Question

find the dual problems for the following primal problem?
maximize z=2x1+10x2 (2)minimize z=5x1+7x2
subject to subjcet
2x1+5x2</=16 x1+2x2</=20
6x1</=30 3x1+x2>/=15
x1,x2>/=0 4x1+3x2</=60
x1,x2>/=0

Answer 1

To find the dual problems for the given primal problems, we will follow the rules for constructing duals from primal linear programming problems.

**Primal Problem 1:**
\[
\text{Maximize } z = 2x_1 + 10x_2
\]
Subject to:
\[
\begin{align*}
2x_1 + 5x_2 & \leq 16 \quad (1) \\
6x_1 & \leq 30 \quad (2) \\
x_1, x_2 & \geq 0
\end{align*}
\]

**Step 1: Identify the coefficients for dual formulation.**
- The objective function has coefficients \(c_1 = 2\) and \(c_2 = 10\).
- The constraints have the right-hand side (RHS) values \(b_1 = 16\) and \(b_2 = 30\).
- The coefficients of the primal constraints form the matrix:
\[
\begin{bmatrix}
2 & 5 \\
6 & 0
\end{bmatrix}
\]

**Dual Problem**
\[
\text{Minimize } w = 16y_1 + 30y_2
\]
Subject to:
\[
\begin{align*}
2y_1 + 6y_2 & \geq 2 \quad (1) \\
5y_1 & \geq 10 \quad (2) \\
y_1, y_2 & \geq 0
\end{align*}
\]

---

**Primal Problem 2:**
\[
\text{Minimize } z = 5x_1 + 7x_2
\]
Subject to:
\[
\begin{align*}
x_1 + 2x_2 & \leq 20 \quad (1) \\
3x_1 + x_2 & \geq 15 \quad (2) \\
4x_1 + 3x_2 & \leq 60 \quad (3) \\
x_1, x_2 & \geq 0
\end{align*}
\]

**Step 2: Identify the coefficients for dual formulation.**
- For the objective function, we have \(c_1 = 5\) and \(c_2 = 7\).
- The constraints have the RHS values \(b_1 = 20\), \(b_2 = 15\), and \(b_3 = 60\).
- The coefficients of the primal constraints form the matrix:
\[
\begin{bmatrix}
1 & 2 \\
3 & 1 \\
4 & 3
\end{bmatrix}
\]

**Dual Problem**
\[
\text{Maximize } w = 20y_1 + 15y_2 + 60y_3
\]
Subject to:
\[
\begin{align*}
y_1 + 3y_2 + 4y_3 & \leq 5 \quad (1) \\
2y_1 + y_2 + 3y_3 & \leq 7 \quad (2) \\
y_1 & \geq 0 \\
y_2 & \text{ free} \\
y_3 & \geq 0
\end{align*}
\]

---

To summarize, the dual problems for the given primal problems are:

**Dual Problem for Primal Problem 1:**
\[
\text{Minimize } w = 16y_1 + 30y_2
\]
Subject to:
\[
\begin{align*}
2y_1 + 6y_2 & \geq 2 \\
5y_1 & \geq 10 \\
y_1, y_2 & \geq 0
\end{align*}
\]

**Dual Problem for Primal Problem 2:**
\[
\text{Maximize } w = 20y_1 + 15y_2 + 60y_3
\]
Subject to:
\[
\begin{align*}
y_1 + 3y_2 + 4y_3 & \leq 5 \\
2y_1 + y_2 + 3y_3 & \leq 7 \\
y_1 & \geq 0 \\
y_2 & \text{ free} \\
y_3 & \geq 0
\end{align*}
\]