Question

1) Solve the following problem graphically. Indicate (a) whether or not the problem is feasible, (b) whether or not the problem has an optimal solution, and (c) whether or not the problem is unbounded. If there is a unique optimal solution, specify the variable values for this solution. If there are 2 alternative optimal solutions, give the values for three different optimal solutions.
max 9x1 + 3x2
s.t. x2 ≤ 125
  − x1 + 2x2 ≤ 170
3x1 + x2 ≤ 300
− x1 + x2 ≥ 20
x1, x2 ≥ 0

2) PART A) Use the graphical approach to verify that the following problem is unbounded.
max 3x1 − x2
s.t. − 2x1 + x2 ≤ 0
  x1 + 2x2 ≥ 4
3x1 − 5x2 ≤ 10
  x1, x2 ≥ 0
PART B) Suppose you change the third constraint to “ax1 − 5x2 ≤ 10,” where a is nonnegative value. For what values of a does the problem (i) remain unbounded, (ii) have an optimal solution, and (iii) become infeasible?

Answer 1

1)

Red Color: x2 <= 125
Green Colour:-x1 + 2x2 <= 170
Blue Colour: 3x1 + x2 <= 300
Brown Colour: -x1 + x2 >= 20

a) The given problem has a Feasible region.

b)  The problem has an optimal solution (But multiple)

c) The problem is bounded.

d) The problem has  2 alternative optimal solutions:

x1=61.43, x2=115.71

x1= 70, x2=90

2) A)

Red Colour: -2x1 + x2 <= 0
Green Colour: x1 + 2x2 >= 4
Blue Coloue : 3x1 - 5x2 <= 10