Question

Consider the following linear programming model

Min 2X1 + 3X2
Subject to:
X1 + X2 ≥ 4
X1 ≥ 2
X1, X2 0
This linear programming model has?
a) No feasible solution
b) Unique optimal solution
c) Multiple optimal solutions
d) Unbounded solution

Answer 1

The linear programming model described has a unique optimal solution (b).

Step-by-step explanation:

The given linear programming model is represented as:

`\[ \text{Minimize } Z = 2X_1 + 3X_2 \]`

`\[ \text{Subject to:} \]`

`\[ X_1 + X_2 \geq 4 \]`

`\[ X_1 \geq 2 \]`

`\[ X_1, X_2 \geq 0 \]`

The constraints define a feasible region, and the objective is to minimize the objective function \( Z = 2X_1 + 3X_2 \) within this feasible region. The feasible region is determined by the overlapping region of the inequalities. Since all constraints are linear, the feasible region is a polygon. The objective is to find the corner point within this polygon where the objective function is minimized.

Considering the given constraints, the feasible region is defined by the area where \( X_1 + X_2 \geq 4 \), \( X_1 \geq 2 \), and \( X_1, X_2 \geq 0 \). The feasible region is a triangular area with vertices at (2, 2), (4, 0), and (4, 4).

To find the optimal solution, we evaluate the objective function at each vertex of the feasible region. Calculating \( Z = 2X_1 + 3X_2 \) for each vertex, we find that the minimum value occurs at (4, 0), resulting in a unique optimal solution. Therefore, the correct answer is option (b) – the linear programming model has a unique optimal solution.