To tackle the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints using mathematical notation.

Let's denote:
- \(x_1\) as 'hours worked by Hank',
- \(x_2\) as 'hours worked by Ringo'.

The objective function is to maximize \(6.31x_1 + 8.64x_2\).

Constraints are as follows:
1. The total combined dollar cost per hour from hours worked by Hank and Ringo must be at least 49: \(17.41x_1 + 10.0x_2 \geq 49\).
2. The total combined likelihood to quit index from hours worked by Hank and Ringo must be at least 32: \(2.79x_1 + 2.13x_2 \geq 32\).
3. \(5x_1 - x_2 \geq 0\).
4. The total combined dollar cost per hour from hours worked by Hank and Ringo should be at most 161: \(17.41x_1 + 10.0x_2 \leq 161\).
5. The total combined likelihood to quit index from hours worked by Hank and Ringo should be at most 110: \(2.79x_1 + 2.13x_2 \leq 110\).

Given that \(x_1\) must be an integer (non-fractional amount of hours worked by Hank) and \(x_2\) can be any real number, we have a mixed-integer linear programming problem.

Here is the symbolic representation in JSON format:
```json
{
    'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Ringo')],
    'objective_function': '6.31*x1 + 8.64*x2',
    'constraints': [
        '17.41*x1 + 10.0*x2 >= 49',
        '2.79*x1 + 2.13*x2 >= 32',
        '5*x1 - x2 >= 0',
        '17.41*x1 + 10.0*x2 <= 161',
        '2.79*x1 + 2.13*x2 <= 110'
    ]
}
```

Now, let's implement this problem in Gurobi Python:
```python
from gurobipy import *

# Create a new 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_Ringo")

# Set objective function
m.setObjective(6.31*x1 + 8.64*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(17.41*x1 + 10.0*x2 >= 49, "dollar_cost_min")
m.addConstr(2.79*x1 + 2.13*x2 >= 32, "quit_index_min")
m.addConstr(5*x1 - x2 >= 0, "hank_ringo_ratio")
m.addConstr(17.41*x1 + 10.0*x2 <= 161, "dollar_cost_max")
m.addConstr(2.79*x1 + 2.13*x2 <= 110, "quit_index_max")

# Optimize model
m.optimize()

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