# Simplex method using two phase [duplicate]

(P) minimize: $$z=x_1+x_2$$ subject to :
\begin{aligned} x_1 + 2 x_2 &\geq 4 & &\text{Eq.1} \\ 2x_1 + x_2 &\geq 6 & &\text{Eq.2} \\ -x_1 + x_2 &\leq 1 & &\text{Eq.3} \\ x_1 &\geq 0 \\ x_2 &\geq 0 \end{aligned}

I'm trying to solve this using two phase method, please review my answer.

2.) For the problem (P), use the nonnegative variable $$x_3$$ for inequality constraint 1 and the nonnegative variable $$x_4$$ for inequality constraint 2 and the nonnegative variable $$x_5$$ for inequality 3 then Show the equation standard form of the problem (P).

standard form $$\min u=x_1 + x_2\quad \text{or} \quad \min u=-x_1 - x_2 \quad \text{(?)}$$

subject to
\begin{aligned} x_1 + 2x_2 - x_3 &=4 \\ 2x_1 + x_2 - x_4 &=6 \\ -x_1 + x_2 + x_5 &=1 \end{aligned}

(3) Find all feasible basis solutions of the equation standard form of the problem (P) obtained in (2).

I'm not sure how to find the feasible basis(?)

$$\begin{bmatrix} 1 & 2 & -1 & 0 & 0 \\ 2 & 1 & 0 & -1 & 0 \\ -1 & 1 & 0 & 0 & 1 \end{bmatrix}$$

am I right?

(4) from the standard form matrix that obtain in number 2, Consider the artificial variable (the problem of the first phase) when applying the two-step method, introduced artificial variable $$v_1$$ and $$v_2$$. Find dictionary for base variable $$v_1,v_2,v_5$$

dictionary: we input $$v_1$$ and $$v_2$$ as artificial variable $$\min u = v_1 + v_2$$ subject to

\begin{aligned} x_1 + 2x_2 - x_3 + v_1 &= 4 \\ 2x_1 + x_2 - x_4 + v_2 &= 6 \\ -x_1 + x_2 + x_5 &=1 \end{aligned}

reason is because if non basic variable are all $$0$$ then the basis variable will produce a feasible solution $$(4,6,1)$$

5) From problem 4, show the optimal dictionary

min

\begin{alignat}{6} u &=& 10&& &{}-{}& 3x_1&& &{}-{}& 3x_2&& {}-x_3-x_4 \\ v_1 &=& 4&& &{}-{}& x_1&& &{}-{}& 2x_2&& {}-x_3 \\ v_2 &=& 6&& &{}-{}& 2x_1&& &{}-{} &x_2&& {}-x_4 \\ x_5 &=& 1&& &{}+{}& x_1&& &{}-{}& x_2 \end{alignat} here I need to find the optimal solution that produces $$z =0$$ ? until artificial variable $${}=0$$?

Am I right?

6. Use the feasible basis solution obtained from the optimal dictionary in (5), find the first dictionary from the standard matrix form (P) and optimal solution of the problem (P),

Is this the two phase ? and solve this using tableau? how to know if the answer is optimize or not?

To optimize number 4, we need to make sure all artificial variables are $$0$$ (?)