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Homework Statement:

(P)
minimize: $$z=x_1+x_2$$
subject to :
$$
x_1+2x_2 >= 4 $$( equation 1)
$$2x_1+x_2>=6$$ (equation 2)
$$x_1+x_2<=1$$ (equation 3)
$$x_1>=0 ,x_2>=0 $$
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).**
answer: standart form
min $$u=x1+x2$$ or $$u=x1x2$$ (?)
subject to <br>
$$x_1+2x_2x_3=4 $$
$$2x_1+x_2 x_4=6 $$
$$x_1+x_2+ x_5=1$$
**(3) Find all feasible basis
solutions of the equation standard form of the problem (P) obtained in (2).**
answer:im 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 twostep method, introduced artificial variable $$v_1$$ and $$v_2$$. find dictionary for base variable $$v_1,v_2,v_5$$**
answer: dictionary
we input v1 and v2 as artificial variable
min $$u=v_1+v_2$$
subject to
$$x_1+2x_2x_3+v_1=4 $$
$$2x_1+x_2 x_4+v_2=6$$
$$x_1+x_2+ x_5=1 $$
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**
answer:
min
$$u=103x_13x_2x_3x_4 $$
$$v_1=4x_12x_2x_3 $$
$$v_2=62x_1x_2x_4$$
$$x_5=1+x_1x_2 $$
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),**
answer:
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(?)
I'm confused, I have read about this but i cant seem understand
Relevant Equations:
 Linear programming , matrices,simplex method
Simplex method