近幾年來，有一些研究提出了只用單一一個管制圖即可同時偵測平均數和變異數。根據此目的，我們提出了加權平均損失管制圖，此管制圖是利用加權平均損失所建立的，在一個製成的目標值和平均數不一定相等時，它可同時監控一個製成的平均數和變異數。此加權平均損失統計量是應用一個加權因子，去調整製程平均和目標值的平方差和變異數的損失比重，所以此管制圖的效能比未經由加權因子調整過的管制圖還好。我們不只建立了固定管制參數(FP)加權平均損失管制圖，也建立了適應性加權平均損失管制圖，包括變動抽樣間隔(VSI)、變動樣本數與抽樣間隔(VSI)、變動管制參數(VP)；我們利用平均連串長度(ARL)來衡量固定管制參數管制圖的偵測績效，利用馬可夫鏈的方法計算偵測出異常訊息所需的平均時間(ATS)來衡量適應性管制圖的績效，並且做比較，我們發現適應性管制圖比固定管制參數管制圖的效能還要好。我們也利用最佳化技術建立最加適應性管制圖，當製成失控時，此最佳化管制圖能使ATS1最小。此外，當平均數和變異數的偏移幅度很小時，我們利用指數加權移動平均法(EWMA)建立EWMA加權平均損失管制圖，使其有較好的偵測力。這些我們所提出的管制圖，是只根據單一一個統計量所建立的，和X bar-S管制圖相比，有較好的效能，且和使用兩個管制圖同時偵測平均數和變異數相比，比較輕易理解且容易執行。 / In recent years, a few researchers had proposed different types of single charts that jointly monitor the process mean and the variation. In this project, we use the weighted average loss (WL) to construct WL control charts for monitoring the process mean and variance simultaneously while the target value may be different from the in-control mean. This statistic WL applied a weighted factor to adjust the weights of the loss due to the square of the deviation of the process mean from the target and the variance change. So the WL charts are more effective than unadjusted loss function charts. We not only construct the fixed parameters (FP) WL chart but also the adaptive WL charts which included variable sampling interval (VSI) WL chart, variable sample size and sampling interval (VSSI) WL chart and variable parameters (VP) WL chart. We calculate the average run length (ARL) for FP WL chart and using Markov chain approach to calculate the average time to signal (ATS) for adaptive WL charts to measure the performance and compare each other. From the comparison, we find the adaptive WL charts are more effective than the FP WL chart. We also proposed the optimal adaptive WL charts using an optimization technique to minimize ATS1 (ARL1) when the process was out-of-control. In addition, in order to detect the small shifts of the process mean and variance effectively, we construct the WL charts using the EWMA scheme. The proposed charts are based on only one statistic and are more effective than the X bar-S chart. And the WL charts are easy to understand and apply than using two charts for detecting the mean and variance shifts simultaneously.

The CUSUM control charts have been widely used in detecting small process shifts since it was first introduced by Page (1954). And recent studies have shown that adaptive charts can improve the efficiency and performance of traditional Shewhart charts. To monitor the process mean and variance in a single chart, the loss function is used as a measure statistic in this article. The loss function can measure the process quality loss while the process mean and/or variance has shifted. This study combines the three features: adaption, CUSUM and the loss function, and proposes the optimal VSSI, VSI, and FP CUSUM Loss chart. The performance of the proposed charts is measured by using Average Time to Signal (ATS) and Average Number of Observations to Signal (ANOS). The ATS and ANOS calculations are based on Markov chain approach. The performance comparisons between the proposed charts and some existing charts, such as X-bar+S^2 charts and CUSUM X-bar+S^2 charts, are illustrated by numerical analyses and some examples. From the results of the numerical analyses, it shows that the optimal VSSI CUSUM Loss chart has better performance than the optimal VSI CUSUM Loss chart, optimal FP CUSUM Loss chart, CUSUM X-bar+S^2 charts and X-bar+S^2 charts. Furthermore, using a single chart to monitor a process is not only easier but more efficient than using two charts simultaneously. Hence, the adaptive CUSUM Loss charts are recommended in real process. / The CUSUM control charts have been widely used in detecting small process shifts since it was first introduced by Page (1954). And recent studies have shown that adaptive charts can improve the efficiency and performance of traditional Shewhart charts. To monitor the process mean and variance in a single chart, the loss function is used as a measure statistic in this article. The loss function can measure the process quality loss while the process mean and/or variance has shifted. This study combines the three features: adaption, CUSUM and the loss function, and proposes the optimal VSSI, VSI, and FP CUSUM Loss chart. The performance of the proposed charts is measured by using Average Time to Signal (ATS) and Average Number of Observations to Signal (ANOS). The ATS and ANOS calculations are based on Markov chain approach. The performance comparisons between the proposed charts and some existing charts, such as X-bar+S^2 charts and CUSUM X-bar+S^2 charts, are illustrated by numerical analyses and some examples. From the results of the numerical analyses, it shows that the optimal VSSI CUSUM Loss chart has better performance than the optimal VSI CUSUM Loss chart, optimal FP CUSUM Loss chart, CUSUM X-bar+S^2 charts and X-bar+S^2 charts. Furthermore, using a single chart to monitor a process is not only easier but more efficient than using two charts simultaneously. Hence, the adaptive CUSUM Loss charts are recommended in real process.