預燒試驗是一道應用於消除產品早夭失效期與剔除不良品之程序，隨著科技進步，產品的可靠度不斷地提升，利用傳統的預燒策略已無法有效地在短時間內進行產品評估，為了縮短預燒時間、提升預燒效率，已有研究指出可以藉由觀察隨著時間發生衰退的產品品質特性作為發展預燒策略考量，一般發展預燒策略研究中，首先會觀察產品品質特性，接著針對所觀察的品質特性進行適合的衰退模型配置，然而使用不同的衰退模型配置於產品品質特性將會發展出不同的預燒策略。
本研究將考量產品無法確定該使用Wiener過程或Gamma過程的衰退模型進行配置的情況下，發展一穩健性預燒程序，並藉此穩健性預燒程序設計減少衰退模型配置錯誤之損失，然而除了以單一觀測時間點所得到的品質特性作為產品分類依據之外，隨後將更進一步考量以多個觀測時間所得到的品質特性作為分類依據，利用期望錯分代價(ECM)方法發展衰退模型符合Wiener過程之分類規則、衰退模型符合Gamma過程之分類規則，接著考量衰退模型可能存在配置錯誤的情況下，發展一穩健於Wiener過程以及Gamma過程之分類規則，並探討其穩健預燒終止時間。
研究最後以模擬的方式分別針對以下兩部分進行比較與分析，一、考量單一觀測時間下穩健性設計對於衰退模型配置錯誤的影響程度，二、考量多個觀測時間下穩健性設計對於衰退模型配置錯誤的影響程度，研究指出無論考量單一觀測時間下的穩健分類切點或考量多個觀測時間下的穩健分類規則皆比使用錯誤的衰退模型進行配置時有更佳的分類結果，然而考量多個觀測時間之穩健性設計又優於考量單一觀測時間下穩健性設計，並由敏感度分析發現產品參數估計對於穩健分類規則存在顯著的影響。

Burn-in is a process applied to eliminate early failure or weak products in the short period of time. Advances in technology, manufacturers used the traditional burn-in policy to collect the time to failure of products becomes inefficient in the short period of time. To shorten the terminate time of burn-in and raise the efficient of burn-in, some studies pointed out that observed the quality characteristic whose degradation over time can enhance the efficient for burn-in policy. If wrong degradation models fit to quality characteristics will affected to result of burn-in, so it’s the most important to use correct degradation model to fit quality characteristics. However, the Wiener process and the Gamma process are common stochastic processes which fit to the quality characteristics of products. Therefore this paper developed the robust burn-in policy under multiple degradation models, and it’s consider not only a single observation time for robust burn-in policy, but also consider multiple observation time for robust burn-in policy. Finally, this paper pointed out that robust burn-in policy have better results than using wrong degradation models to fitted quality characteristics. However, a multiple observation time have better classification than a single observation time, and it’s found that the parameters estimation can’t be neglected for the robust burn-in policy in the sensitivity analysis.

[1] Bebbington, M., Lai, C. D., & Zitikis, R. (2007). Optimum burn-in time for a bathtub shaped failure distribution. Methodology and Computing in Applied Probability, 9(1), 1-20.
[2] Cha, J. H., & Pulcini, G. (2016). Optimal burn-in procedure for mixed populations based on the device degradation process history. European Journal of Operational Research, 251(3), 988-998.
[3] Chhikara, R. S., & Folks, J. L. (1989). The Inverse Gaussian Distribution: Theory. Methodology, and Applications (CRC, New York, 1988).
[4] Çinlar, E. (1980). On a generalization of gamma processes. Journal of Applied Probability, 467-480.
[5] Jensen, F., & Petersen, N. E. (1982) Burn-in: An Engineering Approach to the Design and Analysis of Burn-in Procedures. New York: John Wiley & Sons.
[6] Johnson, R. A., & Wichern, D. W. (2014). Applied multivariate statistical analysis, 6th edition. Pearson New International Edition, 575-670.
[7] Kahle, W., Mercier, S., & Paroissin, C. (2016). Degradation Processes in Reliability. John Wiley & Sons.
[8] Kundu, D., & Manglick, A. (2004). Discriminating between the Weibull and log-normal distributions. Naval Research Logistics, 51(6), 893-905.
[9] Kuo, W. (1984). Reliability enhancement through optimal burn-in. IEEE Transactions on Reliability, R-33(2), 145–156.
[10] Leemis, L. M., & Beneke, M. (1990). Burn-in models and methods: A review. IIE Transactions, 22(2), 172–180.
[11] Liao, C. M., & Tseng, S. T. (2006). Optimal design for step-stress accelerated degradation tests. IEEE Transactions on Reliability, 55(1), 59-66.
[12] Liu, C. C. (1997). A comparison between the Weibull and lognormal models used to analyse reliability data (Doctoral dissertation, University of Nottingham).
[13] Mahmoud, M. A. W., & Moustafa, H. M. (1993). Estimation of a discriminant function from a mixture of two gamma distributions when the sample size is small. Mathematical and computer modelling, 18(5), 87-95.
[14] Meeker, W. Q., & Escobar, L. A. (1998). Statistical methods for reliability data. New York: John Wiley & Sons.
[15] Mi, J. (1995) Bathtub failure rate & upside-down bathtub mean residual life. IEEE Transactions on Reliability, 44(3), 388-391
[16] Nelson, W. (1990). Accelerated testing: statistical models, test plans, and data analysis. John Wiley & Sons.
[17] Pan, Z., & Balakrishnan, N. (2011). Reliability modeling of degradation of products with multiple performance characteristics based on gamma processes. Reliability Engineering & System Safety, 96(8), 949-957.
[18] Pan, Z., Balakrishnan, N., Sun, Q., & Zhou, J. (2013). Bivariate degradation analysis of products based on Wiener processes and copulas. Journal of Statistical Computation and Simulation, 83(7), 1316-1329.
[19] Park, C., & Padgett, W. J. (2005). Accelerated degradation models for failure based on geometric Brownian motion and gamma processes. Lifetime Data Analysis, 11(4), 511-527.
[20] Pascual, F. G., & Montepiedra, G. (2005). Lognormal and Weibull accelerated life test plans under distribution misspecification. IEEE Transactions on Reliability, 54(1), 43-52.
[21] Peng, C. Y., & Tseng, S. T. (2009). Mis-specification analysis of linear degradation models. IEEE Transactions on Reliability, 58(3), 444-455.
[22] Plesser, K. T., & Field, T. O. (1977). Cost-optimized burn-in duration for repairable electronic systems. IEEE Transactions on Reliability, R-26(3), 195–197.
[23] Singpurwalla, N. D. (1995). Survival in dynamic environments. Statistical Science, 86–103.
[24] Singpurwalla, N. D. (2007). Survival in dynamic environments. Reliability and Risk: A Bayesian Perspective: 205-225.
[25] Tsai, C. C., Tseng, S. T., & Balakrishnan, N. (2011). Optimal burn-in policy for highly reliable products using gamma degradation process, IEEE Transactions on Reliability, vol. 60(1), 234-245.
[26] Tsai, C. C., Tseng, S. T., & Balakrishnan, N. (2011). Mis-specification analyses of gamma and Wiener degradation processes. Journal of Statistical Planning and Inference, 141(12), 3725-3735.
[27] Tseng, S. T., Balakrishnan, N., & Tsai, C. C. (2009). Optimal step-stress accelerated degradation test plan for gamma degradation processes. IEEE Transactions on Reliability, 58(4), 611-618.
[28] Tseng, S. T., & Peng, C. Y. (2004). Optimal burn-in policy by using an integrated Wiener process. IIE Transactions, 36(12), 1161–1170.
[29] Tseng, S. T., & Tang, J. (2001). Optimal burn-in time for highly reliable products. International Journal of Industrial Engineering, 8, 329–338.
[30] Tseng, S. T., Tang, J., & Ku, I. H. (2003). Determination of burn-in parameters and residual life for highly reliable products. Naval Research Logistics, 50(1), 1–14.
[31] Tseng, S. T., & Yu, H. F. (1997). A termination rule for degradation experiment. IEEE Transactions on Reliability, R-46(1), 130 –133.
[32] Venturini, S., Dominici, F., & Parmigiani, G. (2008). Gamma shape mixtures for heavy-tailed distributions. The Annals of Applied Statistics, 756-776.
[33] Wang, X., Balakrishnan, N., & Guo, B. (2014). Residual life estimation based on a generalized Wiener degradation process. Reliability Engineering & System Safety, 124, 13-23.
[34] Whitmore, G. A. (1995). Estimating degradation by a Wiener diffusion process subject to measurement error. Lifetime data analysis, 1(3), 307-319.
[35] Wang, X., Balakrishnan, N., & Guo, B. (2014). Residual life estimation based on a generalized Wiener degradation process. Reliability Engineering & System Safety, 124, 13-23.
[36] Ye, Z. S., Xie, M., Tang, L. C., & Shen, Y. (2012). Degradation-based burn-in planning under competing risks. Technometrics, 54(2), 159-168.
[37] Yu, H. F. (2007). Mis-specification analysis between normal and extreme value distributions for a linear regression model. Communications in Statistics—Theory and Methods, 36(3), 499-521.
[38] Zhai, Q., Ye, Z. S., Yang, J., & Zhao, Y. (2016). Measurement errors in degradation-based burn-in. Reliability Engineering & System Safety, 150, 126-135.
[39] Zhang, M., Ye, Z., & Xie, M. (2015). Optimal burn-in policy for highly reliable products using inverse Gaussian degradation process. In Engineering Asset Management-Systems, Professional Practices and Certification (pp. 1003-1011). Springer International Publishing.
[40] Zhou, J., Pan, Z., & Sun, Q. (2010). Bivariate degradation modeling based on gamma process. In Proceedings of the World Congress on Engineering (Vol. 3).