SciELO - Scientific Electronic Library Online

 
vol.41 issue1Estimating dynamic Panel data. A practical approach to perform long panels author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Revista Colombiana de Estadística

Print version ISSN 0120-1751

Rev.Colomb.Estad. vol.41 no.1 Bogotá Jan./June 2018

http://dx.doi.org/10.15446/rce.v41n1.59929 

Artículos originales de investigación

Simultaneous Confidence Bands for the Estimation of Expected Discounted Warranty Costs for Coherent Systems under Minimal Repair

Bandas de confianza simultáneas para la estimación del costo medio de garantía descontado para un sistema coherentes bajo reparo mínimo

Carlos M. Lopera-Gómez1  a  , Nelfi G. González-Álvarez1  b 

1Escuela de Estadística, Facultad de Ciencias, Universidad Nacional de Colombia, Medellín, Colombia

Abstract

This paper develops simultaneous confidence bands using computer intensive methods based on resampling, for the expected discounted warranty costs in coherent systems under physical minimal repair, that is, when the system is observed at its components level and only the component that causes the fault is minimally repaired. For this purpose, it is shown that, under the framework of the Martingale processes and the central limit resampling theorem (CLRT) over stochastic processes, the discounted warranty cost processes satisfy the conditions set by the central limit resampling theorem (CLRT). Additionally, a simulation study is performed on the most relevant variables, such that the finite sample features of the proposed bands may be assessed, based on their actual coverage probabilities. The results in the considered scenarios show that resampling-based simultaneous confidence bands have coverage probabilities that are close to the nominal coverage. In particular, the agreement is good when there are 100 systems or more where a large number of resamples are used for the approximation.

Key words: Central Limit Theorem; Coverage Probability; Point Process; Resampling; Semimartingales; Weak Convergence

Resumen

Este trabajo desarrolla bandas de confianza simultáneas usando métodos computacionales intensivos basados en remuestreo, para el costo medio de garantía descontado en sistemas coherentes bajo reparo mínimo físico, esto es, cuando el sistema es observado al nivel de sus componentes y sólo la componente que causa la falla es reparada mínimamente. Para ello se prueba que, bajo el marco teórico de los procesos martingala, los procesos de costos de garantía descontados cumplen las condiciones del teorema de límite central de remuestreo (CLRT). Un estudio de simulación Monte Carlo se realiza para evaluar, a través de las probabilidades de cobertura alcanzadas, el desempeño de las bandas propuestas en muestras finitas. Los resultados en los escenarios considerados muestran que las bandas de confianza basadas en remuestreo tienen probabilidades de cobertura con valores cercanos a los esperados, en particular para aquellas basadas en muestras con más de 100 sistemas donde el número de remuestras usado para la aproximación es grande.

Palabras-clave: convergencia débil; probabilidad de cobertura; proceso puntual; remuestreo; semimartingalas; teorema de límite central

Full text available only in PDF format.

References

Ascher, H. (1968), 'Evaluation of repairable system reliability using the “bad-asold” concept', IEEE Transactions on Reliability R-17(2), 103-110. [ Links ]

Aven, T. (1983), 'Optimal replacement under a minimal repair strategy: A general failure model', Advances in Applied Probability 15(1), 198-211. [ Links ]

Aven, T. & Castro, I. (2008), 'A minimal repair replacement model with two types of failure and a safety constraint', European Journal of Operational Research 188(2), 506-515. [ Links ]

Aven, T. & Jensen, U. (1999), Stochastic models in reliability, 1 edn, Springer-Verlag Inc, New York. [ Links ]

Bai, J. & Pham, H. (2004), 'Discounted warranty cost of minimally repaired series systems', IEEE Transactions on Reliability 53(1), 37-42. [ Links ]

Bai, J. & Pham, H. (2006), 'Cost analysis on renewable full-service warranties for multi-component systems', European Journal of Operation Research 168(2), 492-508. [ Links ]

Balachandran, K., Maschmeyer, R. & Livingstone, J. (1981), 'Product warranty period: A markovian approach to estimation and analysis of repair and replacement costs', The Accounting Review 56(1), 115-124. [ Links ]

Belyaev, Y. (2000), Computer intensive methods based on resampling in analysis of reliability and survival data, in 'In Recent advances in reliability theory: methodology, practice and inference', Birkhäuser, Boston. [ Links ]

Belyaev, Y. (2007), Resampling for lifetime distributions, in 'In Encyclopedia of Statistics: Quality and Reliability', Wiley, Chichester. [ Links ]

Belyaev, Y. & Seleznjev, O. (2000), 'Approaching in distribution with applications to resampling of stochastic processes', Scandinavian Journal of Statistics 27(2), 371-384. [ Links ]

Blischke, W. & Murthy, D. (1994), Warranty cost analysis, 1 edn, Marcel Dekker Inc, New York. [ Links ]

Block, H., Borges, W. & Savits, T. (1985), 'Age-dependent minimal repair', Journal of Applied Probability 22(2), 370-385. [ Links ]

Chien, Y. (2005), 'Determining optimal warranty periods form the seller's perspective and optimal out-of-warranty replacement age from the buyer's perspective', International Journal of Systems Science 36(10), 631-637. [ Links ]

Chien, Y. (2008), 'A general age-replacement model with minimal repair under renewing free-replacement warranty', European Journal of Operation Research 186(3), 1046-1058. [ Links ]

Chien, Y. (2010), 'Optimal age for preventive replacement under a combined fully renewable free replacement with a pro-rata warranty', International Journal of Production Economics 124(1), 198-205. [ Links ]

Chukova, S. & Dimitrov, B. (1996), Warranty analysis for complex systems, In Product warranty handbook, 1 edn, Marcel Dekker Inc, New York. [ Links ]

Davison, A. & Hinkley, D. (1997), Bootstrap methods and Their Applications, 1 edn, Cambridge University Press, Cambridge. [ Links ]

Duchesne, T. & Marri, F. (2009), 'General distributional properties of discounted warranty costs with risk adjustment under minimal repair', IEEE Transactions on Reliability 58(1), 143-151. [ Links ]

Efron, B. (1979), 'Bootstrap methods: another look at the jacknife', The Annals of Statistics 7(1), 1-26. [ Links ]

Fleming, T. & Harrington, D. (1991), Counting processes and survival analysis, 1 edn, John Wiley and Sons, Inc, New York. [ Links ]

González, N. & Bueno, V. (2011), 'Estimating the discounted warranty cost of a minimally repaired coherent system', Revista Colombiana de Estadística 34(3), 513-543. [ Links ]

Herman, J., Kučera, R. & Šimša, J. (2000), Equations and Inequalities, 1 edn, Springer-Verlag, Inc., New York. [ Links ]

Hussain, A. & Murthy, D. (1998), 'Warranty and redundancy design with uncertain quality', IIE Transactions 30(12), 1191-1199. [ Links ]

Ja, S., Kulkarni, V., Mitra, A. & Patankar, J. (2001), 'A nonrenewable minimalrepair warranty policy with time-dependent costs', IEEE Transactions on Reliability 50(4), 346-352. [ Links ]

Ja, S., Kulkarni, V., Mitra, A. & Patankar, J. (2002), 'Warranty reserves for nonstationary sales processes', Naval Research Logistics 49(5), 499-513. [ Links ]

Jain, M. & Maheshwari, S. (2006), 'Discounted costs for repairable units under hybrid warranty', Applied Mathematics and Computation 173(2), 887-901. [ Links ]

Jung, K., Park, M. & Park, D. (2010), 'System Maintenance Cost Dependent on Life Cycle under Renewing Warranty Policy', Reliability Engineering and System Safety 95(7), 816-821. [ Links ]

Knowles, M. (1988), Simultaneous confidence bands for random functions, Technical Report 113, SIAM Institute for Mathematics and Society. Revista Colombiana de Estadística 41 (2018) 1-30 [ Links ]

Lopera, C. (2014), Bandas de confianza simultáneas para la estimación del costo medio de garantía descontado para un sistema coherentes bajo reparo mínimo, Tesis de doctorado en ciencias, Universidad Nacional de Colombia, Escuela de Estadística, Medellín, Colombia. [ Links ]

Loève, M. (1977), Probability theory, volumes I and II, 4 edn, Springer-Verlag Inc, New York . [ Links ]

Murthy, D. & Djamaludina, I. (2002), 'New product warranty: a literature review', International Journal of Production Economics 79(3), 231-260. [ Links ]

Nguyen, D. & Murthy, D. (1984), 'Cost analysis of warranty policies', Naval Research Logistics 31(4), 525-541. [ Links ]

Ouali, M., Tadj, L., Yacout, S. & Ait-Kadi, D. (2011), A survey of replacement models with minimal repair, in 'Replacement Models with Minimal Repair', Springer-Verlag Inc, London. [ Links ]

Park, M. & Pham, H. (2010), 'Altered quasi-renewal concepts for modeling renewable warranty costs with imperfect repairs', Mathematical and Computer Modelling 52(9-10), 1435-1450. [ Links ]

Patankar, J. & Mitra, A. (1995), 'Effect of warranty execution on warranty reserve costs', Management Science 41(3), 395-400. [ Links ]

Riordan, J. (1937), 'Moment Recurrence Relations for Binomial, Poisson and Hypergeometric Frequency Distributions', The Annals of Mathematical Statistics 8(2), 103-111. [ Links ]

Ritchken, P. & Fuh, D. (1986), 'Optimal replacement policies for irreparable warranty item', IEEE Transactions on Reliability 35(5), 621-624. [ Links ]

Sheu, S., Griffith, W. & Nakagawa, T. (1995), 'Extended optimal replacement model with random minimal repair costs', European Journal of Operational Research 85(3), 636-649. [ Links ]

Su, C. & Shen, J. (2012), 'Analysis of extended warranty policies with different repair options', Engineering Failure Analysis 25, 49-62. [ Links ]

Thomas, M. (1989), 'A prediction model for manufacturer warranty reserves', Management Science 35(12), 1515-1519. [ Links ]

Thomas, M. (2005), 'Engineering economic decisions and warranties', The Engineering Economist 50(4), 307-326. [ Links ]

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License