Question

For a Linear Programming Problem, find min \( Z = 5x + 3y \) (where \( Z \) is the objective function) for the feasible region shaded in the given figure.

Hint:

In Linear Programming Problems, always check the objective function at all the vertices of the feasible region to determine the optimal solution.

Answer 1

We are given the objective function \( Z = 5x + 3y \), and we need to find the minimum value of \( Z \) at the feasible points of the region bounded by the lines: \[ x + y = 5 \quad \text{and} \quad x + 3y = 9 \] From the graph, the feasible region is the area formed by these lines. The vertices of the feasible region are: - \( (0, 5) \), - \( (1, 2) \), - \( (3, 0) \).
Now, evaluate the objective function at each vertex: - At \( (0, 5) \), \( Z = 5(0) + 3(5) = 15 \) - At \( (1, 2) \), \( Z = 5(1) + 3(2) = 5 + 6 = 11 \) - At \( (3, 0) \), \( Z = 5(3) + 3(0) = 15 \)
Thus, the minimum value of \( Z = 11 \) at the point \( (1, 2) \).