Maximize z = x1-x2  subject to constraint: x1+x2 less than or equal to 4 6x1+2x less than or equal to 9 . Find the integer programming problem.Introduce gomorian method 

To solve the given integer programming problem using the Gomory's cutting plane method, we start with the linear programming relaxation of the problem. The problem is to maximize \( z = x_1 - x_2 \) subject to the constraints:

1. \( x_1 + x_2 \leq 4 \)
2. \( 6x_1 + 2x_2 \leq 9 \)
3. \( x_1, x_2 \geq 0 \)
4. \( x_1, x_2 \in \mathbb{Z} \)

### Step-by-Step Solution:

1. **Linear Programming Relaxation:**

First, we solve the linear programming relaxation by ignoring the integer constraints. This means we solve:

\[
\begin{aligned}
\text{Maximize} & \quad z = x_1 - x_2 \\
\text{subject to} & \quad x_1 + x_2 \leq 4 \\
& \quad 6x_1 + 2x_2 \leq 9 \\
& \quad x_1, x_2 \geq 0
\end{aligned}
\]

2. **Solve the Linear Programming Relaxation:**

We can use the Simplex method to solve this. Let's set up the Simplex tableau. To convert the inequalities to equations, we introduce slack variables \( s_1 \) and \( s_2 \):

\[
\begin{aligned}
x_1 + x_2 + s_1 &= 4 \\
6x_1 + 2x_2 + s_2 &= 9 \\
\end{aligned}
\]

Our initial tableau is:

\[
\begin{array}{c|cccccc|c}
\text{Basis} & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\
\hline
s_1 & 1 & 1 & 1 & 0 & 4 \\
s_2 & 6 & 2 & 0 & 1 & 9 \\
\hline
z & -1 & 1 & 0 & 0 & 0 \\
\end{array}
\]

**Simplex Iteration:**

\[
\begin{array}{c|cccc|c}
 & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\
\hline
s_1 & 1 & 1 & 1 & 0 & 4 \\
s_2 & 6 & 2 & 0 & 1 & 9 \\
\hline
z & -1 & 1 & 0 & 0 & 0 \\
\end{array}
\]

**Choosing the pivot column (entering variable):** \( x_1 \) (most negative coefficient in the \( z \)-row)

**Choosing the pivot row (leaving variable):**

\[
\frac{\text{RHS}}{\text{Pivot Column}} = \min \left( \frac{4}{1}, \frac{9}{6} \right) = \frac{9}{6} = 1.5 \implies \text{row 2 is the pivot row}
\]

Pivot element is \( 6 \).

Performing the row operations to get the new tableau:

\[
\begin{array}{c|cccc|c}
 & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\
\hline
s_1 & \frac{1}{6} & \frac{2}{6} & 0 & \frac{1}{6} & \frac{3}{2} \\
x_1 & 1 & \frac{1}{3} & 0 & \frac{1}{6} & \frac{3}{2} \\
\hline
z & 0 & \frac{4}{6} & 0 & \frac{1}{6} & \frac{3}{2} \\
\end{array}
\]

**Next Iteration:**

Pivot element is now \( \frac{2}{6} \) in row 1. After further iterations, you would eventually find the optimal solution of the LP relaxation.

3. **Introduce Gomory's Cutting Plane:**

Once the optimal solution of the relaxed problem is found, if it is not integer, we introduce Gomory’s fractional cuts.

Learn how to solve an integer programming problem using Gomory's cutting plane method. Explore step-by-step solutions and the application of linear programming techniques. Suitable for advanced undergraduate students and professionals.

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