Prove that the dual of dual of a LPP is primal LPP.
To prove that the dual of the dual of a Linear Programming Problem (LPP) is the primal LPP, we can use duality theory in linear programming. The dual of an LPP is obtained by forming a new optimization problem associated with the original problem, and the dual of the dual should take us back to the original problem.
Let's define the primal and dual LPPs and then demonstrate this:
Primal LPP (P):
Maximize: c^T x
Subject to: Ax ≤ b
x ≥ 0
Where:
- c is the vector of coefficients for the objective function.
- x is the vector of decision variables.
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Determine all the basic feasible solutions to the equations
x1 + x2 + 2x3 = 4
2x1 - x2 + x3 = 2
Identify the degenerate basic feasible solutions.
Which of the following statements are true or false? Give reasons for your answer.
i) The number of basic variables in a feasible solution of a transportation problem with m sources and n destinations is mn.
ii) An optimal assignment of the assignment problem with cost matrix C is also an optimal assignment of the assignment problem with cost matrix Ct .
iii) (1, 2) is an optimal solution to the following LPP:
Max Z = 2x + 4x
subject to x1 + 2x2 < 5
x1 + x2 < 4
x1 ,x2 > 0
iv) The intersection of a finite number of convex sets is not a convex set.
v) If value of the 2X2 matrix game is 4, then p > 4.
Solve the following LPP using two-phase method.
Min Z = -3x - 2y - 2z
s.t. 5x + 7y + 4z > 7
- 4x + 7y + 5z > -2
3x + 4y - 6z > 29 / 7
x, y, z > 0.
For the following matrix game, write down the equivalent LPPs for solving the game.
B
Solve the following LPP by simplex method.
Max z = 3x + 2y
s.t. 2x + y < 18
2x + 3y < 42
3x + y < 24
x, y < 0 .
Find an initial basic feasible solution to the following transportation problem by the North-West corner method. Check whether your solution is optimal.
| 6 | 4 | 1 | 5 | 14 |
| 8 | 9 | 2 | 7 | 16 |
| 4 | 3 | 6 | 2 | 5 |
| 6 | 10 | 15 | 4 |
Is the set of vectors{(1,2,3),(3,4,1),(2,3,2)} linearly independent? Give reason for the answer.