Corner points of the feasible region for an LPP are (1, 1), (2, 0), (3, 1), \(\left(\displaystyle\frac{3}{2}, 4\right)\), and (0, 5). Let \(z = px + 4y\), \(p \gt 0\) be the objective function. If maximum of \(z\) occurs at \(\left(\displaystyle\frac{3}{2}, 4\right)\) and (3, 1), then the value of p is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 8
If the maximum occurs at two corners, z has the same value at both corners:
\( \displaystyle z\left(\frac{3}{2}, 4\right) = \frac{3p}{2} + 16 \)
\(z(3, 1) = 3p + 4\)
\( \displaystyle \frac{3p}{2} + 16 = 3p + 4 \)
\( \displaystyle 12 = \frac{3p}{2} \)
\(p = 8\)
Check with z = 8x + 4y at the other corners:
\(z(1, 1) = 12,\quad z(2, 0) = 16,\quad z(0, 5) = 20\)
These are all less than \(z(3, 1) = z\left(\frac{3}{2}, 4\right) = 28\), so 28 is indeed the maximum.