Opinion maximization is a kind of optimization method, which leverages a subset of influential nodes in social networks to spread user opinions towards the target product and eventually obtains the largest opinion propagation. The current propagation models on the opinion maximization mainly focus on the activated nodes and the static opinion formation process. However, they neglect the combination between the activated nodes and the dynamic opinion formation process. Moreover, previous studies are more attentive to the positive relationships among users. In the real scenario, negative relationships among users may damage the product reputation. Therefore, in this paper, we study positive opinion maximization by using an Activated Opinion Maximization Framework (AOMF) in signed social networks. The proposed AOMF is composed of three phases: i) the selection of candidate seed nodes, ii) the activated opinion formation process and iii) the determination of seed nodes. We first use an effective heuristic rule to select candidate seed nodes. To model the activation and dynamic opinion formation process of network nodes, we devise the activated opinion formation model based on the multi-stage linear threshold model and the Degroot model. Then, we calculate the opinion propagation of each candidate seed node by using the activated opinion formation model. Based on the candidate seed nodes and the activated opinion formation process, seed nodes are further determined. Finally, experimental results on six social network datasets demonstrate that the proposed method has superior potential opinions and positive ratio than the chosen benchmarks.