Question

Let \( y = (\alpha, -1)^T \), where \( \alpha \in \mathbb{R} \), be a feasible solution for the dual problem of the linear programming problem
Maximize: \( 5x_1 + 12x_2 \)
subject to: \[ x_1 + 2x_2 + x_3 \leq 10 \] \[ 2x_1 - x_2 + 3x_3 = 8 \] \[ x_1, x_2, x_3 \geq 0 \] Which of the following statements is TRUE?

• A. \( \alpha<3 \)
• B. \( 3 \leq \alpha<5.5 \)
• C. \( 5.5 \leq \alpha<7 \)
• D. \( \alpha \geq 7 \)

Hint:

In linear programming dual problems, feasibility constraints on the dual variables help determine valid ranges for parameters such as \( \alpha \) by ensuring the solution satisfies all necessary conditions.

Answer 1

The Correct Option is D

We are given the dual problem of a linear programming problem with a feasible solution \( y = (\alpha, -1)^T \). To determine the range of \( \alpha \), we analyze the constraints and objective function. Step 1: Analyze the Dual Feasibility
In the dual linear programming problem, the feasible solution must satisfy the dual constraints. In particular, the feasibility of \( y = (\alpha, -1)^T \) will be determined by the conditions on the dual variables and the relationships between them. Step 2: Determine the Conditions on \( \alpha \)
The relationship between \( \alpha \) and the dual constraints suggests that \( \alpha \geq 7 \) for the solution to remain feasible. This is the condition that satisfies all dual constraints and ensures that the dual solution remains valid. Step 3: Conclusion
Thus, the correct answer is \( \alpha \geq 7 \). Final Answer:
\[ \boxed{(D)} \]