Consider the following primal linear programming problem.

Minimize z = 2x_{1} + 3x_{2} + 5x_{3} + 6x_{4} Subject to with x_{1} + 2x_{2} + 3x_{3} + x_{4} >= 2 - 2x_{1} + x_{2} - x_{3} + 3x_{4} <= - 3 x_{1}, x_{2}, x_{3}, x_{4} >= 0


 Utilize the duality principle to find the optimum solution of the primal problem.

To solve this problem using **duality**, we will first express the **primal problem** and then formulate its **dual**. Once we have the dual problem, we will solve it to find the optimal solution for the primal.

### Primal Problem

The primal problem is given as:

\[
\text{Minimize } z = 2x_1 + 3x_2 + 5x_3 + 6x_4
\]

Subject to:

\[
\begin{aligned}
1. & \quad x_1 + 2x_2 + 3x_3 + x_4 \geq 2 \\
2. & \quad -2x_1 + x_2 - x_3 + 3x_4 \leq -3 \\
3. & \quad x_1, x_2, x_3, x_4 \geq 0
\end{aligned}
\]

### Step 1: Write the Dual Problem

To form the dual, we first need to recognize that the primal problem is a **minimization problem** with inequality constraints. For each constraint, we associate a **dual variable**.

- For the first constraint \(x_1 + 2x_2 + 3x_3 + x_4 \geq 2\), we introduce a dual variable \(y_1 \geq 0\), corresponding to a "greater-than-or-equal-to" constraint.
- For the second constraint \(-2x_1 + x_2 - x_3 + 3x_4 \leq -3\), we introduce a dual variable \(y_2 \geq 0\), corresponding to a "less-than-or-equal-to" constraint.

#### The dual problem is a **maximization** problem.

The dual objective function is formed by the right-hand sides of the primal constraints, weighted by their corresponding dual variables. Thus, the dual objective is:

\[
\text{Maximize } w = 2y_1 - 3y_2
\]

#### The dual constraints are formed by the primal variables' coefficients. The primal variables must satisfy the dual constraints. For each \(x_i\), we form a constraint using the coefficients from the primal objective function and the primal constraints.

The dual constraints are:

\[
\begin{aligned}
1. & \quad y_1(-2) + y_2(-2) \leq 2 \quad \text{(from }x_1\text{'s coefficients in the primal)} \\
2. & \quad y_1(3) + y_2(-1) \leq 3x steps as ap

Consider the following primal linear programming problem. Minimize z = 2x_{1} + 3x_{2} + 5x_{3} + 6x_{4} Subject to with x_{1} + 2x_{2} + 3x_{3} + x_{4} >= 2 - 2x_{1} + x_{2} - x_{3} + 3x_{4} <= - 3 x_{1}, x_{2}, x_{3}, x_{4} >= 0. Utilize the duality principle to find the optimum solution of the primal problem.

Solve a linear programming problem using the duality principle. Minimize z = 2x_1 + 3x_2 + 5x_3 + 6x_4 subject to constraints, and utilize duality to find the optimal solution. This detailed solution explores linear programming, optimization, and duality concepts, suitable for students in Grades 11-12 or college.

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