An ARL-unbiased Approach to Setting Control Limits of CCC-r Chart for High-Yield Processes

監控高產出製程之CCC-r管制圖管制界限最佳化之研究

陳佩雯(Pei-Wen Chen) ； 鄭春生(Chuen-Sheng Cheng)

《品質學報》 17卷6期 (2010/12) Pp. 435-452

摘要

由於蕭華特p管制圖並不適用於監控不合格率很低之高產出製程，有學者建議改以累積合格品數管制圖（cumulative count of conforming chart，簡稱CCC管制圖）作爲p管制圖之替代方法。不同於p管制圖著重於管制某特定樣本數之內的不合格品數目，CCC管制圖將檢驗出一個不合格品之前，所需檢驗之累積樣本數（合格品數）作爲管制變數，研究結果顯示當製程不合格率很低時，使用CCC管制圖進行監控之績效優於傳統p管制圖。爲了進一步提升CCC管制圖之靈敏度，有學者進一步提出將管制變數延伸至檢驗出第r個不合格品發生前之累積檢驗樣本數，並稱之爲CCC-r管制圖。由於CCC-r管制圖之判斷依據涵蓋較多組樣本資訊，因此對於製程參數之微量偏移具有較佳的偵測能力。
雖然CCC-r管制圖對於監控高產出製程具有不錯之成效，但當製程不合格率開始偏離目標值時，其平均連串長度（average run length，簡稱ARL）呈現增加之趨勢，表示CCC-r管制圖無法快速偵測出製程不合格率增加之情形。此現象稱之爲ARL-biased。爲解決此問題，本研究推導CCC-r管制圖之管制界限調整係數，並建立其迴歸模型以方便管理者使用。研究結果證實藉由調整係數修正後之管制界限，能夠消除CCC-r管制圖之ARL-biased的現象，同時，能夠有效降低CCC-r管制圖對於偵測製程不合格率增加之反應時間，提升偵測製程退化之敏感度。

關鍵字

CCC-r管制圖 ； 平均連串長度 ； 高產出製程

並列摘要

The cumulative count of conforming (CCC) chart is a new type of statistical process control technique for monitoring high-yield processes. Rather than counting the number of nonconforming items in a fixed sample size, a CCC chart monitors the cumulative number of items inspected until observing one nonconforming item. The CCC chart has shown to be superior to the traditional p chart in monitoring the fraction nonconforming of a high-yield process. The CCC-r chart is an improvement of the CCC chart. It monitors the cumulative number of items inspected until the r(superscript th) nonconforming item is observed based on negative binomial distribution. Due to the skewness of negative binomial distribution, the CCC-r chart usually shows an ARL-biased performance. In this paper, we introduce an ARL-unbiased design of the CCC-r chart. The ARL-unbiased design involves the determination of an adjustment factor of the existing control limits. Extensive numerical work shows that our approach can produce ARL-unbiased or nearly ARL-unbiased performance. In addition, the proposed approach shows more superiority in detecting process deteriorations, which is the major concern in high-yield processes. For simplification, a regression model is derived in this research. An adjustment factor can be calculated by specifying a predetermined false alarm rate α and the in-control nonconforming rate p0 to the regression equation. Then, the optimal control limits of an ARL-unbiased CCC-r chart can be easily obtained. An illustrative example is also presented to illustrate the proposed design procedure.

並列關鍵字

CCC-r chart ； ARL-unbiased ； high-yield processes

參考文獻

Acosta-Mejia, C. A.(1999).Improved p charts to monitor process quality.IIE Transactions.31(6),509-516.

Calvin, T.(1983).Quality control techniques for zero defects.IEEE Transactions on Components Hybrids and Manufacturing Technology.6(3),323-328.

Chan, L. Y.,Xie, M.,Goh, T. N.(2000).Cumulative quantity control charts for monitoring production processes.International Journal of Production Research.38(2),397-408.

Chang, T. C.,Gan, F. F.(2001).Cumulative sum charts for high yield processes.Statistica Sinica.11(3),791-805.

Gan, F. F.(1994).Design of optimal exponential CUSUM control charts.Journal of Quality Technology.26(2),109-124.