Consider the following linear program:

Min Z = 5x₁ + 8x₂

S.t.

4x₁ + 2x₂ 20

-6x₁ + 4x₂ 12

x₁ + x₂ 6

x₂ 7

x₁ 5

x₁, x₂ 0

a) Using the graphical solution method, identify clearly the feasible region.

b) Show the direction of improvement for the objective function.

c) Identify the optimal solution and the corresponding optimal Z value.

d) Are there any redundant constraints? If yes, which one(s)?

e) What are the values of the slack and surplus variables at the optimal solution?

f) If the objective function coefficient for x₂ is kept as is, identify the range within which the x₁ coefficient may be changed without changing the optimal solution found in part (c) above.