The feasible region for a LPP is shown in figure. Evaluate $Z=4x+y$ at each of the corner points of this region. Find the minimum value of Z, if it exists.

3

The correct answer is Option (2) → 3

As per the given figure, ABC is the feasible region which is open unbounded.

Here, we have

$x+y=3$ …(i)

and $x+2y= 4$   ...(ii)

$Z=4x+y$

Solving eq. (i) and (ii), we get

$x=2$ and $y = 1$

So, the corner points are

$A(4,0), B(2, 1)$ and $C(0,3)$

Let us evaluate the value of Z

Now, the minimum value of Z is 3 at (0, 3) but since, the feasible region is open bounded so it may or may not be the minimum value of Z.

Therefore, to face such situation, we draw a graph of $4x + y < 3$ and check whether the resulting open half plane has no point in common with feasible region. Otherwise Z will have no minimum value. From the graph, we conclude that there is no common point with the feasible region.

Hence, Z has the minimum value 3 at (0, 3).