Consider the linear programming problem: Maximise \(Z = 2x + 3y\) subject to \(x + y \le 6\), \(x + 3y \le 12\), \(x \ge 0\), \(y \ge 0\), and the following statements:
(A) \((2, 2)\) is a feasible solution but not an optimal solution.
(B) \((3, 3)\) is the only optimal feasible solution.
(C) \((6, 1)\) is a feasible solution.
(D) The maximum value of Z is 18.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (B) only
The lines \(x + y = 6\) and \(x + 3y = 12\) meet where
\(2y = 6\), so \(y = 3, \; x = 3\)
The region is bounded, with corner points (0, 0), (6, 0), (3, 3) and (0, 4).
\(Z(0, 0) = 0\), \(Z(6, 0) = 12\)
\(Z(3, 3) = 15\), \(Z(0, 4) = 12\)
So the maximum value of Z is 15, reached only at the corner (3, 3), and a point is feasible only if \(x + y \le 6\) and \(x + 3y \le 12\).
Checking the statements
(A) \((2, 2)\) is a feasible solution but not an optimal solution. — Correct. \(4 \le 6\) and \(8 \le 12\), but Z = 10, less than 15.
(B) \((3, 3)\) is the only optimal feasible solution. — Correct. Z = 15 there, and every other corner gives less.
(C) \((6, 1)\) is a feasible solution. — Incorrect. \(x + y = 7 \gt 6\).
(D) The maximum value of Z is 18. — Incorrect. The maximum is 15.