 Self Test Questions
Theory
1. Write the general mathematical formulation of a linear programming
problem.
2. Define the following:
- Slack variable
- Surplus variable
- Artificial variable
3. What do you mean by an optimal basic feasible solution to a linear
programming problem?
4. Explain various steps of the simplex method involved in the computation
of an optimal solution to a linear programming problem.
5. Fill up the blanks:
- .…. variables are introduced to make ……… type inequalities equations.
- A basic solution with m equation and n variables has …… variables
equal to zero.
- A basic feasible solution is a basic solution whose variables are
…..
- The maximum number of basic feasible solutions in a system with
m equations and n variables is ……
- In a linear programming problem every …. point of the Convex set
of feasible solutions is a ….solution of the problem.
- The objective function of a linear programming problem is maximized
or minimized at a …. solution.
Practical
1. Maximize z = 5x + 3y
subject to the constraints:
x + y £ 2
5x + 2y £ 10
3x + 8y £ 12
x, y³ 0
2. Maximize z = 2x1 + 4x2 + x3
+ x4
subject to
x1 + 3x2 + x4 £
4
2x1 + x2 £ 3
x2 + 4x3 + x4 £
3
x1, x2, x3, x4
³ 0
3. Maximize z = 2x + 5y
subject to
x + y £ 600
0 £ x £
400
0 £ y £
300
4. Maximize z = 5x1 + 3x2
subject to
3x1 + 5x2 £ 15
5x1 + 2x2 £ 10
x1, x2 ³ 0
5. Maximize z = x1 - x2 + 3x3
subject to
x1 + x2 + x3 £
10
2x1 - x3 £ 2
2x1 - 2x2 + 3x3 £
0
x1, x2, x3 ³
0
6. Maximize z = x1 - 3x2 + 2x3
subject to
3x1 - x2 + 2x3 £
7
-2x1 + 4x2 £ 12
-4x1 + 3x2 + 8x3 £
10
x1, x2, x3 ³
0
7. Maximize z = -2x1 - x2
subject to
x1 + 2x2 + x3 = 10
x1 + x2+ x4 = 6
x1 - x2 + x5 = 2
x1 - 2x2 + x6 = 1
x1, x2, x3, x4, x5,
x6 ³ 0
8. Minimize z = x1 + x2 + 3x3
subject to
3x1 + 2x2 + x3 £
3
2x1 + x2 + 2x3 ³
2
x1, x2, x3 ³
0
9. Minimize z = x2 - 3x3 + 2x5
subject to
x1 + 3x2 - x3 + 2x5 = 7
-2x2 + 4x3 + x4 = 12
-4x2 + 3x3 + 8x5 + x6 =
10
x1, x2, x3 ³
0
10. Maximize z = 2x1 + 5x2 + 7x3
subject to
3x1 + 2x2 + 4x3 £
100
x1 + 4x2 + 2x3 £
100
x1 + x2 + 3x3 £
100
x1, x2, x3 ³
0
11. Maximize z = 6x + 5y - 3z - 4w
subject to
2x + 3y + 2z - 4w = 24
x + 2y £ 10
x + y + 2z + 3w £ 15
y+ z + w £ 8
x, y, z, w ³ 0
12. Maximize z = 5x - 2y + 3z
subject to
2x + 2y - z ³ 2
3x - 4y £ 3
y + 3z £ 5
x,y,z ³ 0
13. Maximize z = 8x1 + 19x2 + 7x3
subject to
3x1 + 4x2 + x3 £
25
x1 + 3x2 + 3x3 £
50
x1, x2, x3 ³
0
14. Maximize: z = x1 + x2 + x3
subject to
4x1 + 5x2 + 3x3 £
15
10x1 + 7x2 + x3 £
12
x1, x2, x3 ³
0
15. Maximize: z = 3x1 + 4x2
subject to
x1 - x2 £ 1
-x1 + x2 £ 2
x1, x2 ³ 0
16. Maximize z = 3x1 + 5x2 + 4x3
subject to
2x2 + 3x3 £ 18
2x2 + 5x3 £ 18
3x1 + 2x2 + 4x3 £
25
x1, x2, x3 >
10
17. Maximize z = 3x1 + 2x2
subject to
2x1 + x2 £ 40
x1 + x2 £ 24
2x1 + 3x2 £ 60
x1, x2 ³ 0
18. Maximize z = 2x1 + 4x2
subject to
2x1 + 3x2 £ 48
x1 + 3x2 £ 42
x1 + x2 £ 21
x1, x2 ³ 0
19. Minimize z = 4x1 + 8x2 + 3x3
subject to
x1 + x2 ³ 2
2x1 + x3 ³ 5
x1, x2, x3 ³
0
20. Minimize z = x1 + x2 + x3
subject to
x1 - x4 - 2x6 = 5
x2 + 2x4 - 3x5 + x6 = 3
x3 + 2x4 - 5x5 + 6x6 = 5
x1, x2, x3, x4, x5,
x6 ³ 0
21. Minimize z = 2x1 + 9x2 + x3
subject to
x1 + 4x2 + 2x3
³ 5
3x1 + x2 + 2x3
³ 4
x1, x2, x3 ³
0
22. Minimize z = 10x + 12y
subject to
2x + 5y ³ 150
3x + y ³ 120
x, y³ 0
23. Maximize z = 12x1 + 15x2 + 9x3
subject to
8x1 + 16x2 + 12x3 £
250
4x1 + 8x2 + 10x3 ³
80
7x1 + 9x2 + 8x3 =
105
x1, x2, x3 ³
0
24. Maximize z = 4x1 + 14x2
subject to
2x1 + 7x2 £ 21
7x1 + 2x2 £ 21
x1, x2 ³ 0
25. Maximize z = 3x1 + 2x2
subject to
2x1 + x2 £ 2
3x1 + 4x2 ³ 12
x1, x2 ³ 0
26. Maximize z = x1 + x2
subject to
x1 + x2 £ 1
-3x1 + x2 ³ 3
x1, x2 ³ 0
27. Maximize z = 3x1 + 2x2
subject to
x1 - x2 £ 1
x1 + x2 ³ 3
x1, x2 ³ 0
28. Consider the constraints
-x1 + x2 £ 1
6x1 + 4x2 ³ 24
x1 ³ 0, x2
³ 2.
| (a) Minimize x1. |
(f) Maximize x1 + x2. |
| (b) Minimize x2. |
(g) Maximize -x1 + 2x2. |
| (c) Maximize x1. |
(h) Maximize x1 - 2x2. |
| (d) Maximize x2. |
(i) Maximize -3x1 -2x2. |
| (e) Minimize x1 + x2. |
|
29. Consider the constraints
-10x1 - 15x2 ³
-150
5x1 + 10x2 ³ 50
x1 - x2 ³ 0
x1 ³ 2, x2 ³
0.
| (a) Maximize x1 + x2. |
(d) Maximize -2x1 + x2. |
| (b) Minimize x1 + x2. |
(e) Maximize -x1 - 3x2. |
| (c) Maximize x1 + 3x2. |
(f) Maximize -x1 - 2x2. |
|
