$11$ i$)$ Consider the problem:

max,$z=x_1+3x_2$

$s.t.x_1-2x_{2}0$

$,-2x_1+x_{2}4$

$,5x_1+3x_{2}15$

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

a$)$ Draw the feasible region, clearly indicating the constraints, slack variables associated with

each constraint, and normal to the objective function at the origin.

b$)$ Solve the problem graphically. Show the contours of the objective function at the optimum.

c$)$ Solve the problem using the tabular form of the simplex method and write down the basis

matrix $B$ and its inverse $B^{-1}$ at each iteration.

ii$)$ For the linear program in part $($i$),$ associate dual variables to each constraint, and write the

corresponding dual formulation.

a$)$ Solve this dual problem using the two$-$phase method.

b$)$ Verify that the optimal objective function value obtained in part $($i$)$ and $($ii$)($a$)$ are the

same and show that the optimal values of the dual variables can be obtained from the

simplex tableau of part $($i$).$

c$)$ Write the KKT conditions for the linear program in part $($i$)$ and determine the optimal

values to the primal and dual problems by solving the KKT conditions.

d$)$ Verify that the optimal solutions to the primal and dual problems can be obtained directly

from the KKT conditions.