Tasks assignment in grid computing is a challenge for most researchers and developers of these types of systems. This is due to characteristics inherent to grid infrastructures, namely, heterogeneity, dynamicity and scalability. It becomes much more complex when it comes to assign tasks with precedence constraints represented by a Directed Acyclic Graph. The task assignment problem is well known to be NP-complete. In this paper, we present and evaluate a dependent task assignment strategy for Grids. Our goal is two folds: first we reduce, whenever possible, the average response time of tasks submitted to the grid, and secondly, we reduce the transfer cost inducing by the tasks transfer respecting the dependency constraints.