MATH2003J, OPTIMIZATION IN ECONOMICS,
BDIC 2023/2024, SPRING
Problem Sheet 5
Question 1:
Consider the following LP problem:
Maximize z = 7x1 + 4x2
subject to 2x1 + x2 ≤ 20,
x1 + x2 ≤ 18,
x1, x2 ≥ 0.
(I) Solve the above problem using the graphical method.
(II) Solve the above problem using the simplex method.
Question 2:
Consider the following LP problem:
Maximize x1 + 2x2
subject to x1 ≤ 2,
x2 ≤ 2,
x1 + x2 ≤ 3,
x1, x2 ≥ 0.
(I) Solve the above problem using the graphical method.
(II) Solve the above problem using the simplex method.
Question 3:
Consider the following LP problem:
Maximize z = 9x1 + 8x2
subject to x1 + x2 ≤ 6
2x1 + x2 ≤ 8
3x1 + 2x2 ≤ 13
x1, x2 ≥ 0.
(I) Solve the above problem using the simplex method.
(II) Solve the above problem using the graphical method.
Question 4:
Use the simplex method to solve the following LP problem:
Maximize 5x1 + 4x2
subject to −3x1 − 5x2 ≥ −78,
4x1 + x2 ≤ 36,
x1, x2 ≥ 0.
Question 5:
Use the simplex method to solve the following LP problem:
Maximize 4x1 + 2x2
subject to x1 + x2 ≤ 50,
6x1 ≤ 240,
x1 ≥ 0.
Question 6:
Use the simplex method to solve the following LP problem:
Maximize P = 3x + y + 4z
subject to 3x + 5y + 10z ≤ 120,
5x + 5y + 2z ≤ 6,
−8x − 3y − 10z ≥ −105,
x, y, z ≥ 0.
Question 7:
Use the simplex method to solve the following LP problem:
Maximize 5x1 + 6x2 + 4x3
subject to x1 + 2x2 + x3 ≤ 180,
3x1 + x2 + 2x3 ≤ 300,
x1 + 2x2 + 2x3 ≤ 240,
x1, x2, x3 ≥ 0.
Question 8:
Use the simplex method to solve the following LP problem:
Maximize z = 4x1 + 5x2 + 3x3
subject to 3x1 + 2x2 + x3 ≤ 5,
4x1 + 3x2 + 2x3 ≤ 8,
x1 + 4x2 + 2x3 ≤ 11,
x1, x2, x3 ≥ 0.
Question 9:
Use the simplex method to solve the following LP problem:
Maximize z = 5x1 + 4x2 − 6x3
subject to 4x1 + x2 − x3 ≤ 19,
3x1 + 4x2 − 6x3 ≤ 30,
2x1 + 4x2 − x3 ≤ 25,
x1 + x2 − 2x3 ≤ 15,
x1, x2 ≥ 0, x3 ≤ 0.