| 摘要: | 過去製程良率是評估製程品質績效的一項重要依據，然而近二十年來產業已大多採用製程能力指標為量測製程績效的工具。製程能力指標是一無單位的量化值，衡量製程生產出符合要求規格產品的能力，其中，Cpk指標是一個重要且是目前業界常用的能力指標。由於Cpk指標是製程平均數和製程標準差的函數；其中製程平均數與製程標準差大多是未知且必須透過樣本數據予以估計，一般實務應用皆根據樣本數據以點估計的方式來估計Cpk指標值；不過，Cpk指標該視為滿足某種分配性質的隨機變數；而不應只是根據樣本數據以點估計的方式來評估製程的品質績效，不然將會忽略抽樣誤差所造成的效應。又過去的文獻大多探討單一樣組抽樣估計的方式計算Cpk指標估計值；然而製程樣本數據往往可以是以多樣組的型態呈現。另一方面，由於生產過程中投入的原物料或零組件或許由具有不同品質水準的多家供應商所提供，亦或同一家供應商於不同時期所提供的原物料或零組件有著不一致的品質水準等；因此，視製程平均數為一常數並非適當。本文根據樣本數據的收集呈現多樣組的形態時，考慮不同估計製程標準差的方式而定義四種Cpk指標的類似貝氏估計式，其中估計製程標準差的方法包括整體樣本標準差、混合樣本標準差、樣組標準差的平均和樣組全距的平均。本文的具體貢獻主要有三個方面，第一方面討論前兩種Cpk指標估計式的分配和推論統計性質，以說明並確認其估計Cpk指標的適度性。第二方面探討當大樣本時，估計式的漸近分配性質，並建立Cpk指標的近似信賴界限。第三方面根據統計假設檢定的理論考慮抽樣誤差，就四種不同的Cpk指標類似貝氏估計式發展較具有可靠性的評估製程品質績效的方法，決定判定臨界值以評估製程是否能夠滿足品質的要求。其中，利用樣組標準差的平均和樣組全距的平均所建立的Cpk指標估計式無法得到確實的抽樣分配，因此，本文應用Hamaker近似法則以常態逼近方式以避免原本複雜或未知的抽樣分配之使用，而求得近似並保守的判定臨界值和檢定力。再者，本文進一步利用兩定點技巧建構Cpk指標的評估計畫，即決定適當的判定臨界值以及需要達到最少的樣本總數。本研究最後提供實務執行的步驟以供業界在評估製程品質績效時的更有效的決策方式。 Process yield is a reasonable approach used in industry for assessing process performance. There is also another and more common approach to measuring and communicating the assessment of process performance, called process capability indices. Process capability indices have been used widely in the industry to provide a numerical measure to determine whether a process is capable of producing items within the established specification limits present by the customer or designer. Among various capability indices, companies use capability indices to measure process improvement or to compare the processes of vendors and internal suppliers continue to rely heavily on Cpk index. Since the estimator of Cpk index is a random variable with a corresponding distribution, simply reporting from the sample data and then making a conclusion on whether the process is considered capable is not reliable. In practice, process information about process characteristics is often derived from multiple samples rather than from one single sample, particularly, when a daily-based production control plan is implemented for monitoring process stability. Applications in real situation, the production may require multiple supplies with different quality levels on each single shipment of the raw materials or components, or the raw materials or components from supply with unequal performance level on each period. Therefore, the common assumption that the process mean stay as a constant may not be satisfied in real situations. In this dissertation, There are four various standard deviation estimators are substituted for the process standard deviation, including the un-pooled sample standard deviation, the pooled sample standard deviation, the average of the subsample standard deviation and the average of the subsample range. The concrete contributions of this dissertation are threefold. The first is to investigate the distributional and inferential properties of the estimators of Cpk index based on un-pooled and pooled standard deviation estimators. The second is to investigate the asymptotic distributions of these estimators of Cpk index for arbitrary population under fairly general conditions of regularity, assuming that the fourth central moment exists. The third is to ensure the performance assessment reliable, the theory of testing hypothesis using the sampling distributions of these estimators of Cpk index are implemented, which is provided the critical values required for making decisions. Following Hamaker’s approximation, the testing procedures can be adequately derived from normal approximations while avoiding more complicated distributions. By further applying a two-point adjustment, equations to determine the critical values and to estimate the sample sizes necessary to achieve the recommended minimal value for Cpk index are provided. We then develop a step- by-step procedure for testing Cpk index. Practitioners can use the procedure to assess whether their processes meet the quality requirement. |
