## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Jean', 'hours worked by George', and 'hours worked by Bill', which can be represented symbolically as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $7.52x_1 + 9.79x_2 + 6.1x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
1. $3x_1 \geq 3$ (Jean's likelihood to quit index is 3, but this seems to be an individual index, not a product, so it should be directly $x_1 \geq 0$ and $3 \leq 3$ which doesn't limit $x_1$),
2. $2x_2 \geq 2$ (George's likelihood to quit index is 2, similarly, $x_2 \geq 0$ and $2 \leq 2$),
3. $8x_3 \geq 8$ (Bill's likelihood to quit index is 8, $x_3 \geq 0$ and $8 \leq 8$),
4. $3x_1 + 8x_3 \geq 14$,
5. $2x_2 + 8x_3 \geq 19$,
6. $3x_1 + 2x_2 \geq 13$,
7. $3x_1 + 2x_2 + 8x_3 \geq 22$,
8. $3x_1 + 2x_2 + 8x_3 \geq 22$ (duplicate of 7),
9. $9x_1 - 3x_3 \geq 0$,
10. $7x_1 - 2x_2 \geq 0$,
11. $2x_2 + 8x_3 \leq 48$,
12. $3x_1 + 8x_3 \leq 38$,
And the variable constraints:
- $x_1, x_2 \geq 0$ (non-negative hours),
- $x_3$ is an integer.

## 4: Correcting and Clarifying Constraints
Correcting the understanding of individual likelihood indices:
- The problem directly provides individual indices which seem to be constants rather than products with hours worked. Therefore, we focus on the constraints directly related to hours worked.

## 5: Symbolic Representation
The symbolic representation is:
```json
{
'sym_variables': [('x1', 'hours worked by Jean'), ('x2', 'hours worked by George'), ('x3', 'hours worked by Bill')],
'objective_function': '7.52*x1 + 9.79*x2 + 6.1*x3',
'constraints': [
    '3*x1 + 8*x3 >= 14',
    '2*x2 + 8*x3 >= 19',
    '3*x1 + 2*x2 >= 13',
    '3*x1 + 2*x2 + 8*x3 >= 22',
    '9*x1 - 3*x3 >= 0',
    '7*x1 - 2*x2 >= 0',
    '2*x2 + 8*x3 <= 48',
    '3*x1 + 8*x3 <= 38',
    'x3 % 1 == 0'  # x3 is an integer
]
}
```

## 6: Gurobi Code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name="hours_worked_by_Jean", lb=0)  # Non-negative
    x2 = model.addVar(name="hours_worked_by_George", lb=0)  # Non-negative
    x3 = model.addVar(name="hours_worked_by_Bill", lb=0, vtype=gurobi.GRB.INTEGER)  # Non-negative integer

    # Objective function
    model.setObjective(7.52 * x1 + 9.79 * x2 + 6.1 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(3 * x1 + 8 * x3 >= 14)
    model.addConstr(2 * x2 + 8 * x3 >= 19)
    model.addConstr(3 * x1 + 2 * x2 >= 13)
    model.addConstr(3 * x1 + 2 * x2 + 8 * x3 >= 22)
    model.addConstr(9 * x1 - 3 * x3 >= 0)
    model.addConstr(7 * x1 - 2 * x2 >= 0)
    model.addConstr(2 * x2 + 8 * x3 <= 48)
    model.addConstr(3 * x1 + 8 * x3 <= 38)

    # Solve the model
    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Jean: {x1.varValue}")
        print(f"Hours worked by George: {x2.varValue}")
        print(f"Hours worked by Bill: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status.")

solve_optimization_problem()
```