In reference to linear programming problem (LPP), which of the following statement is NOT correct?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → If the feasible region for an LPP is unbounded, then the optimal value of the objective function does not exist.
An unbounded feasible region does not by itself rule out an optimal value. For example, minimising \( z=x+y \) subject to \( x+y\ge 2,\ x,y\ge 0 \) has an unbounded region, yet z has a minimum value 2. The optimal value may or may not exist, and this must be checked with the half-plane test. So the statement in Option (2) is the one that is not correct.
Option (1) In a LPP, linear inequalities on the variables are called linear constraints. — Correct. This is the definition of the linear constraints.
Option (2) If the feasible region for an LPP is unbounded, then the optimal value of the objective function does not exist. — Incorrect. Minimising \( z=x+y \) subject to \( x+y\ge 2,\ x,y\ge 0 \) has an unbounded region, yet the minimum value 2 exists.
Option (3) A corner point of a feasible region is a point which is the intersection of two boundary lines. — Correct. Every corner point is a vertex where two boundary lines meet.
Option (4) In a LPP, if the objective function has the same maximum value on two corner points of the feasible region, then every point on the line segment joining these two points give the same maximum value. — Correct. If z takes the same maximum value M at two corner points, then at every point of the segment joining them z is a weighted average of the two equal values, so it is also M.