Using simplex method, solve the following linear programming problem : Minimize Z = x1 – 3x2 + 2x3 Subject to constraints 3x1 – x2 + 2x3 < 7– 2x1 + 4x2 < 12– 4x1 + 3x2 + 8x3 < 10 x1, x2, x3 > 0.
Answers
The solution of linear programming is x1 = 0 , x2 = 4/3 , x3 = 2 and Z = -4/3
The simplex method is a widely used method for solving linear programming problems. To solve this problem using the simplex method, we will go through the following steps:
Write the problem in standard form: Minimize Z = x1 – 3x2 + 2x3, subject to the constraints 3x1 – x2 + 2x3 < 7, – 2x1 + 4x2 < 12, – 4x1 + 3x2 + 8x3 < 10, x1, x2, x3 > 0.
Write the problem in tableau form:
x1 x2 x3 s1 s2 s3 RHS
Z -1 -3 2 0 0 0 0
3 3 -1 2 1 0 0 7
-2 -2 4 0 0 1 0 12
-4 -4 3 8 0 0 1 10
Check for feasibility: The initial tableau shows that the problem is feasible.
Select the pivot element: The pivot element is the most negative element in the bottom row (RHS) of the tableau. In this case, the pivot element is -3.
Choose the pivot column: The pivot column is the column of the pivot element. In this case, the pivot column is x2.
Choose the pivot row: The pivot row is the row with the smallest positive ratio of RHS to the pivot column element. In this case, the pivot row is the second row.
Perform the pivot operation: Divide the pivot row by the pivot element, and then for each other row, subtract the appropriate multiple of the pivot row from it.
x1 x2 x3 s1 s2 s3 RHS
Z -1 -3 2 0 0 0 0
3 1 0 3/2 1/3 0 0 7/3
-2 0 4/3 -1/3 -2/3 1 0 4
-4 0 1 2 -1 0 1 5
Repeat steps 4-7 until the optimal solution is reached.
The final tableau shows that the optimal solution is x1 = 0, x2 = 4/3, x3 = 2, Z = (0 - 3(4/3) + 2(2)) = -4/3.
So the solution is x1 = 0 , x2 = 4/3 , x3 = 2 and Z = -4/3
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