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

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### 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.

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### 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.

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### 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.

A milk plant manufactures three types of products A, B, and C, and sells each unit of them at a profit of $3, $4, and $1, respectively. Each product is processed on three machines G, H, and P. Type A requires one minute of processing time on G, two minutes on H, and three minutes on P. Type B requires two minutes on G, one minute on H, and one minute on P. Type C requires three minutes on G, one minute on H, and two minutes on P. Machine G is available for not more than one and a half hours, machine H is available for not more than an hour, and machine P can operate for not more than one hour and 20 minutes during any working day. Formulate the problem as a Linear Programming problem and solve for the optimal production quantities and maximum profit.

Solve a linear programming problem to maximize profit for a milk plant producing three products A, B, and C, with machine time constraints. Learn how to formulate and solve using the Simplex method.

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