## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Laura' and 'hours worked by George', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $8.0x_1 + 1.14x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
1. $1x_1 + 9x_2 \geq 37$
2. $1x_1 + 9x_2 \leq 79$
3. $-9x_1 + 6x_2 \geq 0$
4. $x_1$ is an integer
5. $x_2$ is an integer
Additionally, we have the likelihood to quit index constraints for Laura and George:
- Laura's index is 1: $1x_1 = 1x_1$ ( implicit in $x_1$'s definition)
- George's index is 9: $9x_2 = 9x_2$ (implicit in $x_2$'s definition)

## 4: Correct and Simplify Constraints
Given that the likelihood to quit index for Laura is 1 and for George is 9, and they contribute to a combined index, we interpret the constraints as:
- The combined likelihood index is $1x_1 + 9x_2$.
- The constraint that the combined index must be at least 37: $1x_1 + 9x_2 \geq 37$.
- The constraint that the combined index must be at most 79: $1x_1 + 9x_2 \leq 79$.
- And the constraint $-9x_1 + 6x_2 \geq 0$.

## 5: Symbolic Representation
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 + 9*x2 >= 37',
    'x1 + 9*x2 <= 79',
    '-9*x1 + 6*x2 >= 0',
    'x1 % 1 == 0', 
    'x2 % 1 == 0'
]
}
```

## 6: Gurobi Code
To solve this problem using Gurobi, we can use the following Python code:
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name='hours_worked_by_Laura', vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name='hours_worked_by_George', vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(8.0 * x1 + 1.14 * x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x1 + 9 * x2 >= 37)
    model.addConstr(x1 + 9 * x2 <= 79)
    model.addConstr(-9 * x1 + 6 * x2 >= 0)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Optimal solution found.')
        print(f'Hours worked by Laura: {x1.varValue}')
        print(f'Hours worked by George: {x2.varValue}')
        print(f'Objective function value: {model.objVal}')
    else:
        print('No optimal solution found.')

solve_optimization_problem()
```