medium difficulty10 min

Bounded and Unbounded Feasible Regions

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.

In one line: Unbounded means check direction, not automatically no answer.

Practice This Concept

Understand it

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.

Example

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.

Formula and rule

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.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Minimize Z = x + y subject to x + y ≥ 6, x ≥ 0, y ≥ 0.

1The feasible region is unbounded in the first quadrant above the line x + y = 6.
2Boundary corner points are (6,0) and (0,6).
3Z(6,0)=6 and Z(0,6)=6.
4Since 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).

Method you can reuse

1Identify whether the feasible region is enclosed.
2If bounded, apply the corner point table directly.
3If unbounded, compute candidate values at available vertices, then test whether the objective can improve further in the unbounded direction.
4State no finite optimum only when improvement is unlimited.

Visual tip

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.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Can an unbounded feasible region have a finite minimum value? Give a brief answer.

Think first, then show answer

Yes. For example, with x ≥ 0, y ≥ 0, x + y ≥ 4, the objective Z = x + y has minimum value 4 on the line x + y = 4.

Common wrong answer

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

Don't mix this up

Students sometimes assume every unbounded feasible region has no solution. This is false because an unbounded region can still have a finite maximum or minimum depending on the objective function.

Exam tip

For an unbounded region, do not stop after listing corner values. Check whether better objective values are possible in the open direction of the feasible region.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Asked through short-answer reasoning, assertion-reason items, or as the final check in a graphical LPP where the region is open.

1
Enough for 1 mark
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.
2
Add for 2 marks
Example: 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.
3
Add for 3 marks
In a bounded feasible region, every linear objective function that is evaluated over the region has its optimum at a corner point.

MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results