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

The objective is to minimize \( Z = 5x + 3y \) subject to the constraints forming a feasible region bounded by the lines \( x + y = 5 \) and \( x + 3y = 9 \). The vertices of this feasible region are identified as \( (0, 5) \), \( (1, 2) \), and \( (3, 0) \). Evaluating the objective function \( Z \) at each vertex yields: - At \( (0, 5) \), \( Z = 5(0) + 3(5) = 15 \) - At \( (1, 2) \), \( Z = 5(1) + 3(2) = 11 \) - At \( (3, 0) \), \( Z = 5(3) + 3(0) = 15 \). The minimum value of \( Z \) is 11, occurring at the point \( (1, 2) \).