Math Problem Statement

Maximize Z = 10x1 + 20x2 Subject to -x1 + 2x2 ≤ 15 x1 + x2 ≤ 12 5x1 + 3x2 ≤ 45 And X1 ≥ 0, x2 ≥ 0. Solve the problem above by: Using the graphical method -x1 + 2x2 = 15 X1 +x2 = 12 5x1 + 3x2 = 45

could you show me in a graph witch area is correct

Solution

To solve the given linear programming problem using the graphical method, we will follow these steps:

  1. Plot the constraints on a graph:

    • Each inequality represents a half-plane, and we’ll convert them into equations to find the boundary lines.
    • The lines are:
      • x1+2x2=15-x_1 + 2x_2 = 15
      • x1+x2=12x_1 + x_2 = 12
      • 5x1+3x2=455x_1 + 3x_2 = 45
  2. Identify the feasible region:

    • The feasible region is the area where all the inequalities are satisfied simultaneously. This is the intersection of the half-planes defined by each constraint.
  3. Determine the optimal solution:

    • The objective function Z=10x1+20x2Z = 10x_1 + 20x_2 will be maximized at a corner point (vertex) of the feasible region.

Let's now proceed to graph these lines and determine the feasible region, which will include the vertices. I'll then provide the graphical representation for you. numpy as np import matplotlib.pyplot as plt

Define the constraint lines as functions of x1

x1_values = np.linspace(0, 15, 400)

Constraint 1: -x1 + 2x2 = 15 => x2 = (x1 - 15) / 2

x2_1 = (x1_values - 15) / 2

Constraint 2: x1 + x2 = 12 => x2 = 12 - x1

x2_2 = 12 - x1_values

Constraint 3: 5x1 + 3x2 = 45 => x2 = (45 - 5x1) / 3

x2_3 = (45 - 5 * x1_values) / 3

Plot the constraint lines

plt.figure(figsize=(8, 6))

plt.plot(x1_values, x2_1, label=r'$-x_1 + 2x_2 = 15$', color='blue') plt.plot(x1_values, x2_2, label=r'$x_1 + x_2 = 12$', color='green') plt.plot(x1_values, x2_3, label=r'$5x_1 + 3x_2 = 45$', color='red')

Add shading to the feasible region

plt.fill_between(x1_values, np.maximum(0, x2_1), np.minimum(x2_2, x2_3), where=(x1_values <= 12) & (x1_values >= 0), color='gray', alpha=0.5)

Adding constraints for non-negativity of x1 and x2

plt.xlim(0, 12) plt.ylim(0, 12)

Adding labels and title

plt.xlabel(r'$x_1$') plt.ylabel(r'$x_2$') plt.title('Feasible Region for the LP Problem')

Add a legend

plt.legend()

Show the graph

plt.grid(True) plt.show()The graph above represents the constraints and the feasible region of the linear programming problem.

  • The blue line represents the constraint x1+2x2=15-x_1 + 2x_2 = 15.
  • The green line represents the constraint x1+x2=12x_1 + x_2 = 12.
  • The red line represents the constraint 5x1+3x2=455x_1 + 3x_2 = 45.

The shaded area represents the feasible region where all the inequalities are satisfied. The region of interest lies within this shaded area, and it will be bounded by the intersection points of these lines.

Next steps:

  • The corner points of this feasible region (vertices of the shaded area) will be where we will evaluate the objective function Z=10x1+20x2Z = 10x_1 + 20x_2 to find the maximum value.
  • We should now find the exact coordinates of the corner points to evaluate the objective function.

Would you like me to calculate these corner points and evaluate the objective function at each one?

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Graphical Method
Optimization

Formulas

Objective function: Z = 10x1 + 20x2
Constraints: -x1 + 2x2 ≤ 15, x1 + x2 ≤ 12, 5x1 + 3x2 ≤ 45
Non-negativity: x1 ≥ 0, x2 ≥ 0

Theorems

Graphical solution method for linear programming
Feasible region
Corner-point method

Suitable Grade Level

Grades 11-12