this is what i have so far for the following problem but i don't know what to do next:
Use the simplex method to solve the linear programming problem
minimize W = 4 y1 + 2y2
subject to:
3y1+2y2>=60
4y1+y2>=40
y1>=0,y2>=0
so this is how i started it:
i did it in the form of a matrix
3 2 60
4 1 40
------------
4 2 0
then i did the flip on the side
3 4 4
2 1 2
------------
60 40 0
I am just trying to follow the book but after i do the fliping thing i have trouble with the next steps:
After obtaining the flipped matrix, you can continue solving the problem using the simplex method by following these steps:
1. Choose a pivot column: Select the most negative value in the bottom row of the flipped matrix (ignoring the last column). In this case, the most negative value is -2.
2. Choose a pivot row: Divide the rightmost column by the corresponding value in the pivot column. Choose the row with the smallest positive quotient. In this case, the smallest positive quotient is 60/2 = 30. Thus, the pivot row is the first row.
3. Perform row operations to make the pivot element 1 and all other elements in the pivot column zero. To achieve this, divide the pivot row by the pivot element.
New Pivot Row 1: (3/2) 2 2
New Row 2: -4 1 2
4. Perform row operations to make all other elements in the pivot column zero. To do this, we can multiply the new pivot row by suitable constants and subtract it from the corresponding rows.
New Row 2: -4 1 2 - (3/2)(-4 1 2) = -4 1 2 + (6 0 0) = 2 1 -2
5. Repeat steps 1 to 4 until all values in the bottom row (except the last column) are non-negative.
6. Once you have all non-negative values in the bottom row, determine if any values are zero. If all values are non-negative but not zero, the current solution is the optimal solution. If any value is zero, choose the corresponding column as the pivot column and repeat steps 2 to 5.
7. Compute the solution: In this case, the optimal solution is y1 = 10 and y2 = 20, which results in a minimum value for W of 4(10) + 2(20) = 80.
Please note that the steps provided are a generalized description of the simplex method. It is recommended to refer to your textbook or online resources for more detailed explanations and examples.
After performing the matrix transformation correctly, you have reached the initial tableau of the Simplex method. Now, let's proceed with the next steps:
1. Choose a pivot column: Look at the bottom row of the tableau (the objective row). Identify the most negative entry in this row. In your case, the most negative entry is -4.
2. Choose a pivot row: Consider the entries in the rightmost column of the tableau (the ratio column). For each ratio, divide the right-hand side (RHS) entry by the corresponding entry in the pivot column. Choose the smallest positive ratio. If all ratios are non-positive, it means the problem is unbounded, but fortunately, this is not the case here. In your case, the smallest positive ratio is 2.
3. Perform the pivot operation: The pivot operation involves selecting a pivot element and using elementary row operations to make it equal to 1 and eliminate other entries in the pivot column (making them zero).
- Divide the pivot row by the pivot element. In your case, divide row 1 by the pivot element 4: (3 4 4) becomes (3/4 1 1).
- Perform elementary row operations to make other entries in the pivot column zero. To do this, multiply the pivot row by a multiple of -2 and add it to row 2. The goal is to eliminate the 2 in the second row, third column. After performing this operation, row 2 becomes (-1/2 0 0).
4. Update the tableau and repeat steps 1 to 3 until there are no negative entries in the objective row or all ratios in the ratio column are non-positive.
Here's the updated tableau after performing the pivot operation:
0 3/2 2
1 -1/2 0
--------------
-40 0 40
Now, you can proceed with steps 1 to 3 once again to continue the iterations of the Simplex method. Keep following these steps until you reach the optimal solution.