## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Jean', 'hours worked by Peggy', 'hours worked by Laura', 'hours worked by Hank' 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 $7.27x_1 + 5.88x_2 + 2.44x_3 + 4.19x_4$.

## Step 3: List the constraints in symbolic notation
The constraints given are:
1. $4x_1 + 19x_2 \geq 38$
2. $19x_2 + 23x_3 \geq 29$
3. $19x_2 + 22x_4 \geq 23$
4. $23x_3 + 22x_4 \geq 27$
5. $4x_1 + 19x_2 + 23x_3 + 22x_4 \geq 27$
6. $3x_1 - 8x_2 \geq 0$
7. $-2x_1 + 5x_3 \geq 0$
8. $19x_2 + 23x_3 \leq 199$
9. $4x_1 + 23x_3 \leq 169$
10. $4x_1 + 23x_3 + 22x_4 \leq 206$
11. $19x_2 + 23x_3 + 22x_4 \leq 135$
12. $4x_1 + 19x_2 + 23x_3 \leq 108$

## 4: Define the symbolic representation of the problem
The symbolic representation is:
```json
{
'sym_variables': [
    ('x1', 'hours worked by Jean'), 
    ('x2', 'hours worked by Peggy'), 
    ('x3', 'hours worked by Laura'), 
    ('x4', 'hours worked by Hank')
],
'objective_function': '7.27*x1 + 5.88*x2 + 2.44*x3 + 4.19*x4',
'constraints': [
    '4*x1 + 19*x2 >= 38',
    '19*x2 + 23*x3 >= 29',
    '19*x2 + 22*x4 >= 23',
    '23*x3 + 22*x4 >= 27',
    '4*x1 + 19*x2 + 23*x3 + 22*x4 >= 27',
    '3*x1 - 8*x2 >= 0',
    '-2*x1 + 5*x3 >= 0',
    '19*x2 + 23*x3 <= 199',
    '4*x1 + 23*x3 <= 169',
    '4*x1 + 23*x3 + 22*x4 <= 206',
    '19*x2 + 23*x3 + 22*x4 <= 135',
    '4*x1 + 19*x2 + 23*x3 <= 108'
]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='x1', lb=0)  # hours worked by Jean
    x2 = model.addVar(name='x2', lb=0)  # hours worked by Peggy
    x3 = model.addVar(name='x3', lb=0)  # hours worked by Laura
    x4 = model.addVar(name='x4', lb=0)  # hours worked by Hank

    # Define the objective function
    model.setObjective(7.27 * x1 + 5.88 * x2 + 2.44 * x3 + 4.19 * x4, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(4 * x1 + 19 * x2 >= 38)
    model.addConstr(19 * x2 + 23 * x3 >= 29)
    model.addConstr(19 * x2 + 22 * x4 >= 23)
    model.addConstr(23 * x3 + 22 * x4 >= 27)
    model.addConstr(4 * x1 + 19 * x2 + 23 * x3 + 22 * x4 >= 27)
    model.addConstr(3 * x1 - 8 * x2 >= 0)
    model.addConstr(-2 * x1 + 5 * x3 >= 0)
    model.addConstr(19 * x2 + 23 * x3 <= 199)
    model.addConstr(4 * x1 + 23 * x3 <= 169)
    model.addConstr(4 * x1 + 23 * x3 + 22 * x4 <= 206)
    model.addConstr(19 * x2 + 23 * x3 + 22 * x4 <= 135)
    model.addConstr(4 * x1 + 19 * x2 + 23 * x3 <= 108)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('x1: ', x1.varValue)
        print('x2: ', x2.varValue)
        print('x3: ', x3.varValue)
        print('x4: ', x4.varValue)
    else:
        print('The model is infeasible')

optimize_problem()
```