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.
