In Multi-Agent Pathfinding (MAPF) problems, multiple agents move simultaneously to reach their individual destinations without colliding with each other. The computational complexity of the problem has been extensively studied for undirected graphs over the past decades. However, plan existence for Directed MAPF (diMAPF) was only recently studied and was shown to be in PSPACE as well as NP-hard. In this paper, we study the optimization versions (on makespan and on travel distance of agents) of diMAPF problems and show that they remain NP-hard even when various important non-trivial restrictions are imposed (e.g., when considering the problem on directed, acyclic, and planar graphs where the vertex-degrees are bounded). We have also provide membership results, thus presenting the first set of NP-completeness results for various optimal diMAPF variants.