import numpy as np import matplotlib.pyplot as plt from scipy.optimize import linprog

Define the coefficients of the objective function

c = [-5, -3] # Negating for maximization

Define the inequality constraints matrix (Ax <= b)

A = [[3, 6], [2, 2], [1, 0], [0, -1]] b = [30, 12, 4, 0]

Solve the linear programming problem

result = linprog(c, A_ub=A, b_ub=b, method='highs')

Extract the optimal values

optimal_x, optimal_y = result.x

Plotting the feasible region

x = np.linspace(0, 5, 500)

Define inequalities

y1 = (30 - 3 * x) / 6 # From 3x + 6y <= 30 y2 = (12 - 2 * x) / 2 # From 2x + 2y <= 12

Clip negative values as y >= 0

y1 = np.maximum(0, y1) y2 = np.maximum(0, y2)

Define bounds for x and y

plt.plot(x, y1, label=r'$3x + 6y \leq 30$') plt.plot(x, y2, label=r'$2x + 2y \leq 12$') plt.axvline(4, color='r', linestyle='--', label=r'$x \leq 4$') plt.axhline(0, color='g', linestyle='--', label=r'$y \geq 0$')

Feasible region shading

plt.fill_between(x, np.minimum(y1, y2), color='lightgrey', where=(x <= 4))

Plot optimal point

plt.plot(optimal_x, optimal_y, 'ro', label='Optimal Point')

Add labels and legend

plt.xlabel('x') plt.ylabel('y') plt.title('Feasible Region and Optimal Point') plt.legend() plt.grid() plt.show()

Display optimal solution and value of P

optimal_P = 5 * optimal_x + 3 * optimal_y optimal_x, optimal_y, optimal_P <<ImageDisplayed>>(4.0, 2.0, 26.0)(4.0, 2.0, 26.0)successThe solution to the linear programming problem is as follows:

• Optimal Point: $x = 4.0$, $y = 2.0$
• Maximum Value of $P$: $P = 26.0$

The graph above illustrates:

1. The feasible region (shaded area).
2. The constraints.
3. The optimal point marked in red.

Would you like further details on solving or interpreting this problem? Here are some related questions to deepen your understanding:

1. How is the feasible region determined and why is it important?
2. What are the properties of a linear programming problem?
3. How do you verify that the optimal solution lies on a vertex of the feasible region?
4. Why are the inequality constraints converted into equalities to identify corners?
5. How does changing the objective function $P$ affect the solution?

Tip: In linear programming, the optimal solution always occurs at a vertex of the feasible region.