Solving Dual Problem using Simplex Method?

Question

Let's say I have a primal problem as follows:

(P) Max 2x1 + 3x2 + 5x3
s.t
x1 + 2x2 + 3x3 <= 8
x1 - 2x2 + 2x3 <=6
x>=0

Then we obtain the dual as follows:

(D) Min 8w1 + 6w2
s.t
w1 + w2 >= 2
2w1 - 2w2 >= 3
3w1 + 2w2 >= 5
w>=0

After obtaining the dual problem, when I try to apply Simplex method on the dual problem, I have realized that the table is already optimal which leads to the optimal objective value 0. But applying simplex to the primal problem, it is not possible to get the same solution. How is that possible? Is it not possible to apply Simplex method to the dual problem?

Answer 1

Optimal solutions are $x=(7,0.5,0)$ and $w=(1.75,0.25)$, both with objective value $15.5$. Your dual formulation is correct. Yes, you can apply the simplex method to the dual problem, but $0$ is not optimal.