# Expressing all optimal solutions

I have the following optimal tableau:

Thus $x = (0, 1, 0, 4)$ is the optimal solution. However, since non-basic variable $x_{1}$ has reduced cost of 0, then we have multiple optimal solutions.

Since the column $\hat{a}_{1} = [-1 -1]^{T}$, I cannot pivot $x_{1}$ into the basis for an alternative solution.

Given this, how do I express all optimal solutions?

• It seems, that there is only one optimal solution. This is not an extraordinary case. – callculus Sep 27 '15 at 17:21
• Wrong @calculus, see below. – GarryB Sep 29 '15 at 10:02
• @GarryB How did you recreate the original problem ? – callculus Sep 29 '15 at 10:07
• As there are only two constraints and one of the (presumed) slacks is still basic, I assumed that only one iteration had taken place to bring x2 in in place of x3 and just reversed it. The "-z" suggested that a minimisation problem had been converted to a maximisation one. On further thought, I've extended my answer below. – GarryB Sep 30 '15 at 8:40