The provided image contains two mathematical problems involving linear programming and optimization. I will address both parts as follows:

1. Milk Plant Optimization Problem:

a. Formulate the problem as a Linear Programming (LP) problem:

Define the variables:

• $x_A$: Number of units of product A.
• $x_B$: Number of units of product B.
• $x_C$: Number of units of product C.

Objective Function: Maximize profit: $Z = 3x_A + 4x_B + x_C$

Subject to the machine constraints:

1. Machine $G$: $1x_A + 2x_B + 3x_C \leq 90$ (1 hour 30 minutes = 90 minutes)
2. Machine $H$: $2x_A + 1x_B + 1x_C \leq 80$ (1 hour 20 minutes = 80 minutes)
3. Machine $P$: $3x_A + 1x_B + 2x_C \leq 120$ (2 hours = 120 minutes)
4. Non-negativity: $x_A, x_B, x_C \geq 0$

b. Determine the optimal production quantities:

We need to solve the LP problem using methods such as the Simplex or graphical method.

c. Maximum profit:

This is the value of $Z$ at the optimal solution.

d. Dual Formulation:

The dual problem will assign a shadow price (dual variables) to each constraint related to the machine limits. Define $y_1, y_2, y_3$ as the dual variables for the constraints associated with machines $G, H, P$, respectively. The dual problem is then structured to minimize the total shadow prices subject to conditions derived from the primal problem.

2. Simplex Method Problem:

Problem Statement:

Minimize: $Z = 3x_1 + 2.5x_2$ Subject to the constraints:

1. $2x_1 + 4x_2 \geq 40$
2. $-5x_1 - 2x_2 \leq -50$ (Rewritten: $5x_1 + 2x_2 \geq 50$)
3. $x_1, x_2 \geq 0$

Steps:

1. Convert the inequalities to standard form.
2. Set up the initial Simplex tableau.
3. Apply the Simplex algorithm to find the optimal solution.

Next Steps:

Would you like me to:

1. Solve the primal linear programming problem for Question 1 using specific methods?
2. Formulate the dual problem explicitly for Question 1(d)?
3. Solve the minimization problem in Question 2 using the Simplex method?
4. Provide detailed explanations for setting up the Simplex tableau?
5. Explain the process of converting the dual problem for Question 1?

Tip:

For linear programming problems, always double-check the constraints' units (time, resources) to ensure consistency in formulation.