1Department of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran
2Department of Industrial & Material Engineering, Sadjad University of Technology, Mashhad, Iran
Manufacturers need to evaluate the reliability of their products in order to increase the customer satisfaction. Proper analysis of reliability also requires an effective study of the failure process of a product, especially its failure time. So, the Failure Process Modeling (FPM) plays a key role in the reliability analysis of the system that has been less focused on. This paper introduces a framework defining an approach for the failure process modeling with censored data in Constant Stress Accelerated Life Tests (CSALTs). For the first time, various types of censoring schemes are considered in this study. Usually, in data analysis, it is impossible to get closed form of estimates of the unknown parameter due to complex and nonlinear likelihood equations. As a new approach, a mathematical programming problem is formed and the Maximum Likelihood Estimation (MLE) of parameters is obtained to maximize the likelihood function. A case study in red Light- Emitting Diode (LED) lamps is also presented. The MLE of parameters is obtained using genetic algorithm (GA). Furthermore, the Fisher information matrix is obtained for constructing the asymptotic variances and the approximate confidence intervals of estimates of the parameters.
 Ascher, H. and Feingold, H. “Repairable systems reliability: Modeling, inference, misconceptions and their causes,” 1th ed., Marcel Dekker, New York, 1984.  Vaurio, J.K. “Identification of process and distribution characteristics by testing monotonic and non-monotonic trends in failure intensities and hazard rates,” Reliability Engineering and System Safety, vol. 64, no.3, pp. 345–357, 1999.  Louit, D.M.,Pascual, R. and Jardine, A.K.S. “A practical procedure for the selection of time-to-failure models based on the assessment of trends in maintenance data,” Reliability Engineering and System Safety, vol. 94, no. 10, pp.1618-1628, 2009.  Regattieri, A.,Manzini, R. and Battini, D. “Estimating reliability characteristics in the presence of censored data: A case study in a light commercial vehicle manufacturing system,” Reliability Engineering and System Safety, vol. 95, pp.1093–1102, 2010.  Nelson WB. Accelerated Testing: Statistical Models, Test Plans, and Data Analyses, John Wiley & Sons, New York, 1990.  Chernoff, H. “Optimal accelerated life designs for estimation,” Technometrics, vol. 4, no. 3, pp. 381–408, 1962.  Bessler, S.,Chernoff, H. and Marshall, A.W. “An optimal sequential accelerated life test,” Technometrics, vol. 4, no. 3, pp. 367–379, 1962.  Chenhua L. “Optimal step-stress plans for accelerated life testing considering reliability/life prediction,” PhD Thesis, Department of Mechanical and Industrial Engineering, Northeastern University, USA, 2009.  Bai, D. and Chung, S. “An accelerated life test model with the inverse power law,” Reliability Engineering and System Safety, vol. 24, no. 3, pp. 223-230, 1989.  Teng, N.H. and Kolarik, W.J. “On/off cycling under multiple stresses,” IEEE Trans. Reliability, vol. 38, no. 4, pp. 494–498, 1989.  Fan, T.H. and Yu, C.H.“Statistical inference on constant stress accelerated life tests under generalized Gamma lifetime distributions,” Quality and Reliability Engineering International, vol. 29, no. 5, pp. 631-638, 2013.  Zhang, J., Liu, C., Cheng, G.,Chen, X., Wu, J., Zhu, Q. and Zhang, L. “Constant-stress accelerated life test of white organic light-emitting diode based on least square method under Weibull distribution,” Journal of Information Display, vol. 15, no. 2, PP. 71-75, 2014.  Guan, Q., Tang, Y., Fu, J. and Xu, A. “Optimal multiple constant-stress accelerated life tests for generalized Exponential distribution,” Communications in Statistics - Simulation and Computation, vol. 43, no. 8, PP.1852-1865, 2014.  Nelson WB. Applied Life Data Analysis, John Wiley & Sons, New York (1982).  Yang, G.B. “Optimum constant-stress accelerated life test plans,” IEEE Trans. Reliability, vol. 43, no. 4, PP. 575–581, 1994.  Mettas, A. “Modeling and analysis for multiple stress-type accelerated life data,” in Proceedings of Annual Reliability and Maintainability Symposium, Los Angeles, CA, pp. 138–143, 2000.  Zhou, K., Shi, Y. and Sun, T. “Reliability analysis for accelerated life-test with progressive hybrid censored data using geometric process,” Journal of Physical Sciences, vol. 16, PP.133-143, 2012.  Bai, D. and Kim, M. “Optimum simple step-stress accelerated life tests for Weibull distribution and type-I censoring,” Naval Res. Logist, vol. 40, no. 2, PP. 193–210, 1993.  Xiong, C. “Inference on a simple step-stress model with type-II censored Exponential data,” IEEE Trans. Reliability, vol. 47, no. 2, PP. 142–146, 1998.  Tang, L.C., Goh, T.N., Sun, Y.S. and Ong, H.L. “Planning accelerated life tests for censored two-parameter Exponential distributions,” Nav Res Logistic, vol. 46, no. 2, PP. 169–186, 1999.  Abdel-Ghaly, A., Attia, A. and Abdel-Ghani, M. “The maximum likelihood estimates in step partially accelerated life tests for the Weibull parameters in censored data,” Commun. Statist.Theory Meth, vol. 31, no. 4, PP. 551–573, 2002.
 Balakrishnan, N., Kundu, D., Tony Ng, H.K. and Kannan, N.“Point and interval estimation for a simple step-stress model with type-II censoring,” J. Qual. Technol, vol. 39, no. 1, pp. 35–47, 2007.  Balakrishnan, N. and Xie, Q. “Exact inference for a simple step-stress model with type-I hybrid censored data from the Exponential distribution,” J. Statist. Plann. Inference, vol. 137, no. 11, PP. 3268–3290, 2007.  Li, C. and Fard, N. “Optimum bivariate step-stress accelerated life test for censored data,” IEEE Trans. Reliability, vol. 56, no. 1, PP. 77–84, 2007.  Kateri, M. and Balakrishnan, N. “Inference for a simple step-stress model with type-II censoring, and Weibull distributed lifetimes,” IEEE Trans. Reliability, vol. 57, no. 4, PP. 616-626, 2008.  Watkins, A. and John, A. “On constant stress accelerated life tests terminated by type II censoring at one of the stress levels,” Journal of statistical Planning and Inference, vol. 138, no. 3, PP. 768-786, 2008.  Abdel-Hamid, A. and AL-Hussaini, E. “Estimation in step-stress accelerated life tests for the exponentiated Exponential distribution with type-I censoring,” Computational Statistics & Data Analysis, vol. 53, no. 4, PP. 1328-1338, 2009.  Guan, v. and Tang, Y. “Optimal step-stress test under type-I censoring for multivariate Exponential distribution,” Journal of Statistical Planning and Inference, vol. 142, no. 7, PP. 1908-1923, 2012.  Ling, L., Xu, W. and Li, M. “Optimal bivariate step-stress accelerated life test for type-I hybrid censored data,” Journal of Statistical Computation and Simulation, vol. 81, no. 9, PP. 1175–1186, 2010.  Attia, A.F.,Aly, H.M. and Bleed, S.O. “Estimating and planning accelerated life test using constant stress for generalized Logistic distribution under type-I censoring,” ISRN Applied Mathematics; 203618, 15 pages, 2011.  Lee, J. and Pan, R. “Bayesian analysis of step-stress accelerated life test with Exponential distribution,” Quality and Reliability Engineering International, vol. 28, no.3, PP. 353-361, 2012.  Wang, R.,Xu, X., Pan, R. and Sha, N. “On parameter inference for step-stress accelerated life test with Geometric distribution,” Communications in Statistics - Theory and Methods, vol. 41, no. 10, pp.1796-1812, 2012.  Srivastava, P.W. and Mittal, N. “Optimum multi-objective modified constant-stress accelerated life test plan for the Burr type-XII distribution with type-I censoring, ” Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, vol. 227, no. 2, PP. 132-143, 2013.  Kamal, M. andZarrin, S. “Design of accelerated life testing using geometric process for Pareto distribution withtype-I censoring,” Journal of global research in mathematical archives, vol. 1, no. 8, PP. 59-66, 2013.  Aly, H.M. and Bleed, S.O. “Bayesian estimation for the generalized Logistic distribution type-II censored accelerated life testing,” Int. J. Contemp. Math. Sciences, vol. 8, no. 20, PP. 969 – 986, 2013.  Shi, Y.M., Jin, L., Wei, C. and Yue, H.B. “Constant-stress accelerated life test with competing risks under progressive type-II hybrid censoring,” Advanced Materials Research, vol. 712 – 715, pp. 2080-2083, 2013.  Aly, H.M. and Bleed, S.O. “Estimating and planning step stress accelerated life test for generalized Logistic distribution under type-I censoring,” International Journal of Applied Mathematics & Statistical Sciences, vol. 2, no. 2, pp. 1-16, 2013.  Asser, S. and Abd EL-Maseh, M.“Estimation of the parameters of the bivariate generalized Exponential distribution using accelerated life testing with censoring data,” International Journal of Advanced Statistics and Probability, vol. 2, no. 2, pp. 77-83, 2014.  Amal, S.H., Assar, S.M. and Shelbaia, A. “Optimum inspection times of step stress accelerated life tests with progressively type-I interval censored,” Australian Journal of Basic and Applied Sciences, vol. 8, no. 17, pp. 282-292, 2014.  Saleem, Z.A. “Effect of progressive type-I right censoring on Bayesian statistical inference of simple step-stress acceleration life testing plan under Weibull life distribution,” International Journal of Mechanical, Aerospace, Industrial and Mechatronics Engineering, vol. 8, no. 2, pp. 325-329 , 2014.  Zhao, J., Shi, Y. and Yan, W. “Inference for constant-stress accelerated life test with type-I progressively hybrid censored data from Burr-XII distribution,” Journal of Systems Engineering and Electronics, vol. 25, no. 2, pp. 340-348, 2014.  Cox, D.R. “Regression models andlife tables,” Springer-Verlag, New York, pp. 527-541, 1992.  Holland, J.H. Adaptation in natural and artificial systems: An introductory analysis with Applications to Biology, Control, and Artificial Intelligence, 1th ed., MIT Press, Cambridge, 1992.
 Wang, F.K., Cheng, Y.F. and Lu, W.L., ”Partially accelerated life tests for the weibull distribution under multiply censored data,” Communications in Statistics-Simulation and computation, vol. 41, no. 9, pp. 1667-1678.