### Symbolic Representation of the Problem

To convert the problem description into a symbolic representation, we define variables as follows:

* $x_1$: hours worked by Bill
* $x_2$: hours worked by Laura
* $x_3$: hours worked by Peggy
* $x_4$: hours worked by John
* $x_5$: hours worked by Dale
* $x_6$: hours worked by Paul

The objective function is not explicitly defined in the problem description. However, we can assume that the goal is to minimize or maximize one of the given constraints.

```json
{
  'sym_variables': [
    ('x1', 'hours worked by Bill'),
    ('x2', 'hours worked by Laura'),
    ('x3', 'hours worked by Peggy'),
    ('x4', 'hours worked by John'),
    ('x5', 'hours worked by Dale'),
    ('x6', 'hours worked by Paul')
  ],
  'objective_function': 'Not explicitly defined',
  'constraints': [
    # Paperwork competence constraints
    'x1 + x2 >= 124',
    'x3^2 + x6^2 >= 149',
    # Likelihood to quit index constraints
    'x1 * x6 <= 103',
    'x3 * x4 <= 249',
    'x1 * x2 <= 229',
    'x2 * x5 <= 268',
    'x3 * x5 <= 336',
    'x4 * x5 <= 172',
    'x2 + x4 <= 140',
    'x2 + x6 <= 164',
    # Productivity rating constraints
    'x3 + x6 <= 897',
    'x1^2 + x4^2 <= 400',
    'x1^2 + x6^2 <= 650',
    'x4 * x6 <= 1179',
    'x2 + x5 <= 1163',
    'x1 * x5 <= 605',
    # Dollar cost per hour constraints
    'x1 + x6 >= 124',
    'x3^2 + x6^2 >= 149'
  ]
}
```

### Gurobi Code

```python
from gurobipy import *

# Create a model
m = Model("Optimization_Problem")

# Define variables
x1 = m.addVar(name="hours_worked_by_Bill", lb=0)
x2 = m.addVar(name="hours_worked_by_Laura", lb=0)
x3 = m.addVar(name="hours_worked_by_Peggy", lb=0)
x4 = m.addVar(name="hours_worked_by_John", lb=0)
x5 = m.addVar(name="hours_worked_by_Dale", lb=0)
x6 = m.addVar(name="hours_worked_by_Paul", lb=0)

# Define constraints
m.addConstr(x1 + x2 >= 124, name="paperwork_competence_constraint_1")
m.addConstr(x3**2 + x6**2 >= 149, name="paperwork_competence_constraint_2")
m.addConstr(x1 * x6 <= 103, name="likelihood_to_quit_index_constraint_1")
m.addConstr(x3 * x4 <= 249, name="likelihood_to_quit_index_constraint_2")
m.addConstr(x1 * x2 <= 229, name="likelihood_to_quit_index_constraint_3")
m.addConstr(x2 * x5 <= 268, name="likelihood_to_quit_index_constraint_4")
m.addConstr(x3 * x5 <= 336, name="likelihood_to_quit_index_constraint_5")
m.addConstr(x4 * x5 <= 172, name="likelihood_to_quit_index_constraint_6")
m.addConstr(x2 + x4 <= 140, name="likelihood_to_quit_index_constraint_7")
m.addConstr(x2 + x6 <= 164, name="likelihood_to_quit_index_constraint_8")

# Since there is no objective function defined in the problem,
# we will use a simple objective function for demonstration purposes
m.setObjective(x1 + x2 + x3 + x4 + x5 + x6, GRB.MINIMIZE)

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