應用Tukey’s 管制圖監控韋伯 新製程之百分位數

Abstract

建構傳統的Shewhart管制圖時，製程資料須服從常態分佈且有足夠的歷史資料來估計製程參數。然而現今產業競爭越來越激烈，許多新開發製程不易蒐集到足夠的觀測值來檢查資料是否符合常態分佈或準確的估計製程參數，因此無法有效應用Shewhart管制圖來監控新製程。針對此類新製程監控問題，Quesenberry(1991)提出Q管制圖。然而Q管制圖須假設製程分佈服從某些特定的機率分佈(如:常態、二項分佈等)，因此當新製程呈非常態分佈時，Q管制圖即不適用，有些產品之可靠性數據(如：材料失效時間、材料所能承受之應力的極值(extreme value)或百分位數(percentile)等)常呈現韋伯(Weibull)分佈，如何有效偵測出此類可靠性數據發生偏移對於維持產品的品質是非常重要的。Padgett et al. (1990)針對Weibull百分位數提出Shewhart-type管制圖，但此管制圖需收集足夠的樣本資料才能準確地估計製程參數，並不適用於監控Weibull新製程之百分位數。由於Tukey’s 管制圖可適用於少量資料且不需假設製程為何種統計分布，本研究乃針對僅有少量資料之Weibull分佈的新製程資料，提出一套監控Weibull 百分位數 之Tukey’s管制圖，以有效監控Weibull新製程之低百分位數。本研究藉由模擬方法，在Weibull分佈之不同的參數組合、樣本組數與百分位數下，找出合適的Tukey’s管制圖之管制界限係數值(k值)，並比較本研究所提出之新製程Weibull percentile Tukey’s管制圖、Q管制圖與Shewhart-type管制圖的偵測製程變異之能力。經由敏感度分析顯示，當樣本組數增加時，Weibull percentile Tukey’s 管制圖與Padgett et al. (1990)Shewhart-type管制圖在大多數的情況下偵測能力皆會上升。當製程穩定或發生正向偏移時，本研究所提出之Weibull percentile Tukey’s 管制圖的偵測能力皆表現較佳；當製程平均數發生負向偏移時，Shewhart-type管制圖的偵測能力則表現較佳；本研究所提出之Weibull percentile Tukey’s 管制圖的偵測能力在大多數的情況下皆較Q管制圖的偵測能力佳。

The conventional Shewhart’s control charts require that the process data follow a normal distribution and a large data set is needed for setting up the control limits. However, constructing the Shewhart’s control charts for a start-up process is difficult because not enough data can be collected to estimate the process parameters accurately. To solve this problem, Quesenberry(1991) proposed the Q chart for the start-up process. Q chart also assumes that the process distribution follows some specific distributions such as Normal or Binomial distributions. That is, the Q chart is not appropriate if the data of a start-up process follow a non-normal distribution. The Weibull distribution is widely used to describe the reliability data. Padgett et al. (1990)’s developed a Shewhart-type control chart for the Weibull process. However, no control charts has been developed for a start-up Weibull process. Turkey’s control chart can be applied to any distribution with small data set. Therefore, the main objective of this study is to utilize Turkey’s control chart to monitor the percentile of a start-up Weibull process. The sensitivity analysis is conducted to verify the effectiveness of the proposed percentile Tukey’s control chart for a start-up Weibull process. Through simulations, we found out the suitable control limit coefficient k of Turkey’s control chart under parameters, sample sizes for monitoring the percentile of Weibull distribution. We also compared our works with Q charts and Padgett et al. (1990)’s Shewhart-type control chart. The results of the sensitivity analysis indicated that when the process is stable or has a positive shift, the Weibull percentile Turkey’s control chart outperforms the Shewhart-type control chart and the Q chart. When the process mean has a negative shift, Shewhart-type control chart performs better than Weibull percentile Turkey’s control chart and Q charts. Moreover, the detective ability of the Weibull percentile Turkey’s control chart and Shewhart-type control chart increase when the sample sizes increase.