## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill', 'hours worked by John', 'hours worked by Laura', 'hours worked by Jean', which can be represented symbolically as $x_1, x_2, x_3, x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $7.73x_1 + 6.89x_2 + 8.6x_3 + 6.81x_4$.

## Step 3: List the constraints in symbolic notation
The constraints are:
1. $9x_1 \leq 174$ and $x_1 \geq 0$ is not explicitly stated but implied as a non-negative variable, however, the work quality rating for Bill is given as 9, which seems to be a fixed attribute rather than a constraint on $x_1$. Similar for others.
2. $8x_2 \leq 174$
3. $14x_3 \leq 174$
4. $13x_4 \leq 174$
5. $8x_2 + 14x_3 \geq 38$
6. $9x_1 + 13x_4 \geq 24$
7. $9x_1 + 8x_2 \geq 43$
8. $9x_1 + 8x_2 + 14x_3 + 13x_4 \geq 43$
9. $-x_1 + 2x_2 \geq 0$
10. $-6x_1 + 8x_3 \geq 0$
11. $8x_2 + 14x_3 \leq 174$
12. $9x_1 + 13x_4 \leq 121$
13. $9x_1 + 8x_2 \leq 119$
14. $8x_2 + 13x_4 \leq 118$
15. $9x_1 + 8x_2 + 14x_3 \leq 72$
16. $9x_1 + 8x_2 + 13x_4 \leq 100$
17. $8x_2 + 14x_3 + 13x_4 \leq 173$

## 4: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Bill'), 
        ('x2', 'hours worked by John'), 
        ('x3', 'hours worked by Laura'), 
        ('x4', 'hours worked by Jean')
    ], 
    'objective_function': '7.73*x1 + 6.89*x2 + 8.6*x3 + 6.81*x4', 
    'constraints': [
        '8*x2 + 14*x3 >= 38',
        '9*x1 + 13*x4 >= 24',
        '9*x1 + 8*x2 >= 43',
        '9*x1 + 8*x2 + 14*x3 + 13*x4 >= 43',
        '-x1 + 2*x2 >= 0',
        '-6*x1 + 8*x3 >= 0',
        '8*x2 + 14*x3 <= 174',
        '9*x1 + 13*x4 <= 121',
        '9*x1 + 8*x2 <= 119',
        '8*x2 + 13*x4 <= 118',
        '9*x1 + 8*x2 + 14*x3 <= 72',
        '9*x1 + 8*x2 + 13*x4 <= 100',
        '8*x2 + 14*x3 + 13*x4 <= 173'
    ]
}
```

## 5: Implement the optimization 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_Bill', lb=0)  # Bill
    x2 = model.addVar(name='hours_worked_by_John', lb=0)   # John
    x3 = model.addVar(name='hours_worked_by_Laura', lb=0)  # Laura
    x4 = model.addVar(name='hours_worked_by_Jean', lb=0)   # Jean

    # Objective function
    model.setObjective(7.73 * x1 + 6.89 * x2 + 8.6 * x3 + 6.81 * x4, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(8 * x2 + 14 * x3 >= 38)
    model.addConstr(9 * x1 + 13 * x4 >= 24)
    model.addConstr(9 * x1 + 8 * x2 >= 43)
    model.addConstr(9 * x1 + 8 * x2 + 14 * x3 + 13 * x4 >= 43)
    model.addConstr(-x1 + 2 * x2 >= 0)
    model.addConstr(-6 * x1 + 8 * x3 >= 0)
    model.addConstr(8 * x2 + 14 * x3 <= 174)
    model.addConstr(9 * x1 + 13 * x4 <= 121)
    model.addConstr(9 * x1 + 8 * x2 <= 119)
    model.addConstr(8 * x2 + 13 * x4 <= 118)
    model.addConstr(9 * x1 + 8 * x2 + 14 * x3 <= 72)
    model.addConstr(9 * x1 + 8 * x2 + 13 * x4 <= 100)
    model.addConstr(8 * x2 + 14 * x3 + 13 * x4 <= 173)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Hours worked by Bill: ', x1.varValue)
        print('Hours worked by John: ', x2.varValue)
        print('Hours worked by Laura: ', x3.varValue)
        print('Hours worked by Jean: ', x4.varValue)
    else:
        print('No optimal solution found')

solve_optimization_problem()
```