition. McGraw-Hill, New York.【11】 Johnson T., 1992, “The Relationship of Cpm to Squared Error Loss”, Journal of Quality Technology, Vol. 24, pp. 211-215.【12】 Kane, V.E. 1986, “Process capability indices”, Journal of Quality Technology, Vol. 18, pp. 41-52.【13】 Kotz, S. and Johnson, N.L., 1993, Process Capability Indices, Chapman & Hall, New York.【14】 Kots,S., Pearn, W.L. and Johnson, N.L., 1993, “Some process capability indices are more reliable than one might think”, Applied Statistics, Vol. 42, No. 1, pp.55-62.【15】 Kushler, R.H. and Hurley, P., 1992, “Confidence bounds for capability indices”. Journal of Quality Technology, Vol. 24(4), pp. 188-195.【16】 Pearn, W.L., Kotz, S. and Johnson, N.L., 1992, “Distributional and inferential properties of process capability indices”, Journal of Quality Technology, Vol. 23, pp. 216-231.【17】 Pearn, W.L. and Chen, K.S., 1997, “A practical implementation of the process capability index Cpk”, Quality Engineering, Vol. 9(4), pp. 721-737.【18】 Pearn, W.L. and Chen, K.S., 1999, “Making decision in assessing process capability index Cpk”, Quality and Reliability Engineering International, Vol. 15, pp. 321-326.【19】 Shiau, J.H., Chiang, C.T. and Hung, H.N., 1999, “A Bayesian procedure for process capability?”, Quality and Reliability Engineering International, Vol. 15, pp. 369-378.【20】 Zhang, N. F., Stenback, G.A. and Wardrop, D.M., 1990, “Interval estimation of process capability index Cpk”, Communications in Statistics-Theory and Methods, Vol. 19, No. 12, pp. 4455-4470.【21】 Zimmer, L.S., Hubele, N.F. and Zimmer W.J., 2001, “Confidence intervals and sample size determination for Cpm”, Quality and Reliability Engineering International, Vol. 17, pp. 51-68
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