Question

Consider the Linear Programming Problem \( P \): \[ \text{Maximize } 2x_1 + 3x_2 \]  subject to \[ 2x_1 + x_2 \leq 6, \] \[ -x_1 + x_2 \leq 1, \] \[ x_1 + x_2 \leq 3, \] \[ x_1 \geq 0 \text{ and } x_2 \geq 0. \] Then the optimal value of the dual of \( P \) is equal to \(\underline{\hspace{1cm}}\).

Hint:

To solve for the optimal value of the dual, form the dual problem by introducing variables for the primal constraints and solve the system of dual constraints.

Answer 1

Correct Answer: 8

The dual of the given linear programming problem can be formed by introducing dual variables for each constraint. Solving for the dual problem yields: \[ \text{Dual Problem:} \text{Minimize } 6y_1 + y_2 + 3y_3 \] subject to \[ 2y_1 - y_2 + y_3 = 2, \] \[ y_1 + y_2 + y_3 = 3, \] \[ y_1, y_2, y_3 \geq 0. \] Solving the dual gives the optimal value of the dual as \( \boxed{8} \).