## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Jean', 'hours worked by Laura', and 'hours worked by Mary'. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $6x_1 + 2x_2 + 8x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
- $1.81x_1 \geq 0$ (implicit, as there's no upper bound given for Jean's likelihood to quit index but an upper bound for the sum)
- $2.04x_2 \geq 0$ (implicit, similar reasoning)
- $2.11x_3 \geq 0$ (implicit, similar reasoning)
- $1.81x_1 + 2.04x_2 \geq 141$
- $2.04x_2 + 2.11x_3 \leq 327$
- $1.81x_1 + 2.04x_2 \leq 321$
- $1.81x_1 + 2.04x_2 + 2.11x_3 \leq 321$

## 4: Consider the upper bound for the likelihood to quit index
Given that the upper bound for the likelihood to quit index $r0$ is 487, and individual indices are $x_0=1.81$, $x_1=2.04$, $x_2=2.11$, we need to ensure that $x_1 \leq \frac{487}{1.81}$, $x_2 \leq \frac{487}{2.04}$, and $x_3 \leq \frac{487}{2.11}$.

## 5: Calculate the upper bounds for $x_1$, $x_2$, and $x_3$
- $x_1 \leq \frac{487}{1.81} \approx 269.06$
- $x_2 \leq \frac{487}{2.04} \approx 238.73$
- $x_3 \leq \frac{487}{2.11} \approx 230.81$

## 6: Formulate the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'hours worked by Jean'), ('x2', 'hours worked by Laura'), ('x3', 'hours worked by Mary')],
'objective_function': '6*x1 + 2*x2 + 8*x3',
'constraints': [
    '1.81*x1 + 2.04*x2 >= 141',
    '2.04*x2 + 2.11*x3 <= 327',
    '1.81*x1 + 2.04*x2 <= 321',
    '1.81*x1 + 2.04*x2 + 2.11*x3 <= 321',
    'x1 >= 0',
    'x2 >= 0',
    'x3 >= 0',
    'x1 <= 269.06',
    'x2 <= 238.73',
    'x3 <= 230.81'
]
}
```

## 7: Implement the problem using Gurobi
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='hours_worked_by_Jean', lb=0, ub=269.06)
    x2 = model.addVar(name='hours_worked_by_Laura', lb=0, ub=238.73)
    x3 = model.addVar(name='hours_worked_by_Mary', lb=0, ub=230.81)

    # Define the objective function
    model.setObjective(6 * x1 + 2 * x2 + 8 * x3, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(1.81 * x1 + 2.04 * x2 >= 141)
    model.addConstr(2.04 * x2 + 2.11 * x3 <= 327)
    model.addConstr(1.81 * x1 + 2.04 * x2 <= 321)
    model.addConstr(1.81 * x1 + 2.04 * x2 + 2.11 * x3 <= 321)

    # Optimize the model
    model.optimize()

    # Print the status of the model
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print('Optimal solution found.')
        print('Hours worked by Jean: ', x1.varValue)
        print('Hours worked by Laura: ', x2.varValue)
        print('Hours worked by Mary: ', x3.varValue)
        print('Objective function value: ', model.objVal)
    else:
        print('No optimal solution found.')

solve_optimization_problem()
```