1. The Surfs-Up Company manufactures sailboats. The sailboats are made at plants I and II; the outputs of each plant are at most 400 and 650 per month respectively. The sailboats are shipped to three marinas: A, B, and C. In order to meet customer demands, marina A must receive exactly 300 boats, and marinas B and C at least 100 and 350 boats respectively, per month. Shipping costs from plant I to marinas A, B, and C are $50, $60, and $70 per boat, respectively; and the shipping costs from plant II to marinas A, B, and C are $75, $65, and $55 per boat, respectively. Set-up the linear programming structure to determine if the Surfs-Up Co. satisfies the requirements of the marinas while keeping its shipping costs to a minimum? A. Define the fewest number of variables necessary to solve this application. B. Objective Function C. Constraints
1. The Surfs-Up Company manufactures sailboats. The sailboats are made at plants I and II; the outputs of each plant are at most 400 and 650 per month respectively. The sailboats are shipped to three marinas: A, B, and C. In order to meet customer demands, marina A must receive exactly 300 boats, and marinas B and C at least 100 and 350 boats respectively, per month. Shipping costs from plant I to marinas A, B, and C are $50, $60, and $70 per boat, respectively; and the shipping costs from plant II to marinas A, B, and C are $75, $65, and $55 per boat, respectively. Set-up the linear programming structure to determine if the Surfs-Up Co. satisfies the requirements of the marinas while keeping its shipping costs to a minimum? A. Define the fewest number of variables necessary to solve this application. B. Objective Function C. Constraints
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
1. The Surfs-Up Company manufactures sailboats. The sailboats are made at plants I
and II; the outputs of each plant are at most 400 and 650 per month respectively. The
sailboats are shipped to three marinas: A, B, and C. In order to meet customer
demands, marina A must receive exactly 300 boats, and marinas B and C at least
100 and 350 boats respectively, per month. Shipping costs from plant I to marinas A,
B, and C are $50, $60, and $70 per boat, respectively; and the shipping costs from
plant II to marinas A, B, and C are $75, $65, and $55 per boat, respectively.
Set-up the linear programming structure to determine if the Surfs-Up Co.
satisfies the requirements of the marinas while keeping its shipping costs to a
minimum?
A. Define the fewest number of variables necessary to solve this application.
B. Objective Function C. Constraints
![## Linear Programming for Sailboat Distribution
### Problem Statement
The Surfs-Up Company manufactures sailboats. The sailboats are produced at two plants: Plant I and Plant II. The maximum outputs of each plant are 400 and 650 sailboats per month, respectively. These sailboats are distributed to three marinas: A, B, and C. The requirements for distribution are as follows:
- Marina A must receive exactly 300 boats.
- Marina B must receive at least 100 boats.
- Marina C must receive at least 350 boats.
### Shipping Costs
The shipping costs per boat from each plant to each marina are given below:
- **From Plant I:**
- To Marina A: $50
- To Marina B: $60
- To Marina C: $70
- **From Plant II:**
- To Marina A: $75
- To Marina B: $65
- To Marina C: $55
### Objective
The goal is to set up a linear programming structure to determine if the Surfs-Up Company can meet the marina requirements while minimizing shipping costs.
### Steps to Set Up Linear Programming Structure
#### Step A: Define the Fewest Number of Variables
Define variables to represent the number of boats shipped from each plant to each marina. Let:
- \( x_{IA} \) be the number of boats shipped from Plant I to Marina A.
- \( x_{IB} \) be the number of boats shipped from Plant I to Marina B.
- \( x_{IC} \) be the number of boats shipped from Plant I to Marina C.
- \( x_{IIA} \) be the number of boats shipped from Plant II to Marina A.
- \( x_{IIB} \) be the number of boats shipped from Plant II to Marina B.
- \( x_{IIC} \) be the number of boats shipped from Plant II to Marina C.
#### Step B: Objective Function
Formulate the objective function to minimize the total shipping cost:
\[ \text{Minimize } Z = 50x_{IA} + 60x_{IB} + 70x_{IC} + 75x_{IIA} + 65x_{IIB} + 55x_{IIC} \]
#### Step C: Constraints
Based on the requirements and capacities, establish the constraints:
1. **Demand Constraints for Marinas:**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff1799000-33ab-4b28-9bb2-f03cfa8defbf%2F201ce9aa-ace2-4f3e-8bde-b07200bc8813%2Fmmlt3n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Linear Programming for Sailboat Distribution
### Problem Statement
The Surfs-Up Company manufactures sailboats. The sailboats are produced at two plants: Plant I and Plant II. The maximum outputs of each plant are 400 and 650 sailboats per month, respectively. These sailboats are distributed to three marinas: A, B, and C. The requirements for distribution are as follows:
- Marina A must receive exactly 300 boats.
- Marina B must receive at least 100 boats.
- Marina C must receive at least 350 boats.
### Shipping Costs
The shipping costs per boat from each plant to each marina are given below:
- **From Plant I:**
- To Marina A: $50
- To Marina B: $60
- To Marina C: $70
- **From Plant II:**
- To Marina A: $75
- To Marina B: $65
- To Marina C: $55
### Objective
The goal is to set up a linear programming structure to determine if the Surfs-Up Company can meet the marina requirements while minimizing shipping costs.
### Steps to Set Up Linear Programming Structure
#### Step A: Define the Fewest Number of Variables
Define variables to represent the number of boats shipped from each plant to each marina. Let:
- \( x_{IA} \) be the number of boats shipped from Plant I to Marina A.
- \( x_{IB} \) be the number of boats shipped from Plant I to Marina B.
- \( x_{IC} \) be the number of boats shipped from Plant I to Marina C.
- \( x_{IIA} \) be the number of boats shipped from Plant II to Marina A.
- \( x_{IIB} \) be the number of boats shipped from Plant II to Marina B.
- \( x_{IIC} \) be the number of boats shipped from Plant II to Marina C.
#### Step B: Objective Function
Formulate the objective function to minimize the total shipping cost:
\[ \text{Minimize } Z = 50x_{IA} + 60x_{IB} + 70x_{IC} + 75x_{IIA} + 65x_{IIB} + 55x_{IIC} \]
#### Step C: Constraints
Based on the requirements and capacities, establish the constraints:
1. **Demand Constraints for Marinas:**
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