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Disjoint Products and Efficient Computation of Reliability

Author

Listed:
  • Michael O. Ball

    (University of Maryland, College Park, Maryland)

  • J. Scott Provan

    (University of North Carolina, Chapel Hill, North Carolina)

Abstract

In this paper, we analyze the problem of computing the probability of the union of a set of events, where each event is given as the product of a set of Boolean variables. Each Boolean variable represents the operation or failure of a particular component. The problem has direct applications to the reliability analysis of complex systems as well as more general applications. After showing that the problem is NP-hard in general, we define simple, computationally efficient procedures for generating upper and lower bounds on the required probability. We show that these procedures produce the exact answer, i.e., the upper bound equals the lower bounds, for two classes of systems, matroids and threshold systems. These results draw on the relationship between this problem and the notion of shellability studied in the context of simplicial polytopes. Shellability is shown to have a very interesting and useful interpretation in this problem setting.

Suggested Citation

  • Michael O. Ball & J. Scott Provan, 1988. "Disjoint Products and Efficient Computation of Reliability," Operations Research, INFORMS, vol. 36(5), pages 703-715, October.
    • Handle: RePEc:inm:oropre:v:36:y:1988:i:5:p:703-715
    DOI: 10.1287/opre.36.5.703
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    Citations

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    Cited by:

    1. Ferdinando Cicalese & Martin Milanič, 2011. "Competitive evaluation of threshold functions in the priced information model," Annals of Operations Research, Springer, vol. 188(1), pages 111-132, August.
    2. Jørund Gåsemyr & Bent Natvig, 2001. "Bayesian inference based on partial monitoring of components with applications to preventive system maintenance," Naval Research Logistics (NRL), John Wiley & Sons, vol. 48(7), pages 551-577, October.
    3. 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.
    4. Esha Datta & Neeraj Kumar Goyal, 2017. "Sum of disjoint product approach for reliability evaluation of stochastic flow networks," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 8(2), pages 1734-1749, November.
    5. Bigatti, A.M. & Pascual-Ortigosa, P. & Sáenz-de-Cabezón, E., 2021. "A C++ class for multi-state algebraic reliability computations," Reliability Engineering and System Safety, Elsevier, vol. 213(C).
    6. Molinero, Xavier & Riquelme, Fabián & Serna, Maria, 2015. "Forms of representation for simple games: Sizes, conversions and equivalences," Mathematical Social Sciences, Elsevier, vol. 76(C), pages 87-102.

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