This is a linear programming (LP) problem that can be solved using methods such as the Simplex method or computational tools. The objective is to maximize:

$Z = 5x_1 + 4x_2 + 3x_3$

Subject to the constraints:

1. $2x_1 + 3x_2 + x_3 \leq 5$
2. $4x_1 + x_2 + 2x_3 \leq 11$
3. $3x_1 + 4x_2 + 2x_3 \leq 8$
4. $x_1, x_2, x_3 \geq 0$

Steps to Solve:

1. Represent the problem in standard form if using the Simplex method.
2. Define slack variables for the inequalities.
3. Solve iteratively or use import linprog

Coefficients of the objective function (maximize 5x1 + 4x2 + 3x3)

c = [-5, -4, -3] # Negated for minimization

Coefficients of the inequality constraints

A = [ [2, 3, 1], # 2x1 + 3x2 + x3 <= 5 [4, 1, 2], # 4x1 + x2 + 2x3 <= 11 [3, 4, 2] # 3x1 + 4x2 + 2x3 <= 8 ]

Right-hand side of the inequality constraints

b = [5, 11, 8]

Bounds for the variables x1, x2, x3 (non-negative constraints)

x_bounds = [(0, None), (0, None), (0, None)]

Solve the linear programming problem

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

The optimal solution to the given linear programming problem is:

$x_1 = 2, \quad x_2 = 0, \quad x_3 = 1$

Objective Function Value:

$Z = 5x_1 + 4x_2 + 3x_3 = 5(2) + 4(0) + 3(1) = 13$

Additional Details:

• All constraints are satisfied.
• The values of the variables are non-negative, as required.

Would you like a detailed explanation of the solving process or a graphical representation of the feasible region?

Related Questions:

1. How do we derive the solution using the Simplex method manually?
2. What is the geometric interpretation of this optimization problem?
3. How do we handle equality constraints in linear programming?
4. Can we use duality to verify the solution in this problem?
5. What changes in the solution if the objective function or constraints are altered slightly?

Tip: Always verify if the constraints' slack values and objective coefficients align with your interpretation when solving LP