Consider the following minimum problem: Minimize: \[ C=2 x_{1}+x_{2}+x_{3} \] Subject to the constraints: \[ \left\{\begin{array}{l} 2 x_{1}+x_{2}+x_{3} \geq 4 \\ x_{1}+2 x_{2}+x_{3} \geq 6 \\ x_{1} \geq 0 \\ x_{2} \geq 0 \\ x_{3} \geq 0 \end{array}\right. \] Write the dual problem for the above minimum problem by selecting the appropriate nur each blank box shown below (Do not solve the dual problem). \[ P=[\text { Select }] \quad \hat{\star} y_{1}+[\text { Select }] \quad \hat{\star} y_{2} \] [ Select ] \( y_{1}+ \) \( \square \) [ Select ] \( y_{2} \leq 2 \) [ Select ] \( y_{1}+ \) \( \square \) [Select ] \( y_{2} \leq 1 \) [ Select ] \( y_{1}+ \) \( \square \) [Select ] \( y_{2} \leq 1 \) \[ y_{1} \geq 0 \quad y_{2} \geq 0 \]

To formulate the dual problem from the given primal problem, we need to follow the rules of duality in linear programming. The primal problem is: Minimize: \[ C = 2x_1 + x_2 + x_3 \] Subject to the constraints: \[ \begin{align*} 1. & \quad 2x_1 + x_2 + x_3 \geq 4 \quad \text{(Constraint 1)} \\ 2. & \quad x_1 + 2x_2 + x_3 \geq 6 \quad \text{(Constraint 2)} \\ 3. & \quad x_1 \geq 0 \\ 4. & \quad x_2 \geq 0 \\ 5. & \quad x_3 \geq 0 \end{align*} \] ### Step 1: Identify the dual variables Let \( y_1 \) and \( y_2 \) be the dual variables corresponding to the two constraints in the primal problem. ### Step 2: Write the dual objective function The dual problem will maximize the right-hand side of the primal constraints, which are 4 and 6. Therefore, the dual objective function is: \[ \text{Maximize } 4y_1 + 6y_2 \] ### Step 3: Write the dual constraints The coefficients of \( x_1, x_2, \) and \( x_3 \) in the primal objective function will form the right-hand side of the dual constraints. 1. For \( x_1 \): \[ 2y_1 + y_2 \leq 2 \] 2. For \( x_2 \): \[ y_1 + 2y_2 \leq 1 \] 3. For \( x_3 \): \[ y_1 + y_2 \leq 1 \] ### Step 4: Non-negativity constraints The dual variables must also be non-negative: \[ y_1 \geq 0, \quad y_2 \geq 0 \] ### Final Dual Problem Putting it all together, the dual problem can be written as: \[ D: \quad \text{Maximize } 4y_1 + 6y_2 \] Subject to: \[ \begin{align*} 2y_1 + y_2 & \leq 2 \\ y_1 + 2y_2 & \leq 1 \\ y_1 + y_2 & \leq 1 \\ y_1 & \geq 0 \\ y_2 & \geq 0 \end{align*} \] ### Filling in the blanks: Now, we can fill in the blanks as follows: \[ P=[\text{Maximize }] \quad 4y_{1}+[\text{Maximize }] \quad 6y_{2} \] \[ 2y_{1}+ \quad [\text{Select }] \quad y_{2} \leq 2 \] \[ y_{1}+ \quad [\text{Select }] \quad 2y_{2} \leq 1 \] \[ y_{1}+ \quad [\text{Select }] \quad y_{2} \leq 1 \] \[ y_{1} \geq 0 \quad y_{2} \geq 0 \] This completes the formulation of the dual problem.