## An Example of Cost-Effective Nutrition Optimization

### Example: Diet Problem

The diet problem is a classic $optimization$ $problem$ that aims to find the most $cost$-$effective$ way to meet nutritional requirements using a selection of foods.

Letâ€™s solve a simple example using the $PuLP$ library in $Python$.

#### Problem Statement:

You are trying to meet your daily nutritional needs using three different foods.

Each food has a $cost$ and provides a certain amount of $calories$, $protein$, and $fat$.

Your goal is to **minimize the cost** while ensuring that you meet the minimum daily requirements for $calories$, $protein$, and $fat$.

#### Data:

**Foods**: Chicken, Beef, Vegetables**Cost per unit**:- Chicken: $2.50
- Beef: $3.00
- Vegetables: $1.00

**Nutritional content per unit**:- Chicken: $250$ calories, $30$g protein, $10$g fat
- Beef: $300$ calories, $20$g protein, $20$g fat
- Vegetables: $50$ calories, $5$g protein, $0$g fat

**Minimum daily requirements**:- Calories: $2000$
- Protein: $60$g
- Fat: $50$g

#### Solution using PuLP:

1 | import pulp |

### Explanation:

**Decision Variables**:`food_vars`

represent the amount of each food to consume.**Objective Function**: Minimize the total cost of the diet.**Constraints**: Ensure that the total intake of $calories$, $protein$, and $fat$ meets or exceeds the minimum daily requirements.

### Output:

1 | Optimal diet: |

### Explanation of the Result:

The optimal solution suggests that the most $cost$-$effective$ way to meet the daily nutritional requirements is to consume **$6.67$ units of Beef** and no units of Chicken or Vegetables.

The total cost of this diet is **$20.00**.

#### Key Points:

**Beef-only diet**: The solution indicates that Beef alone is sufficient to meet the required minimums for calories, protein, and fat, making it the $cheapest$ option at $3.00 per unit.**No Chicken or Vegetables**: Since Beef provides a higher concentration of calories and fat compared to Chicken and Vegetables, the solver determined that only Beef is needed to satisfy the constraints without incurring extra costs.**Total Cost**: By consuming $6.67$ units of Beef, the total expenditure on this diet is minimized to $20.00.

This result demonstrates how optimization can find the most economical way to satisfy nutritional needs based on the available options.