The manager of a Burger Doodle franchise wants to determine how many sausage biscuits and ham biscuits to prepare each morning for breakfast customers. The two types of biscuits require the following resources. The franchise has 6 hours of labor available each morning. The manager has a contract with a local grocer for 30 pounds of sausage and 30 pounds of ham each morning. The manager also purchases 16 pounds of flour. The profit for a sausage biscuit is $0.60; the profit for a ham biscuit is $0.50. The manager wants to know the number of each type of biscuit to prepare each morning in order to maximize profit. Formulate a linear programming model for this problem. Define x₁ as the number of sausage biscuits to make, x2 as the number of ham biscuits to make, and Z as the total profit each morning. Which of the following model formulations is correct? Biscuit Labor (hr.) Sausage (ib.) Ham (lb.) Flour (lb.) Sausage 0.010 0.10 0.04 Ham 0.024 0.15 0.04
The manager of a Burger Doodle franchise wants to determine how many sausage biscuits and ham biscuits to prepare each morning for breakfast customers. The two types of biscuits require the following resources. The franchise has 6 hours of labor available each morning. The manager has a contract with a local grocer for 30 pounds of sausage and 30 pounds of ham each morning. The manager also purchases 16 pounds of flour. The profit for a sausage biscuit is $0.60; the profit for a ham biscuit is $0.50. The manager wants to know the number of each type of biscuit to prepare each morning in order to maximize profit. Formulate a linear programming model for this problem. Define x₁ as the number of sausage biscuits to make, x2 as the number of ham biscuits to make, and Z as the total profit each morning. Which of the following model formulations is correct? Biscuit Labor (hr.) Sausage (ib.) Ham (lb.) Flour (lb.) Sausage 0.010 0.10 0.04 Ham 0.024 0.15 0.04
Practical Management Science
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Just need help finding the sausage constraint quantity value
![### Linear Programming Model for Franchise Biscuit Production
The manager of a Burger Doodle franchise wants to determine how many sausage biscuits and ham biscuits to prepare each morning for breakfast customers. The two types of biscuits require the following resources:
- **Labor**: 6 hours available each morning.
- **Sausage**: 30 pounds available each morning.
- **Ham**: 30 pounds available each morning.
- **Flour**: 16 pounds available each morning.
#### Profits:
- **Sausage Biscuit**: $0.60 profit per biscuit.
- **Ham Biscuit**: $0.50 profit per biscuit.
#### Objective:
Formulate a linear programming model to maximize the total profit (Z) each morning.
#### Variables:
- \( x_1 \): Number of sausage biscuits to make.
- \( x_2 \): Number of ham biscuits to make.
#### Resource Usage per Biscuit:
| Biscuit | Labor (hr.) | Sausage (lb.) | Ham (lb.) | Flour (lb.) |
|------------------|-------------|---------------|-----------|-------------|
| Sausage | 0.010 | 0.10 | — | 0.04 |
| Ham | 0.024 | — | 0.15 | 0.04 |
#### Constraints:
1. **Labor Constraint**:
\[ 0.010x_1 + 0.024x_2 \leq 6 \]
2. **Sausage Constraint**:
\[ 0.10x_1 \leq 30 \]
3. **Ham Constraint**:
\[ 0.15x_2 \leq 30 \]
4. **Flour Constraint**:
\[ 0.04x_1 + 0.04x_2 \leq 16 \]
#### Objective Function:
\[ Z = 0.60x_1 + 0.50x_2 \]
Where \( Z \) represents the total daily profit from the production of sausage and ham biscuits.
This model will help determine the optimal number of sausage and ham biscuits that should be produced each morning to maximize profits while adhering to resource constraints.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0506d55b-bb84-43f9-9218-6fc41e3ed9b6%2Fbb824d20-94b3-41fe-9275-83e26c674e1a%2Frxb04t9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Programming Model for Franchise Biscuit Production
The manager of a Burger Doodle franchise wants to determine how many sausage biscuits and ham biscuits to prepare each morning for breakfast customers. The two types of biscuits require the following resources:
- **Labor**: 6 hours available each morning.
- **Sausage**: 30 pounds available each morning.
- **Ham**: 30 pounds available each morning.
- **Flour**: 16 pounds available each morning.
#### Profits:
- **Sausage Biscuit**: $0.60 profit per biscuit.
- **Ham Biscuit**: $0.50 profit per biscuit.
#### Objective:
Formulate a linear programming model to maximize the total profit (Z) each morning.
#### Variables:
- \( x_1 \): Number of sausage biscuits to make.
- \( x_2 \): Number of ham biscuits to make.
#### Resource Usage per Biscuit:
| Biscuit | Labor (hr.) | Sausage (lb.) | Ham (lb.) | Flour (lb.) |
|------------------|-------------|---------------|-----------|-------------|
| Sausage | 0.010 | 0.10 | — | 0.04 |
| Ham | 0.024 | — | 0.15 | 0.04 |
#### Constraints:
1. **Labor Constraint**:
\[ 0.010x_1 + 0.024x_2 \leq 6 \]
2. **Sausage Constraint**:
\[ 0.10x_1 \leq 30 \]
3. **Ham Constraint**:
\[ 0.15x_2 \leq 30 \]
4. **Flour Constraint**:
\[ 0.04x_1 + 0.04x_2 \leq 16 \]
#### Objective Function:
\[ Z = 0.60x_1 + 0.50x_2 \]
Where \( Z \) represents the total daily profit from the production of sausage and ham biscuits.
This model will help determine the optimal number of sausage and ham biscuits that should be produced each morning to maximize profits while adhering to resource constraints.
![## Sensitivity Analysis in Linear Programming
In a previously formulated model for the Burger Doodle franchise, the sensitivity analysis for the sausage constraint's quantity value was conducted using Excel and POM-QM software. The sensitivity analysis helps determine how changes in the constraint's right-hand side affect the optimal solution.
### Sensitivity Range for the Sausage Constraint Quantity Value
The sensitivity range, also known as the allowable range, indicates the range within which the right-hand side value of the constraint can change without altering the optimal solution. For the sausage constraint quantity value, the sensitivity range was determined to be:
- \([25.71, 30]\)
### Options for Sensitivity Range
The options considered were:
- \([30, 40]\)
- **\([25.71, 30]\)** (The correct answer)
- \([25.71, 40]\)
- \([30, 34.29]\)
The correct sensitivity range for the sausage constraint quantity, as calculated, is \( [25.71, 30] \). This means any value within this range will not affect the optimal solution derived from the linear programming model.
Understanding and identifying the correct sensitivity range is crucial for decision-making, allowing the management to understand how flexible their constraints are in the business operations and planning.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0506d55b-bb84-43f9-9218-6fc41e3ed9b6%2Fbb824d20-94b3-41fe-9275-83e26c674e1a%2Fn7j9rpc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Sensitivity Analysis in Linear Programming
In a previously formulated model for the Burger Doodle franchise, the sensitivity analysis for the sausage constraint's quantity value was conducted using Excel and POM-QM software. The sensitivity analysis helps determine how changes in the constraint's right-hand side affect the optimal solution.
### Sensitivity Range for the Sausage Constraint Quantity Value
The sensitivity range, also known as the allowable range, indicates the range within which the right-hand side value of the constraint can change without altering the optimal solution. For the sausage constraint quantity value, the sensitivity range was determined to be:
- \([25.71, 30]\)
### Options for Sensitivity Range
The options considered were:
- \([30, 40]\)
- **\([25.71, 30]\)** (The correct answer)
- \([25.71, 40]\)
- \([30, 34.29]\)
The correct sensitivity range for the sausage constraint quantity, as calculated, is \( [25.71, 30] \). This means any value within this range will not affect the optimal solution derived from the linear programming model.
Understanding and identifying the correct sensitivity range is crucial for decision-making, allowing the management to understand how flexible their constraints are in the business operations and planning.
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