Could you please help me to solve those 2 questions? The first one should be talking something in general, about the theory, not solve a specific question or LLP. **Primal and Dual Infeasibility Analysis**In linear programming, we often deal with primal and dual problem formulations. Given a primal problem in standard form, with all right-hand side constraints being positive, it’s crucial to analyze the implications of its dual problem being infeasible.**Problem Statement:**- We have a primal and its corresponding dual.- The primal is in standard form with the positive right-hand side for all constraints.- The dual is infeasible.**Analysis:**If the dual is infeasible, it suggests that there is no solution that satisfies all the dual constraints. According to the strong duality theorem, if a primal linear programming problem has an optimal solution, then its dual must also have an optimal solution. Therefore, if the dual is infeasible, the primal problem must be unbounded.**Conclusion:**For a feasible primal linear programming problem with positive constraint values, the infeasibility of the dual indicates the primal problem is unbounded. This means the objective function of the primal problem can be increased indefinitely without violating any constraints.**Implication:**Understand that dual infeasibility provides critical insights into the nature of the primal problem, specifically pointing towards its unboundedness under given conditions. Consider the following Linear Programming Problem (LPP):\[\begin{align*}\text{max} \ & 3x_1 + 2x_2 \\\text{s.t.} \ & x_1 + x_2 \leq 2 \\& 2x_1 - x_2 \leq 3 \\& x_1, x_2 \geq 0\end{align*}\]Suppose the z-line optimal simplex table of this problem is $ z + \frac{7}{3}w_1 + \frac{1}{3}w_2 = ? $. Find the optimal value for the problem without solving it or its dual. Show your work.

Answered: We have a primal and its corresponding…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Could you please help me to solve those 2 questions? The first one should be talking something in general, about the theory, not solve a specific question or LLP.

**Primal and Dual Infeasibility Analysis**

In linear programming, we often deal with primal and dual problem formulations. Given a primal problem in standard form, with all right-hand side constraints being positive, it’s crucial to analyze the implications of its dual problem being infeasible.

**Problem Statement:**
- We have a primal and its corresponding dual.
- The primal is in standard form with the positive right-hand side for all constraints.
- The dual is infeasible.

**Analysis:**
If the dual is infeasible, it suggests that there is no solution that satisfies all the dual constraints. According to the strong duality theorem, if a primal linear programming problem has an optimal solution, then its dual must also have an optimal solution. Therefore, if the dual is infeasible, the primal problem must be unbounded.

**Conclusion:**
For a feasible primal linear programming problem with positive constraint values, the infeasibility of the dual indicates the primal problem is unbounded. This means the objective function of the primal problem can be increased indefinitely without violating any constraints.

**Implication:**
Understand that dual infeasibility provides critical insights into the nature of the primal problem, specifically pointing towards its unboundedness under given conditions.

Consider the following Linear Programming Problem (LPP):

\[
\begin{align*}
\text{max} \ & 3x_1 + 2x_2 \\
\text{s.t.} \ & x_1 + x_2 \leq 2 \\
& 2x_1 - x_2 \leq 3 \\
& x_1, x_2 \geq 0
\end{align*}
\]

Suppose the z-line optimal simplex table of this problem is $ z + \frac{7}{3}w_1 + \frac{1}{3}w_2 = ? $. Find the optimal value for the problem without solving it or its dual. Show your work.

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We have a primal and its corresponding dual. The primal is in standard form and its right hand side on all the constraints is positive. The dual is infeasible. We have to explain about the primal.

The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However in general the optimal values of the primal and dual problems need not be equal. their difference is called the duality gap.

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