For the linear programming problem (LPP):
Maximize \(Z = x + 1.5y\), subject to constraints,
\(x + 2y \le 40, 2x + y \le 40, x + y \le 25, x \ge 0, y \ge 0\).
Which of the following is NOT correct?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → The optimal value of the objective function is attained at the point (15, 10).
The corner points of the feasible region are where the boundary lines meet:
\(x + 2y = 40 \text{ and } x = 0:\quad (0, 20)\)
\(2x + y = 40 \text{ and } y = 0:\quad (20, 0)\)
\(x + 2y = 40 \text{ and } x + y = 25:\quad y = 15,\ x = 10\)
\(2x + y = 40 \text{ and } x + y = 25:\quad x = 15,\ y = 10\)
Together with the origin, the corners are (0, 0), (20, 0), (15, 10), (10, 15) and (0, 20); each satisfies all the constraints. The values of Z at the corners:
\(Z(0, 0) = 0\)
\(Z(20, 0) = 20\)
\(Z(15, 10) = 15 + 15 = 30\)
\(Z(10, 15) = 10 + 22.5 = 32.5\)
\(Z(0, 20) = 30\)
The maximum is 32.5, at (10, 15) only. Checking each statement:
Option (1) The feasible region is bounded. — Correct. The region is a closed pentagon with corners (0, 0), (20, 0), (15, 10), (10, 15) and (0, 20).
Option (2) The corner points of the feasible region are (0, 0), (20, 0), (15, 10), (10, 15) and (0, 20) — Correct. They are where the boundary lines meet, e.g. \(2x + y = 40\) and \(x + y = 25\) give (15, 10).
Option (3) The optimal value of the objective function is attained at the point (15, 10). — Incorrect. Z(15, 10) = 30 is less than Z(10, 15) = 32.5, so this is the statement that is NOT correct.
Option (4) The LPP has a unique optimal solution. — Correct. The maximum value 32.5 occurs only at the corner (10, 15); the neighbouring corners (15, 10) and (0, 20) give smaller values, so no edge gives the maximum.