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

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $2x_0 + 4x_1 + 7x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $9x_0 \leq 407$
- $1x_0 \leq 305$
- $14x_0 \leq 423$
- $3x_1 \leq 407$
- $11x_1 \leq 305$
- $15x_1 \leq 423$
- $15x_2 \leq 407$
- $5x_2 \leq 305$
- $2x_2 \leq 423$
- $9x_0 + 3x_1 \geq 132$
- $9x_0 + 3x_1 + 15x_2 \geq 132$
- $11x_1 + 5x_2 \geq 33$
- $1x_0 + 11x_1 \geq 85$
- $1x_0 + 11x_1 + 5x_2 \geq 85$
- $14x_0 + 2x_2 \geq 110$
- $14x_0 + 15x_1 + 2x_2 \geq 98$
- $-4x_0 + 9x_1 \geq 0$
- $9x_0 + 15x_2 \leq 179$
- $3x_1 + 15x_2 \leq 304$
- $11x_1 + 5x_2 \leq 232$
- $x_0 \in \mathbb{Z}$
- $x_1 \in \mathbb{Z} \times \mathbb{N}$ or simply $x_1 \in \mathbb{Z}$ and $x_1 \geq 0$ but to reflect "Do not have a fractional amount" it implies $x_1$ is an integer, 
- $x_2 \in \mathbb{Z}$

However, given $x_1$ should not have fractional hours and combining with integrality requirements:
- $x_0, x_2 \in \mathbb{Z}$ and $x_1 \in \mathbb{Z}$, $x_1 \geq 0$.


## 4: Provide the symbolic representation of the problem
```json
{
'sym_variables': [('x0', 'hours worked by Ringo'), ('x1', 'hours worked by Peggy'), ('x2', 'hours worked by Paul')],
'objective_function': '2*x0 + 4*x1 + 7*x2',
'constraints': [
'9*x0 <= 407', '1*x0 <= 305', '14*x0 <= 423',
'3*x1 <= 407', '11*x1 <= 305', '15*x1 <= 423',
'15*x2 <= 407', '5*x2 <= 305', '2*x2 <= 423',
'9*x0 + 3*x1 >= 132', '9*x0 + 3*x1 + 15*x2 >= 132',
'11*x1 + 5*x2 >= 33', '1*x0 + 11*x1 >= 85',
'1*x0 + 11*x1 + 5*x2 >= 85', '14*x0 + 2*x2 >= 110',
'14*x0 + 15*x1 + 2*x2 >= 98', '-4*x0 + 9*x1 >= 0',
'9*x0 + 15*x2 <= 179', '3*x1 + 15*x2 <= 304',
'11*x1 + 5*x2 <= 232'
]
}
```

## 5: Write the Gurobi code for the optimization problem
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="hours_worked_by_Ringo", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="hours_worked_by_Peggy", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="hours_worked_by_Paul", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(2*x0 + 4*x1 + 7*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(9*x0 <= 407)
    model.addConstr(1*x0 <= 305)
    model.addConstr(14*x0 <= 423)
    model.addConstr(3*x1 <= 407)
    model.addConstr(11*x1 <= 305)
    model.addConstr(15*x1 <= 423)
    model.addConstr(15*x2 <= 407)
    model.addConstr(5*x2 <= 305)
    model.addConstr(2*x2 <= 423)
    model.addConstr(9*x0 + 3*x1 >= 132)
    model.addConstr(9*x0 + 3*x1 + 15*x2 >= 132)
    model.addConstr(11*x1 + 5*x2 >= 33)
    model.addConstr(1*x0 + 11*x1 >= 85)
    model.addConstr(1*x0 + 11*x1 + 5*x2 >= 85)
    model.addConstr(14*x0 + 2*x2 >= 110)
    model.addConstr(14*x0 + 15*x1 + 2*x2 >= 98)
    model.addConstr(-4*x0 + 9*x1 >= 0)
    model.addConstr(9*x0 + 15*x2 <= 179)
    model.addConstr(3*x1 + 15*x2 <= 304)
    model.addConstr(11*x1 + 5*x2 <= 232)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Ringo: {x0.varValue}")
        print(f"Hours worked by Peggy: {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()
```