Reconstruction of rooted phylogenetic networks opens exciting new possibilities for determining the evolution process of organisms and genes. While most of the presented methods produce rooted phylogenetic networks just using the set of reliable triplets, category of unlikely relationships between taxa will improve the quality and reliability of these networks. Here we present two approaches to produce a rooted phylogenetic network from a given set of reliable triplets and a given set of forbidden triplets, which is known as mixed triplets problem or MT. The first approach produces a rooted phylogenetic network containing the whole set of reliable triplets and inconsistent with any triplet of forbidden set. The second one is an efficient heuristic method for constructing rooted phylogenetic networks consistent with the maximum number of reliable triplets and avoiding the most probable part of forbidden triplets. We show that our heuristic method provides optimum rooted phylogenetic networks in all cases via some experiments using simulated and real data.