In parts of this problem $\text{'}$ Solve LP$\text{'}$ must be answered using graphical method with

some reasoning $-$ do not simply write final answer. Recall that this question has three

possible outcomes:

a$)$ You either give the optimal $z$ value and optimal solution$($s$).$

b$)$ State that LP is infeasible $($SFS is empty$)$

c$)$ State that no optimal value exists because objective function is unbounded $($in the

direction of optimization$).$

Consider the LP$1$:

Maximize $z=x+2y$

Subject to

$-x+y1$

$x+y2$

$2x+y8$

$x,y0$

For this question, the notation with a prime will denote a minimization problem so

LP$1\text{'}$ is basically same as this, but with minimization objective.

A$.$ Sketch the set of feasible solutions $($SFS$).$ This will be a convex region bounded

by polygon. Compute all vertices.

B$.$ To identify the pattern, sketch the level sets of the objective function at $z=-1,z$

$=0,z=1,z=5,z=9.$

C$.$ Solve LP$1.$

D$.$ Solve minimization problem LP$1\text{'}.$

E$.$ Reverse the direction of inequality in the third constraint of LP$1.$ Call this LP$2.$

Solve LP$2.$

F$.$ Solve the minimization problem LP$2\text{'}.$

G$.$ Reverse the direction of inequalities in both the second and third constraints of

the original LP$1-$ call this new LP$3.$ Solve LP$3.$

H$.$ Solve LP$3\text{'}$