In recent years, the use of directional sensor networks (DSNs) has continued to rise increasingly, which is due to their extensive use in many situations. In such networks, the main problem is how to cover the targets distributed in a defined area and simultaneously prolong the network lifetime as much as possible. The reason of this problem is the limitation of both sensing angle and energy resource of sensors in such networks. This problem gets more complex in cases where targets need to be covered by more than one sensor direction (i.e., each target needs to be monitored for at least k times). This problem is generally known as target k-coverage problem which its NP-completeness has been already proved in the literature. The k-coverage problem can be considered in over-provisioned and under-provisioned environments. In both of these environments, especially in the latter, it is important to create a balanced coverage, as these environments do not have enough sensors to monitor all targets for k times. Thus, in this paper, we proposed a genetic-based algorithm to solve the problem in over-provisioned environments, then developed the proposed algorithm in another way to solve the problem in under-provisioned networks so that it uses the minimum number of sensors. many experiments were performed to test the efficiency of the proposed algorithm, and the obtained results showed the high capacity of the proposed algorithm in solving the k-coverage problem in both environments.