A condition-based imperfect replacement policy for a periodically inspected system with two dependent wear indicators
A two‐component system is considered, which is subject to accumulative deterioration. Because of common stress, the components are dependent. Their joint deterioration is modelled with a bivariate nondecreasing Lévy process. The deterioration level of both components is known only through perfect an...
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| Published in | Applied stochastic models in business and industry Vol. 30; no. 6; pp. 766 - 782 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Blackwell Publishing Ltd
01.11.2014
Wiley |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1524-1904 1526-4025 |
| DOI | 10.1002/asmb.2011 |
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| Abstract | A two‐component system is considered, which is subject to accumulative deterioration. Because of common stress, the components are dependent. Their joint deterioration is modelled with a bivariate nondecreasing Lévy process. The deterioration level of both components is known only through perfect and periodic inspections. By an inspection, components with deterioration level beyond a specific threshold are instantaneously replaced by new ones (corrective or preventive replacements). Otherwise, they are left as they are. Between inspections, failures remain unrevealed. This replacement policy is classical in a univariate setting, with deterioration modelled by a Gamma process. In the bivariate case, it leads to imperfect repairs at the system level, which highly complicates the study. The replacement policy is assessed through cost functions on both finite and infinite horizons, which take into account some economical dependence between components. Markov renewal theory is used to study the behaviour of the system, in a continuous and bivariate setting. Numerical experiments illustrate the study, considering a specific Lévy process with univariate Gamma processes as margins. Although technical details are not provided here for the numerical computations, the paper shows that there is a technical gap between the traditional one‐dimensional studies and the present two‐dimensional one, especially for the computation of the asymptotic distribution of the underlying Markov chain. Hence, there is a need for further development in the bivariate (or multivariate) setting. Copyright © 2014 John Wiley & Sons, Ltd. |
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| AbstractList | A two‐component system is considered, which is subject to accumulative deterioration. Because of common stress, the components are dependent. Their joint deterioration is modelled with a bivariate nondecreasing Lévy process. The deterioration level of both components is known only through perfect and periodic inspections. By an inspection, components with deterioration level beyond a specific threshold are instantaneously replaced by new ones (corrective or preventive replacements). Otherwise, they are left as they are. Between inspections, failures remain unrevealed. This replacement policy is classical in a univariate setting, with deterioration modelled by a Gamma process. In the bivariate case, it leads to imperfect repairs at the system level, which highly complicates the study. The replacement policy is assessed through cost functions on both finite and infinite horizons, which take into account some economical dependence between components. Markov renewal theory is used to study the behaviour of the system, in a continuous and bivariate setting. Numerical experiments illustrate the study, considering a specific Lévy process with univariate Gamma processes as margins. Although technical details are not provided here for the numerical computations, the paper shows that there is a technical gap between the traditional one‐dimensional studies and the present two‐dimensional one, especially for the computation of the asymptotic distribution of the underlying Markov chain. Hence, there is a need for further development in the bivariate (or multivariate) setting. Copyright © 2014 John Wiley & Sons, Ltd. A two-component system is considered, which is subject to accumulative deterioration. Because of common stress, the components are dependent. Their joint deterioration is modelled with a bivariate nondecreasing Lévy process. The deterioration level of both components is known only through perfect and periodic inspections. By an inspection, components with deterioration level beyond a specific threshold are instantaneously replaced by new ones (corrective or preventive replacements). Otherwise, they are left as they are. Between inspections, failures remain unrevealed. This replacement policy is classical in a univariate setting, with deterioration modelled by a Gamma process. In the bivariate case, it leads to imperfect repairs at the system level, which highly complicates the study. The replacement policy is assessed through cost functions on both finite and infinite horizons, which take into account some economical dependence between components. Markov renewal theory is used to study the behaviour of the system, in a continuous and bivariate setting. Numerical experiments illustrate the study, considering a specific Lévy process with univariate Gamma processes as margins. Although technical details are not provided here for the numerical computations, the paper shows that there is a technical gap between the traditional one-dimensional studies and the present two-dimensional one, especially for the computation of the asymptotic distribution of the underlying Markov chain. Hence, there is a need for further development in the bivariate (or multivariate) setting. A two-component system is considered, which is subject to accumulative deterioration. Because of common stress, the components are dependent. Their joint deterioration is modelled with a bivariate nondecreasing Levy process. The deterioration level of both components is known only through perfect and periodic inspections. By an inspection, components with deterioration level beyond a specific threshold are instantaneously replaced by new ones (corrective or preventive replacements). Otherwise, they are left as they are. Between inspections, failures remain unrevealed. This replacement policy is classical in a univariate setting, with deterioration modelled by a Gamma process. In the bivariate case, it leads to imperfect repairs at the system level, which highly complicates the study. The replacement policy is assessed through cost functions on both finite and infinite horizons, which take into account some economical dependence between components. Markov renewal theory is used to study the behaviour of the system, in a continuous and bivariate setting. Numerical experiments illustrate the study, considering a specific Levy process with univariate Gamma processes as margins. Although technical details are not provided here for the numerical computations, the paper shows that there is a technical gap between the traditional one-dimensional studies and the present two-dimensional one, especially for the computation of the asymptotic distribution of the underlying Markov chain. Hence, there is a need for further development in the bivariate (or multivariate) setting. Copyright copyright 2014 John Wiley & Sons, Ltd. |
| Author | Mercier, Sophie Pham, Hai Ha |
| Author_xml | – sequence: 1 givenname: Sophie surname: Mercier fullname: Mercier, Sophie email: Correspondence to: Sophie Mercier, Université de Pau et des Pays de l'Adour, Laboratoire de Mathématiques et de leurs Applications-Pau (UMR CNRS 5142), Bâtiment IPRA, Avenue de l'Université, F-64013 Pau cedex, France., sophie.mercier@univ-pau.fr organization: Université de Pau et des Pays de l'Adour, Laboratoire de Mathématiques et de leurs Applications-Pau (UMR CNRS 5142), Bâtiment IPRA, Avenue de l'Université, F-64013 Pau cedex, France – sequence: 2 givenname: Hai Ha surname: Pham fullname: Pham, Hai Ha organization: Université de Pau et des Pays de l'Adour, Laboratoire de Mathématiques et de leurs Applications-Pau (UMR CNRS 5142), Bâtiment IPRA, Avenue de l'Université, F-64013 Pau cedex, France |
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| References | Mercier S, Pham HH. A preventive maintenance policy for a continuously monitored system with correlated wear indicators. European Journal of Operational Research 2012; 222(2):263-272. Bertoin J. Lévy Processes, Cambridge Tracts in Mathematics, Vol. 121. Cambridge University Press: Cambridge, 1996. Abdel-Hameed M. A Gamma wear process. IEEE Transactions on Reliability 1975; 24(2):152-153. Castanier B, Grall A, Bérenguer C. A condition-based maintenance policy with non-periodic inspections for a two-unit series system. Reliability Engineering & System Safety 2005; 87(1):109-120. Zabreyko PP, Koshelev AI, Krasnosel'skii MA, Mikhlin SG, Rakovshchik LS, Stetsenko Y. Integral Equations-A Reference Text. Noordhoff International Publishing Company: Leyden, Holland, 1975. Bäuerle N, Blatter A, Müller A. Dependence properties and comparison results for Lévy processes. Mathematical Methods of Operations Research 2008; 67: 161-186. Asmussen S. Applied Probability and Queues, second edition, Applications of Mathematics, Vol. 51. Springer-Verlag: New York, 2003. Mercier S, Meier-Hirmer C, Roussignol M. Bivariate Gamma wear processes for track geometry modelling, with application to intervention scheduling. Structure and Infrastructure Engineering 2012; 8(4):357-366. Ebrahimi N. Indirect assessment of the bivariate survival function. Annals of the Institute of Statistical Mathematics 2004; 56(3):435-448. Kress R. Linear Integral Equations, Applied Mathematical Sciences, Vol. 82. Springer Verlag: New York, 1999. Cocozza-Thivent C. Convergence de Fonctionnelles de Processus Semi-régénératifs, Prépublications de l'Université de Marne la Vallée: Champs-sur-Marne, France 02/2000 (in French), 2000. Singpurwalla N. Survival in dynamic environments. Statistical Science 1995; 10(1):86-103. Van Noortwijk JM. A survey of the application of Gamma processes in maintenance. Reliability Engineering & System Safety 2009; 94(1):2-21. 2012; 222 2000 2009; 94 2004; 56 1995; 10 1975; 24 1975 2008; 67 2005; 87 2005 1996; 121 1999; 82 2003; 51 2012; 8 e_1_2_7_6_1 e_1_2_7_4_1 Bertoin J (e_1_2_7_9_1) 1996 Cocozza‐Thivent C (e_1_2_7_12_1) 2000 e_1_2_7_3_1 Buijs FA (e_1_2_7_5_1) 2005 e_1_2_7_8_1 e_1_2_7_7_1 e_1_2_7_2_1 e_1_2_7_15_1 e_1_2_7_14_1 e_1_2_7_13_1 e_1_2_7_10_1 Asmussen S (e_1_2_7_11_1) 2003 |
| References_xml | – reference: Singpurwalla N. Survival in dynamic environments. Statistical Science 1995; 10(1):86-103. – reference: Bäuerle N, Blatter A, Müller A. Dependence properties and comparison results for Lévy processes. Mathematical Methods of Operations Research 2008; 67: 161-186. – reference: Mercier S, Pham HH. A preventive maintenance policy for a continuously monitored system with correlated wear indicators. European Journal of Operational Research 2012; 222(2):263-272. – reference: Abdel-Hameed M. A Gamma wear process. IEEE Transactions on Reliability 1975; 24(2):152-153. – reference: Ebrahimi N. Indirect assessment of the bivariate survival function. Annals of the Institute of Statistical Mathematics 2004; 56(3):435-448. – reference: Castanier B, Grall A, Bérenguer C. A condition-based maintenance policy with non-periodic inspections for a two-unit series system. Reliability Engineering & System Safety 2005; 87(1):109-120. – reference: Cocozza-Thivent C. Convergence de Fonctionnelles de Processus Semi-régénératifs, Prépublications de l'Université de Marne la Vallée: Champs-sur-Marne, France 02/2000 (in French), 2000. – reference: Zabreyko PP, Koshelev AI, Krasnosel'skii MA, Mikhlin SG, Rakovshchik LS, Stetsenko Y. Integral Equations-A Reference Text. Noordhoff International Publishing Company: Leyden, Holland, 1975. – reference: Kress R. Linear Integral Equations, Applied Mathematical Sciences, Vol. 82. Springer Verlag: New York, 1999. – reference: Bertoin J. Lévy Processes, Cambridge Tracts in Mathematics, Vol. 121. Cambridge University Press: Cambridge, 1996. – reference: Van Noortwijk JM. A survey of the application of Gamma processes in maintenance. Reliability Engineering & System Safety 2009; 94(1):2-21. – reference: Mercier S, Meier-Hirmer C, Roussignol M. Bivariate Gamma wear processes for track geometry modelling, with application to intervention scheduling. Structure and Infrastructure Engineering 2012; 8(4):357-366. – reference: Asmussen S. Applied Probability and Queues, second edition, Applications of Mathematics, Vol. 51. Springer-Verlag: New York, 2003. – volume: 222 start-page: 263 issue: 2 year: 2012 end-page: 272 article-title: A preventive maintenance policy for a continuously monitored system with correlated wear indicators publication-title: European Journal of Operational Research – volume: 67 start-page: 161 year: 2008 end-page: 186 article-title: Dependence properties and comparison results for Lévy processes publication-title: Mathematical Methods of Operations Research – volume: 24 start-page: 152 issue: 2 year: 1975 end-page: 153 article-title: A Gamma wear process publication-title: IEEE Transactions on Reliability – volume: 51 year: 2003 – volume: 94 start-page: 2 issue: 1 year: 2009 end-page: 21 article-title: A survey of the application of Gamma processes in maintenance publication-title: Reliability Engineering & System Safety – volume: 82 year: 1999 – volume: 87 start-page: 109 issue: 1 year: 2005 end-page: 120 article-title: A condition‐based maintenance policy with non‐periodic inspections for a two‐unit series system publication-title: Reliability Engineering & System Safety – year: 2000 – year: 1975 – volume: 8 start-page: 357 issue: 4 year: 2012 end-page: 366 article-title: Bivariate Gamma wear processes for track geometry modelling, with application to intervention scheduling publication-title: Structure and Infrastructure Engineering – start-page: 2209 year: 2005 end-page: 2216 – volume: 10 start-page: 86 issue: 1 year: 1995 end-page: 103 article-title: Survival in dynamic environments publication-title: Statistical Science – volume: 121 year: 1996 – volume: 56 start-page: 435 issue: 3 year: 2004 end-page: 448 article-title: Indirect assessment of the bivariate survival function publication-title: Annals of the Institute of Statistical Mathematics – ident: e_1_2_7_7_1 doi: 10.1016/j.ejor.2012.05.011 – ident: e_1_2_7_15_1 doi: 10.1080/15732479.2011.563090 – ident: e_1_2_7_4_1 doi: 10.1016/j.ress.2007.03.019 – ident: e_1_2_7_6_1 doi: 10.1007/BF02530535 – ident: e_1_2_7_3_1 doi: 10.1214/ss/1177010132 – ident: e_1_2_7_14_1 doi: 10.1007/978-94-010-1909-5 – ident: e_1_2_7_10_1 doi: 10.1007/s00186-007-0185-6 – volume-title: Convergence de Fonctionnelles de Processus Semi‐régénératifs year: 2000 ident: e_1_2_7_12_1 – volume-title: Applied Probability and Queues year: 2003 ident: e_1_2_7_11_1 – volume-title: Lévy Processes year: 1996 ident: e_1_2_7_9_1 – ident: e_1_2_7_8_1 doi: 10.1016/j.ress.2004.04.013 – ident: e_1_2_7_13_1 doi: 10.1007/978-1-4612-0559-3 – ident: e_1_2_7_2_1 doi: 10.1109/TR.1975.5215123 – start-page: 2209 volume-title: Safety and Reliability of Engineering Systems and Structures; Proceedings of ICOSSAR 9 year: 2005 ident: e_1_2_7_5_1 |
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| Snippet | A two‐component system is considered, which is subject to accumulative deterioration. Because of common stress, the components are dependent. Their joint... A two-component system is considered, which is subject to accumulative deterioration. Because of common stress, the components are dependent. Their joint... |
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| SubjectTerms | Asymptotic properties Cost function dependent wear indicators Deterioration Economics Failure Gamma processes Inspection Markov renewal theory Mathematics multivariate Lévy processes Numerical analysis Policies Probability reliability |
| Title | A condition-based imperfect replacement policy for a periodically inspected system with two dependent wear indicators |
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