Q.A linear programming problem (LPP) along with the graph of its constraints is shown below. The corresponding objective function is: $Z=18x+10y$, which has to be minimized. The smallest value of the objective function $Z$ is $134$ and is obtained at the corner point $(3,8)$. (Note: The figure is not to scale.) The optimal solution of the above linear programming problem __________.
(A) does not exist as the feasible region is unbounded.
(B) does not exist as the inequality $18x+10y<134$ does not have any point in common with the feasible region.
(C) exists as the inequality $18x+10y<134$ has infinitely many points in common with the feasible region.
(D) exists as the inequality $18x+10y<134$ does not have any point in common with the feasible region.

Concept: Unbounded Feasible Region — when the feasible region is unbounded, a minimum may still exist if the half-plane of values smaller than the optimum has no intersection with the feasible region.

Reasoning:

1. The feasible region is unbounded (extends infinitely in some direction), so we must check whether $Z$ can go below 134.
2. The corner point $(3,8)$ gives $Z = 18(3) + 10(8) = 54 + 80 = 134$, stated as the smallest value. …

When the feasible region of an LPP is unbounded, the minimum (or maximum) value of the objective function may or may not exist. The key test is this: if you can keep reducing $Z$ indefinitely while staying inside the feasible region, then no optimal solution exists. But if there is a lower bound — a value below which no feasible point lies — then the minimum exists and is attained at a corner point.

In this problem, we are told that the smallest value of $Z$ is $134$, achieved at $(3,8)$. The question is whether this is truly the minimum, or whether we could go even lower.

1. Understand the geometry of the test.

   Consider the inequality $18x + 10y < 134$. This represents the open half-plane strictly below the line $18x+10y = 134$ (the line passing through the optimal corner). If any point in the feasible region satisfies this inequality, then we can get a value of $Z$ smaller than $134$ — meaning the claimed minimum is not actually the minimum.

   Conversely, if no point in the feasible region satisfies $18x+10y < 134$, then $134$ is indeed the smallest possible value, and the optimal solution exists.

2. Check the given information.

   The problem states that the smallest value of $Z$ is $134$ and is obtained at $(3,8)$. This is a fact provided to us. The graph (though not to scale) shows an unbounded feasible region. The only way this fact can be true is if the half-plane $18x+10y < 134$ lies entirely outside the feasible region.

3. Interpret the options.

   • Option (A) says the optimal solution does not exist because the region is unbounded. This is false — an unbounded region can still have a minimum if the objective function is bounded below within it.
   • Option (B) says the solution does not exist because $18x+10y<134$ has no point in common with the feasible region. This is self-contradictory: if that half-plane has no common point, then $134$ is the minimum, so the solution does exist. …

Method: Existence Test for an Optimum on an Unbounded Region

Use this when the feasible region is unbounded and you must decide whether a candidate optimum (found at a corner) is genuinely attained.

Steps

Step 1: Evaluate $Z$ at the corner points to get a candidate.

Compute $Z=ax+by$ at each vertex. For a minimisation, let $m$ be the smallest corner value (for a maximisation, let $M$ be the largest).

Step 2: Draw the "better-than-candidate" open half-plane.

Through the optimal corner draw the objective line $ax+by=m$. The region that would beat the candidate is the open half-plane

Step 3: Check for common points. …

Common Mistakes

Mistake 1: Assuming an unbounded region has no optimum.

Why it's wrong: unboundedness only threatens the optimum; here the minimum $134$ genuinely exists because $Z$ cannot go lower within the region. Choosing option (A) on the reflex "unbounded ⇒ no solution" is the error. Correct approach: run the existence test on the open half-plane below the optimum.

Mistake 2: Confusing the two "no common point" options. …