## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill', 'hours worked by Jean', and 'hours worked by Paul', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the objective function into symbolic notation
The objective function to minimize is $2x_0 + 9x_1 + 4x_2$.

## Step 3: List the constraints in symbolic notation
The constraints given are:
- $7x_1 + 4x_2 \geq 13$
- $8x_0 + 4x_2 \geq 11$
- $8x_0 + 7x_1 + 4x_2 \geq 11$
- $x_1 + 3x_2 \geq 6$
- $x_0 + 3x_2 \geq 12$
- $x_0 + x_1 + 3x_2 \geq 12$
- $-3x_0 + 8x_1 \geq 0$
- $-2x_0 + 6x_2 \geq 0$
- $7x_1 + 4x_2 \leq 25$
- $x_0 \in \mathbb{Z}$
- $x_1 \in \mathbb{Z}$
- $x_2 \in \mathbb{Z}$

## 4: Create a symbolic representation of the problem
```json
{
'sym_variables': [('x0', 'hours worked by Bill'), ('x1', 'hours worked by Jean'), ('x2', 'hours worked by Paul')],
'objective_function': '2*x0 + 9*x1 + 4*x2',
'constraints': [
    '7*x1 + 4*x2 >= 13',
    '8*x0 + 4*x2 >= 11',
    '8*x0 + 7*x1 + 4*x2 >= 11',
    'x1 + 3*x2 >= 6',
    'x0 + 3*x2 >= 12',
    'x0 + x1 + 3*x2 >= 12',
    '-3*x0 + 8*x1 >= 0',
    '-2*x0 + 6*x2 >= 0',
    '7*x1 + 4*x2 <= 25',
    'x0 ∈ ℤ',
    'x1 ∈ ℤ',
    'x2 ∈ ℤ'
]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="x0", vtype=gurobi.GRB.INTEGER)  # hours worked by Bill
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Jean
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # hours worked by Paul

    # Define the objective function
    model.setObjective(2 * x0 + 9 * x1 + 4 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(7 * x1 + 4 * x2 >= 13)
    model.addConstr(8 * x0 + 4 * x2 >= 11)
    model.addConstr(8 * x0 + 7 * x1 + 4 * x2 >= 11)
    model.addConstr(x1 + 3 * x2 >= 6)
    model.addConstr(x0 + 3 * x2 >= 12)
    model.addConstr(x0 + x1 + 3 * x2 >= 12)
    model.addConstr(-3 * x0 + 8 * x1 >= 0)
    model.addConstr(-2 * x0 + 6 * x2 >= 0)
    model.addConstr(7 * x1 + 4 * x2 <= 25)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Bill: {x0.varValue}")
        print(f"Hours worked by Jean: {x1.varValue}")
        print(f"Hours worked by Paul: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```