[SOLVED] MATH2003J OPTIMIZATION IN ECONOMICS BDIC 2023/2024 SPRING Problem Sheet 5 C/C

MATH2003J, OPTIMIZATION IN ECONOMICS,

BDIC 2023/2024, SPRING

Problem Sheet 5

Question 1:

Consider the following LP problem:

Maximize z = 7×1 + 4×2

subject to 2×1 + 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 + 2×2

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 = 9×1 + 8×2

subject to x1 + x2 ≤ 6

2×1 + x2 ≤ 8

3×1 + 2×2 ≤ 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 5×1 + 4×2

subject to −3×1 − 5×2 ≥ −78,

4×1 + x2 ≤ 36,

x1, x2 ≥ 0.

Question 5:

Use the simplex method to solve the following LP problem:

Maximize 4×1 + 2×2

subject to x1 + x2 ≤ 50,

6×1 ≤ 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 5×1 + 6×2 + 4×3

subject to x1 + 2×2 + x3 ≤ 180,

3×1 + x2 + 2×3 ≤ 300,

x1 + 2×2 + 2×3 ≤ 240,

x1, x2, x3 ≥ 0.

Question 8:

Use the simplex method to solve the following LP problem:

Maximize z = 4×1 + 5×2 + 3×3

subject to 3×1 + 2×2 + x3 ≤ 5,

4×1 + 3×2 + 2×3 ≤ 8,

x1 + 4×2 + 2×3 ≤ 11,

x1, x2, x3 ≥ 0.

Question 9:

Use the simplex method to solve the following LP problem:

Maximize z = 5×1 + 4×2 − 6×3

subject to 4×1 + x2 − x3 ≤ 19,

3×1 + 4×2 − 6×3 ≤ 30,

2×1 + 4×2 − x3 ≤ 25,

x1 + x2 − 2×3 ≤ 15,

x1, x2 ≥ 0, x3 ≤ 0.