The objective function of an LPP is $z = αx + βy, (α,β > 0)$ that has to be maximized/minimized subject to constraints $x + y ≤2, x≥0, y ≥0$, Then $\text{max (z)- min (z)}$ is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $2\, max\{α,β\}$
Given objective function:
$z = \alpha x + \beta y, \; (\alpha, \beta > 0)$
Subject to constraints:
$x + y \le 2, \; x \ge 0, \; y \ge 0$
The feasible region is bounded by the triangle with vertices:
$(0,0), (2,0), (0,2)$
Compute $z$ at each vertex:
At $(0,0)$: $z_1 = 0$
At $(2,0)$: $z_2 = 2\alpha$
At $(0,2)$: $z_3 = 2\beta$
Maximum value: $\max(2\alpha, 2\beta)$
Minimum value: $0$
Hence,
$\max(z) - \min(z) = 2 \max(\alpha, \beta)$
Therefore, $\max(z) - \min(z) = 2\max(\alpha, \beta)$