To solve the optimization problem described, 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.

Given:
- Variables: hours worked by Laura (let's denote this as \(x_1\)), hours worked by George (denoted as \(x_2\))
- Objective Function: Minimize \(8.0x_1 + 1.14x_2\)
- Constraints:
  1. \(x_1\) and \(x_2\) must be integers.
  2. Likelihood to quit index for Laura is \(1\), for George is \(9\).
  3. Combined likelihood to quit index: \(1x_1 + 9x_2 \geq 37\)
  4. Combined likelihood to quit index must be exactly at least 37 (which seems redundant with the previous point but implies \(1x_1 + 9x_2 = 37\) or more is acceptable, however given as "at minimum" it suggests a lower bound without an upper limit directly from this statement alone).
  5. \(-9x_1 + 6x_2 \geq 0\)
  6. Combined likelihood to quit index has an upper bound: \(1x_1 + 9x_2 \leq 79\)

The symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'hours worked by Laura'), ('x2', 'hours worked by George')],
  'objective_function': '8.0*x1 + 1.14*x2',
  'constraints': [
    'x1 == int(x1)', 
    'x2 == int(x2)', 
    '1*x1 + 9*x2 >= 37', 
    '-9*x1 + 6*x2 >= 0', 
    '1*x1 + 9*x2 <= 79'
  ]
}
```
Note: The integer constraints for \(x_1\) and \(x_2\) are represented in a way that indicates they should be integers, but in the actual Gurobi model, we will use the `GRB.INTEGER` attribute to enforce this.

Here is the Gurobi code that solves the problem:
```python
from gurobipy import *

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

# Add variables
x1 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Laura")
x2 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_George")

# Set the objective function
m.setObjective(8.0*x1 + 1.14*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(1*x1 + 9*x2 >= 37, "Combined_Likelihood_Lower_Bound")
m.addConstr(-9*x1 + 6*x2 >= 0, "Additional_Constraint")
m.addConstr(1*x1 + 9*x2 <= 79, "Combined_Likelihood_Upper_Bound")

# Optimize the model
m.optimize()

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