The corner points of the bounded feasible region for an LPP are (0, 10), (3, 12), (6, 8), and (0, 15). The objective function is \(Z= \alpha x + \beta y\), where \(\alpha, \beta \gt 0\). If the maximum of Z occurs at the corner points (3, 12) and (6, 8), then the relationship between \(\alpha\) and \(\beta\) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\alpha =\frac{4}{3}\beta\)
For the maximum to occur at both (3, 12) and (6, 8), Z must take equal values there:
\(3\alpha + 12\beta = 6\alpha + 8\beta\)
\(4\beta = 3\alpha\)
\(\alpha = \frac{4}{3}\beta\)
Check that this common value is the largest: with \(\alpha = \frac{4}{3}\beta\),
\(Z(3, 12) = Z(6, 8) = 16\beta,\quad Z(0, 15) = 15\beta,\quad Z(0, 10) = 10\beta\)
So the maximum, \(16\beta\), occurs at (3, 12) and (6, 8).
Note: It seems that there is a mistake in the original NTA question, which gives the first corner as (0, 20). With that corner, \(\alpha = \frac{4}{3}\beta\) gives \(Z(0, 20) = 20\beta\), which is larger than \(16\beta\), so the maximum could never occur at (3, 12) and (6, 8). Also, (3, 12) lies inside the triangle formed by (0, 15), (0, 20) and (6, 8), so it could not be a corner. For the benefit of students and to ensure that the question is correct and has a single valid answer, the corner (0, 20) has been changed to (0, 10), so that Option (4) gives the maximum at the two points.