Although enterprises make profits by satisfying customers' needs, which is most often facilitated by information systems and network infrastructures, it is difficult to keep the systems and the network components regularly operating without any downtime. As a result, it is an important and critical issue for the system administrator or defender to efficiently evaluate network survivability. In this paper, a network optimization problem of attacker and defender behavior is considered. Both actors utilize their resources: the attackers launching attacks try to maximize the damage of the network, while the defenders deploy resources on the nodes and recover the compromised nodes. This problem is considering with survivability with a metric, Degree of Disconnectivity (DOD), which is used to evaluate the damage level of the network. The Lagrangean Relaxation Method is then applied to obtain optimal solutions for the proposed problem.