Problem $2(75$ points$)$

Consider the following linear programming problem $P.$

Minimize $Z=,x_1,-2x_2$

Subject $to{x}_1+x_{2}2$

$-x_1+x_{2}1$

$x_{2}3$

$x_2,0$

$x_{\text{I}}$ unconstrained $in$ sign

Let the surplus of constraint $(1)$ and $(2)$ be $x_3$ and $x_4,$ respectively, and the slack of constraint $(3)$

be $x.$ Answer the following independent questions:

$1$ Solve the problem graphically:

Identify the feasible region by its corner points $($coordinates $x_1$ and $x_2)$ and shade it$.$ Find the

optimal point on the graph and write the optimal values of the variables and $z,$ below.

$2$ Determine the optimal solution, if instead of minimization the objective was maximization.

$3$ Write an objective function that has multiple optima on the feasible region of problem $P.$

$4$ Determine a right hand side value of constraint $(3)$ that renders the problem infeasible. Note

that there are many values with that property.

$5$ What is the optimal solution if constraint $(3)$ is removed from the formulation?

$6$ Construct the initial basic solution by adding artificial variables and making the necessary

variable transformations so that you can apply the Big M method to Problem P$.$ Set up Big

M method iteration $(0)$ tableau. Indicate the entering and leaving variable and perform a

single iteration. Write the resulting basic solution of iteration $(1)$ and indicate whether it is

feasible or infeasible to Problem $P.$ Indicate on the graph the point this basic solution

corresponds to and the constraints $($if any$)$ that are violated.