The objective function of an LPP is \(z = \alpha x + \beta y, (\alpha,\beta \gt 0)\) that has to be maximized/minimized subject to constraints \(x + y \le 2, x \ge 0, y \ge 0\), Then \(\text{max (z)- min (z)}\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(2\, \max\{\alpha,\beta\}\)
The feasible region is the triangle with corner points (0, 0), (2, 0) and (0, 2). The values of z there are
\((0, 0): z = 0\)
\((2, 0): z = 2\alpha\)
\((0, 2): z = 2\beta\)
Since \(\alpha, \beta \gt 0\), the smallest value is 0 and the largest is the bigger of \(2\alpha\) and \(2\beta\):
\(\max(z) = 2\max\{\alpha, \beta\},\quad \min(z) = 0\)
\(\max(z) - \min(z) = 2\max\{\alpha, \beta\}\)