List of top Linear Programming Questions on Optimality Conditions and Duality asked in GATE MA

Let \( \alpha, \beta \in \mathbb{R} \). If \( (4, 0, 2, \beta) \) is an optimal solution of the Linear Programming Problem:
\[ \text{minimize} \quad x_1 + 3x_2 + 2x_3 - \alpha x_4 \]
subject to
\[ 4x_1 + x_2 + x_3 = 18 \]
\[ -3x_1 + 2x_3 + x_4 = 2 \]
\[ x_1, x_2, x_3, x_4 \geq 0, \]
then the maximum value of \( 22(\alpha + \beta) \) is equal to ______.