We will explain a definition of open Gromov-Witten invariants on the rational elliptic surfaces and explain the connection of the invariants with tropical geometry. For certain rational elliptic surfaces coming from meromorphic Hitchin system, we will show that the open Gromov-Witten invariants with boundary conditions near infinity (up to some transformation) coincide with the closed geodesic counting invariants defined by Gaiotto-Moore-Neitzke, which are integer-valued.