New Bounds for the k-out-of-n Type Probabilities and Their Applications
Binmahfoudh, Ahmed Mohammed
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The contribution of the shape information of the underlying distribution in probability bounding problem is investigated and an efficient linear programming based bounding methodology, which takes advantage of the advanced optimization techniques, probability theory, and the state-of-the-art tools, to obtain robust and efficiently computable bounds for the probabilities that at least k and exactly k-out-of-n events occur is developed. The k-out-of-n type probability bounding problem is formulated as linear programs under the assumption that the probability distribution is unimodal. The dual feasible bases structures of the relaxed versions of linear programs involved are fully described. The bounds for the probability that at least k and exactly k-out-of-n events occur are obtained in the form of formulas. A dual based linear programming algorithm is proposed to obtain bounds as the customized algorithmic solutions of the LP’s formulated. Numerical examples are presented to show that the use of shape constraint significantly improves on the bounds for the probabilities that at least k and exactly k-out-of-n events occur when only first a few binomial moments are known. An application in PERT, where the shape of the underlying probability distribution can be used to obtain bounds for the distribution of the critical path length, is presented.