Question $2(9$ points; $2,2,1,1,1,1,1)$

Consider the following linear programming model:

Min$5x_1+20x_2$

$s.t.x_1+x_{2}12$

$2x_1+5x_{2}40$

$x_1+x_{2}15$

$x_1,x_{2}0$

$(a)$ Manually draw the constraints and identify the feasible region.

$(b)$ Use the graphical method $(manually)to$ find the optimal solution and the optimal objective function value.

$(c)$ List all the binding constraints.

$(d)Is$ there a constraint such that $if$ you remove that constraint form this $LP$ model, the solution would become

unbounded? $If$ yes, which one $isit?$

$(e)$ Write $an$ objective function for which the original $LP$ problem would have multiple optimal solutions.

$(f)$ Using the graphical method $(manually),$ find the range $of$ feasibility for the first constraint. Explain your

approach.

$(g)$ What $is$ the reduced cost for $x_1?$ What does this reduced cost mean $(provide$ managerial insights$)?$