Which of the following is incorrect about the Linear Programming Problem (LPP)?

Statement 1 is correct.

If the feasible region is bounded (a closed shape like a triangle or quadrilateral), the objective function has both a maximum and minimum value. These values occur at the corner points of the region. A finite area guarantees both a highest and lowest point.

Statement 2 is correct.

An LPP can have no solution when constraints contradict each other (such as $x > 5$ and $x < 3$ simultaneously).

An LPP can have multiple optimal solutions when the objective function line is parallel to a constraint line. If two corner points both give the optimal answer, then every point on the line segment joining them is also optimal.

Statement 3 is correct.

When the objective function is parallel to a boundary line, two corner points can both be optimal. If point $A$ gives the optimal value and point $B$ gives the same optimal value, then any point between $A$ and $B$ also gives that optimal value. This occurs because the objective function is linear and changes at a constant rate.

Statement 4 is incorrect.

An unbounded region extends infinitely in some direction. This does not guarantee a minimum value exists.

Consider minimizing $Z = x + y$ subject to $x \geq 0, y \geq 0$ (first quadrant, unbounded).

Moving towards the origin: $(0.1, 0.1), (0.01, 0.01), (0.001, 0.001)...$

The value of $Z$ approaches $0$ but never reaches a minimum point. The minimum does not exist.

For an unbounded region, a minimum or maximum may or may not exist depending on the direction of the region and the objective function. In contrast, a bounded region always guarantees both minimum and maximum exist.

Therefore, Statement 4 is incorrect.