製造業者在蒐集產品製程中所感興趣的關鍵品質特性資料時，也會蒐集一些與製程有關的其他輔助訊息。Cochran(1940) 提出當兩個品質特性為正相關時， Ratio estimator 能較精確地估計出品質特性之值；另一方面，Robson(1957) 提出 Product estimator，可用來估計品質特性與輔助訊息呈負相關時的情況。近年來陸續有學者針對感興趣的品質特性與輔助變數間的關係進行研究。Adegoke et al.(2017) 以及 Sanusi et al.(2017) 曾分別運用 Product estimator 及 Ratio estimator 提出指數加權移動平均管制圖。然而彼等所提出之指數加權移動平均管制圖僅在輔助變數的變異係數與兩倍品質特性的變異係數之比值小於或等於相關係數的絕對值時，此管制圖的偵測能力才優於傳統的指數加權移動平均管制圖。
本研究係藉由加入快速初始反應值的想法，並根據相關係數的轉換取得新的快速初始反應值之計算方式，建立了加入快速初始反應的新指數加權移動平均管制圖 (Fast Initial Response-Exponentially Weighted Moving Average control chart, FIR-EWMA chart)。接著，我們使用平均連串長度 (ARL) 以模擬的方式來比較本研究所提出的 FIR-EWMA chart 與 Adegoke et al.(2017) 之 EWMA_P chart 以及 Sanusi et al.(2017) 之 MrEWMA chart 在偵測能力上的優劣。結果發現當關鍵品質特性的製程平均發生改變時，FIR-EWMA chart 較能及早偵測到製程的異常。最後，我們以 Marlin(2000) 文中非等溫連續攪拌釜化學反應器 (CSTR) 的資料集進行數值實例的驗證與說明，本研究結果未來可供製造業者在監控關鍵品質特性時的參考。

The SPC control charts play an important role in monitoring and improving the process quality. Generally speaking, when the key quality characteristics of interest are measured, auxiliary information other than the key quality characteristics are also collected. Cochran(1940) pointed out that ratio estimator can accurately estimate the value of quality characteristics when two quality characteristics are positively correlated. Robson(1957) found that product estimator which can be used to estimate the quality characteristics that has a negative correlation coefficient between auxiliary information. Recently, some researchers have studied the control chart for monitoring the quality characteristics of interest with auxiliary variables. The EWMA_p (proposed by Adegoke et al.,2017) and MrEWMA charts (proposed by Sanusi et al.,2017) are based on product estimator and ratio estimator respectively. However, EWMA_p and MrEWMA charts perform better than the classical EWMA only when the ratio of the coefficient of variation of the auxiliary information to the coefficient of variation of the double quality characteristic is less than or equal to the absolute value of the correlation coefficient.
Thus, we adopt the idea of adding the fast initial response value to establish a new Fast Initial Response-Exponentially Weighted Moving Average control chart (FIR-EWMA chart). A simulation study has been conducted to evaluate the detecting ability of our proposed FIR-EWMA chart. We compare the detecting performance of our proposed FIR-EWMA chart with that of both EWMA_p and MrEWMA charts by using the out-of-control average run length (ARL_1). Finally, a CSTR (nonisothermal continuous stirred tank chemical reactor) data set is given to demonstrate the usefulness of our proposed FIR-EWMA chart. Both the simulation results and numerical example show the detecting ability of our proposed FIR-EWMA charts outperforms EWMA_p and MrEWMA charts when a process shift occurs.

1.Abbasi, S. A., & Riaz, M. (2013). “On enhanced control charting for process monitoring.” International Journal of Physical Sciences Full Length Research Paper, 8(17), 759-775.
2.Abbas, N., Riaz, M., & Does, R. J. (2014). “An EWMA-type control chart for monitoring the process mean using auxiliary information.” Communications in Statistics-Theory and Methods, 43(16), 3485-3498.
3.Adegoke, N. A., Riaz, M., Sanusi, R. A., Smith, A. N., & Pawley, M. D. (2017). “EWMA-type scheme for monitoring location parameter using auxiliary information.” Computers & Industrial Engineering, 114, 114-129.
4.Ahmad, S., Lin, Z., Abbasi, S. A., & Riaz, M. (2012). “On efficient monitoring of process dispersion using interquartile range.” Open Journal of Applied Sciences, 2(4B), 39-43.
5.Cochran, W.G. (1940). “The estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce.” The journal of agricultural science, 30(2), 262-275.
6.Hunter, J. S. (1986). “The exponentially weighted moving average.” Journal of Quality Technology, 18(4), 203-210.
7.Knoth, S. (2005). “Fast initial response features for EWMA control charts.” Statistical Papers, 46(1), 47-64.
8.Lucas, J. M., & Crosier, R. B. (1982). “Fast initial response for CUSUM quality control schemes.” Technometrics, 24(3), 199-205.
9.Lucas, J. M., & Saccucci, M. S. (1990). “Exponentially weighted moving average control schemes: properties and enhancements.” Technometrics, 32(1), 1-12.
10.Marlin, T. E. (1995). Process control: Designing processes and control systems for dynamic performance. New York: McGraw-Hill.
11.Montgomery, D. C. (2009). Introduction to statistical quality control. New York: John Wiley & Sons.
12.Murthy, M.N. (1964). “Product method of estimation.” Sankhyā: The Indian Journal of Statistics, Series A, 26(1), 69-74.
13.Page, E. S. (1954). “Continuous inspection schemes.” Biometrika, 41(1/2), 100-115.
14.Rhoads, T. R., Montgomery, D. C., & Mastrangelo, C. M. (1996). “A fast initial response scheme for the exponentially weighted moving average control chart.” Quality Engineering, 9(2), 317-327.
15.Riaz, M. (2008). “Monitoring process variability using auxiliary information.” Computational Statistics, 23(2), 253-276.
16.Roberts, S.W. (1959). “Control chart tests based on geometric moving averages.” Technometrics, 42(1): 239-250.
17.Robson, D. S. (1957). “Applications of multivariate polykays to the theory of unbiased ratio-type estimation.” Journal of the American Statistical Association, 52(280), 511-522.
18.Sanusi, R. A., Riaz, M., Adegoke, N. A., & Xie, M. (2017). “An EWMA monitoring scheme with a single auxiliary variable for industrial processes.” Computers & Industrial Engineering, 114, 1-10.
19.Shabbir, J., & Awan, W. H. (2016). “An efficient shewhart‐type control chart to monitor moderate size shifts in the process mean in phase II.” Quality and Reliability Engineering International, 32(5), 1597-1619.
20.Shewhart, W. A. (1924). “Some applications of statistical methods to the analysis of physical and engineering data.” Bell Labs Technical Journal, 3(1), 43-87.
21.Shi, X., Lv, Y., Fei, Z., & Liang, J. (2013). “A multivariable statistical process monitoring method based on multiscale analysis and principal curves.” International Journal of Innovative Computing, Information and Control, 9(4), 1781-1800.
22.Singh, R., & Mangat, N. S. (2013). Elements of survey sampling (Vol. 15). 出版地: Springer Science & Business Media.
23.Steiner, S. H. (1999). “EWMA control charts with time-varying control limits and fast initial response.” Journal of Quality Technology, 31(1), 75-86.
24.Yang, S.F. & Arnold, B. C. (2014). “A simple approach for monitoring business service time variation.” The Scientific World Journal, Vol. 2014, Article ID 238719, 16 pages. DOI: 10.1155/2014/238719
25.Yoon, S., & MacGregor, J. F. (2001). “Fault diagnosis with multivariate statistical models part I: using steady state fault signatures.” Journal of process control, 11(4), 387-400.