QUESTION $1.(70$ Points$)$

A linear programming model is presented below.

Minz$=5x_1+2x_2,+4x_3,+4x_4$

$s.t.$

$2x_1+x_2,+2x_3,+x_{4}4$

$\{\begin{array}{ccc}\text{:}[2x_2,& +x_3,& -x_{4}6]\end{array}$

$6x_1+3x_2+5x_3+2x_{4}10$

$\{\begin{array}{ccc}\text{:}[3x_1,& -x_2,+x_3,& =0]\end{array}$

$x_1,x_2,x_3,x_{4}0$

a$)$ Solve the given model using the Dual Simplex method.

b$)$ What is the solution when the following constraints are added to the model? Add new constraints to the

optimal table you found in option a and find the solution with Dual Simplex.

Additional Constraints:

$x_1,-3x_2+x_3,+3x_4,2$

$-x_1,+2x_3,-x_4,5$

c$)$ What is the solution when the following constraints are added to the model? Add new constraints to the

optimal table you found in option a and find the solution with Dual Simplex. $($Additional constraints will

be appended to the initial model. This option is independent of option b$).$

Additional Constraints:

$4x_2,-3x_3-x_4,7$

$x_1+x_2,+x_3+x_4,3$

QUESTION $2.(50$ Points$)$

Solve the following LP using the revised simplex method. Employ the Big M method to find an initial

basic feasible solution.