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A general weapon-target assignment (WTA) problem is to find a proper assignment of weapons to targets with the objective of minimizing the expected damage of own-force asset. Genetic algorithms (GAs) are widely used for solving complicated optimization problems, such as WTA problems. In this paper, a novel GA with greedy eugenics is proposed. Eugenics is a process of improving the quality of offspring. The proposed algorithm is to enhance the performance of GAs by introducing a greedy reformation scheme so as to have locally optimal offspring. This algorithm is successfully applied to general WTA problems. From our simulations for those tested problems, the proposed algorithm has the best performance when compared to other existing search algorithms.