### Symbolic Representation of the Problem

The problem can be represented symbolically as follows:

```json
{
  'sym_variables': [
    ('x1', 'hours worked by John'),
    ('x2', 'hours worked by Laura'),
    ('x3', 'hours worked by Bobby'),
    ('x4', 'hours worked by Ringo'),
    ('x5', 'hours worked by Mary'),
    ('x6', 'hours worked by George')
  ],
  'objective_function': 'maximize x1 + x2 + x3 + x4 + x5 + x6',
  'constraints': [
    '11*x1 + 12*x2 + 13*x3 + 14*x4 + 15*x5 + 16*x6 >= 200', # Productivity rating constraint
    '21*x1 + 22*x2 + 23*x3 + 24*x4 + 25*x5 + 26*x6 <= 300', # Cost per hour constraint
    '31*x1 + 32*x2 + 33*x3 + 34*x4 + 35*x5 + 36*x6 >= 400', # Productivity rating constraint
    '41*x1 + 42*x2 + 43*x3 + 44*x4 + 45*x5 + 46*x6 <= 500', # Likelihood to quit index constraint
    'x1 >= 0', 'x2 >= 0', 'x3 >= 0', 'x4 >= 0', 'x5 >= 0', 'x6 >= 0' # Non-negativity constraints
  ]
}
```

Note that the coefficients in the constraints are arbitrary and may need to be adjusted based on the actual problem requirements.

### Gurobi Code

```python
from gurobipy import *

# Create a model
m = Model("Optimization Problem")

# Define variables
x1, x2, x3, x4, x5, x6 = m.addVars(6, name=['hours worked by John', 'hours worked by Laura', 
                                         'hours worked by Bobby', 'hours worked by Ringo', 
                                         'hours worked by Mary', 'hours worked by George'])

# Define objective function
m.setObjective(x1 + x2 + x3 + x4 + x5 + x6, GRB.MAXIMIZE)

# Define constraints
m.addConstr(11*x1 + 12*x2 + 13*x3 + 14*x4 + 15*x5 + 16*x6 >= 200, name='Productivity rating constraint')
m.addConstr(21*x1 + 22*x2 + 23*x3 + 24*x4 + 25*x5 + 26*x6 <= 300, name='Cost per hour constraint')
m.addConstr(31*x1 + 32*x2 + 33*x3 + 34*x4 + 35*x5 + 36*x6 >= 400, name='Productivity rating constraint')
m.addConstr(41*x1 + 42*x2 + 43*x3 + 44*x4 + 45*x5 + 46*x6 <= 500, name='Likelihood to quit index constraint')

# Optimize the model
m.optimize()
```