利用貝氏方法評估製程能力

Abstract

製程能力分析的目的是對製程的產出績效提供一個數值的衡量方法，根據這些數值生產者可來作為衡量製程能力好壞及接單與否的重要參考，對消費者而言，它可作為採購契約中對產品品質要求的重要標準，因此逐漸受到學術研究的重視以及業界的廣泛應用。然而，大部分製程能力分析的文獻都是基於傳統頻率學派方法所作的推論，但由於估計式之抽樣分配往往太過複雜而使得準確的信賴區間並不易求得。 Cheng與Spiring (1989) 提出利用貝氏方法來評估製程能力指標 Cp。 Chan等人 (1988) 也應用了類似的方法於指標Cpm 但假設了製程平均數等於目標值。 接著，Shiau等人 (1999a) 則針對指標Cpm發展出一貝氏評估程序，另外也針對Cpk指標但限制在製程平均數等於規格中心點的情形做分析。然而這樣的假設對大部分的實際應用是不能滿足的，因為在這種情形Cpk會簡化成Cp。 因此，在本博士論文中，我們首先將利用貝氏方法針對指標Cpk在沒有對製程平均數做任何限制下建構出一個評估的準則。 接著對於單邊規格製程之產品，我們也推導出單邊指標CPU和CPL的驗後機率以供製程做績效檢定。 不幸地，在實務應用上，資料之抽樣收集常常是在不同時間下所抽取的多個樣本而非一次抽取的單一樣本，尤其製程能力分析往往是針對“統計製程管制下”的資料去做分析。雖然製程能力指標估計式之統計性質已被廣泛地討論研究但多基於單一抽樣樣本所建構的，而不是針對多重樣本。而多個樣本之製程能力估計與檢定與單一樣本之製程能力估計檢定是不一樣的，錯誤的使用可能造成錯誤的決策。 因此，在本論文中我們將進一步利用貝氏方法探討如何在多重樣本下估計及檢定 Cp, Cpk, CPU, CPL及Cpm這些指標並推導出其驗後機率以及檢定之臨界值。對於Cp和Cpm所得到的結果可分別視為Cheng 與 Spiring (1989) 和Shiau等人 (1999a) 之推論在多重樣本下的推廣。 如此一來，操作人員可以根據所提出的檢定程序簡單地判斷他們的製程是否達到指定所要求的製程能力水準，而作出正確的決策。

The purpose of process capability analysis is to provide numerical measures for determine whether a process is capable of reproducing items meeting the manufacturing specifications, which have received considerable research attention and increased usage in process assessments and purchasing decisions. Most existing research works on capability analysis have focused on the traditional distribution frequency approach. However, the sampling distributions are usually so complicated, this makes establishing the exact confidence interval very difficult. Cheng and Spiring (1989) proposed a Bayesian procedure for assessing process capability index Cp. Chan et al. (1988) applied a similar Bayesian approach to index Cpm under the assumption that the process mean is equal to the target value T. However, the restriction of mu = T is not a practical assumption for most factory applications. Shiau et al. (1999a) developed a Bayesian approach to index Cpm without the restriction on the process mean and index Cpk but under the restriction that the process mean equals to the midpoint of the two specification limits, M. We note that in this case Cpk will reduce to Cp. In this dissertation, we first consider a Bayesian procedure for the capability index Cpk relaxing the restriction. The posterior probability, p, for which the process under investigation is capable, is derived. For processes with unilateral specifications, an accordingly Bayesian procedure for capability testing based on the one-sided indices CPU and CPL, is obtained. For applications where a routine-based data collection plans are implemented, a common practice on process control is to estimate the process capability by analyzing past “in control” data. Unfortunately, statistical properties of those PCI estimators based on one single sample, have been investigated extensively, but not for multiple samples. To use estimators based on multiple samples and then interpret the results as if they were based on a single sample may result in incorrect conclusions. Therefore, the manufacturing information regarding product quality characteristic should be derived from multiple samples rather than one single sample. In this dissertation we further consider the problem of estimating and testing Cp, Cpk, CPU, CPL and Cpm with multiple samples based on Bayesian approach. The results obtained for Cp and Cpm with multiple samples in our investigation, are generalizations of those obtained in Cheng and Spiring (1989) and Shiau et al. (1999a) from one single sample case to multiple samples case based on control chart data. Practitioners can easily use the proposed procedure to determine whether their manufacturing processes are capable of reproducing products satisfying the preset capability requirement.