Formulating a linear program s.t. only extreme point solutions are found
Perturb the vector of the objective function slightly. If you got multiple optimal solutions than this vector is orthogonal to a facet $F$ of the polytope that defines the feasible area of the LP. Notice that $F$ could be an edge, but also a higher dimensional flat. By changing the objective function slightly you will get an extreme point that lies on the boundary of $F$.

You might want to check that the perturbation was small enough. So check the computed point with the original objective function, if it produces the optimum.