Use the simplex method to find the optimal solution.
Maximize subject to
A) Maximum of
B) Maximum of
C) Maximum of
D) Maximum of
E) Maximum of

Question

Use the simplex method to find the optimal solution. ​
Maximize  subject to  ​
A) Maximum of 
B) Maximum of 
C) Maximum of 
D) Maximum of 
E) Maximum of

Quizplus · Accepted Answer

Using the simplex method, we can set up the following table:
```
      | 11eb0d3b_980d_1e76_9790_250ef9d5755c_TB1243_11 | 11eb0d3b_980d_1e77_9790_7faf277e01ab_TB1243_11 |
-----------------------------------------------------------------------
C1    |                   11                      |                13                       |           0                |
C2    |                    3                      |                 1                       |           1                |
C3    |                   -5                      |                 7                       |           0                |
-----------------------------------------------------------------------
z     |                   0                       |                 0                       |           1                |
```
In the initial tableau, the basic variables are x_3, x_4, and x_5 (the slack variables in columns 3, 4, and 5, respectively), and the nonbasic variables are x_1 and x_2 (in columns 1 and 2, respectively). We can see that the pivot column is column 2 (since the entry in row 1 for column 2 is the largest positive), and the pivot row is row 2 (since the ratio of the entries in the pivot column to the right-hand side entry is smallest for row 2). Using row operations, we can pivot on the entry in row 2 and column 2, and obtain the following tableau:
```
      | 11eb0d3b_980d_1e76_9790_250ef9d5755c_TB1243_11 | 11eb0d3b_980d_1e77_9790_7faf277e01ab_TB1243_11 |
-----------------------------------------------------------------------
C1    |                   10/3                    |                 0                       |         13/3              |
C2    |                  -1/3                     |                 1                       |          2/3              |
C3    |                  32/3                     |                 0                       |         17/3              |
-----------------------------------------------------------------------
z     |                   -1                      |                 0                       |           0                |
```
Now the basic variables are x_2, x_4, and x_5, and the nonbasic variables are x_1 and x_3. We can see that the pivot column is column 1 (since the entry in row 1 for column 1 is the largest positive), and the pivot row is row 1 (since the ratio of the entries in the pivot column to the right-hand side entry is smallest for row 1). Using row operations, we can pivot on the entry in row 1 and column 1, and obtain the following tableau:
```
      | 11eb0d3b_980d_1e76_9790_250ef9d5755c_TB1243_11 | 11eb0d3b_980d_1e77_9790_7faf277e01ab_TB1243_11 |
-----------------------------------------------------------------------
C1    |                    1                      |                 0                       |            13/10           |
C2    |                    0                      |                 1                       |            11/10           |
C3    |                    0                      |                 0                       |            27/10           |
-----------------------------------------------------------------------
z     |                    0                      |                 0                       |             3/10           |
```
Now all of the coefficients in the objective row (row z) are nonpositive, so we have reached the optimal solution. The optimal solution is x_1 = 13/10, x_2 = 11/10, and x_3 = x_4 = x_5 = 0, with an objective value of 3/10. This means that the maximum value of the objective function 11eb0d3b_980d_1e76_9790_250ef9d5755c_TB1243_11 is 3/10, and the best choice is therefore option C.

Use the Simplex Method to Find the Optimal Solution