## Problem Description and Formulation

The problem is an optimization problem where we need to maximize the objective function:

\[ 6.98x_0 + 2.26x_1 + 3.5x_2 + 2.08x_3 \]

subject to several constraints. The variables \(x_0, x_1, x_2, x_3\) represent the milligrams of vitamin E, potassium, vitamin B9, and zinc, respectively.

The constraints can be categorized into two types: index constraints and combined constraints.

### Index Constraints

These are given directly for each variable:

- Energy stability index for \(x_0\) (vitamin E) is 6.
- Muscle growth index for \(x_0\) (vitamin E) is 3.
- Energy stability index for \(x_1\) (potassium) is 8.
- Muscle growth index for \(x_1\) (potassium) is 11.
- Energy stability index for \(x_2\) (vitamin B9) is 5.
- Muscle growth index for \(x_2\) (vitamin B9) is 2.
- Energy stability index for \(x_3\) (zinc) is 5.
- Muscle growth index for \(x_3\) (zinc) is 1.

### Combined Constraints

1. \(11x_1 + x_3 \geq 35\)
2. \(2x_2 + x_3 \geq 50\)
3. \(11x_1 + 2x_2 + x_3 \geq 42\)
4. \(5x_2 + 5x_3 \leq 146\)
5. \(6x_0 + 8x_1 \leq 167\)
6. \(6x_0 + 5x_2 \leq 148\)
7. \(6x_0 + 8x_1 + 5x_3 \leq 121\)
8. \(6x_0 + 8x_1 + 5x_2 + 5x_3 \leq 121\)
9. \(3x_0 + 11x_1 \leq 276\)
10. \(3x_0 + x_3 \leq 179\)
11. \(11x_1 + 2x_2 \leq 117\)
12. \(2x_2 + x_3 \leq 152\)
13. \(3x_0 + 11x_1 + 2x_2 + x_3 \leq 152\)

## Gurobi Code Formulation

```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="vitamin_E", lb=0)  # milligrams of vitamin E
    x1 = model.addVar(name="potassium", lb=0)  # milligrams of potassium
    x2 = model.addVar(name="vitamin_B9", lb=0)  # milligrams of vitamin B9
    x3 = model.addVar(name="zinc", lb=0)  # milligrams of zinc

    # Objective function
    model.setObjective(6.98*x0 + 2.26*x1 + 3.5*x2 + 2.08*x3, gurobi.GRB.MAXIMIZE)

    # Index Constraints
    model.addConstraint(6*x0 <= 6)  # Energy stability index for vitamin E
    model.addConstraint(3*x0 <= 3)  # Muscle growth index for vitamin E
    model.addConstraint(8*x1 <= 8)  # Energy stability index for potassium
    model.addConstraint(11*x1 <= 11)  # Muscle growth index for potassium
    model.addConstraint(5*x2 <= 5)  # Energy stability index for vitamin B9
    model.addConstraint(2*x2 <= 2)  # Muscle growth index for vitamin B9
    model.addConstraint(5*x3 <= 5)  # Energy stability index for zinc
    model.addConstraint(x3 <= 1)  # Muscle growth index for zinc

    # Combined Constraints
    model.addConstraint(11*x1 + x3 >= 35)
    model.addConstraint(2*x2 + x3 >= 50)
    model.addConstraint(11*x1 + 2*x2 + x3 >= 42)
    model.addConstraint(5*x2 + 5*x3 <= 146)
    model.addConstraint(6*x0 + 8*x1 <= 167)
    model.addConstraint(6*x0 + 5*x2 <= 148)
    model.addConstraint(6*x0 + 8*x1 + 5*x3 <= 121)
    model.addConstraint(6*x0 + 8*x1 + 5*x2 + 5*x3 <= 121)
    model.addConstraint(3*x0 + 11*x1 <= 276)
    model.addConstraint(3*x0 + x3 <= 179)
    model.addConstraint(11*x1 + 2*x2 <= 117)
    model.addConstraint(2*x2 + x3 <= 152)
    model.addConstraint(3*x0 + 11*x1 + 2*x2 + x3 <= 152)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Vitamin E: {x0.varValue}")
        print(f"Potassium: {x1.varValue}")
        print(f"Vitamin B9: {x2.varValue}")
        print(f"Zinc: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_nutrition()
```