Questions: Consider the following minimum problem: Minimize: C=3x1+2x2 Subject to the constraints: - 2x1+x2 ≥ 6 - x1+x2 ≥ 4 - x1 ≥ 0 - x2 ≥ 0 Write the dual problem for the above minimum problem by selecting the appropriate number for each blank box shown below (Do not solve the dual problem). P= [Select ] y1+ [Select ] y2 [Select ] y1+ [Select] y2 ≤ 3 [Select ] y1+ [Select ] y2 ≤ 2 - y1 ≥ 0 ; y2 ≥ 0

Solution Steps

To write the dual of a linear programming problem, we need to follow these steps:

1. Identify the primal problem's objective function and constraints.
2. For each constraint in the primal problem, there will be a corresponding variable in the dual problem.
3. The coefficients of the objective function in the primal become the right-hand side constants in the dual constraints.
4. The right-hand side constants of the primal constraints become the coefficients in the dual objective function.
5. The inequality direction is reversed in the dual problem.

The primal problem is defined as follows: \[ \text{Minimize: } C = 3x_1 + 2x_2 \] Subject to the constraints: \[ \begin{align_} 2x_1 + x_2 & \geq 6 \\ x_1 + x_2 & \geq 4 \\ x_1 & \geq 0 \\ x_2 & \geq 0 \end{align_} \]

To formulate the dual problem, we identify the coefficients and constraints from the primal problem. The dual problem is given by: \[ P = 6y_1 + 4y_2 \] Subject to the constraints: \[ \begin{align_} 2y_1 + 1y_2 & \leq 3 \\ 1y_1 + 1y_2 & \leq 2 \end{align_} \] with the non-negativity conditions: \[ y_1 \geq 0, \quad y_2 \geq 0 \]

Final Answer