## Problem Description and Formulation

The problem is an optimization problem where we need to maximize an objective function subject to several constraints. The objective function to be maximized is:

\[ 4.44 \times \text{milligrams of vitamin B12} + 1.36 \times \text{milligrams of vitamin B9} + 2.79 \times \text{grams of fat} + 9.58 \times \text{milligrams of vitamin B5} \]

The variables are:
- \(x_0\): milligrams of vitamin B12
- \(x_1\): milligrams of vitamin B9
- \(x_2\): grams of fat
- \(x_3\): milligrams of vitamin B5

The constraints are based on a "muscle growth index" for each variable:
- \(r_0 = \{7, 1, 5, 16\}\) for \(\{x_0, x_1, x_2, x_3\}\) respectively.

## Constraints

1. The muscle growth index of milligrams of vitamin B12 is 7.
2. Milligrams of vitamin B9 each have a muscle growth index of 1.
3. Grams of fat have a muscle growth index of 5.
4. Milligrams of vitamin B5 each have a muscle growth index of 16.
5. The total combined muscle growth index from grams of fat and milligrams of vitamin B5 must be 97 or more: \(5x_2 + 16x_3 \geq 97\).
6. The total combined muscle growth index from milligrams of vitamin B12 and milligrams of vitamin B5 should be 339 or less: \(7x_0 + 16x_3 \leq 339\).
7. The total combined muscle growth index from milligrams of vitamin B12 and milligrams of vitamin B9 should be no more than 182: \(7x_0 + x_1 \leq 182\).
8. The total combined muscle growth index from milligrams of vitamin B12 and grams of fat should be 168 or less: \(7x_0 + 5x_2 \leq 168\).
9. The total combined muscle growth index from milligrams of vitamin B9, grams of fat, and milligrams of vitamin B5 must be 233 at a maximum: \(x_1 + 5x_2 + 16x_3 \leq 233\).
10. The total combined muscle growth index from all variables has to be 233 or less: \(7x_0 + x_1 + 5x_2 + 16x_3 \leq 233\).

## Gurobi Code

```python
import gurobi

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

    # Define variables
    x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="milligrams of vitamin B12")
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="milligrams of vitamin B9")
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="grams of fat")
    x3 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="milligrams of vitamin B5")

    # Objective function
    model.setObjective(4.44 * x0 + 1.36 * x1 + 2.79 * x2 + 9.58 * x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(5 * x2 + 16 * x3 >= 97, name="constraint_1")
    model.addConstr(7 * x0 + 16 * x3 <= 339, name="constraint_2")
    model.addConstr(7 * x0 + x1 <= 182, name="constraint_3")
    model.addConstr(7 * x0 + 5 * x2 <= 168, name="constraint_4")
    model.addConstr(x1 + 5 * x2 + 16 * x3 <= 233, name="constraint_5")
    model.addConstr(7 * x0 + x1 + 5 * x2 + 16 * x3 <= 233, name="constraint_6")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin B12: {x0.varValue}")
        print(f"Milligrams of vitamin B9: {x1.varValue}")
        print(f"Grams of fat: {x2.varValue}")
        print(f"Milligrams of vitamin B5: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_muscle_growth()
```