How to find the number of possible solutions of LP problems? Let us assume that we have a linear optimization problem (LP) that has multiple optimal solutions.
I would like to know if there is a solver or an algorithm that can provide the number of optimal solutions. I would also like to know if there has been any research on this question.
 A: Yes there are already results on this question. There are two cases to consider:
If variables $\in \mathbb{R}+ $:
Since the problem is linear, the constraints form a polygon, and if there is more than one solution, than it means that there is an infinite amount of them, all lie on an edge of the polygon. In this case it is clear that it is impossible to output all of them.
If variables $\in \mathbb{N} $: In this case, you can sequentially generate all solutions by adding new constraints. See for example this paper or this one…or this one.
A: In the LP case (opposed to MIP) you most likely want to enumerate corner points or basic solutions. There is a low level pivoting algorithm described in Ralph E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, 1986. 
The paper Sangbum Lee, Chan Phalakornkule, Michael M. Domach, Ignacio E Grossmann, Recursive MILP model for finding all the alternate optima in LP models for metabolic networks, Computers & Chemical Engineering, Volume 24, Issues 2–7, 15 July 2000, Pages 711-716 is also interesting. It enumerates all optimal bases using a MIP formulation. See also here for a GAMS implementation. This method can be optimized somewhat using the Cplex solution pool technology, see here. 
