The Ace Manufacturing Company has orders for three similar products. Product A Min B C 1 2 Machine 3 A Three machines are available for the manufacturing operations. All three machines can produce all the products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are shown below. B A C B с с 1 Product 12 3 Orders (units) 1,900 500 1,100 Capacity (units) 1,300 1,600 800 1 $1.00 $1.30 $1.10 $1.20 $1.40 $1.00 $0.90 $1.20 $1.20 (a) Develop the linear programming formulation of this problem. (Let XA1 be the number of units of product A produced by machine 1, X;; be the number of units of product i produced by machine j, etc.) Machine 2 3
The Ace Manufacturing Company has orders for three similar products. Product A Min B C 1 2 Machine 3 A Three machines are available for the manufacturing operations. All three machines can produce all the products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are shown below. B A C B с с 1 Product 12 3 Orders (units) 1,900 500 1,100 Capacity (units) 1,300 1,600 800 1 $1.00 $1.30 $1.10 $1.20 $1.40 $1.00 $0.90 $1.20 $1.20 (a) Develop the linear programming formulation of this problem. (Let XA1 be the number of units of product A produced by machine 1, X;; be the number of units of product i produced by machine j, etc.) Machine 2 3
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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![### The Ace Manufacturing Company Production Problem
The Ace Manufacturing Company has received orders for three similar products. The order quantities for these products are as follows:
| Product | Orders (units) |
|---------|----------------|
| A | 1,900 |
| B | 500 |
| C | 1,100 |
Three machines are available for the manufacturing operations. All machines can produce all the products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are shown below.
#### Machine Capacities
| Machine | Capacity (units) |
|---------|------------------|
| 1 | 1,300 |
| 2 | 1,600 |
| 3 | 800 |
#### Unit Costs for Products by Machine
| Product | Machine 1 | Machine 2 | Machine 3 |
|---------|-----------|-----------|-----------|
| A | $1.00 | $1.30 | $1.10 |
| B | $1.20 | $1.40 | $1.00 |
| C | $0.90 | $1.20 | $1.20 |
### Problem Formulation
(a) Develop the linear programming formulation of this problem.
- Let \( x_{A1} \) be the number of units of product A produced by machine 1.
- Let \( x_{ij} \) be the number of units of product \( i \) produced by machine \( j \), etc.
Minimize:
\[ \text{Min } \underline{\hspace{30px}} \]
Fill in your linear programming constraints and objective function in the space provided. This problem aims to minimize the total production cost while meeting the order demands and respecting the machine capacities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862c5bc2-e09a-4feb-bd28-f7638a835094%2Feafbaabb-fe5b-4fa8-811f-a3e9560215bb%2Fpfblqo8_processed.png&w=3840&q=75)
Transcribed Image Text:### The Ace Manufacturing Company Production Problem
The Ace Manufacturing Company has received orders for three similar products. The order quantities for these products are as follows:
| Product | Orders (units) |
|---------|----------------|
| A | 1,900 |
| B | 500 |
| C | 1,100 |
Three machines are available for the manufacturing operations. All machines can produce all the products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are shown below.
#### Machine Capacities
| Machine | Capacity (units) |
|---------|------------------|
| 1 | 1,300 |
| 2 | 1,600 |
| 3 | 800 |
#### Unit Costs for Products by Machine
| Product | Machine 1 | Machine 2 | Machine 3 |
|---------|-----------|-----------|-----------|
| A | $1.00 | $1.30 | $1.10 |
| B | $1.20 | $1.40 | $1.00 |
| C | $0.90 | $1.20 | $1.20 |
### Problem Formulation
(a) Develop the linear programming formulation of this problem.
- Let \( x_{A1} \) be the number of units of product A produced by machine 1.
- Let \( x_{ij} \) be the number of units of product \( i \) produced by machine \( j \), etc.
Minimize:
\[ \text{Min } \underline{\hspace{30px}} \]
Fill in your linear programming constraints and objective function in the space provided. This problem aims to minimize the total production cost while meeting the order demands and respecting the machine capacities.
![### Production Optimization Problem
**Constraints (s.t.)**
- **Machine Capacities:**
- Machine 1 Capacity: [Input Field]
- Machine 2 Capacity: [Input Field]
- Machine 3 Capacity: [Input Field]
- **Product Orders:**
- Product A Orders: [Input Field]
- Product B Orders: [Input Field]
- Product C Orders: [Input Field]
- **Non-negativity:**
- \( x_{ij} \geq 0 \) for all \( i, j \).
---
**Problem Statement:**
**(b)** Solve the transportation model to determine the minimum cost production schedule for the products and machines. Follow these steps:
1. **Production Schedule:**
- \( (x_{A1}, x_{A2}, x_{A3}, x_{B1}, x_{B2}, x_{B3}, x_{C1}, x_{C2}, x_{C3}) = \) [Input Field]
2. **Cost Calculation:**
- Determine the total cost (in dollars) of the production schedule.
- Total = $ [Input Field]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862c5bc2-e09a-4feb-bd28-f7638a835094%2Feafbaabb-fe5b-4fa8-811f-a3e9560215bb%2Fdnn6h1w_processed.png&w=3840&q=75)
Transcribed Image Text:### Production Optimization Problem
**Constraints (s.t.)**
- **Machine Capacities:**
- Machine 1 Capacity: [Input Field]
- Machine 2 Capacity: [Input Field]
- Machine 3 Capacity: [Input Field]
- **Product Orders:**
- Product A Orders: [Input Field]
- Product B Orders: [Input Field]
- Product C Orders: [Input Field]
- **Non-negativity:**
- \( x_{ij} \geq 0 \) for all \( i, j \).
---
**Problem Statement:**
**(b)** Solve the transportation model to determine the minimum cost production schedule for the products and machines. Follow these steps:
1. **Production Schedule:**
- \( (x_{A1}, x_{A2}, x_{A3}, x_{B1}, x_{B2}, x_{B3}, x_{C1}, x_{C2}, x_{C3}) = \) [Input Field]
2. **Cost Calculation:**
- Determine the total cost (in dollars) of the production schedule.
- Total = $ [Input Field]
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