Given the following linear program:

min$-5x_1-5x_2-3x_3,$

$s.t.x_1+3x_2+x_{3}3$

$-,x_1+3x_{3}2$

$2x_1-x_2+2x_3,4$

$2x_1+3x_2-x_3,2$

$x_{1}0,x_{2}0,x_{3}0$

i$-,$ State the problem in Standard Form

ii$-$ In class we generalized the set of equality constraints of an LP in standard form as $Ax=b.$

What is A and $b$ in this case?

iii$-$ Enumerate all the vertices of the feasible region $($i$.$e$.$ all basic feasible solutions$).$ Note: You

can write a script in Matlab to facilitate the enumeration.

iv$-$ By considering the result in iii, what is the optimal solution? Justify.

v$-,$ Solve the linear program using the simplex method. Describe each iteration.

vi$-,$ Suppose the matrix $A$ is replaced with a new matrix with $10$ rows and $100$ columns and the

vector $b$ is replaced with a new vector of $10$ components. What is the maximum number of

vertices you may have in this case? Why is it possible to have a fewer number of vertices

than the maximum number?