Consider the following statements about solving a linear programming problem (LPP) by the graphical method:
(A) The graphical method is used for an LPP with two decision variables.
(B) If the feasible region is bounded, the objective function has both a maximum and a minimum value, and each occurs at a corner point of the region.
(C) If the feasible region is unbounded, the objective function never has an optimal value.
(D) Every point on the boundary of the feasible region is a corner point.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
The graphical method draws the constraints in the xy-plane, so it works with two variables. A corner point is a point of the region where two boundary lines meet. On a bounded region both optimal values occur at corner points; on an unbounded region an optimal value may or may not exist.
For example, \(Z = x + y\) on the unbounded region \(x + y \ge 4\), \(x \ge 0\), \(y \ge 0\) has minimum 4 but no maximum.
So the method is for two variables, a bounded region always has both optimal values at corners, and an unbounded region may still have one.
Checking the statements
(A) The graphical method is used for an LPP with two decision variables. — Correct. The constraints are drawn as lines in the xy-plane.
(B) If the feasible region is bounded, the objective function has both a maximum and a minimum value, and each occurs at a corner point of the region. — Correct. This is the corner-point theorem for a bounded region.
(C) If the feasible region is unbounded, the objective function never has an optimal value. — Incorrect. \(Z = x + y\) on \(x + y \ge 4\) has minimum 4.
(D) Every point on the boundary of the feasible region is a corner point. — Incorrect. Only points where two boundary lines meet are corners; (2, 0) on the edge from (0, 0) to (4, 0) is not.