# Simplex Method: 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:

1. .. variables are introduced to make type inequalities equations.
2. A basic solution with m equation and n variables has variables equal to zero.
3. A basic feasible solution is a basic solution whose variables are ..
4. The maximum number of basic feasible solutions in a system with m equations and n variables is
5. In a linear programming problem every . point of the Convex set of feasible solutions is a .solution of the problem.
6. 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, y ≥ 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 ≥ 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 ≥ 0

5. Maximize z = x1 - x2 + 3x3

subject to

x1 + x2 + x3 ≤ 10
2x1 - x3 ≤ 2
2x1 - 2x2 + 3x3 ≤ 0

x1, x2, x3 ≥ 0

6. Maximize z = x1 - 3x2 + 2x3

subject to

3x1 - x2 + 2x3 ≤ 7
-2x1 + 4x2 ≤ 12
-4x1 + 3x2 + 8x3 ≤ 10

x1, x2, x3 ≥ 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 ≥ 0

8. Minimize z = x1 + x2 + 3x3

subject to

3x1 + 2x2 + x3 ≤ 3
2x1 + x2 + 2x3 ≥ 2

x1, x2, x3 ≥ 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 ≥ 0

10. Maximize z = 2x1 + 5x2 + 7x3

subject to

3x1 + 2x2 + 4x3 ≤ 100
x1 + 4x2 + 2x3 ≤ 100
x1 + x2 + 3x3 ≤ 100

x1, x2, x3 ≥ 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 ≥ 0

12. Maximize z = 5x - 2y + 3z

subject to

2x + 2y - z ≥ 2
3x - 4y ≤3
y + 3z ≤ 5

x,y,z ≥ 0

13. Maximize z = 8x1 + 19x2 + 7x3

subject to

3x1 + 4x2 + x3 ≤ 25
x1 + 3x2 + 3x3 ≤ 50

x1, x2, x3 ≥ 0

14. Maximize: z = x1 + x2 + x3

subject to

4x1 + 5x2 + 3x3 ≤ 15
10x1 + 7x2 + x3 ≤ 12

x1, x2, x3 ≥ 0

15. Maximize: z = 3x1 + 4x2

subject to

x1 - x2 ≤ 1
-x1 + x2 ≤ 2

x1, x2 ≥ 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 ≥ 0

18. Maximize z = 2x1 + 4x2

subject to

2x1 + 3x2 ≤ 48
x1 + 3x2 ≤ 42
x1 + x2 ≤ 21

x1, x2 ≥ 0

19. Minimize z = 4x1 + 8x2 + 3x3

subject to

x1 + x2 ≥ 2
2x1 + x3 ≥ 5

x1, x2, x3 ≥ 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≥ 0

21. Minimize z = 2x1 + 9x2 + x3

subject to

x1 + 4x2 + 2x3 ≥ 5
3x1 + x2 + 2x3 ≥ 4

x1, x2, x3 ≥ 0

22. Minimize z = 10x + 12y

subject to

2x + 5y ≥ 150
3x + y ≥ 120

x, y≥ 0

23. Maximize z = 12x1 + 15x2 + 9x3

subject to

8x1 + 16x2 + 12x3 ≤ 250
4x1 + 8x2 + 10x3 ≥ 80
7x1 + 9x2 + 8x3 = 105

x1, x2, x3≥ 0

24. Maximize z = 4x1 + 14x2

subject to

2x1 + 7x2 ≤ 21
7x1 + 2x2 ≤ 21

x1, x2 ≥ 0

25. Maximize z = 3x1 + 2x2

subject to

2x1 + x2 ≤ 2
3x1 + 4x2 ≥ 12

x1, x2 ≥ 0

26. Maximize z = x1 + x2

subject to

x1 + x2 ≤ 1
-3x1 + x2 ≥ 3

x1, x2 ≥ 0

27. Maximize z = 3x1 + 2x2

subject to

x1 - x2 ≤ 1
x1 + x2 ≥ 3

x1, x2 ≥ 0

28. Consider the constraints

-x1 + x2 ≤ 1
6x1 + 4x2 ≥ 24

x1 ≥ 0, x2 ≥ 2.

 (a) Minimize x1. (b) Minimize x2. (c) Maximize x1. (d) Maximize x2. (e) Minimize x1 + x2. (f) Maximize x1 + x2. (g) Maximize -x1 + 2x2. (h) Maximize x1 - 2x2. (i) Maximize -3x1 -2x2.

29. Consider the constraints

-10x1 - 15x2 ≥ -150
5x1 + 10x2 ≥ 50
x1 - x2 ≥ 0

x1 ≥ 2, x2 ≥ 0.

 (a) Maximize x1 + x2. (b) Minimize x1 + x2. (c) Maximize x1 + 3x2. (d) Maximize -2x1 + x2. (e) Maximize -x1 - 3x2. (f) Maximize -x1 - 2x2.

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