Bounded and Unbounded Feasible Regions

Identify whether the feasible region is enclosed. If bounded, apply the corner point table directly. If unbounded, compute candidate values at available vertices, then test whether the objective can improve further in the unbounded direction. State no finite optimum only when improvement is unlimited.

A graph is useful when deciding boundedness. The open direction of the feasible region should be visible, and arrows may be used to show that the region extends indefinitely.

Writing no optimum merely because the feasible region is unbounded is a wrong application of the condition.

A feasible region is bounded if it can be enclosed within a finite part of the plane; it is unbounded if it extends indefinitely in at least one direction.

A feasible region is bounded if it can be enclosed within a finite part of the plane; it is unbounded if it extends indefinitely in at least one direction. Condition for bounded region: feasible region is closed and enclosed by constraint boundaries. For unbounded region: corner point candidates must be checked against the direction in which the feasible region extends. If Z can improve indefinitely, the required optimum does not exist. The constraints x >= 0, y >= 0, and x + y >= 4 form an unbounded feasible region in the first quadrant outside the line x + y = 4. A minimum of x + y is 4 on the boundary, but a maximum of x + y does not exist because x and y can grow without bound.

In a bounded feasible region, every linear objective function that is evaluated over the region has its optimum at a corner point. In an unbounded feasible region, an optimum may or may not exist. If the objective value can keep increasing in a maximization problem, no finite maximum exists. If it can keep decreasing in a minimization problem, no finite minimum exists. Condition for bounded region: feasible region is closed and enclosed by constraint boundaries. For unbounded region: corner point candidates must be checked against the direction in which the feasible region extends. If Z can improve indefinitely, the required optimum does not exist. Minimize Z = x + y subject to x + y >= 6, x >= 0, y >= 0. The feasible region is unbounded in the first quadrant above the line x + y = 6. Boundary corner points are (6,0) and (0,6). Z(6,0)=6 and Z(0,6)=6. Since x + y cannot be less than 6 due to the constraint x + y >= 6, the minimum is 6 at every feasible point on the segment from (6,0) to (0,6). Asked through short-answer reasoning, assertion-reason items, or as the final check in a graphical LPP where the region is open. Writing no optimum merely because the feasible region is unbounded is a wrong application of the condition.