考慮韋伯分配在非機遇性原因下之ARMA管制圖經濟性設計

Consideration of Weibull Distribution under the Assignable Causes for Economic Design of the ARMA Control Chart

駱景堯(Chinyao Low)；林文儀(Wen-Yi Lin)

經濟性設計 ； 自我相關 ； ARMA管制圖 ； 韋伯分配 ； 基因演算法 ； economic design ； autocorrelation ； ARMA control chart ； Weibull distribution ； genetic algorithm

品質學報

17卷5期（2010 / 10 / 31）

365 - 387

繁體中文

品質管制在現今的製造業中是相當重要的，如何使用一套快速又經濟的製程管制手法，便是一門相當重要的課題。在連續型生產製程的產品品質特性間存在著自我相關性，目前傳統的修華特管制圖偵測自我相關製程，會因爲平均連串長度(average run length, ARL)縮短，造成管制圖誤警率偏高。因此，本研究採用自我迴歸移動平均(autoregressive moving average, ARMA)管制圖來改善此缺點，並加入經濟性設計之探討，期望在最低成本下能達到管制圖的偵測能力。由於Duncan在管制圖的經濟性設計中，提到當製程非機遇性原因發生，導致製程偏移發生的時間符合指數分配。但是韋伯分配被廣泛地使用在可靠度工程上，如電機、機械系統的衰減時間模式。這些例子包括在發電裝置的記憶單位；機械方面，如使用在飛機或汽車上的軸承的部份。因此，本研究將以韋伯分配(Weibull)來呈現當製程非機遇性原因發生，導致製程產生偏移的時間，並利用模擬方式來推估各種參數組合下的平均連串長度值。隨後以基因演算法(genetic algorithm, GA)求其ARMA管制圖的經濟設計之參數（抽樣樣本數n、抽樣間隔h、管制界限寬度k、ARMA管制圖參數θ及φ)最適組合。最後，並以敏感度分析來探討模式參數值對ARMA管制圖參數的影響。

Control charting is very important in the manufacturing nowadays and it will be significant to use the faster and economical manufacture technique. In a continuous manufacturing process, there exist autocorrelations among the observations of quality characteristics that cause the reduction of average run length (ARL). This results in higher false alarm probability in traditional Shewhart control chart which is capable for monitoring manufacturing process with autocorrelation. In this paper, we develop an economic design of the autoregressive moving average (ARMA) control chart which is determined by the parameter set that minimizes the total control chart cost. In economical design control charts, Duncan mentioned that the time that the process shifts which is caused by the occurrence of assignable causes conforms exponential distribution. The Weibull distribution has been used extensively in reliability engineering as a model of time to failure for electrical and mechanical components and systems. Examples of situations in which the Weibull has been used include electronic devices such as memory elements, mechanical components such as bearing, and structural elements in aircraft and automobiles. In this research, the Weibull distribution is used to present the time that the process shifts which is caused by the occurrence of assignable causes and the simulation technique is also used to assess the ARL values under every parameter set. Also, genetic algorithm is used to search the optimal parameter set (the sample size n, the sampling interval h, the control limit k, the autoregressive parameter φ, and the moving average parameter θ) for ARMA control chart. Finally, sensitivity analysis is then carried out to investigate the effects of model parameters on the solutions of the parameters of ARMA control chart.

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