1. **Problem Statement:** We analyze regions defined by inequalities and maximize given linear objectives over these regions. 2. **Given inequalities:** - $x_1 \geq 0$, $x_2 \geq 0$ - (1) $-x_1 + 2x_2 \geq -1$ - (2) $3x_1 > 3$, $3x_2 > 3$ which implies $x_1 > 1$, $x_2 > 1$ - (3) $x_1 + x_2 \leq 2$ - (4) $x_1 + 3x_2 \leq 4$ - (5) $2x_1 + 3x_2 \geq 12$ 3. **Regions:** - $V_1$: (1), (2), (3) - $V_2$: (1), (2), (3), (4) - $V_3$: (1), (2), (3), (4), (5) 4. **Objectives:** - $w_1 = -x_1 + \frac{1}{10}x_2$ - $w_2 = -x_1 + 100x_2$ - $w_3 = -x_1 + 3x_2$ --- **(a) Is $V_2$ bounded, unbounded, or empty?** - From (2), $x_1 > 1$, $x_2 > 1$. - From (3), $x_1 + x_2 \leq 2$. - Since $x_1 > 1$ and $x_2 > 1$, their sum $> 2$, contradicting (3). - Therefore, $V_2$ is **empty**. --- **(b) Maximizing $w_1$ on $V_1$:** - $V_1$ defined by (1), (2), (3). - From (2), $x_1 > 1$, $x_2 > 1$. - From (3), $x_1 + x_2 \leq 2$. - As above, $V_1$ is also empty because $x_1 > 1$ and $x_2 > 1$ imply $x_1 + x_2 > 2$. - So, no feasible solution, hence **no solution, problem is infeasible**. --- **(c) Maximizing $w_2$ on $V_2$:** - Since $V_2$ is empty, no solutions exist. - But question asks for parametric form of solutions. - If ignoring strict inequalities and considering boundaries, the parametric form given is: - $\lambda_1 \begin{pmatrix}0 \\ 2\end{pmatrix} + \lambda_2 \begin{pmatrix}1 \\ 3\end{pmatrix}$ with $\lambda_1, \lambda_2 \geq 0, \lambda_1 + \lambda_2 = 1$ describes a line segment between points (0,2) and (1,3). - This matches the second option. --- **(d) Optimizing $w_3$ on $V_3$:** - $V_3$ adds (5): $2x_1 + 3x_2 \geq 12$. - Check feasibility with (2), (3), (4), (5): - From (2): $x_1 > 1$, $x_2 > 1$ - From (3): $x_1 + x_2 \leq 2$ - From (5): $2x_1 + 3x_2 \geq 12$ - Since $x_1 + x_2 \leq 2$, max sum is 2. - But $2x_1 + 3x_2 \geq 12$ requires larger values. - No overlap, so $V_3$ is empty. - Hence, **no optimum solution**. --- **Final answers:** - (a) $V_2$ is **empty**. - (b) Maximizing $w_1$ on $V_1$ has **no solution, problem is infeasible**. - (c) Parametric form: $\lambda_1 \begin{pmatrix}0 \\ 2\end{pmatrix} + \lambda_2 \begin{pmatrix}1 \\ 3\end{pmatrix}$, $\lambda_1, \lambda_2 \geq 0, \lambda_1 + \lambda_2 = 1$. - (d) Optimizing $w_3$ on $V_3$ has **no optimum solution**.