品質管控技術和新技術進步的進展導致了高質量的缺陷發生的過程。然而，在處理高質量的流程時，現有的控制製圖可能會面臨一些困難。事件間隔時間（TBE）圖可以克服傳統管製圖的困難，並且在事件發生次數較少時特別適合。韋伯分佈是用於分析可靠性問題的最廣泛使用的統計模型之一。由於其靈活性，它可以假設與其他一些分佈類似的形狀，如指數和常態。
在本研究中，我們設計了混合累積和指數加權平均管製圖(mixed cumulative sum-exponentially weighted average control chart, MCE)，用於單次測量韋伯分佈的TBE，並與現有的管製圖比較，像是Weibull CUSUM（WCUSUM）、Weibull EWMA（WEWMA ）和混合EWMA-CUSUM（MEC）。本研究的目的是為韋伯分佈的TBE設計一個混合的CUSUM-EWMA管制圖，並用具有固定形狀參數的韋伯分佈的不同尺度參數來監測TBE平均尺度變化。通常通過分析其連串長度（RL）分佈的性質，如平均連串長度(ARL)和連串長度的標準偏差(SDRL)來評估控製圖的性能。這兩個指標有助於我們描述圖表的RL的整個概況，並使用蒙特卡羅模擬進行104次迭代計算。相對平均指數（RMI）也用於測量本研究提出的控製圖和其他三個已經存在控製圖的總體表現。 RMI值越小表示控製圖的性能越好，反之亦然。
根據結果，本研究提出的管制圖在迅速檢測向下的尺度變化中比WCUSUM和WEWMA管制圖更有效率但比MEC差，而WEWMA和MEC管制圖比其他管制圖在檢測向上的尺度變化中表現較好。在案例分析中，提供了兩個闡述性範例說明WCUSUM、WEWMA、MEC以及本研究提供的管制圖在監控韋伯分佈的TBE上的應用。

The progress of the quality control techniques and new technological developments have led to high-quality processes in which small amount of defects occur. However, when dealing with high-quality processes, the existing control charting schemes may face some difficulties. Time-between-events (TBE) charts can overcome the difficulties with traditional attributes control chart, and they are particularly suitable when the events rarely occur. The Weibull Distribution is one of the most widely used statistical models for the analysis of reliability problems. Because of its flexibility, it may assume shapes similar to some other distributions, such as exponential and normal.
In this research, we have designed mixed cumulative sum (CUSUM)-exponentially weighted average (EWMA) control chart (MCE) for monitoring Weibull distributed TBE with individual measurements and compare it with existing control charts, such as Weibull CUSUM (WCUSUM), Weibull EWMA (WEWMA) and mixed EWMA-CUSUM (MEC). The objective of this study is to design a mixed CUSUM-EWMA control chart for Weibull distributed TBE and to monitor the TBE mean scale change with a different scale parameters of Weibull distribution with fixed shape parameters. The performance of a control chart is usually evaluated by analyzing the properties of its run length (RL) distribution such as Average run length (ARL) and the standard deviation of the run lengths (SDRL). These two metrics help us to describe the entire profile of RL of the chart and they are computed using Monte Carlo simulation with 104 iterations. Relative mean index (RMI) is also utilized to measure the overall performance of the proposed control chart and other three existing control charts.
According to the results, the proposed control chart is more efficient than the WCUSUM and the WEWMA charts next to MEC charts to detect downward scale changes swiftly, whereas the WEWMA and the MEC chart performs better than other charts in detecting upward scale changes swiftly. From real data, two illustrative examples are provided to show the application of existing (WCUSUM, WEWMA, MEC) and the proposed control charts for monitoring Weibull distributed TBE.

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