## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' for 'hours worked by Hank'
- 'x2' for 'hours worked by Bobby'
- 'x3' for 'hours worked by Paul'
- 'x4' for 'hours worked by George'
- 'x5' for 'hours worked by Ringo'

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is: $9x_1 + 6x_2 + 1x_3 + 9x_4 + 8x_5$

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $13x_1 \geq 0$ (Implicit, as $x_1$ can be fractional and no direct constraint is given for Hank's rating)
- $12x_2 \geq 0$ (Implicit, as $x_2$ can be fractional and no direct constraint is given for Bobby's rating)
- $14x_3 \geq 0$ (Implicit, as $x_3$ can be fractional and no direct constraint is given for Paul's rating)
- $4x_4 \geq 0$ (Implicit, as $x_4$ can be fractional and no direct constraint is given for George's rating)
- $7x_5 \geq 0$ (Implicit, as $x_5$ must be an integer but no direct constraint is given for Ringo's rating)
- $12x_2 + 7x_5 \geq 31$
- $14x_3 + 7x_5 \geq 41$
- $13x_1 + 14x_3 \geq 25$
- $13x_1 + 12x_2 + 14x_3 + 4x_4 + 7x_5 \geq 25$
- $-2x_1 + 1x_5 \geq 0$
- $12x_2 + 4x_4 \leq 170$
- $14x_3 + 4x_4 \leq 154$
- $13x_1 + 7x_5 \leq 137$
- $14x_3 + 7x_5 \leq 160$
- $13x_1 + 12x_2 \leq 156$
- $12x_2 + 7x_5 \leq 125$
- $13x_1 + 4x_4 \leq 75$
- $12x_2 + 14x_3 \leq 79$

## 4: Formulate the problem in Gurobi

```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name='x1', lb=0)  # hours worked by Hank
x2 = m.addVar(name='x2', lb=0)  # hours worked by Bobby
x3 = m.addVar(name='x3', lb=0)  # hours worked by Paul
x4 = m.addVar(name='x4', lb=0)  # hours worked by George
x5 = m.addVar(name='x5', lb=0, vtype='I')  # hours worked by Ringo

# Objective function
m.setObjective(9*x1 + 6*x2 + x3 + 9*x4 + 8*x5, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(12*x2 + 7*x5 >= 31)
m.addConstr(14*x3 + 7*x5 >= 41)
m.addConstr(13*x1 + 14*x3 >= 25)
m.addConstr(13*x1 + 12*x2 + 14*x3 + 4*x4 + 7*x5 >= 25)
m.addConstr(-2*x1 + x5 >= 0)
m.addConstr(12*x2 + 4*x4 <= 170)
m.addConstr(14*x3 + 4*x4 <= 154)
m.addConstr(13*x1 + 7*x5 <= 137)
m.addConstr(14*x3 + 7*x5 <= 160)
m.addConstr(13*x1 + 12*x2 <= 156)
m.addConstr(12*x2 + 7*x5 <= 125)
m.addConstr(13*x1 + 4*x4 <= 75)
m.addConstr(12*x2 + 14*x3 <= 79)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.Status.OPTIMAL:
    print('Objective: ', m.objVal)
    print('Hours worked by Hank: ', x1.varValue)
    print('Hours worked by Bobby: ', x2.varValue)
    print('Hours worked by Paul: ', x3.varValue)
    print('Hours worked by George: ', x4.varValue)
    print('Hours worked by Ringo: ', x5.varValue)
else:
    print('The model is infeasible')
```

## 5: Symbolic Representation

```json
{
    'sym_variables': [
        ('x1', 'hours worked by Hank'),
        ('x2', 'hours worked by Bobby'),
        ('x3', 'hours worked by Paul'),
        ('x4', 'hours worked by George'),
        ('x5', 'hours worked by Ringo')
    ],
    'objective_function': '9*x1 + 6*x2 + 1*x3 + 9*x4 + 8*x5',
    'constraints': [
        '12*x2 + 7*x5 >= 31',
        '14*x3 + 7*x5 >= 41',
        '13*x1 + 14*x3 >= 25',
        '13*x1 + 12*x2 + 14*x3 + 4*x4 + 7*x5 >= 25',
        '-2*x1 + 1*x5 >= 0',
        '12*x2 + 4*x4 <= 170',
        '14*x3 + 4*x4 <= 154',
        '13*x1 + 7*x5 <= 137',
        '14*x3 + 7*x5 <= 160',
        '13*x1 + 12*x2 <= 156',
        '12*x2 + 7*x5 <= 125',
        '13*x1 + 4*x4 <= 75',
        '12*x2 + 14*x3 <= 79'
    ]
}
```