Use the simplex meth to solve the linear programming problem
Maximize z = 8x1 + 2x2
Subject to: 4x1 + 3x2 < 11
X1 + 4x2 < 7
With x1 > 0, x2> 0
I get z(2.75,0) = 22
Jdj
. Solve the following L.P.P. by simplex method:
Minimize 8x1 2x2
To solve this linear programming problem using the simplex method, follow these steps:
Step 1: Convert the problem into standard form by introducing slack variables:
Maximize z = 8x1 + 2x2
Subject to: 4x1 + 3x2 + s1 = 11
x1 + 4x2 + s2 = 7
x1, x2, s1, s2 > 0
Step 2: Set up the initial simplex tableau:
| 4 3 1 0 0 11 |
| 1 4 0 1 0 7 |
|-8 -2 0 0 1 0 |
Step 3: Select the pivot column:
In this case, the pivot column will be the one with the most negative coefficient in the last row. In the initial tableau, the pivot column is column 1.
Step 4: Determine the pivot row:
To find the pivot row, divide the elements in the rightmost column (the constant column) by their corresponding values in the pivot column. The pivot row will be the one with the smallest positive ratio.
For row 1: 11/4 = 2.75
For row 2: 7/1 = 7
The smallest positive ratio is 2.75, so row 1 is the pivot row.
Step 5: Perform row operations:
Divide row 1 by the pivot element (4 in this case) to make the pivot element equal to 1:
| 1 0.75 0.25 0 0 2.75 |
| 1 4 0 1 0 7 |
|-8 -2 0 0 1 0 |
To make the other elements in the pivot column equal to 0, use row operations:
{-4 -3 1 0 0 11 }
{ 1 4 0 1 0 7 }
{-8 -2 0 0 1 0 }
Step 6: Repeat steps 3-5 until the bottom row contains all non-negative coefficients:
- Select the pivot column (column 2 since it has the most negative coefficient)
- Determine the pivot row (row 1 or 2 depending on the ratio)
- Perform row operations to make the pivot element 1 and all other elements in the pivot column 0
Step 7: Read the final solution from the simplex tableau:
In the final simplex tableau:
z = 2.75
x1 = 2.75
x2 = 0
s1 = 0
s2 = 4.25
So the maximum value of z is 2.75, and the values of x1, x2, s1, and s2 that correspond to this maximum are 2.75, 0, 0, and 4.25 respectively.