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Question
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.
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Solution
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 equation (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
| Corner points | Z = 4x + y | |
| A(4, 0) | Z = 4(4) + (0) = 16 | |
| B(2, 1) | Z = 4(2) + (1) = 9 | |
| C(0, 3) | Z = 4(0) + (3) = 3 | ← Minimum |
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.
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