## Problem Description and Formulation

The problem is a linear programming (LP) optimization problem. The goal is to minimize the cost of purchasing two powders, Alpha and Beta, that contain iron and biotin, while meeting the daily recommended intake of these nutrients.

### Decision Variables

Let $x$ be the number of scoops of Alpha and $y$ be the number of scoops of Beta.

### Objective Function

The cost per scoop of Alpha is $1, and the cost per scoop of Beta is $2. The objective is to minimize the total cost:

$$\min \quad 1x + 2y$$

### Constraints

1. Iron intake: A scoop of Alpha contains 5 grams of iron, and a scoop of Beta contains 10 grams of iron. The doctor recommends at least 50 grams of iron daily:

$$5x + 10y \geq 50$$

2. Biotin intake: A scoop of Alpha contains 20 grams of biotin, and a scoop of Beta contains 3 grams of biotin. The doctor recommends at least 40 grams of biotin daily:

$$20x + 3y \geq 40$$

3. Non-negativity: The number of scoops of Alpha and Beta cannot be negative:

$$x \geq 0, \quad y \geq 0$$

## Gurobi Code

```python
import gurobi

def solve_supplement_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define decision variables
    x = model.addVar(name="Alpha", lb=0, obj=1)
    y = model.addVar(name="Beta", lb=0, obj=2)

    # Define constraints
    iron_constraint = model.addConstr(5 * x + 10 * y >= 50, name="Iron_Intake")
    biotin_constraint = model.addConstr(20 * x + 3 * y >= 40, name="Biotin_Intake")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal Solution:")
        print(f"Alpha: {x.varValue}")
        print(f"Beta: {y.varValue}")
        print(f"Cost: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_supplement_problem()
```