Exact optimal solutions of the minisum facility and transfer points location problems on a network

We consider hierarchical facility location problems on a network called Multiple Location of Transfer Points (MLTP) and Facility and Transfer Points Location Problem (FTPLP), where q facilities and p transfer points are located and each customer goes to one of the facilities directly or via one of the transfer points. In FTPLP, we need to find an optimal location of both the facilities and the transfer points while the location of facilities is given in MLTP. Although good heuristics have been proposed for the minisum MLTP and FTPLP, no exact optimal solution has been tested due to the size of the problems. We show that the minisum MLTP can be formulated as a p-median problem, which leads to obtaining an optimal solution.