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The problem in the images you uploaded is a transportation problem, a type of linear programming problem where the objective is to minimize shipping costs while satisfying supply and demand constraints. Let's break down the information and structure a solution.

### Problem Summary:
- There are 3 plants (suppliers) and 3 distribution centers (demand points).
- The company wants to minimize the total shipping cost while meeting the demands of each distribution center.
- Supply and demand:
  - **Supply** (available units per month):
    - Houston: 850 units
    - Phoenix: 650 units
    - Memphis: 300 units
  - **Demand** (required units per month):
    - Dallas: 800 units
    - Atlanta: 600 units
    - Denver: 200 units

### Shipping Costs (per unit):
- From **Houston**:
  - To Dallas: $8
  - To Atlanta: $12
  - To Denver: $10
- From **Phoenix**:
  - To Dallas: $10
  - To Atlanta: $14
  - To Denver: $9
- From **Memphis**:
  - To Dallas: $11
  - To Atlanta: $8
  - To Denver: $12

### Objective:
- Determine how many units to ship from each plant to each regional center while minimizing the total cost.
  
### Steps to Solve:
1. **Formulate the decision variables**:
   Let:
   - $ x_{HD} $ = units shipped from Houston to Dallas
   - $ x_{HA} $ = units shipped from Houston to Atlanta
   - $ x_{HN} $ = units shipped from Houston to Denver
   - $ x_{PD} $ = units shipped from Phoenix to Dallas
   - $ x_{PA} $ = units shipped from Phoenix to Atlanta
   - $ x_{PN} $ = units shipped from Phoenix to Denver
   - $ x_{MD} $ = units shipped from Memphis to Dallas
   - $ x_{MA} $ = units shipped from Memphis to Atlanta
   - $ x_{MN} $ = units shipped from Memphis to Denver

2. **Set up the constraints**:
   - **Supply Constraints**:
     - $ x_{HD} + x_{HA} + x_{HN} \leq 850 $ (Houston supply)
     - $ x_{PD} + x_{PA} + x_{PN} \leq 650 $ (Phoenix supply)
     - $ x_{MD} + x_{MA} + x_{MN} \leq 300 $ (Memphis supply)
   - **Demand Constraints**:
     - $ x_{HD} + x_{PD} + x_{MD} = 800 $ (Dallas demand)
     - $ x_{HA} + x_{PA} + x_{MA} = 600 $ (Atlanta demand)
     - $ x_{HN} + x_{PN} + x_{MN} = 200 $ (Denver demand)

3. **Set up the objective function** (minimizing total cost):
   \[
   \text{Minimize } Z = 8x_{HD} + 12x_{HA} + 10x_{HN} + 10x_{PD} + 14x_{PA} + 9x_{PN} + 11x_{MD} + 8x_{MA} + 12x_{MN}
   \]

4. **Solve the system** using a linear programming method such as the **Simplex algorithm**.

I can help you calculate the solution step-by-step, either manually or using software like Python to solve the linear programming problem. Let me know how you'd like to proceed!

Would you like the detailed steps for solving this, or would you prefer I use a tool to calculate the exact solution? 

### Follow-up Questions:
1. How familiar are you with linear programming and optimization problems?
2. Have you used the Simplex method before?
3. Would you like an explanation of how to set up and solve these types of problems in Excel?
4. Do you need clarification on the shipping cost setup?
5. Are you interested in learning how these problems are typically solved in industries?

### Tip:
In transportation problems, ensuring that supply equals demand (or setting up a dummy variable if they're unequal) is crucial to solving the problem effectively.

An air-conditioning manufacturer produces room air conditioners at plants in Houston, Phoenix, and Memphis. They are sent to regional distributors in Dallas, Atlanta, and Denver. The shipping costs vary, and the company would like to find the least-cost way to meet the demands at each of the distribution centers. Dallas needs to receive 800 air conditioners per month, Atlanta needs 600, and Denver needs 200. Houston has 850 air conditioners available each month, Phoenix has 650, and Memphis has 300. Shipping costs per unit are provided in the problem. How many units should be shipped from each plant to each regional distribution center? What is the total cost for this?

Solve an air-conditioning manufacturer's transportation problem by minimizing shipping costs across multiple plants and distribution centers. Learn how to apply linear programming, including setting up supply and demand constraints and calculating the total cost of shipping. Ideal for students in advanced high school or college.

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