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The operation of a system, such as a vehicle, communication network or automatic process, heavily depends on the correct operation of its components. A Stochastic Binary System (SBS) mathematically models the behavior of on-off systems, where the components are subject to probabilistic failures. Our goal is to understand the reliability of the global system. The reliability evaluation of an SBS belongs to the class of NP-Hard problems, and the combinatorics of SBS imposes several challenges. In a previous work by the same authors, a special sub-class of SBSs called "separable systems" was introduced. These systems accept an efficient representation by a linear inequality on the binary states of the components. However, the reliability evaluation of separable systems is still hard. A theoretical contribution in the understanding of separable systems is given. We fully characterize separable systems under the all-terminal reliability model, finding that they admit efficient reliability evaluation in this relevant context.

This paper addresses a lot-sizing and scheduling problem variant arising from the study of the curing process of a tire factory. The aim is to find the minimum makespan needed for producing enough tires to meet the demand requirements on time, considering the availability and compatibility of different resources involved. To solve this problem, we suggest a hybrid approach that consists in first applying a heuristic to obtain an estimated value of the makespan and then solving a mathematical model to determine the minimum value. We note that the size of the model (number of variables and constraints) depends significantly on the estimated makespan. Extensive numerical experiments over different instances based on real data are presented to evaluate the effectiveness of the hybrid procedure proposed. From the results obtained we can note that the hybrid approach is able to achieve the optimal makespan for many of the instances, even large ones, since the results provided by the heuristic allow to reduce significantly the size of the mathematical model.

We review the state of the art in the problem of counting the number open knight tours, since the publication in internet of a computation of this quantity., Comment: 3 pages

We give an estimate of the number of geometrically distinct open tours $\G$ for a knight on a chessboard. We use a randomization of Warnsdorff rule to implement importance sampling in a backtracking scheme, correcting the observed bias of the original rule, according to the proposed principle that ``most solutions follow Warnsdorff rule most of the time''. After some experiments in order to test this principle, and to calibrate a parameter, interpreted as a distance of a general solution from a Warnsdorff solution, we conjecture that $\G=1.22\times 10^{15}$., Comment: 8 pages. See also http://www.cmat.edu.uy/~mordecki/articles