Consider the following \( \underline{\text { minimum }} \) 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 number for each blank box shown below (Do not solve the dual problem). \[ P=[\text { Select }] \quad \hat{v} y_{1}+[\text { Select }] \quad \hat{v} y_{2} \] [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. ### Primal Problem The primal problem is given as: Minimize: \[ C = 2x_1 + x_2 + x_3 \] Subject to the constraints: \[ \begin{align*} 1. & \quad 2x_1 + x_2 + x_3 \geq 4 \quad \text{(Constraint 1)} \\ 2. & \quad x_1 + 2x_2 + x_3 \geq 6 \quad \text{(Constraint 2)} \\ 3. & \quad x_1 \geq 0 \\ 4. & \quad x_2 \geq 0 \\ 5. & \quad x_3 \geq 0 \end{align*} \] ### Dual Problem The dual problem will have the following structure: Maximize: \[ D = 4y_1 + 6y_2 \] Subject to the constraints derived from the coefficients of the primal objective function and the right-hand side of the primal constraints: 1. From the first constraint \(2x_1 + x_2 + x_3\), we have: \[ y_1 + y_2 \leq 2 \] 2. From the second constraint \(x_1 + 2x_2 + x_3\), we have: \[ y_1 + 2y_2 \leq 1 \] 3. From the third constraint \(x_1\), we have: \[ y_1 + y_2 \leq 1 \] ### Final Dual Problem Putting it all together, the dual problem can be expressed as: Maximize: \[ D = 4y_1 + 6y_2 \] Subject to: \[ \begin{align*} y_1 + y_2 & \leq 2 \\ y_1 + 2y_2 & \leq 1 \\ y_1 + y_2 & \leq 1 \\ y_1 & \geq 0 \\ y_2 & \geq 0 \end{align*} \] ### Filling in the Blanks Now, we can fill in the blanks as follows: \[ P=[4] \quad \hat{v} y_{1}+[6] \quad \hat{v} y_{2} \] \[ [1] \quad y_{1}+ \quad [1] \quad y_{2} \leq 2 \] \[ [1] \quad y_{1}+ \quad [2] \quad y_{2} \leq 1 \] \[ [1] \quad y_{1}+ \quad [1] \quad y_{2} \leq 1 \] \[ y_{1} \geq 0 \quad y_{2} \geq 0 \] Thus, the selections are: - First blank: 4 - Second blank: 6 - Third blank: 1 - Fourth blank: 1 - Fifth blank: 1 - Sixth blank: 2 This completes the dual formulation.