Let's say I've got a linear programming problem $(P)$ which consits in maximizing $z=c^tx$ subject to $Ax=b$, and in the process, I need to add an artificial variable $x_a$ to get a feasible solution.
When applying the Big-M method to the problem, that is, maximizing $z=c^tx-Mx_a$, I get an optimal solution $\overline{x}^*$ with objective function value of $\overline{z}$ (the final value for $x_a$ is $0$). My question then is,
Can I get more optimal solutions for the problem $(P)$? How can I tell if $(P)$ has more than one solution by looking only at the simplex algorithm using the Big-M method?
Thanks in advance!
The simplex algorithm (whether using the big M method or not) gives you only one solution. Having several optimal solutions would mean you have a facet of solutions on the polyhedron $\{x:Ax=b\}$. Once you have one optimal solution, this can be seen by calculating the dimension of $\{x:Ax=b,c^tx=\bar z\}$ (check the rank of the matrix $(c,A^t)^t$)
Following comment : what you are describing is a special case of having many optimal solutions. It could be more difficult to see, hence the dimension estimation.