IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v204y2010i3p485-495.html
   My bibliography  Save this article

The reliability importance of components and prime implicants in coherent and non-coherent systems including total-order interactions

Author

Listed:
  • Borgonovo, E.

Abstract

In the management of complex systems, knowledge of how components contribute to system performance is essential to the correct allocation of resources. Recent works have renewed interest in the properties of the joint (J) and differential (D) reliability importance measures. However, a common background for these importance measures has not been developed yet. In this work, we build a unified framework for the utilization of J and D in both coherent and non-coherent systems. We show that the reliability function of any system is multilinear and its Taylor expansion is exact at an order T. We then introduce a total order importance measure (DT) that coincides with the exact portion of the change in system reliability associated with any (finite or infinitesimal) change in component reliabilities. We show that DT synthesizes the Birnbaum, joint and differential importance of all orders in one unique indicator. We propose an algorithm that enables the numerical estimation of DT by varying one probability at a time, making it suitable in the analysis of complex systems. Findings demonstrate that the simultaneous utilization of DT and J provides reliability analysts with a complete dissection of system performance.

Suggested Citation

  • Borgonovo, E., 2010. "The reliability importance of components and prime implicants in coherent and non-coherent systems including total-order interactions," European Journal of Operational Research, Elsevier, vol. 204(3), pages 485-495, August.
    • Handle: RePEc:eee:ejores:v:204:y:2010:i:3:p:485-495
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377-2217(09)00789-9
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ali Dogramaci & Nelson M. Fraiman, 2004. "Replacement Decisions with Maintenance Under Uncertainty: An Imbedded Optimal Control Model," Operations Research, INFORMS, vol. 52(5), pages 785-794, October.
    2. Castro, I.T., 2009. "A model of imperfect preventive maintenance with dependent failure modes," European Journal of Operational Research, Elsevier, vol. 196(1), pages 217-224, July.
    3. Emad El-Neweihi, 1980. "A Relationship Between Partial Derivatives of the Reliability Function of a Coherent System and its Minimal Path (Cut) Sets," Mathematics of Operations Research, INFORMS, vol. 5(4), pages 553-555, November.
    4. Grabisch, Michel & Labreuche, Christophe & Vansnick, Jean-Claude, 2003. "On the extension of pseudo-Boolean functions for the aggregation of interacting criteria," European Journal of Operational Research, Elsevier, vol. 148(1), pages 28-47, July.
    5. Hanif D. Sherali & Patrick J. Driscoll, 2002. "On Tightening the Relaxations of Miller-Tucker-Zemlin Formulations for Asymmetric Traveling Salesman Problems," Operations Research, INFORMS, vol. 50(4), pages 656-669, August.
    6. Michel Grabisch & Jean-Luc Marichal & Marc Roubens, 2000. "Equivalent Representations of Set Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 157-178, May.
    7. Michael O. Ball & J. Scott Provan, 1988. "Disjoint Products and Efficient Computation of Reliability," Operations Research, INFORMS, vol. 36(5), pages 703-715, October.
    8. Zio, Enrico & Podofillini, Luca, 2006. "Accounting for components interactions in the differential importance measure," Reliability Engineering and System Safety, Elsevier, vol. 91(10), pages 1163-1174.
    9. Do Van, Phuc & Barros, Anne & Bérenguer, Christophe, 2008. "Reliability importance analysis of Markovian systems at steady state using perturbation analysis," Reliability Engineering and System Safety, Elsevier, vol. 93(11), pages 1605-1615.
    10. Süleyman Özekici, 1988. "Optimal Periodic Replacement of Multicomponent Reliability Systems," Operations Research, INFORMS, vol. 36(4), pages 542-552, August.
    11. George S. Fishman, 1989. "Estimating the s − t Reliability Function Using Importance and Stratified Sampling," Operations Research, INFORMS, vol. 37(3), pages 462-473, June.
    12. Stephan Foldes & Peter L. Hammer, 2005. "Submodularity, Supermodularity, and Higher-Order Monotonicities of Pseudo-Boolean Functions," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 453-461, May.
    13. Avinash Agrawal & Richard E. Barlow, 1984. "A Survey of Network Reliability and Domination Theory," Operations Research, INFORMS, vol. 32(3), pages 478-492, June.
    14. C. Russell Philbrick & Peter K. Kitanidis, 2001. "Improved Dynamic Programming Methods for Optimal Control of Lumped-Parameter Stochastic Systems," Operations Research, INFORMS, vol. 49(3), pages 398-412, June.
    15. Young H. Chun, 2008. "Bayesian Analysis of the Sequential Inspection Plan via the Gibbs Sampler," Operations Research, INFORMS, vol. 56(1), pages 235-246, February.
    16. Theodore J. Lambert & Marina A. Epelman & Robert L. Smith, 2005. "A Fictitious Play Approach to Large-Scale Optimization," Operations Research, INFORMS, vol. 53(3), pages 477-489, June.
    17. M. A. Kubzin & V. A. Strusevich, 2006. "Planning Machine Maintenance in Two-Machine Shop Scheduling," Operations Research, INFORMS, vol. 54(4), pages 789-800, August.
    18. Smith, Curtis & Knudsen, James & Kvarfordt, Kellie & Wood, Ted, 2008. "Key attributes of the SAPHIRE risk and reliability analysis software for risk-informed probabilistic applications," Reliability Engineering and System Safety, Elsevier, vol. 93(8), pages 1151-1164.
    19. Robert F. Bordley & Craig W. Kirkwood, 2004. "Multiattribute Preference Analysis with Performance Targets," Operations Research, INFORMS, vol. 52(6), pages 823-835, December.
    20. Richard E. Barlow & Frank Proschan, 1976. "Theory of Maintained Systems: Distribution of Time to First System Failure," Mathematics of Operations Research, INFORMS, vol. 1(1), pages 32-42, February.
    21. Gao, Xueli & Cui, Lirong & Li, Jinlin, 2007. "Analysis for joint importance of components in a coherent system," European Journal of Operational Research, Elsevier, vol. 182(1), pages 282-299, October.
    22. Craig W. Kirkwood & Rakesh K. Sarin, 1980. "Preference Conditions for Multiattribute Value Functions," Operations Research, INFORMS, vol. 28(1), pages 225-232, February.
    23. Borgonovo, E., 2007. "Differential, criticality and Birnbaum importance measures: An application to basic event, groups and SSCs in event trees and binary decision diagrams," Reliability Engineering and System Safety, Elsevier, vol. 92(10), pages 1458-1467.
    24. Alonso-Meijide, J.M. & Casas-Mendez, B. & Holler, M.J. & Lorenzo-Freire, S., 2008. "Computing power indices: Multilinear extensions and new characterizations," European Journal of Operational Research, Elsevier, vol. 188(2), pages 540-554, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Rocco, Claudio M. & Hernandez-Perdomo, Elvis & Mun, Johnathan, 2021. "Application of logic regression to assess the importance of interactions between components in a network," Reliability Engineering and System Safety, Elsevier, vol. 205(C).
    2. Sedlacek, Peter & Zaitseva, Elena & Levashenko, Vitaly & Kvassay, Miroslav, 2021. "Critical state of non-coherent multi-state system," Reliability Engineering and System Safety, Elsevier, vol. 215(C).
    3. Jorge Navarro, 2016. "Stochastic comparisons of generalized mixtures and coherent systems," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 150-169, March.
    4. Borgonovo, Emanuele & Aliee, Hananeh & Glaß, Michael & Teich, Jürgen, 2016. "A new time-independent reliability importance measure," European Journal of Operational Research, Elsevier, vol. 254(2), pages 427-442.
    5. Dui, Hongyan & Zhang, Chi & Tian, Tianzi & Wu, Shaomin, 2022. "Different costs-informed component preventive maintenance with system lifetime changes," Reliability Engineering and System Safety, Elsevier, vol. 228(C).
    6. Aliee, Hananeh & Borgonovo, Emanuele & Glaß, Michael & Teich, Jürgen, 2017. "On the Boolean extension of the Birnbaum importance to non-coherent systems," Reliability Engineering and System Safety, Elsevier, vol. 160(C), pages 191-200.
    7. La Rovere, Stefano & Vestrucci, Paolo, 2012. "Investigation of the structure of a networked system," Reliability Engineering and System Safety, Elsevier, vol. 107(C), pages 214-223.
    8. Di Maio, Francesco & Baronchelli, Samuele & Zio, Enrico, 2014. "Hierarchical differential evolution for minimal cut sets identification: Application to nuclear safety systems," European Journal of Operational Research, Elsevier, vol. 238(2), pages 645-652.
    9. Lu, H.W. & Pan, H.Y. & He, L. & Zhang, J.Q., 2016. "Importance analysis of off-grid wind power generation systems," Renewable and Sustainable Energy Reviews, Elsevier, vol. 60(C), pages 999-1007.
    10. Wu, Shaomin & Coolen, Frank P.A., 2013. "A cost-based importance measure for system components: An extension of the Birnbaum importance," European Journal of Operational Research, Elsevier, vol. 225(1), pages 189-195.
    11. Toppila, Antti & Salo, Ahti, 2017. "Selection of risk reduction portfolios under interval-valued probabilities," Reliability Engineering and System Safety, Elsevier, vol. 163(C), pages 69-78.
    12. Zaitseva, Elena & Levashenko, Vitaly & Kostolny, Jozef, 2015. "Importance analysis based on logical differential calculus and Binary Decision Diagram," Reliability Engineering and System Safety, Elsevier, vol. 138(C), pages 135-144.
    13. Jiaqi Zhang & Li He & Hongwei Lu & Jing Li, 2014. "Importance Analysis of Groundwater Remediation Systems," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 28(1), pages 115-129, January.
    14. Roy Cerqueti, 2022. "A new concept of reliability system and applications in finance," Annals of Operations Research, Springer, vol. 312(1), pages 45-64, May.
    15. Borgonovo, Emanuele & Plischke, Elmar, 2016. "Sensitivity analysis: A review of recent advances," European Journal of Operational Research, Elsevier, vol. 248(3), pages 869-887.
    16. Zhai, Qingqing & Yang, Jun & Xie, Min & Zhao, Yu, 2014. "Generalized moment-independent importance measures based on Minkowski distance," European Journal of Operational Research, Elsevier, vol. 239(2), pages 449-455.
    17. Do Van, Phuc & Barros, Anne & Bérenguer, Christophe, 2010. "From differential to difference importance measures for Markov reliability models," European Journal of Operational Research, Elsevier, vol. 204(3), pages 513-521, August.
    18. Zaitseva, Elena & Levashenko, Vitaly & Sedlacek, Peter & Kvassay, Miroslav & Rabcan, Jan, 2021. "Logical differential calculus for calculation of Birnbaum importance of non-coherent system," Reliability Engineering and System Safety, Elsevier, vol. 215(C).
    19. Borgonovo, E. & Smith, C.L., 2012. "Composite multilinearity, epistemic uncertainty and risk achievement worth," European Journal of Operational Research, Elsevier, vol. 222(2), pages 301-311.
    20. Vaurio, Jussi K., 2016. "Importances of components and events in non-coherent systems and risk models," Reliability Engineering and System Safety, Elsevier, vol. 147(C), pages 117-122.
    21. Dutuit, Yves & Rauzy, Antoine, 2015. "On the extension of Importance Measures to complex components," Reliability Engineering and System Safety, Elsevier, vol. 142(C), pages 161-168.
    22. Dui, Hongyan & Wu, Shaomin & Zhao, Jiangbin, 2021. "Some extensions of the component maintenance priority," Reliability Engineering and System Safety, Elsevier, vol. 214(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. E. Borgonovo & C. L. Smith, 2011. "A Study of Interactions in the Risk Assessment of Complex Engineering Systems: An Application to Space PSA," Operations Research, INFORMS, vol. 59(6), pages 1461-1476, December.
    2. Borgonovo, E. & Smith, C.L., 2012. "Composite multilinearity, epistemic uncertainty and risk achievement worth," European Journal of Operational Research, Elsevier, vol. 222(2), pages 301-311.
    3. Xiaoyan Zhu & Way Kuo, 2014. "Importance measures in reliability and mathematical programming," Annals of Operations Research, Springer, vol. 212(1), pages 241-267, January.
    4. Wu, Shaomin & Chen, Yi & Wu, Qingtai & Wang, Zhonglai, 2016. "Linking component importance to optimisation of preventive maintenance policy," Reliability Engineering and System Safety, Elsevier, vol. 146(C), pages 26-32.
    5. Borgonovo, Emanuele & Plischke, Elmar, 2016. "Sensitivity analysis: A review of recent advances," European Journal of Operational Research, Elsevier, vol. 248(3), pages 869-887.
    6. Navarro, Jorge & Rychlik, Tomasz, 2010. "Comparisons and bounds for expected lifetimes of reliability systems," European Journal of Operational Research, Elsevier, vol. 207(1), pages 309-317, November.
    7. Do Van, Phuc & Barros, Anne & Bérenguer, Christophe, 2010. "From differential to difference importance measures for Markov reliability models," European Journal of Operational Research, Elsevier, vol. 204(3), pages 513-521, August.
    8. Michel Grabisch & Christophe Labreuche, 2010. "A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid," Annals of Operations Research, Springer, vol. 175(1), pages 247-286, March.
    9. Hans Kellerer & Vitaly A. Strusevich, 2016. "Optimizing the half-product and related quadratic Boolean functions: approximation and scheduling applications," Annals of Operations Research, Springer, vol. 240(1), pages 39-94, May.
    10. Dui, Hongyan & Wu, Shaomin & Zhao, Jiangbin, 2021. "Some extensions of the component maintenance priority," Reliability Engineering and System Safety, Elsevier, vol. 214(C).
    11. Dui, Hongyan & Si, Shubin & Yam, Richard C.M., 2017. "A cost-based integrated importance measure of system components for preventive maintenance," Reliability Engineering and System Safety, Elsevier, vol. 168(C), pages 98-104.
    12. Shumin Li & Shubin Si & Liudong Xing & Shudong Sun, 2014. "Integrated importance of multi-state fault tree based on multi-state multi-valued decision diagram," Journal of Risk and Reliability, , vol. 228(2), pages 200-208, April.
    13. Zhu, Xiaoyan & Boushaba, Mahmoud & Coit, David W. & Benyahia, Azzeddine, 2017. "Reliability and importance measures for m-consecutive-k, l-out-of-n system with non-homogeneous Markov-dependent components," Reliability Engineering and System Safety, Elsevier, vol. 167(C), pages 1-9.
    14. Marichal, Jean-Luc & Mathonet, Pierre, 2011. "Weighted Banzhaf power and interaction indexes through weighted approximations of games," European Journal of Operational Research, Elsevier, vol. 211(2), pages 352-358, June.
    15. Aizpurua, J.I. & Catterson, V.M. & Papadopoulos, Y. & Chiacchio, F. & D'Urso, D., 2017. "Supporting group maintenance through prognostics-enhanced dynamic dependability prediction," Reliability Engineering and System Safety, Elsevier, vol. 168(C), pages 171-188.
    16. Zhai, Qingqing & Yang, Jun & Xie, Min & Zhao, Yu, 2014. "Generalized moment-independent importance measures based on Minkowski distance," European Journal of Operational Research, Elsevier, vol. 239(2), pages 449-455.
    17. Kojadinovic, Ivan, 2007. "A weight-based approach to the measurement of the interaction among criteria in the framework of aggregation by the bipolar Choquet integral," European Journal of Operational Research, Elsevier, vol. 179(2), pages 498-517, June.
    18. Dui, Hongyan & Tian, Tianzi & Zhao, Jiangbin & Wu, Shaomin, 2022. "Comparing with the joint importance under consideration of consecutive-k-out-of-n system structure changes," Reliability Engineering and System Safety, Elsevier, vol. 219(C).
    19. Zhang, Chi & Ramirez-Marquez, José Emmanuel & Wang, Jianhui, 2015. "Critical infrastructure protection using secrecy – A discrete simultaneous game," European Journal of Operational Research, Elsevier, vol. 242(1), pages 212-221.
    20. Rostami, Borzou & Malucelli, Federico & Belotti, Pietro & Gualandi, Stefano, 2016. "Lower bounding procedure for the asymmetric quadratic traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 253(3), pages 584-592.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:204:y:2010:i:3:p:485-495. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.