To solve this linear programming problem using the Gomory's Cutting Plane Method, we first need to rewrite the problem as follows:

Problem:

Maximize $z = x_1 + x_2$

Subject to:

1. $2x_1 + 5x_2 \leq 16$
2. $6x_1 + 5x_2 \leq 30$

We need to find the optimal integer solution to this problem.

Steps to solve using the Gomory Cutting Plane Method:

1. Solve the Linear Program as a Relaxed LP: Ignore the integer constraint initially (i.e., allow the variables to take fractional values) and solve the linear programming problem.

2. Find the Optimal Solution of the Relaxed LP: From the LP, we calculate the fractional values of $x_1$ and $x_2$.

3. Generate a Cutting Plane: If the solution is fractional (not integers), we create a cutting plane by adding a constraint that eliminates the fractional solution and refines the feasible region.

4. Repeat the process until an integer solution is obtained.

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Let’s solve the relaxed LP first.

Step 1: Solve the Relaxed LP

The linear program is:

$\text{Maximize} \quad z = x_1 + x_2$

Subject to:

1. $2x_1 + 5x_2 \leq 16$
2. $6x_1 + 5x_2 \leq 30$
3. $x_1 \geq 0, x_2 \geq 0$

We will solve this using the Simplex Method. Let me compute this for you. scipy.optimize import linprog

Coefficients of the objective function (for maximization we negate them)

c = [-1, -1] # z = x1 + x2, so we negate to convert to minimization

Coefficients of the inequality constraints

A = [ [2, 5], # 2x1 + 5x2 <= 16 [6, 5] # 6x1 + 5x2 <= 30 ]

b = [16, 30] # Right-hand side of the inequalities

Bounds for the variables x1, x2 (non-negative)

x_bounds = (0, None) y_bounds = (0, None)

Solve the relaxed LP using linprog (minimization version)

result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method='highs')

Results

result.x, result.funThe solution to the relaxed linear programming problem is:

• $x_1 = 3.5$
• $x_2 = 1.8$

The maximum value of the objective function is $z = 5.3$.

Since the solution contains fractional values, we now need to apply Gomory's Cutting Plane Method to find an integer solution.

Would you like me to proceed with applying the cutting plane method, or would you like further details on any part of the process?