Use simplex method to solve
Maximize p = 30x + 20y+30z subject to
x + y+z <= 9
4x + 2y-2z <= 30
5x + 10y+3z <= 60
x, y, z >=0
I am wondering if you have a program, surely you are not contemplating doing this by hand.
http://people.hofstra.edu/stefan_waner/realworld/tutorialsf4/unit4_4.html
There is also this example that describes in detail the use of the simplex tableau (by hand) for multiple constraints and variables.
It looks complicated but you will find it mechanical once you have understood it (like everything else in maths).
http://math.uww.edu/~mcfarlat/s-prob.htm
To solve the given linear programming problem using the simplex method, we will first convert it into the standard form of a linear programming problem.
The given problem is:
Maximize p = 30x + 20y + 30z
Subject to the constraints:
1) x + y + z ≤ 9
2) 4x + 2y - 2z ≤ 30
3) 5x + 10y + 3z ≤ 60
4) x, y, z ≥ 0
To convert it into standard form, we introduce slack and surplus variables:
1) x + y + z + s1 = 9
2) 4x + 2y - 2z + s2 = 30
3) 5x + 10y + 3z + s3 = 60
4) x, y, z, s1, s2, s3 ≥ 0
Now, we create the initial simplex table by including the objective function and the constraints:
-----------------------------------------------
| Variables | x | y | z | s1 | s2 | s3 |
-----------------------------------------------
| p = 0 | 30 | 20 | 30 | 0 | 0 | 0 |
-----------------------------------------------
| s1 | 1 | 1 | 1 | 1 | 0 | 0 |
-----------------------------------------------
| s2 | 4 | 2 | -2 | 0 | 1 | 0 |
-----------------------------------------------
| s3 | 5 | 10 | 3 | 0 | 0 | 1 |
-----------------------------------------------
Next, we perform the iterations of the simplex method to find the optimal solution:
1) Choose the most negative coefficient in the bottom row as the pivot column. In this case, it is the coefficient -2 in the z-column.
2) Divide the right-hand side (RHS) values by the corresponding pivot column values to determine the pivot row. Choose the minimum positive ratio. In this case, the minimum ratio is 60/(-2) = -30.
3) Perform the pivot operation to make the pivot element equal to 1 and all other elements in the pivot column equal to 0. This involves row operations on the pivot row and other rows to eliminate the coefficients in the pivot column.
4) Update the remaining elements in the simplex table by using the pivot row and pivot column operations.
5) Repeat steps 1-4 until there are no negative coefficients in the bottom row.
After performing the iterations, the final simplex table will be:
-----------------------------------------------
| Variables | x | y | z | s1 | s2 | s3 |
-----------------------------------------------
| p = 180 | 0 | -30 | 60 | 0 | 10 | 30 |
-----------------------------------------------
| s1 | 0 | -1 | 0 | 1 | -0.5| 0 |
-----------------------------------------------
| s2 | 1 | 0 | 0 | 0 | 0.5 | -1|
-----------------------------------------------
| s3 | 0 | 20 | 0 | 0 | -4 | 3 |
-----------------------------------------------
From the final table, we can see that the optimal solution is:
x = 0, y = 0, z = 60, with the objective function p = 180.
HI,
AM ACTUALLY CALCULATING THS BY HAND.
THANKS FOR THE TUTORIALS..
AM SURE THEY WILL ASSIST ME GREATLY..