多品質特性製程的製程能力指標 CpkT 的分布和推論

Abstract

與良率有關的製程能力指標Cpk 在製造業中已經廣泛的被使用來評量製程之表現。此方面多數的研究都著重在單一品質特性的製程，然而在實際的應用上，一個製程常常是具有多個品質特性，而每個特性都有不同的規格。在這一篇論文中，我們研究將Cpk推廣為多個獨立的品質特性製程的Cpk，因此新定義一個指標 。 我們證明了 像Cpk一樣與製程良率的上限跟下限有個一對一對應的關係。我們也以理論證明的方式找出估計量 的常態近似分配，透過常態近似分配，統計假設檢定、信賴區間、信賴下限都可以用來檢驗製程是否是有達到特定的標準。其中信賴下限在實務上特別重要，因為信賴下限可以用來估計製程的最小能力，與品質保證有很大的相關性；而精確性 (R) 則是估計量與信賴下限的比值，定義這個值是為了方便工程師每天量測製程的最小能力。接著透過資料模擬的方式，檢驗常態近似分配逼近估計量真正分配的準確性。 最後，我們用一個實際的製程－雙蕊光纖(a dual-fiber tip process) 作為例子，說明如何將新提出的指標及本文所提出之統計推論方法應用到實際的製程上。

Process capability index Cpk has been popularly used in the manufacturing industry for measuring process performance based on yield (proportion of conformities). Most researches on Cpk focus on processes with single quality characteristic; but in many real applications, a process often has multiple quality characteristics. In this study, we extend Cpk to a new index CpkT for processes with multiple characteristics. We prove that the inequalities that link Cpk to the yield also hold for the new index. A natural estimator of CpkT_hat is provided and a normal approximation to its distribution is derived. With this normal approximation, standard processes for statistical inferences such as hypothesis testing and confidence interval are developed for testing whether the process is capable and providing an interval estimate on CpkT, respectively. More importantly, we can obtain a confidence lower bound for CpkT, which measures the minimum process capability and is directly linked to quality assurance of products. The accuracy of the normal approximation is studied by simulation. Finally, we demonstrate how the new index CpkT as well as the inferential procedures developed in this study can be used with a real example of a dual-fiber tip process.