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), optimal solutions are found at the vertices of the feasible region. The provided constraints are: 1. \(x + y \leq 4\), defining a line through \((4, 0)\) and \((0, 4)\). 2. \(3x + 3y \geq 18\), defining a line through \((6, 0)\) and \((0, 6)\). 3. \(x, y \geq 0\), confining the solution to the first quadrant. The feasible region is the intersection of these constraints. The solution lies within the combined area of $\triangle AOB$ and the unbounded shaded region. Consequently, option (4) is the correct choice.