在可靠度分析中經常使用Weibull分佈來描述產品壽命或是工業材料的耐力與阻力等，因此監控Weibull分佈的尺度及形狀參數相當於監控產品製程的可靠與穩定度，可以幫助生產者提升產品的品質。通常在進行製程的第二階段線上監控時，我們會需要假設製程管制狀態下的參數值為已知，而在實際的情形下則是利用大量的第一階段管制狀態下的樣本資料來估計參數值。本文利用廣義概似比檢定統計量建構改變點偵測管制圖來同時監控Weibull製程的尺度及形狀雙參數，並且將EWMA機制引入改變點偵測管制圖，使得在監控製程的小幅度改變時亦能維持高監控效率。本文所提出的改變點偵測管制圖不僅不需要知道製程管制狀態下的參數值，還能在管制圖發出失控警訊的同時估計出製程改變點的位置。統計模擬的結果顯示我們提出的改變點偵測管制圖在大部分的參數改變情形下皆比Huwang、Wu和Lee (2021)所提出的EWMA管制圖有更高的監控效率。最後，我們利用碳纖維強度資料來說明如何使用所提出的改變點偵測管制圖，並討論未來可能的研究方向。

In reliability analysis, Weibull distribution is often used to describe the product lifetime or the endurance and resistance of industrial materials. Therefore, monitoring the scale and shape parameters of Weibull distribution is equivalent to monitoring the reliability and stability of the production, which can improve the quality of products. Traditionally, the in-control parameters of Weibull distribution are often assumed to be known on Phase II online monitoring. However, in fact we need to use a large amount of Phase I in-control data to estimate the in-control parameters when we have insufficient knowledge of Weibull distribution. In this article, based on the generalized likelihood ratio test statistic, we propose a change-point detection control chart to monitor Weibull scale and shape parameters simultaneously. Moreover, exponentially weighted moving average（EWMA）mechanism is incorporated into the change-point detection control chart, so that we can maintain high monitoring efficiency when the shift of the process is relatively small. The proposed control charts can not only be conducted without knowing the in-control parameters but also give the estimate of the unknown change-point at the same time when the proposed control charts trigger a signal. According to the simulation results, the proposed control charts have higher monitoring efficiency than the EWMA control chart proposed by Huwang, Wu and Lee (2021) in most of the out-of-control scenarios considered. Finally, we use a set of carbon fiber strength data to demonstrate how to implement the proposed control charts and discuss some future research directions.

1. Faraz, A., Saniga, E. M., and Heuchenne, C. (2015). Shewhart control charts for
monitoring reliability with Weibull lifetimes. Quality and Reliability Engineering
International, 31(8):1565–1574.
2. Guo, B. and Wang, B. X. (2014). Control charts for monitoring the Weibull shape
parameter based on type-II censored sample. Quality and Reliability Engineering
International, 30(1):13–24.
3. Guo, B., Wang, B. X., and Xie, M. (2014). ARL-unbiased control charts for the
monitoring of exponentially distributed characteristics based on type-II censored
samples. Journal of Statistical Computation and Simulation, 84(12):2734–2747.
4. Hawkins, D. M. and Zamba, K. (2005a). A change-point model for a shift in variance.
Journal of Quality Technology, 37(1):21–31.
5. Hawkins, D. M. and Zamba, K. (2005b). Statistical process control for shifts in mean
or variance using a changepoint formulation. Technometrics, 47(2):164–173.
6. Huwang, L. and Lin, L.-W. (2020). New EWMA control charts for monitoring
the Weibull shape parameter. Quality and Reliability Engineering International,
36(6):1872–1894.
7. Huwang, L., Wu, C.-H., and Lee, Y.-R. (2021). EWMA and adaptive EWMA variable sampling intervals charts for simultaneous monitoring of Weibull parameters.
Quality Technology & Quantitative Management, 18(5):552–575.
8. Jones, L. A., Champ, C. W., and Rigdon, S. E. (2001). The performance of exponentially weighted moving average charts with estimated parameters. Technometrics,
43(2):156–167.
9. Kim, N. (2016). On the maximum likelihood estimators for parameters of a Weibull
distribution under random censoring. Communications for Statistical Applications
and Methods, 23(3):241–250.
10. Mahmoud, M. A., Parker, P. A., Woodall, W. H., and Hawkins, D. M. (2007). A
change point method for linear profile data. Quality and Reliability Engineering
International, 23(2):247–268.
11. Padgett, W. and Spurrier, J. D. (1990). Shewhart-type charts for percentiles of
strength distributions. Journal of Quality Technology, 22(4):283–288.
12. Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41(1/2):100–115.
13. Pascual, F. (2010). EWMA charts for the Weibull shape parameter. Journal of
Quality Technology, 42(4):400–416.
14. Pascual, F. and Li, S. (2012). Monitoring the Weibull shape parameter by control
charts for the sample range of type II censored data. Quality and Reliability Engineering International, 28(2):233–246.
15. Ramalhoto, M. and Morais, M. (1999). Shewhart control charts for the scale parameter
of a Weibull control variable with fixed and variable sampling intervals. Journal of
Applied Statistics, 26(1):129–160.
16. Roberts, S. (1959). Control chart tests based on geometric moving averages. Technometrics, 42(1):97–101.
17. Zhang, C., Tsung, F., and Xiang, D. (2016). Monitoring censored lifetime data with
a weighted-likelihood scheme. Naval Research Logistics (NRL), 63(8):631–646.
18. Zhang, C. W., Ye, Z., and Xie, M. (2017). Monitoring the shape parameter of a
Weibull renewal process. IISE Transactions, 49(8):800–813.
19. Zou, C., Zhang, Y., and Wang, Z. (2006). A control chart based on a change-point
model for monitoring linear profiles. IIE transactions, 38(12):1093–1103.