Question
An example of a linear problem
Answers
GPT-4o mini
A linear problem typically involves finding the best outcome (such as maximum profit or minimum cost) within a linear framework, often constrained by various conditions. One common type of linear problem is a Linear Programming problem.
**Example: Diet Problem**
Suppose a nutritionist wants to determine the optimal combination of two foods to meet daily nutritional requirements at the minimum cost.
**Variables:**
- Let \( x_1 \) be the number of servings of Food A.
- Let \( x_2 \) be the number of servings of Food B.
**Objective Function:**
Minimize total cost:
\[ \text{Cost} = 2x_1 + 3x_2 \]
(where Food A costs $2 per serving and Food B costs $3 per serving)
**Constraints:**
1. **Caloric Intake:** The total calories should be at least 2000 calories.
\[ 400x_1 + 500x_2 \geq 2000 \]
2. **Protein Intake:** The total protein should be at least 100 grams.
\[ 30x_1 + 20x_2 \geq 100 \]
3. **Carbohydrate Intake:** The total carbohydrates should be at least 300 grams.
\[ 50x_1 + 60x_2 \geq 300 \]
4. **Non-negativity:** Both variables must be non-negative.
\[ x_1 \geq 0, \quad x_2 \geq 0 \]
In this example, the problem is to minimize the cost while satisfying the nutritional constraints. This is a linear problem because both the objective function and constraints are linear equations.
**Example: Diet Problem**
Suppose a nutritionist wants to determine the optimal combination of two foods to meet daily nutritional requirements at the minimum cost.
**Variables:**
- Let \( x_1 \) be the number of servings of Food A.
- Let \( x_2 \) be the number of servings of Food B.
**Objective Function:**
Minimize total cost:
\[ \text{Cost} = 2x_1 + 3x_2 \]
(where Food A costs $2 per serving and Food B costs $3 per serving)
**Constraints:**
1. **Caloric Intake:** The total calories should be at least 2000 calories.
\[ 400x_1 + 500x_2 \geq 2000 \]
2. **Protein Intake:** The total protein should be at least 100 grams.
\[ 30x_1 + 20x_2 \geq 100 \]
3. **Carbohydrate Intake:** The total carbohydrates should be at least 300 grams.
\[ 50x_1 + 60x_2 \geq 300 \]
4. **Non-negativity:** Both variables must be non-negative.
\[ x_1 \geq 0, \quad x_2 \geq 0 \]
In this example, the problem is to minimize the cost while satisfying the nutritional constraints. This is a linear problem because both the objective function and constraints are linear equations.