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Optimal testing and repairing a failed series system

Author

Listed:
  • Mikhail Y. Kovalyov

    (Belarusian State University
    National Academy of Sciences of Belarus)

  • Marie-Claude Portmann

    (Ecole des Mines de Nancy, Parc de Saurupt)

  • Ammar Oulamara

    (Ecole des Mines de Nancy, Parc de Saurupt)

Abstract

We consider a series repairable system that includes n components and assume that it has just failed because exactly one of its components has failed. The failed component is unknown. Probability of each component to be responsible for the failure is given. Each component can be tested and repaired at given costs. Both testing and repairing operations are assumed to be perfect, that is, the result of testing a component is a true information that this component is failed or active (not failed), and the result of repairing is that the component becomes active. The problem is to find a sequence of testing and repairing operations over the components such that the system is always repaired and the total expected cost of testing and repairing the components is minimized. We show that this problem is equivalent to minimizing a quadratic pseudo-boolean function. Polynomially solvable special cases of the latter problem are identified and a fully polynomial time approximation scheme (FPTAS) is derived for the general case. Computer experiments are provided to demonstrate high efficiency of the proposed FPTAS. In particular, it is able to find a solution with relative error ɛ = 0.1 for problems with more than 4000 components within 5 minutes on a standard PC with 1.2 Mhz processor.

Suggested Citation

  • Mikhail Y. Kovalyov & Marie-Claude Portmann & Ammar Oulamara, 2006. "Optimal testing and repairing a failed series system," Journal of Combinatorial Optimization, Springer, vol. 12(3), pages 279-295, November.
    • Handle: RePEc:spr:jcomop:v:12:y:2006:i:3:d:10.1007_s10878-006-9633-0
    DOI: 10.1007/s10878-006-9633-0
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    References listed on IDEAS

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    1. Janiak, Adam & Kovalyov, Mikhail Y. & Kubiak, Wieslaw & Werner, Frank, 2005. "Positive half-products and scheduling with controllable processing times," European Journal of Operational Research, Elsevier, vol. 165(2), pages 416-422, September.
    2. T. Badics & E. Boros, 1998. "Minimization of Half-Products," Mathematics of Operations Research, INFORMS, vol. 23(3), pages 649-660, August.
    3. Wayne E. Smith, 1956. "Various optimizers for single‐stage production," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 59-66, March.
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    Cited by:

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    2. Justkowiak, Jan-Erik & Kovalev, Sergey & Kovalyov, Mikhail Y. & Pesch, Erwin, 2023. "Single machine scheduling with assignable due dates to minimize maximum and total late work," European Journal of Operational Research, Elsevier, vol. 308(1), pages 76-83.

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