How can I turn this into canonical form and then use the two-phase simplex method to solve it? Would I need to add slack variable AND surplus variables?
$$\max\quad -2x_1-x_2-x_3 \\ \text{subject to: }-x_1-x_3 \le -1 \\ -x_1+x_2 \le -2 \\ -x_2+x_3 \le -1 \\ x_1,x_2,x_3 \ge 0 $$
When I put it into the tableau with only slack variables there are no negative coefficients in the bottom room to find the column to pivot around.
Yes, in order to put this model into standard form, we must add slack and artificial variables. For simplicity, we'll make the righthand-side non-negative.
Thus, given the model,
$$\max\quad z=-2x_1-x_2-x_3$$ Subject to, $$x_1+x_3\ge1$$ $$x_1-x_2\ge2$$ $$x_2-x_3\ge1$$ $$x_1,x_2,x_3\ge0$$
Put in it into standardized form (Big-M) as such:
$$\max\quad z+2x_1+x_2+x_3+M(a_1+a_2+a_3)=0$$ Subject to, $$x_1+x_3-e_1+a_1=1$$ $$x_1-x_2-e_2+a_2=2$$ $$x_2-x_3-e_3+a_3=1$$ $$x_1,x_2,x_3,e_1,e_2,e_3,a_1,a_2,a_3\ge0$$
Where $M$ is an arbitrary "largest" number in $\Bbb R$.
This standard form will look like the following in the initial tableau: \begin{array} {|c|c|} \hline BV & z & x_1 & x_2 & x_3 & e_1 & e_2 & e_3 & a_1 & a_2 & a_3 & RHS & RT \\ \hline z & 1 & 2 & 1 & 1 & 0 & 0 & 0 & M & M & M & 0 & - \\ \hline ? & 0 & 1 & 0 & 1 & -1 & 0 & 0 & 1 & 0 & 0 & 1 & - \\ ? & 0 & 1 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 2 & - \\ ? & 0 & 0 & 1 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & - \\ \hline \end{array}
Since there are no basic variables in this tableau, we'll need to make $a_1,a_2,$ and $a_3$ our initial basic variables as shown in the following tableau: \begin{array} {|c|c|} \hline BV & z & x_1 & x_2 & x_3 & e_1 & e_2 & e_3 & a_1 & a_2 & a_3 & RHS & RT \\ \hline z & 1 & -2M+2 & 1 & 1 & M & M & M & 0 & 0 & 0 & -4M & - \\ \hline a_1 & 0 & 1 & 0 & 1 & -1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\ a_2 & 0 & 1 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 2 & 1 \\ a_3 & 0 & 0 & 1 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & \infty \\ \hline \end{array}
From here, the $x_1$ column has the most negative $C^\pi_j$ for this maximization problem, thus we'll pivot and carry on like any other model via the Simplex method.