# Duality & Sensitivity Analysis: Self Test Questions

##### Theory

1. Define the dual of linear programming problem.

2. Explain the primal-dual relationship.

3. Write the algorithm of the dual simplex method.

4. Discuss differences between the standard simplex method and the dual simplex method.

5. How the dual can be useful in management decision making? Discuss.

6. Write the advantages of duality.

##### Practical

Use duality to solve the following LP problems.

1. Minimize z = 3x1 + x2

subject to

2x1 + 3x2 ≥ 2
x1 + x2 ≥ 1

x1, x2 ≥ 0

2. Minimize z = 50x1 - 80x2 - 140x3

subject to

x1 - x2 - 3x3 ≥ 4
x1 - 2x2 - 2x3 ≥ 3

x1, x2, x3 ≥ 0

3. Maximize z = 3x1+ 4x2

subject to

2x1+ 3x2 ≤ 16
5x1+ 2x2 ≥ 20

x1, x2 ≥ 0

4. Minimize z = x1 - x2

subject to

2x1 - x2 ≥ 2
-x1 - x2 ≥ 1

x1, x2 ≥ 0

5. Maximize z = x1 + 5x2

subject to

3x1 + 4x2 ≤ 6
x1 + 3x2 ≤ 2

x1, x2 ≥ 0

6. Maximize z = x1 + x2

subject to

2x1 + x2 ≥ 4
x1 + 7x2 ≥ 7

x1, x2 ≥ 0

7. Minimize z = 10y1 + 6y2 + 2y3

-y1 + y2 + y3 ≥ 1
3y1 + y2 - y3 ≥ 2

y1, y2, y3 ≥ 0

8. Minimize z = 50x1 - 80x2 - 140x3

subject to

x1 - x2 - 3x3 ≥ 4
x1 - 2x2 - 2x3 ≥ 3

x1, x2, x3 ≥ 0

9. Minimize z = 8x1 - 2x2 - 4x3

subject to

x1 - 4x2 - 2x3 ≥ 2
x1 + x2 - 3x3 ≥ -1

x1, x2, x3 ≥ 0

10. Minimize z = (15/2)x1 - 3x2

subject to

3x1 - x2 - x3 ≥ 3
x1 - x2 + x3 ≥ 2

x1, x2, x3 ≥ 0

11. Minimize z = x3 + x4 + x5

subject to

x1 - x3 + x4 - x5 = -2
x2 - x3 - x4 + x5 = 1

x1, x2, x3, x4, x5 ≥ 0

12. Minimize z = x1 + x2 + x3

subject to

x1 - 3x2 + 4x3 = 5
x1 - 2x2 ≤ 3
2x2 - x3 ≥ 4

x1, x2 ≥ 0, x3 unrestricted.

13. Maximize z = 6x1 + 4x2 + 6x3+ x4

subject to

4x1 + 4x2 + 4x3+ 8x4 = 21
3x1 + 17x2 + 80x3+ 2x4 ≤ 48

x1, x2 ≥ 0, x3, x4 unrestricted.