Question

In a Linear Programming Problem (LPP), the objective function $Z = 2x + 5y$ is to be maximized under the following constraints:

\[ x + y \leq 4, \quad 3x + 3y \geq 18, \quad x, y \geq 0. \] Study the graph and select the correct option.

• A. The solution of the given LPP lies in the shaded unbounded region.
• B. The solution lies in the shaded region $\triangle AOB$.
• C. The solution does not exist.
• D. The solution lies in the combined region of $\triangle AOB$ and unbounded shaded region.

Hint:

In an LPP, the optimal solution is always found at one of the corner points of the feasible region.

Answer 1

The Correct Option is D

In Linear Programming Problems (LPPs), the solution lies at one of the corner points of the feasible region. The given constraints are: 1. $x + y \leq 4$ (represents a line passing through $(4, 0)$ and $(0, 4)$). 2. $3x + 3y \geq 18$ (represents a line passing through $(6, 0)$ and $(0, 6)$). 3. $x, y \geq 0$ (restricts the solution to the first quadrant). From the graph, the feasible region is the region where these constraints overlap. The correct solution lies in the combined region formed by $\triangle AOB$ and the unbounded shaded area. Therefore, the correct option is (4).