統計製程管制(Statistical Process Control, SPC)是現代生產製造業中一項不可或缺的重要方法，若能善用此工具於監控產品及製程特性，則可大幅提升生產品良率。隨著時代的進步，現今產品的製造過程愈趨精密、複雜，多數的產品往往需要經過多個階段的製造流程，才能順利完成。例如：半導體製造業、電路印刷版、化學工業、航太工業、電信產業等許多工業領域之製程，皆屬於多階段製造系統之範疇。而上述高科技工業，各個階段的製程資料常具有自我相關的性質。Pan et al.(2016)已建構出多階段系統之殘差及管制圖，其中之指數加權移動平均( Exponentially Weighted Moving Average, EWMA )管制圖及累和( Cumulative Sum, CUSUM )管制圖之偵測能力均優於一般傳統的EWMA及CUSUM管制圖，但仍有進一步改善的空間。
在多階段系統的製程中，本研究擬採用Pan et al.(2016)多階段系統之模型，先取得各階段模型的殘差值，再將所取得之殘差值代入混和指數加權移動平均與累和(Mixed EWMA-CUSUM)的管制圗中並考慮快速初始反應值(Fast initial Response, FIR)與Headstart，以期建立多階段殘差RMECFIRHS (Revised Mixed EWMA-CUSUM with modified FIR and Headstart )管制圗。接著，我們利用整體平均串長度(Overall Average Run Length, OARL)作為在各種參數組合下多階段系統新的殘差混和指數平滑與累和管制圗與Pan et al.(2016)的多階段系統殘差管制圗偵測能力的比較基準。最後，我們以模擬結果發現新的多階段殘差RMECFIRHS管製圖的偵測能力均優於Pan et al.(2016)的多階段系統殘差管製圖。

With the advent of modern technology, manufacturing processes have become very sophisticated; most industries require multiple process stages to complete their final products. In this research, we develop a new control chart model suitable for monitoring the process quality of multistage manufacturing systems. Considering the correlation often occurs among various stages in a manufacturing system, we first adopt the idea of multistage manufacturing systems model proposed by Pan et al.(2016) to establish a multistage residual RMECFIRHS (Revised Mixed EWMA-CUSUM with modified Fast Initial Response and Headstart) control chart. Then, a simulation study is conduct to evaluate the detecting ability of our proposed multistage residual RMECFIRHS control chart under various combinations of process shifts. The overall average run length (OARL) is used to compare the detecting performances of our proposed multistage residual RMECFIRHS control chart with those of multistage residual EWMA and CUSUM control charts.

Finally, a numerical example of a three-stage with chemical manufacturing process is given to demonstrate the usefulness of our proposed multistage residual RMECFIRHS control chart. Both the simulation results and numerical example show that the detecting ability of our proposed multistage residual RMECFIRHS control chart outperforms the multistage residual EWMA and CUSUM control charts when a process shift occurs. Hopefully, the results of this research can be served as a useful guideline for monitoring the process quality of multistage systems.

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