Xbar 與R 管制圖的聯合經濟統計模式設計 應用在服從非常態分配的品質特性

The Joint Economic-Statistical Design of Xbar and R Control Charts for Nonnormal Data

包益坤(Yi-Kun Pao)

指導教授 ： 陳慧芬

中原大學/工學院/工業工程研究所/碩士(2004年)

https://doi.org/10.6840/cycu200400309

摘要

本論文研究探討X 與R 管制圖的經濟統計性設計，我們假設品質特性測量值(也就是觀察值)服從非常態分配且系統在製程管制內時間服從韋伯分配。當我們設計一個管制圖時，必須決定四個設計參數：樣本數n、抽樣間隔時間h、X 管制圖上下限的距離因素1 k 及R 管制圖
上下限的距離因素2 k ，若使用經濟統計性設計，這四個參數必須滿足在型一與型二誤差限制下使得期望單位時間成本最低，其中成本函數是修改Costa 成本模式計算得來的。
在實際的生產線上，並不只有一種原因會造成不良品的產生。因此，若只有考量平均值的位移並用X 管制圖來偵測，在實際的生產情況上並不是十分的合理。因為平均值與變異數有可能同時發生位移，所以必須同時使用X 與R 管制圖來進行管制。
本論文建立一個成本模式是根據Costa 的經濟模式修改而成可適用於非常態品質特性且系統在製程管制內時間服從韋伯分配。選擇經濟統計性設計模式是因為可以控制型一與型二誤差同時成本只會微量增加。我們假設樣本觀察直是獨立從一組Johnson 分配抽樣出來。Johnson
分配家族可涵蓋所有可能的偏度與峰度值。型一與型二誤差難以計算必須經由模擬實驗求得。
我們提出敏感度分析研究非常態資料與不同程度的平均值和變異數位移所造成的影響。利用格點搜尋法找出最佳解{ n, h, k_1 ,k_2} 。結論如下所示:
當峰態值遞增時，樣本數n、k_1 與成本遞增，抽樣時間h 遞減。當平均值與變異數位移增加時，會造成 k_1 、 k_2 與成本的增加但是n、h 是遞減的。當變異數位移發生較平均數頻繁時，X 管制圖的1 k 是當作不必要的。當平均數位移發生較變異數頻繁時， k_2 是不必要的當品質特性是Johnson 無界分配時， k_2 保持不變當品質特性是Johnson 有界分配。當變異數位移發生
較平均數頻繁時，會造成成本增加。

關鍵字

經濟統計性設計 ； X 與R 管制圖 ； 韋伯分配 ； 期望單位時間成本 ； 格點法 ； 非常態性

並列摘要

This thesis considers the economic-statistical design of ¯ X and R control chart assuming that the quality characteristic measurement (i.e. observations) are nonnormal and the in control time is Weibull. When designing a control chart, four parameters–the sample size n, successive sampling time h, control limit factor k1 for
¯ X chart, and control limit factor k2 for R chart–must be determined. In economicstatistical
design, the four parameters are chosen so that the expected cost per hour is minimized under constraints on Type I and Type II error probabilities. The expected cost function is computed by the modfy Costa cost model.
In reality manufacture process, there is not only one cause to produce the fail products. Hence, if we only consider the shift in mean and use ¯ X chart to detect, it is not reasonable in reality manufacture process. Because both the process mean and variance may change simultaneoisly during a production cycle, therefore both ¯ X and R charts are employed to monitor the variations in the process parameters
simultaneously.
In this study, we propose a cost model which can be used in nonnormal data and Weibull in-control time. We modify the Costa fully economic design model to conform the general case. We choose the economic-statistical design because it controls the probability of having Type I and II errors probabilities and the expected cost increases marginally. We assume that the quality characteristic measurement are sampled independently from a Johnson distribution. The Johnson distribution is general in that it can be modeled to fit all possible values of the skewness andkurtosis. Type I and Type II error probabilities are difficult to compute and need to be estimated via simulation experiment.
We perform the sensitivity analysis to study the effects of nonnormality and the amount of shift in mean and variance. The optimal value of {n, h, k1, k2} are computed by the grid search method. The results show that:
When the kurtosis increases, the sample size n, control limit k1 for ¯ X chart, and the expected cost decrease and sampling interval h increaes. When the amount shift of mean or variance increases, the sample size n and sampling interval h decrease but control limit k1 for ¯ X chart and control limit k2 for R chart increases for Johnson unbounded distribution. When variance shift occurs frequently, the control limit coefficient for ¯ X chart, k1, tends to values that render the ¯ X chart unnecessary. When mean shift occurs frequently, the control limit coe?cient for R chart, k2, tends to unnecessary for Johnson unbounded distribution but still remains for Johnson bounded
distribution. When the variance shifts more frequently than the mean, the expected
cost increases.

並列關鍵字

Economic-Statistical Design ； Grid Search ； Expected Cost Per Hour

參考文獻

[1] Arnold, B. F. and E. V. Collani. (1987). Economic Process Control. Statistica

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Journal of Quality Technology 27, 120—131.

Under Weibull Shock Model. Technometrics 30, 407—414.

[4] Chan, L. K. and H. J. Chi. (2003). Skewness Correction ¯ X and R Charts Skeweed