Solving the given linear programming problem

The given linear programming problem is to maximize the objective function z = X - 2X + 3X subject to the constraints -2X + X2 + 3X = 2, 2X + 3X2 - 4X = 1, and X1, X2, X3 ≥ 0. Let us solve the problem using the simplex method.

Step 1: Formulating the initial simplex tableau

We can write the problem in the standard form as follows:

Maximize z = X1 - 2X2 + 3X3

Subject to:

-2X1 + X2 + 3X3 = 2
2X1 + 3X2 - 4X3 = 1
X1, X2, X3 ≥ 0

The initial simplex tableau can be written as follows:

| BV | X1 | X2 | X3 | RHS |
| --- | -- | -- | -- | --- |
| X2 | 1 | -2 | 3 | 0 |
| X1 | -2 | 1 | 3 | 2 |
| X3 | 2 | 3 | -4 | 1 |
| z | 1 | -2 | 3 | 0 |

Where BV denotes the basic variables.

Step 2: Implementing the simplex method

We will use the simplex method to solve the problem by finding an optimal solution or determining that the problem is infeasible or unbounded.

- Pivot on the element in row 1 and column 2, which is -2.
- Perform row operations to make the pivot element 1 and other elements in column 2 zero.
- The resulting tableau is:

| BV | X1 | X2 | X3 | RHS |
| --- | -- | --- | -- | --- |
| X2 | 0 | 1 | 1 | 2 |
| X1 | 1 | -0.5| -1.5| -1 |
| X3 | 0 | 5.5 | -6 | 5 |
| z | 0 | 0.5 | 4.5| 1 |

- Pivot on the element in row 2 and column 3, which is -1.5.
- Perform row operations to make the pivot element 1 and other elements in column 3 zero.
- The resulting tableau is:

| BV | X1 | X2 | X3 | RHS |
| --- | --- | -- | -- | --- |
| X2 | 0 | 1 | 0 | 3 |
| X3 | 0.25| 0 | 1 | 0.25|
| X1 | 1 | 0 | 0 | 0.25|
| z | 0.75| 0 | 0 | 3.75|

- The optimal solution is X1 = 0.25, X2 = 3, X3 = 0.25, and the maximum value of z is