To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables for each object of interest (e.g., hours worked by Ringo and Mary), formulating the objective function using these variables, and then expressing all constraints in terms of these variables as well.

Given:
- Let `x1` represent the hours worked by Ringo.
- Let `x2` represent the hours worked by Mary.

The objective function aims to maximize: `2*x1 + 8*x2`.

Constraints are as follows:
1. Likelihood to quit index for Ringo: `15`.
2. Likelihood to quit index for Mary: `4`.
3. Combined likelihood to quit index from both should be greater than or equal to `23`: `15*x1 + 4*x2 >= 23`.
4. `-8*x1 + 3*x2 >= 0`.
5. The total combined likelihood to quit index should not exceed `84`: `15*x1 + 4*x2 <= 84`.

Symbolic representation:
```json
{
    'sym_variables': [('x1', 'hours worked by Ringo'), ('x2', 'hours worked by Mary')],
    'objective_function': '2*x1 + 8*x2',
    'constraints': [
        '15*x1 + 4*x2 >= 23',
        '-8*x1 + 3*x2 >= 0',
        '15*x1 + 4*x2 <= 84'
    ]
}
```

To implement this optimization problem in Gurobi, we use Python. We'll define the model, variables, objective function, and constraints according to the symbolic representation provided above.

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, name="hours_worked_by_Ringo")
x2 = m.addVar(lb=0, name="hours_worked_by_Mary")

# Set the objective function
m.setObjective(2*x1 + 8*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(15*x1 + 4*x2 >= 23, name="Combined_Quit_Index_Min")
m.addConstr(-8*x1 + 3*x2 >= 0, name="Ringo_Mary_Balance")
m.addConstr(15*x1 + 4*x2 <= 84, name="Combined_Quit_Index_Max")

# Optimize model
m.optimize()

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