To tackle this problem, we first need to understand the objective function and the constraints provided. The goal is to maximize the value of `6.54 * hours worked by Hank + 6.17 * hours worked by Dale`, subject to several constraints involving their likelihood to quit index, work quality rating, and specific conditions on their working hours.

Let's define:
- `x1` as the hours worked by Hank
- `x2` as the hours worked by Dale

The symbolic representation of the problem is:

```json
{
  'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Dale')],
  'objective_function': '6.54*x1 + 6.17*x2',
  'constraints': [
    '18*x1 + 7*x2 >= 57', 
    '8*x1 + 16*x2 >= 67', 
    '4*x1 - x2 >= 0', 
    '18*x1 + 7*x2 <= 123', 
    '8*x1 + 16*x2 <= 70',
    'x1 == int(x1)'
  ]
}
```

Now, let's implement this problem using Gurobi in Python. Note that we need to ensure `x1` is an integer variable, while `x2` can be a continuous variable.

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Hank")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Dale", lb=0)

# Set the objective function
m.setObjective(6.54*x1 + 6.17*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(18*x1 + 7*x2 >= 57, "likelihood_to_quit_index_min")
m.addConstr(8*x1 + 16*x2 >= 67, "work_quality_rating_min")
m.addConstr(4*x1 - x2 >= 0, "hours_worked_constraint")
m.addConstr(18*x1 + 7*x2 <= 123, "likelihood_to_quit_index_max")
m.addConstr(8*x1 + 16*x2 <= 70, "work_quality_rating_max")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Hank: {x1.x}")
    print(f"Hours worked by Dale: {x2.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```