Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T03:55:57.314Z Has data issue: false hasContentIssue false
  • English
  • Français

BIRNBAUM CRITICALITY AND IMPORTANCE MEASURES FOR MULTISTATE SYSTEMS WITH REPAIRABLE COMPONENTS

Published online by Cambridge University Press:  01 June 2020

Arne Bang Huseby Open the ORCID record for Arne Bang Huseby [Opens in a new window] ,
Martyna Kalinowska  and
Tobias Abrahamsen
Arne Bang Huseby
Affiliation:
University of Oslo, Oslo, Norway E-mail: arne@math.uio.no
Martyna Kalinowska
Affiliation:
University of Oslo, Oslo, Norway E-mail: arne@math.uio.no
Tobias Abrahamsen
Affiliation:
University of Oslo, Oslo, Norway E-mail: arne@math.uio.no
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We suggest four new measures of importance for repairable multistate systems based on the classical Birnbaum measure. Periodic component life cycles and general semi-Markov processes are considered. Similar to the Birnbaum measure, the proposed measures are generic in the sense that they only depend on the probabilistic properties of the components and the system structure. The multistate system model encodes physical properties of the components and the system directly into the structure function. As a result, calculating importance is easy, especially in the asymptotic case. Moreover, the proposed measures are composite measures, combining importance for all component states into a unified quantity. This simplifies ranking of the components with respect to importance. The proposed measures can be characterized with respect to two features: forward-looking versus backward-looking and with respect to how criticality is measured. Forward-looking importance measures focus on the next component states, while backward-looking importance measures focus on the previous component states. Two approaches to measuring criticality are considered: probability of criticality versus expected impact. Examples show that the different importance measures may result in unequal rankings.

Keywords

Birnbaum importancecomponent criticalitymultistate systemsrepairable componentssemi-Markov processes
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 1 , January 2022 , pp. 66 - 86
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

References

Amrutkar, K.P. & Kamalja, K.K. (2017). An overview of various importance measures of reliability system. International Journal of Mathematical, Engineering and Management Sciences 2(3): 150–171.CrossRefGoogle Scholar
Barlow, R.E. & Proschan, F. (1975). Importance of system components and fault tree events. Stochastic Processes and their Applications 3: 153–173.CrossRefGoogle Scholar
Birnbaum, Z.W. (1969). On the importance of different components in a multicomponent system. In P.R. Krishnaia (ed.), Multivariate analysis – II. New York: Academic Press, pp. 581–592.Google Scholar
Borgonovo, E. & Apostolakis, G.E. (2001). A new importance measure for risk-informed decision making. Reliability Engineering & System Safety 72: 193–212.CrossRefGoogle Scholar
Cai, Z., Si, S., Sun, S., & Li, C. (2016). Optimization of linear consecutive-k-out-of-n system with a Birnbaum importance-based genetic algorithm. Reliability Engineering & System Safety 152: 248–258.CrossRefGoogle Scholar
Cai, Z., Si, S., Liu, Y., & Zhao, J. (2018). Maintenance optimization of continuous state systems based on performance improvement. IEEE Transactions on Reliability 67(2): 651–665.10.1109/TR.2017.2743225CrossRefGoogle Scholar
Dinic, E.A. (1970). Algorithm for solution of a problem of maximum flow in a network with power estimation. Soviet Mathematics – Doklady 11: 1277–1280.Google Scholar
Dui, H., Si, S., Zuo, M.J., & Sun, S. (2015). Semi-Markov process-based integrated importance measures for multi-state systems. IEEE Transactions on Reliability 64(2): 754–765.CrossRefGoogle Scholar
Dui, H., Li, S., Xing, L., & Liu, H. (2019). System performance-based joint importance analysis guided maintenance for repairable systems. Reliability Engineering & System Safety 186: 162–175.CrossRefGoogle Scholar
Dui, H., Zhang, C., & Zheng, X. (2020). Component joint importance measures for maintenances in submarine blowout preventer system. Journal of Loss Prevention in the Process Industries 63: 1–10.CrossRefGoogle Scholar
Ford, L.R. & Fulkerson, D.R. (1956). Maximal flow through a network. Canadian Journal of Mathematics 8: 399–404.CrossRefGoogle Scholar
Fussell, J.B. & Vesely, W.E. (1972). A new methodology for obtaining cut sets for fault trees. Transactions of the American Nuclear Society 15: 262–263.Google Scholar
Griffith, W. (1980). Multi-state reliability models. Journal of Applied Probability 17: 735–744.CrossRefGoogle Scholar
Hosseini, S., Barker, K., & Ramirez-Marquez, J.E. (2016). A review of definitions and measures of system resilience. Reliability Engineering & System Safety 145: 47–61.CrossRefGoogle Scholar
Huseby, A.B. & Natvig, B. (2010). Advanced discrete simulation methods applied to repairable multistate systems. In Bris, R., Soares, C.G., & Martorell, S. (eds), Reliability, risk and safety. Theory and applications, vol. 1. London: CRC Press, pp. 659–666.Google Scholar
Huseby, A.B. & Natvig, B. (2012). Discrete event simulation methods applied to advanced importance measures of repairable components in multistate network flow systems. Reliability Engineering & System Safety 119: 186–198.CrossRefGoogle Scholar
Levitin, G. & Lisnianski, A. (1999). Importance and sensitivity analisis of multistate systems using the universal generating function. Reliability Engineering & System Safety 65: 271–282.CrossRefGoogle Scholar
Levitin, G., Podofillini, L., & Zio, E. (2003). Generalised importance measures for multistate elements based on performance level restrictions. Reliability Engineering & System Safety 82: 287–298.CrossRefGoogle Scholar
Natvig, B. (1979). A suggestion for a new measure of importance of system components. Stochastic Processes and their Applications 9: 319–330.CrossRefGoogle Scholar
Natvig, B. (1985). New light on measures of importance of system components. Scandinavian Journal of Statistics 12: 43–54.Google Scholar
Natvig, B. (2011). Measures of component importance in nonrepairable and repairable multistate strongly coherent systems. Methodology and Computing in Applied Probability 13: 523–547.CrossRefGoogle Scholar
Natvig, B. (2011). Multistate systems reliability theory with applications. New York, USA: John Wiley and Sons, Inc.CrossRefGoogle Scholar
Natvig, B. & Gåsemyr, J. (2009). New results on the Barlow-Proschan and Natvig measures of component importance in nonrepairable and repairable systems. Methodology and Computing in Applied Probability 11: 603–620.CrossRefGoogle Scholar
Natvig, B., Eide, K.A., Gåsemyr, J., Huseby, A.B., & Isaksen, S.L. (2009). Simulation based analysis and an application to an offshore oil and gas production system of the Natvig measures of component importance in repairable systems. Reliability Engineering & System Safety 94: 1629–1638.CrossRefGoogle Scholar
Natvig, B., Huseby, A.B., & Reistadbakk, M. (2011). Measures of component importance in repairable multistate systems: a numerical study. Reliability Engineering & System Safety 96: 1680–1690.CrossRefGoogle Scholar
Ramirez-Marquez, J.E. & Coit, D.W. (2005). Composite importance measures for multi-state systems with multistate components. IEEE Transactions on Reliability 54: 517–529.CrossRefGoogle Scholar
Ramirez-Marquez, J.E. & Coit, D.W. (2007). Multi-state component criticality analysis for reliability improvement in multi-state systems. Reliability Engineering & System Safety 92: 1608–1619.CrossRefGoogle Scholar
Ramirez-Marquez, J.E., Rocco, C.M., Gebre, B.A., Coit, D.W., & Tortorella, M. (2006). New insights on multi-state component criticality and importance. Reliability Engineering & System Safety 91: 894–904.CrossRefGoogle Scholar
Ross, S. (2014). Introduction to probability models, 11th ed. San Diego, USA: Academic Press.Google Scholar
Si, S., Dui, H., Zhao, X., Zhang, S., & Sun, S. (2012). Integrated importance measure of component states based on loss of system performance. IEEE Transactions on Reliability 61(1): 192–202.CrossRefGoogle Scholar
Si, S., Dui, H., Cai, Z., & Sun, S. (2012). The integrated importance measure of multistate coherent systems for maintenance processes. IEEE Transactions on Reliability 61(2): 266–273.CrossRefGoogle Scholar
Si, S., Levitin, G., Dui, H., & Sun, S. (2013). Component state-based integrated importance measure for multi-state systems. Reliability Engineering & System Safety 116: 75–83.CrossRefGoogle Scholar
Si, S., Liu, M., Jiang, Z., & Jin, T. (2019). System reliability allocation and optimization based on generalized Birnbaum importance measure. IEEE Transactions on Reliability 68(3): 831–843.CrossRefGoogle Scholar
Skutlaberg, K. & Natvig, B. (2016). Minimization of the expected total net loss in a stationary multistate flow network system. Applied Mathematics 7: 793–817.CrossRefGoogle Scholar
Todinov, M.T. (2013). Flow networks. Oxford, UK: Elsevier Insights.Google Scholar
Wu, S. & Coolen, F. (2013). A cost-based importance measure for system components: an extension of the Birnbaum importance. European Journal of Operational Research 225: 189–195.CrossRefGoogle Scholar
Wu, S., Chen, Y., Wu, Q., & Wang, Z. (2016). Linking component importance to optimisation of preventive maintenance policy. Reliability Engineering & System Safety 146: 26–32.CrossRefGoogle Scholar
Zhu, X., Fu, Y., Yuan, T., & Wu, X. (2017). Birnbaum importance based heuristics for multi-type component assignment problems. Reliability Engineering & System Safety 165: 209–221.CrossRefGoogle Scholar
Zio, E. & Podofillini, L. (2003). Monte-Carlo simulation analysis of the effects on different system performance levels on the importance on multistate components. Reliability Engineering & System Safety 82: 63–73.CrossRefGoogle Scholar
Zio, E. & Podofillini, L. (2006). Accounting for components interactions in the differential importance measure. Reliability Engineering & System Safety 91: 1163–1174.CrossRefGoogle Scholar
Zio, E., Podofillini, L., & Levitin, G. (2004). Estimation of the importance measures of multistate elements by monte carlo simulation. Reliability Engineering & System Safety 86: 191–204.CrossRefGoogle Scholar
Zio, E., Marella, M., & Podofillini, L. (2007). Importance measures-based prioritization for improving the performance of multi-state systems: Application to the railway industry. Reliability Engineering & System Safety 92: 1303–1314.CrossRefGoogle Scholar