Minimum Weight Directed Subgraph ensuring all pairs reachability?
This problem is a generalized version of the famous Minimum Spanning Strong Subdigraph or MSSS [1]. This problem is NP-hard and there are several heuristic algorithms for it. You may look at the reference given in this solution and the related papers to find out more about this problem.

Note that this problem itself is a simplified version of Minimum Equivalent Graph of a Digraph [2]. Therefore, there are some algorithms that you can use as a heuristic for this problem.

[1] Bang-Jensen, Jorgen, Jing Huang, and Anders Yeo. "Strongly connected spanning subdigraphs with the minimum number of arcs in quasi-transitive digraphs." SIAM Journal on Discrete Mathematics 16.2 (2003): 335-343.

[2] Martello, Silvano, and Paolo Toth. "Finding a minimum equivalent graph of a digraph." Networks 12.2 (1982): 89-100.