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The objective of this paper is to optimize the amount of spare capacity reserved for the backup virtual paths (BVPs) in multicast asynchronous transfer mode networks. In this paper, we study the capacity and routing assignment problem arising in the design of self-healing networks using the VP concept. A major contribution of this work is to apply a genetic algorithm (GA) to the backup path search process instead of the trivial exhaustive search method. Experimental results indicate that both approaches have very comparable results in finding the multicast backup paths. Further, it also indicated that using a GA approach has a number of advantages over the exhaustive search approach, such as the computational requirement for a GA in finding good BVPs is small when compared to the exhaustive search method. A major drawback of our approach is that we cannot guarantee the finding of global optimum in real time.