良率指標Spk的抽樣性質與樣本資訊下估計良率的精準度

Abstract

Boyles在1994年提出一個良率指標取名為Spk。該指標為常態製程提供一個精準正確的良率衡量。Lee et al.在2002年提出一個常態近似的方法來估計製程的Spk值。在本論文，我們延伸擴展前人的成就，在三種不同的情況下考量運用良率指標的抽樣分配：(i)多重抽樣樣本下(ii)摺合逼近法的抽樣分配(iii)允收抽樣情況。我們在多重抽樣樣本下推導出良率指標的抽樣分配，並且發現針對相同的Spk值，製程平均值落在規格上下限中心時良率指標估計量的變異程度最大。針對一些常用的指標需求，Spk的信賴下限已列成表。為了讓使用者更易於接受推導出來的常態逼近分配，我們檢查了真實的型I誤差並與一開始就設定好的顯著水準比較。我們計算了該常態逼近會收斂到與真值只有特定差異的需求樣本數。最後，一個可再充電鋰電池製程實例被呈現並說明從業者如何應用Spk的信賴下限到多重抽樣的樣本。 接著，我們使用摺合逼近法來估計製程的Spk值，並與常態逼近法做比較。比較的結果顯示摺合的方法的確比常態逼近法在估計製程良率及Spk值上更為準確。根據摺合的方法，我們建構一個逐步且有效率的步驟來說明如何估計製程良率。我們亦研究了摺合方法的準確性，提供在特定檢定力需求下，以及在特定收斂需求下所需要的採集的樣本個數。本論文最後一個部份，我們考慮根據Spk值來做允收與否的決策。允收抽樣計畫是一個提供買賣雙方對於檢驗貨品是否符合產品品質需求的決策法則。我們提出一個以Spk為依據的計數值抽樣計畫來處理不良率為極小PPM的製程。我們根據同時解一對非線性的方程式，建立一個有效率的方法來決定所需抽樣的個數以及接受貨品與否的臨界值。根據我們設計出來的抽樣計畫，從業者可以決定檢驗貨批所需的數量和相對應的允收決策值。

The yield index Spk proposed by Boyles (1994) provides an exact measure on the production yield of normal processes. Lee et al. (2002) considered a normal approximation for estimating Spk. In the thesis, we extend the results and consider the sampling distribution of the yield index in three conditions (i) for multiple samples, (ii) in the convolution method, and (iii) for acceptance sampling. Under multiple samples, we derive the sampling distribution for the estimator of Spk, and observe that for the same Spk, the variance of would be largest when the process mean is on the center of specification limits. Lower bounds of Spk are tabulated for some commonly used capability requirements. To assess the normally approximated distribution of , we also check out the actual type I error and compare with the preset significant level. We also compute the sample sizes required for the normal approximation to converge to the actual Spk within a designated accuracy. Then, a real-world application of the one-cell rechargeable Li-ion battery packs is presented to illustrate how practitioners can apply the lower bounds to actual data collected in multiple samples. Next, we consider a convolution approximation for estimating Spk, and compare with the normal approximation. The comparison results show that the convolution method does provide a more accurate estimation to Spk as well as the production yield than the normal approximation. An efficient step-by-step procedure based on the convolution method is developed to illustrate how to estimate the production yield. Also investigated is the accuracy of the convolution method which provides useful information about sample size required for designated power levels, and for convergence. Finally, we consider the acceptance determination based on the Spk index. Acceptance sampling plans provide the vendor and the buyer decision rules for lot sentencing to meet their product quality needs. A variables sampling plan based on the index Spk is proposed to handle processes requiring very low PPM fraction of defectives. We develop an effective method for obtaining the required sample sizes n and the critical acceptance value c0 by solving simultaneously two nonlinear equations. Based on the designed sampling plan, the practitioners can determine the number of production items to be sampled for inspection and the corresponding critical acceptance value for lot sentencing.