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Optimal Classification Policy and Comparisons for Highly Reliable Products

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Abstract

In the current competitive marketplace, manufacturers need to classify products in a short period of time, according to market demand. Hence, it is a challenge for manufacturers to implement a classification test that can distinguish the different grades of a product quickly and efficiently. For highly reliable products, if quality characteristics exist whose degradation over time can be related to the lifetime of the product, the degradation model can then be constructed based on the degradation data. In this study, we propose a general degradation model using a Gaussian mixture process that simultaneously considers unit-to-unit variability, within-unit variability and measurement error. Then, by adopting the concept of linear discriminant analysis, we propose a three-step classification policy to determine the optimal coefficients, the optimal cutoff points and the optimal test stopping time. In addition, we use an analytic approach to compare the efficiency of our proposed procedure with the methods that are reported in the previous literature under small sample size cases. Analytical comparisons provide functional equations under different assumptions. The solutions are found to elucidate the foundation between different methods proposed in recent studies. Finally, several data sets are used to illustrate the proposed classification procedure.

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References

  • Bian, L. and Gebraeel, N. (2012). Computing and Updating First-Passage Time Distributions for Randomly Evolving Degradation Signals. IIE Transactions 44, 974– 987.

    Article  Google Scholar 

  • Boldea, O. and Magnus, J. R. (2009). Maximum Likelihood Estimation of the Multivariate Normal Mixture Model. Journal of the American Statistical Association 104, 1539–1549.

    Article  MathSciNet  MATH  Google Scholar 

  • Chao, M. T. (1999). Degradation Analysis and Related Topics: Some Thoughts and a Review. The Proceedings of the National Science Council, Part A 23, 555–566.

    Google Scholar 

  • Cheng, Y. S. and Peng, C. Y. (2012). Integrated Degradation Models in R Using iDEMO. Journal of Statistical Software 49, 2, 1–22.

    Article  Google Scholar 

  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society. B 39, 1–22.

    MathSciNet  MATH  Google Scholar 

  • Feng, Q., Peng, H. and Coit, D. W. (2010). A Degradation-Based Model for Joint Optimization of Burn-in, Quality Inspection, and Maintenance: A Light Display Device Application. The International Journal of Advanced Manufacturing Technology 50, 801–808.

    Article  Google Scholar 

  • Hamada, M. S., Wilson, A. G., Reese, C. S. and Martz, H. F. (2008). Bayesian Reliability. Springer, New York.

    Book  MATH  Google Scholar 

  • Johnson, R. A. and Wichern, D. W. (2007). Applied Multivariate Statistical Analysis, 6th ed. Prentice Hall, New Jersey.

    MATH  Google Scholar 

  • Lai, C. D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.

    MATH  Google Scholar 

  • Lawless, J. and Crowder, M. (2004). Covariates and Random Effects in a Gamma Process Model with Application to Degradation and Failure. Lifetime Data Analysis 10, 213–227.

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, C. J. and Meeker, W. Q. (1993). Using Degradation Measures to Estimate a Time-to-Failure Distribution. Technometrics 35, 161–174.

    Article  MathSciNet  MATH  Google Scholar 

  • McLachlan, G. J. and Krishnan, T. (2008). The EM Algorithm and Extensions, 2nd ed. Wiley, New York.

    Book  MATH  Google Scholar 

  • Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data. Wiley, New York.

    MATH  Google Scholar 

  • Nelson, W. (1990). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis. Wiley, New York.

    Book  Google Scholar 

  • Peng, C. Y. (2012). A Note on Optimal Allocations for the Second Elementary Symmetric Function with Applications for Optimal Reliability Design. Naval Research Logistics 59, 278–284.

    Article  MathSciNet  Google Scholar 

  • Peng, C. Y. (2015). Inverse Gaussian processes with random effects and explanatory variables for degradation data. Technometrics 57, 100–111.

    Article  MathSciNet  Google Scholar 

  • Peng, C. Y. and Hsu, S. C. (2012). A Note on a Wiener Process with Measurement Error. Applied Mathematics Letters 25, 729–732.

    Article  MathSciNet  MATH  Google Scholar 

  • Peng, C. Y. and Tseng, S. T. (2009). Mis-specification Analysis of Linear Degradation Models. IEEE Transactions on Reliability 58, 444–455.

    Article  Google Scholar 

  • Peng, C. Y. and Tseng, S. T. (2013). Statistical Lifetime Inference with Skew-Wiener Linear Degradation Models. IEEE Transactions on Reliability 62, 338–350.

    Article  Google Scholar 

  • Schott, J. R. (2005). Matrix analysis for statistics, 2nd ed. John Wiley & Sons, New York.

  • Singpurwalla, N. D. (1995). Survival in Dynamic Environments. Statistical Science 10, 86–103.

    Article  MATH  Google Scholar 

  • Tseng, S. T. and Peng, C. Y. (2004). Optimal Burn-in Policy by Using an Integrated Wiener Process. IIE Transactions 36, 1161–1170.

    Article  Google Scholar 

  • Tseng, S. T. and Tang, J. (2001). Optimal Burn-in Time for Highly Reliable Products. International Journal of Industrial Engineering 8, 329–338.

    Google Scholar 

  • Tseng, S. T., Tang, J. and Ku, I. H. (2003). Determination of Optimal Burn-in Parameter and Residual Life for Highly Reliable Products. Naval Research Logistics 50, 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  • Tsai, C. C., Tseng, S. T. and Balakrishnan, N. (2011). Optimal Burn-in Policy for Highly Reliable Products Using Gamma Degradation Process. IEEE Transactions on Reliability 60, 234–245.

    Article  Google Scholar 

  • van Noortwijk, J. M. (2009). A Survey of the Application of Gamma Processes in Maintenance. Reliability Engineering and System Safety 94, 2–21.

    Article  Google Scholar 

  • Whitmore, G. A. (1995). Estimating Degradation by a Wiener Diffusion Process Subject to Measurement Error. Lifetime Data Analysis 1, 307–319.

    Article  MATH  Google Scholar 

  • Wu, W. F. and Ni, C. C. (2003). A Study of Stochastic Fatigue Crack Growth Modeling Through Experimental Data. Probabilistic Engineering Mechanics 18, 107–118.

    Article  Google Scholar 

  • Wu, S. and Xie, M. (2007). Classifying Weak, and Strong Components Using ROC Analysis with Application to Burn-in. IEEE Transactions on Reliability 56, 552–561.

    Article  Google Scholar 

  • Xiang, Y., Coit, D. W. and Feng, Q. (2013). n Subpopulations Experiencing Stochastic Degradation: Reliability Modeling, Burn-in, and Preventive Replacement Optimization. IIE Transactions 45, 391–408.

    Article  Google Scholar 

  • Yang, G. B. (2012). Optimum Degradation Tests for Comparison of Products. IEEE Transactions on Reliability 61, 220–226.

    Article  Google Scholar 

  • Yu, H. F. (2003). Optimal Classification of Highly Reliable Products Whose Degradation Paths Satisfy Wiener Processes. Engineering Optimization 35, 313–324.

    Article  MathSciNet  Google Scholar 

  • Yu, H. F. and Yu, W. C. (2006). Optimal Classification of Highly Reliable Products with a Linearized Degradation Model. Journal of Industrial and Production Engineering 23, 382–392.

    Google Scholar 

  • Zhou, R. R., Serban, N. and Gebraeel, N. (2011). Degradation Modeling and Lifetime Monitoring Using Functional Data Analysis. Annals of Applied Statistics 5, 1586–1610.

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Chien-Yu Peng.

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Peng, CY. Optimal Classification Policy and Comparisons for Highly Reliable Products. Sankhya B 77, 321–358 (2015). https://doi.org/10.1007/s13571-015-0097-z

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  • DOI: https://doi.org/10.1007/s13571-015-0097-z

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AMS (2000) subject classification

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