MAXIMIZE: Z = 3 X1 + 1 X2 + 3 X3 + X4 + X5 + X6 |
MAXIMIZE: Z = 3 X1 + 1 X2 + 3 X3 + 0 X4 + 0 X5 + 0 X6 + 0 X7 + 0 X8 + 0 X9 |
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subject to 2 X1 + 1 X2 + 1 X3 + 1 X4 + 0 X5 + 0 X6 = 2 |
subject to 2 X1 + 1 X2 + 1 X3 + 1 X4 + 1 X9 = 2 |
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X1, X2, X3, X4, X5, X6 ≥ 0 |
X1, X2, X3, X4, X5, X6, X7, X8, X9 ≥ 0 |
Tableau 1 |
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-1 |
-1 |
-1 |
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Cb |
P0 |
P1 |
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P9 |
-1 |
2 |
2 |
1 |
1 |
1 |
0 |
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1 |
P8 |
-1 |
5 |
1 |
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3 |
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2 |
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P7 |
-1 |
6 |
2 |
2 |
1 |
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3 |
1 |
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0 |
Z |
-13 |
-5 |
-5 |
-5 |
-1 |
-2 |
-3 |
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Tableau 2 |
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Base |
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P9 |
P1 |
0 |
1 |
1 |
0.5 |
0.5 |
0.5 |
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0 |
0 |
0.5 |
P8 |
-1 |
4 |
0 |
1.5 |
2.5 |
-0.5 |
2 |
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1 |
-0.5 |
P7 |
-1 |
4 |
0 |
1 |
0 |
-1 |
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3 |
1 |
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-1 |
Z |
-8 |
0 |
-2.5 |
-2.5 |
1.5 |
-2 |
-3 |
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2.5 |
Tableau 3 |
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P9 |
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1 |
0.5 |
0.5 |
0.5 |
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0 |
0 |
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0.5 |
P8 |
-1 |
4 |
0 |
1.5 |
2.5 |
-0.5 |
2 |
0 |
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1 |
-0.5 |
P6 |
0 |
1.33 |
0 |
0.333 |
0 |
-0.333 |
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1 |
0.333 |
0 |
-0.333 |
Z |
-4 |
0 |
-1.5 |
-2.5 |
0.5 |
-2 |
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1 |
0 |
1.5 |
Tableau 4 |
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P8 |
P9 |
P1 |
0 |
0.2 |
1 |
0.2 |
0 |
0.6 |
-0.4 |
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0 |
-0.2 |
0.6 |
P3 |
0 |
1.6 |
0 |
0.6 |
1 |
-0.2 |
0.8 |
0 |
0 |
0.4 |
-0.2 |
P6 |
0 |
1.333 |
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0.333 |
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-0.333 |
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1 |
0.333 |
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-0.333 |
Z |
0 |
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1 |
1 |
1 |
Tableau 1 |
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P1 |
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P4 |
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P6 |
P1 |
3 |
0.2 |
1 |
0.2 |
0 |
0.6 |
-0.4 |
0 |
P3 |
3 |
1.6 |
0 |
0.6 |
1 |
-0.2 |
0.8 |
0 |
P6 |
0 |
1.333 |
0 |
0.333 |
0 |
-0.333 |
0 |
1 |
Z |
5.4 |
0 |
1.4 |
0 |
1.2 |
1.2 |
0 |
The optimal solution value is Z = 5.4
X1 = 0.2
X2 = 0
X3 = 1.6
X4 = 0
X5 = 0
X6 = 1.3333333333333
Herein I used simplex method to derive optimal solution, I am not 100% sure whether this optimum solution is right but I did my best.
Problem 1 (20 pts) Consider the mathematical program max 3x1+x2 +3x3 s.t. 2x1 +x2 + x3...
Problem 1 (20 pts) Consider the mathematical program max 3x1+x2 +3x3 s.t. 2x1 +x2 + x3 +x2 x1 + 2x2 + 3x3 +2xs 5 2x 2x2 +x3 +3x6-6 Xy X2, X3, X4, Xs, X620 Three feasible solutions ((a) through (c)) are listed below. (0.3, 0.1, 0.4, 0.9, 1.65, 1.6) (c) x Please choose one appropriate interior point from the list, and use the Karmarkar's Method at the interior point and determine the optimal solution. 25 Problem 1 (20 pts) Consider...
Consider the mathematical program max 3x1 x2 +3x3 s.t. 2X1 + X2 + X3 +X4-2 x1 + 2x2 + 3x3 + 2xs 5 2x1 + 2x2 + x3 + 3x6 = 6 Three feasible solutions ((a) through (c)) are listed below. (b) xo) (0.9, 0, 0, 0.2,2.05, 1.4) (c) xo) (0.3, 0.1, 0.4, 0.9, 1.65, 1.6) Please choose one appropriate interior point from the list, and use the Karmarkar's Method at the interior point and determine the optimal solution.
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1. (20 pts.) Consider the following linear program: max 4x4 +xz+5x3 +3x4 s.t. *1 -X2 -X3 +3X, 51 5x +xz+3X3 +8X555 -X2 +2x2+3x3 -5x53 It is claimed that the solution x* = (0,14,0,5) is an optimal solution to the problem. Give a proof of the claim. Do not use the simplex method to solve this problem.
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samplex Problem1: Solve the following problem using simplex method: Max. z = 2 x1 + x2 – 3x3 + 5x4 S.t. X; + 7x2 + 3x3 + 7x, 46 (1) 3x1 - x2 + x3 + 2x, 38 .(2) 2xy + 3x2 - x3 + x4 S 10 (3) E. Non-neg. x > 0, x2 > 0, X3 > 0,44 20 Problem2: Solve the following problem using big M method: Max. Z = 2x1 + x2 + 3x3 s.t. *+...
Consider the following LP problem max z = x1 +2x2 + x3 + x4 s.t. x1 + 2x2 + x3 く2 +2x3 く! X1, x2, x3, x4 20 a) Obtain the dual formulation of the LP.
Consider the following LP problem: Minimize Cost = 3x1 + 2x2 s.t. 1x1 + 2x2 ≤ 12 2x1 + 3 x2 = 12 2 x1 + x2 ≥ 8 x1≥ 0, x2 ≥ 0 A) What is the optimal solution of this LP? Give an explanation. (4,0) (2,3) (0,8) (0,4) (0,6) (3,2) (12,0) B)Which of the following statements are correct for a linear programming which is feasible and not unbounded? 1)All of the above. 2)Only extreme points may be optimal....
Use the simplex algorithm to find all optimal solutions to the following LP. max z=2x1+x2 s.t. 4x1 + 2x2 ≤ 4 −2x1 + x2 ≤ 2 x1 ≥1 x1,x2 ≥0