統計製程管制已廣泛應用於製造過程中。然而，針對多項製程的監控方法在文獻中 較少被探討。現有方法多數利用皮爾森適合度統計量作為監控統計量建構管制圖。然而， 此類方法仰賴大樣本近似，在樣本數有限時，其誤警率與偵測能力可能變得不穩健。在 實務上，管制圖通常應用於小樣本情境，使得上述限制更加顯著。更重要的是，現有多 項製程之管制圖多未考量各類別之損失差異。然而在實務應用中，不同類別往往對應到 不同程度的成本損失。為解決上述問題，本研究提出一種精確加權損失管制圖，用於監 控多項製程。所提出之方法不僅避免依賴大樣本近似，亦將各類別之損失資訊納入監控 統計量之建構中。本研究透過數值模擬評估管制圖之偵測能力，並與多種現有多項製程 監控方法進行比較。結果顯示，在多數情境下，所提出之方法具有較佳且穩健之偵測能 力，尤其當失控偏移發生在高損失類別時，其優勢更為顯著。此外，本研究將所提出之 方法應用於新北市 PM2.5 空氣品質監測，以說明其實務應用方式與偵測能力。此外， 也發展一套診斷方法，用以識別造成製程失控之關鍵類別。

Statistical process control (SPC) has been widely applied in manufacturing processes. However, methods for monitoring multinomial processes have been less studied in the literature. Existing approaches for multinomial monitoring primarily rely on the Pearson goodness-of-fit statistic to construct control chart. However, these methods depend on large sample sizes approximations, and their false alarm rates and detection performance become unreliable when the sample size is limited. In practice, SPC is often applied in situation with small sample sizes, making this limitation more pronounced. More importantly, most existing control charts for multinomial processes do not take into account the differences in loss associated with each category. In practical applications, different categories often correspond to different levels of loss. To address these issues, this study proposes an exact weighted loss control chart for monitoring multinomial processes. The proposed method not only avoids reliance on large sample sizes approximations but also incorporates category specific loss information into the monitoring procedure. The detection performance of the proposed control chart is evaluated through numerical studies and compared with several existing methods for multinomial process monitoring. The results show that the proposed control chart demonstrates more robust and superior detection performance in the majority of scenarios, particularly when shifts occur in categories associated with higher loss. In addition, the proposed method is applied to monitor PM2.5 in New Taipei City to demonstrate its practical utility and detection capability. Furthermore, a diagnostic procedure is developed to identify which categories are responsible for the out-of-control signals.

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