Q.State whether the following statement is True or False: If the feasible region for a LPP is unbounded, maximum or minimum of the objective function $Z = ax + by$ may or may not exist.

Why this is true — the core idea

The key is the word bounded. A bounded feasible region means the entire region can be enclosed inside some large circle (or rectangle). It doesn't stretch off to infinity in any direction. When the region is bounded, the objective function $Z = ax + by$ — which is a linear function — is continuous. A famous theorem from calculus (the Extreme Value Theorem) says that a continuous function on a closed and bounded set always attains both a maximum and a minimum value.

In linear programming, the feasible region is always closed (it includes its boundary lines). So if it's also bounded, both extreme values exist.

Step-by-step reasoning

1. What does "bounded" mean here?

   A region is bounded if you can draw a big enough circle that completely covers it. For example, a triangle or a pentagon is bounded. A region that goes on forever (like a half-plane or an unbounded ray) is unbounded.

2. What does the objective function look like?

   $Z = ax + by$ is a linear function. Its graph is a plane tilted in 3D. On the 2D feasible region, it produces a "height" at every point. As you move across the region, this height changes linearly.

3. What happens on a bounded region? …

Concept and Intuition

In Linear Programming, the feasible region is the set of all points that satisfy every constraint — including non-negativity conditions. If this region is empty, it means the constraints are contradictory; no point in the plane (or space) can meet all of them simultaneously. Naturally, if there’s no point to evaluate the objective function on, the problem cannot have a solution — neither a maximum nor a minimum. This is a fundamental logical consequence, not a subtle technicality.

The Reason (R) simply defines what a feasible region is. That definition is the very basis for why an empty region leads to no solution. So (R) is not just true — it’s the direct explanation of (A).

Step-by-step reasoning

1. Understand Assertion (A):

   “If the feasible region is empty in an LPP, then the LPP has no solution.”

   This is a standard result. An LPP’s solution (optimal value) must lie within the feasible region. If the region has no points, there is no candidate for an optimal solution. Hence the LPP is infeasible — it has no solution.

2. Understand Reason (R):

   “Feasible region is the region in which all the constraints are satisfied.”

   This is the textbook definition. Every constraint — including non-negativity — must hold. The feasible region is the intersection of all half-planes (or half-spaces) defined by the constraints.

3. Check the truth of (A):

   True. An empty feasible region implies no point satisfies all constraints, so no feasible solution exists. Therefore the LPP has no solution.

4. Check the truth of (R):

   True. The definition is correct.

5. Check if (R) correctly explains (A): …

Understanding the Feasible Region

When you set up a linear programming problem, you're essentially drawing boundaries on a coordinate plane. Each constraint — whether it's an inequality like $2x + 3y \leq 12$ or a non-negativity condition like $x \geq 0$ — carves out a half-plane. The region where all these half-planes overlap is special: it contains every point that respects every single rule you've laid down.

This overlap region has a name that captures its essence: it's the set of all feasible solutions, meaning solutions that are actually allowed by the problem's constraints. Not all of these solutions are optimal (that's what we're trying to find), but they're all valid candidates.

Why Each Term Means What It Does

Let me walk through what each option actually describes:

1. Feasible region (Option D): This is the technical term for the common region determined by all constraints. A point is feasible if and only if it satisfies every constraint simultaneously. This is the fundamental concept in linear programming — before we can maximize or minimize an objective function, we need to know where we're allowed to look.

2. Bounded vs. Unbounded (Options A and C): These are properties that a feasible region might have, not names for the region itself. A feasible region is bounded if it fits inside some large enough circle — think of a triangle or polygon. It's unbounded if it stretches infinitely in some direction — imagine the region $x \geq 0, y \geq 0, x + y \geq 5$, which extends forever to the upper-right. Both types are still called feasible regions; bounded/unbounded just describes their shape. …