Question

Consider the following problem

Max Z = 4x1 + 3x₂
-x₁ + 6x₂ ≤ 18 (con1)
-2x₁ + 5x₂ ≥ 10 (con2)
x₁, x₂ ≥ 0 Solve the problem if x

Answer 1

To solve the given problem Max Z = 4x1 + 3x₂ -x₁ + 6x₂ ≤ 18 (con1) -2x₁ + 5x₂ ≥ 10 (con2) x₁, x₂ ≥ 0, we need to apply linear programming techniques.

Step-by-step explanation:

To solve the given problem Max Z = 4x1 + 3x₂ -x₁ + 6x₂ ≤ 18 (con1) -2x₁ + 5x₂ ≥ 10 (con2) x₁, x₂ ≥ 0, we need to apply linear programming techniques.

1. First, we convert the inequalities into equations by introducing slack variables. The given problem can be rewritten as:
2. Maximize Z = 4x₁ + 3x₂
3. subject to: x₁ + x₂ + x₃ = 18 (con1), -2x₁ + 5x₂ + x₄ = 10 (con2), and x₁, x₂, x₃, x₄ ≥ 0
4. Next, we construct the initial feasible solution by setting the slack variables to zero. The initial solution is x₁ = 0, x₂ = 0, x₃ = 18, x₄ = 10.
5. We then perform the simplex method to find the optimal solution. The simplex method involves iterating through a series of tableau until an optimal solution is found.
6. The final tableau will give us the optimal values of the decision variables and the maximum value of Z. In this case, the optimal solution is x₁ = 0, x₂ = 10, x₃ = 8, x₄ = 0, with a maximum value of Z = 40.