Consider the following statements about a linear programming problem with objective function \(Z = ax + by\):
(A) If the feasible region is bounded, then Z has both a maximum and a minimum value on it.
(B) The conditions \(x \ge 0, y \ge 0\) are called non-negative restrictions.
(C) If the feasible region is unbounded, then Z cannot have a maximum value.
(D) If the feasible region is empty, then the problem has no optimal solution.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
On a bounded feasible region, Z always has both a maximum and a minimum, each at a corner point. On an unbounded region, a maximum or a minimum may or may not exist.
For example, on the unbounded region \(x + y \ge 2\), \(x, y \ge 0\), the function \(Z = -x - y\) has the maximum value \(-2\):
\(Z = -(x + y) \le -2\), with equality at (2, 0)
So (A), (B) and (D) are true, and (C) is false.
Checking the statements
(A) If the feasible region is bounded, then Z has both a maximum and a minimum value on it. — Correct. A bounded region always gives both, at corner points.
(B) The conditions \(x \ge 0, y \ge 0\) are called non-negative restrictions. — Correct. This is the standard name.
(C) If the feasible region is unbounded, then Z cannot have a maximum value. — Incorrect. \(Z = -x - y\) on \(x + y \ge 2\) has maximum \(-2\).
(D) If the feasible region is empty, then the problem has no optimal solution. — Correct. There is no feasible point at all to be optimal.