Optimal Testing and repairing a Failed series system

Rapport interne.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 failed 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 $\varepsilon=0.1$ for problems with more than 4000 components within 5 minutes on a standard PC