4. Solve the linear programming problem below. Show graph with
feasible region, identify corners and Maximize P.
Maximize: 𝑃 = 5𝑥 + 3𝑦
Subject to: 3𝑥 + 6𝑦 ≤ 30
2𝑥 + 2𝑦 ≤ 12
𝑥 ≤ 4, 𝑦 ≥ 0, 𝑥 ≥ 0

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.

Solve the linear programming problem below. Show graph with feasible region, identify corners and maximize P.
Maximize: P = 5x + 3y
Subject to:
3x + 6y ≤ 30
2x + 2y ≤ 12
x ≤ 4, y ≥ 0, x ≥ 0

Learn how to solve a linear programming problem to maximize P = 5x + 3y. This step-by-step solution includes the feasible region graph, corner points, and the optimal solution: x = 4, y = 2, with P = 26. Understand constraints like 3x + 6y ≤ 30 and explore linear programming fundamentals. Suitable for high school and college students.

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