Reliability of repairable systems with Non-Central Gamma frailty
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Abstract
Maintenance actions on industrial equipment are essential to reduce expenses associated with equipment failures. Based on a well-fitted model, it is possible, through the estimated parameters, to predict several functions of interest, such as the cumulative average and reliability functions. In this paper, a new frailty model is proposed to analyze failure times of repairable systems subject to unobserved heterogeneity actions. The Non-Central Gamma distribution is assumed to the frailty random variable effect. The class of minimal repair models for repairable systems is explored considering an approach that includes the frailty term to estimate the unobserved heterogeneity over the systems’ failure process. Classical inferential methods were used to parameter estimation and define the reliability prediction functions. A simulation study was conducted to confirm the properties expected in the estimators. Two real-world data known in literature were used to illustrate the estimation procedures and validate the proposed model as a viable alternative to those already established in the literature. The results obtained highlight the potential of our proposed approach, particularly for industries dealing with such systems, where unquantifiable factors may impact equipment failure times.
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References
Almeida, M. P., Paixão, R. S., Ramos, P. L., Tomazella, V., Louzada, F. & Ehlers, R. S. Bayesian non-parametric frailty model for dependent competing risks in a repairable systems framework. Reliability Engineering & System Safety 204, 107145 (2020).
Ascher, H. & Feingold, H. Repairable systems reliability: modeling, inference, misconceptions and their causes (Marcel Dekker Inc, New York, 1984).
Barlow, R. E. & Proschan, F. Mathematical theory of reliability (SIAM, 1996).
Brito, É. S., Tomazella, V. L. & Ferreira, P. H. Statistical modeling and reliability analysis of multiple repairable systems with dependent failure times under perfect repair. Reliability Engineering & System Safety 222, 108375 (2022).
Burnham, K. P. & Anderson, D. R. Multimodel inference: understanding AIC and BIC in model selection. Sociological methods & research 33, 261–304 (2004).
Coque Junior, M. A. Modelo de confiabilidade para sistemas reparáveis considerando diferentes condições de manutenção preventiva imperfeita. PhD thesis (Universidade de São Paulo, 2016).
Crow, L. H. Reliability analysis for complex, repairable systems tech. rep. (Army Materiel Systems Analysis Activityaberdeen Proving Ground Md, 1975).
De Oliveira, I. R. C. & Ferreira, D. F. Computing the Noncentral Gamma distribution, its inverse and the noncentrality parameter. Computational Statistics 28, 1663–1680 (2013).
De Toledo, M. L. G., Freitas, M. A., Colosimo, E. A. & Gilardoni, G. L. ARA and ARI imperfect repair models: Estimation, goodness-of-fit and reliability prediction. Reliability Engineering & System Safety 140, 107–115 (2015).
Dias De Oliveira, M., Colosimo, E. A. & Gilardoni, G. L. Power law selection model for repairable systems. Communications in Statistics-Theory and Methods 42, 570–578 (2013).
D’Andrea, A., Feitosa, C. C., Tomazella, V. & Vieira, A. M. C. Frailty modeling for repairable systems with Minimum Repair: An application to dump truck data of a Brazilian Mining Company. J Math Stat Sci 6, 179–198 (2017).
D’Andrea, A. M., Tomazella, V. L., Aljohani, H. M., Ramos, P. L., Almeida, M. P., Louzada, F., Verssani, B. A., Gazon, A. B. & Afify, A. Z. Objective bayesian analysis for multiple repairable systems. Plos one 16, e0258581 (2021).
Elbers, C. & Ridder, G. True and spurious duration dependence: The identifiability of the proportional hazard model. The Review of Economic Studies 49, 403–409 (1982).
Gilardoni, G. L. & Colosimo, E. A. Optimal maintenance time for repairable systems. Journal of quality Technology 39, 48–53 (2007).
Hougaard, P. Life table methods for heterogeneous populations: distributions describing the heterogeneity. Biometrika 71, 75–83 (1984).
Knüsel, L. & Bablok, B. Computation of the noncentral gamma distribution. SIAM Journal on Scientific Computing 17, 1224–1231 (1996).
Mettas, A. & Zhao, W. Modeling and analysis of repairable systems with general repair in Annual Reliability and Maintainability Symposium, 2005. Proceedings. (2005), 176–182.
Onchere, W,Weke, P, Jam, O & Carolyne, O. Non-Central Gamma Frailty with application to life term assurance data. Advances and Applications in Statistics 67, 237–253 (2021).
Proschan, F. Theoretical explanation of observed decreasing failure rate. Technometrics 5, 375– 383 (1963).
R Core Team. R: A Language and Environment for Statistical Computing R Foundation for Statistical Computing (Vienna, Austria, 2021). https://www.R-project.org/.
Slimacek, V. & Lindqvist, B. H.Nonhomogeneous Poisson process with nonparametric frailty. Reliability Engineering & System Safety 149, 14–23 (2016).
Tomazella, V. L. D. Modelagem de dados de eventos recorrentes via processo de Poisson com termo de fragilidade. PhD thesis (Universidade de São Paulo, 2003).
Vaupel, J. W., Manton, K. G. & Stallard, E. The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16, 439–454 (1979).
Verssani, B. A. W. Modelo de regressão para sistemas reparáveis: um estudo da confiabilidade de colhedoras de cana-de-açúcar PhD thesis (Universidade de São Paulo, 2018).
Wang, H. A survey of maintenance policies of deteriorating systems. European journal of operational research 139, 469–489 (2002).
Wienke, A. Frailty models in survival analysis (CRC press, 2010).