Consider the following network representation of a transportation problem. 60 30 Jefferson City Omaha 15 59 co 00 24 Des Moines Kansas City St. Louis 25 20 45 Supplies The supplies, demands, and transportation costs per unit are shown on the network. (a) Develop a linear programming model for this problem; be sure to define the variables in your Let Demands
Consider the following network representation of a transportation problem. 60 30 Jefferson City Omaha 15 59 co 00 24 Des Moines Kansas City St. Louis 25 20 45 Supplies The supplies, demands, and transportation costs per unit are shown on the network. (a) Develop a linear programming model for this problem; be sure to define the variables in your Let Demands
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
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![## Network Representation of a Transportation Problem
### Diagram Overview
The diagram depicts a network model for a transportation problem, which involves two supply nodes and three demand nodes:
- **Supply Nodes:**
- Jefferson City (Supply: 60 units)
- Omaha (Supply: 30 units)
- **Demand Nodes:**
- Des Moines (Demand: 25 units)
- Kansas City (Demand: 20 units)
- St. Louis (Demand: 45 units)
### Transportation Costs
The costs per unit of transportation between supply and demand nodes are labeled on the graph's connecting arrows:
- **Jefferson City to:**
- Des Moines: 15
- Kansas City: 9
- St. Louis: 8
- **Omaha to:**
- Des Moines: 8
- Kansas City: 11
- St. Louis: 24
### Linear Programming Model
The task is to develop a linear programming model to minimize transportation costs.
#### Variables Definition
- \( x_{11} \): Amount shipped from Jefferson City to Des Moines
- \( x_{12} \): Amount shipped from Jefferson City to Kansas City
- \( x_{13} \): Amount shipped from Jefferson City to St. Louis
- \( x_{21} \): Amount shipped from Omaha to Des Moines
- \( x_{22} \): Amount shipped from Omaha to Kansas City
- \( x_{23} \): Amount shipped from Omaha to St. Louis
#### Objective Function
Minimize the total transportation cost:
\[
15x_{11} + 9x_{12} + 8x_{13} + 8x_{21} + 11x_{22} + 24x_{23}
\]
This model helps determine the most cost-effective shipping strategy to meet demands while considering the available supply and associated transportation costs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862c5bc2-e09a-4feb-bd28-f7638a835094%2F5fa70692-6a31-40de-a853-8e9dc66af708%2Fgg1vo3v_processed.png&w=3840&q=75)
Transcribed Image Text:## Network Representation of a Transportation Problem
### Diagram Overview
The diagram depicts a network model for a transportation problem, which involves two supply nodes and three demand nodes:
- **Supply Nodes:**
- Jefferson City (Supply: 60 units)
- Omaha (Supply: 30 units)
- **Demand Nodes:**
- Des Moines (Demand: 25 units)
- Kansas City (Demand: 20 units)
- St. Louis (Demand: 45 units)
### Transportation Costs
The costs per unit of transportation between supply and demand nodes are labeled on the graph's connecting arrows:
- **Jefferson City to:**
- Des Moines: 15
- Kansas City: 9
- St. Louis: 8
- **Omaha to:**
- Des Moines: 8
- Kansas City: 11
- St. Louis: 24
### Linear Programming Model
The task is to develop a linear programming model to minimize transportation costs.
#### Variables Definition
- \( x_{11} \): Amount shipped from Jefferson City to Des Moines
- \( x_{12} \): Amount shipped from Jefferson City to Kansas City
- \( x_{13} \): Amount shipped from Jefferson City to St. Louis
- \( x_{21} \): Amount shipped from Omaha to Des Moines
- \( x_{22} \): Amount shipped from Omaha to Kansas City
- \( x_{23} \): Amount shipped from Omaha to St. Louis
#### Objective Function
Minimize the total transportation cost:
\[
15x_{11} + 9x_{12} + 8x_{13} + 8x_{21} + 11x_{22} + 24x_{23}
\]
This model helps determine the most cost-effective shipping strategy to meet demands while considering the available supply and associated transportation costs.
![### Linear Programming Problem for Shipping Optimization
#### Variables
- \( x_{11} \): Amount shipped from Jefferson City to Des Moines
- \( x_{12} \): Amount shipped from Jefferson City to Kansas City
- \( x_{13} \): Amount shipped from Jefferson City to St. Louis
- \( x_{21} \): Amount shipped from Omaha to Des Moines
- \( x_{22} \): Amount shipped from Omaha to Kansas City
- \( x_{23} \): Amount shipped from Omaha to St. Louis
#### Objective
Minimize the cost:
\[ 15x_{11} + 9x_{12} + 8x_{13} + 8x_{21} + 11x_{22} + 24x_{23} \]
#### Constraints
1. From Jefferson City:
\[ x_{11} + x_{12} + x_{13} \leq 60 \]
2. From Omaha:
\[ x_{21} + x_{22} + x_{23} \leq 30 \]
3. To Des Moines:
\[ x_{11} + x_{21} = 25 \]
4. To Kansas City:
\[ x_{12} + x_{22} = 20 \]
5. To St. Louis:
\[ x_{13} + x_{23} = 45 \]
6. Non-negativity:
\[ x_{11}, x_{12}, x_{13}, x_{21}, x_{22}, x_{23} \geq 0 \]
#### Task
(b) Solve the linear program to determine the optimal solution.
#### Input Table
The problem requires filling out the following table with optimal amounts and costs:
| | Amount | Cost |
|-----------------------|--------|-------|
| Jefferson City – Des Moines | | |
| Jefferson City – Kansas City | | |
| Jefferson City – St. Louis | | |
| Omaha – Des Moines | | |
| Omaha – Kansas City | | |
| Omaha – St. Louis | | |
| Total | | |
This problem involves setting up and solving a linear programming model to minimize shipping costs while meeting specific demand constraints.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862c5bc2-e09a-4feb-bd28-f7638a835094%2F5fa70692-6a31-40de-a853-8e9dc66af708%2Fglhi4wx_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Programming Problem for Shipping Optimization
#### Variables
- \( x_{11} \): Amount shipped from Jefferson City to Des Moines
- \( x_{12} \): Amount shipped from Jefferson City to Kansas City
- \( x_{13} \): Amount shipped from Jefferson City to St. Louis
- \( x_{21} \): Amount shipped from Omaha to Des Moines
- \( x_{22} \): Amount shipped from Omaha to Kansas City
- \( x_{23} \): Amount shipped from Omaha to St. Louis
#### Objective
Minimize the cost:
\[ 15x_{11} + 9x_{12} + 8x_{13} + 8x_{21} + 11x_{22} + 24x_{23} \]
#### Constraints
1. From Jefferson City:
\[ x_{11} + x_{12} + x_{13} \leq 60 \]
2. From Omaha:
\[ x_{21} + x_{22} + x_{23} \leq 30 \]
3. To Des Moines:
\[ x_{11} + x_{21} = 25 \]
4. To Kansas City:
\[ x_{12} + x_{22} = 20 \]
5. To St. Louis:
\[ x_{13} + x_{23} = 45 \]
6. Non-negativity:
\[ x_{11}, x_{12}, x_{13}, x_{21}, x_{22}, x_{23} \geq 0 \]
#### Task
(b) Solve the linear program to determine the optimal solution.
#### Input Table
The problem requires filling out the following table with optimal amounts and costs:
| | Amount | Cost |
|-----------------------|--------|-------|
| Jefferson City – Des Moines | | |
| Jefferson City – Kansas City | | |
| Jefferson City – St. Louis | | |
| Omaha – Des Moines | | |
| Omaha – Kansas City | | |
| Omaha – St. Louis | | |
| Total | | |
This problem involves setting up and solving a linear programming model to minimize shipping costs while meeting specific demand constraints.
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