## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Laura', 'hours worked by John', 'hours worked by George', and 'hours worked by Hank', which can be represented symbolically as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $3.4x_0 + 3.75x_1 + 9.58x_2 + 9.08x_3$.

## Step 3: List the constraints in symbolic notation
1. $8x_1 + 6x_3 \geq 45$
2. $28x_0 + 8x_1 \geq 55$
3. $28x_0 + 8x_1 + 27x_2 + 6x_3 \geq 55$
4. $4x_0 + 2x_1 \geq 30$
5. $4x_0 + 11x_3 \geq 44$
6. $6x_2 + 11x_3 \geq 31$
7. $4x_0 + 6x_2 \geq 30$
8. $4x_0 + 6x_2 + 11x_3 \geq 32$
9. $4x_0 + 2x_1 + 11x_3 \geq 32$
10. $4x_0 + 6x_2 + 11x_3 \geq 45$
11. $4x_0 + 2x_1 + 11x_3 \geq 45$
12. $4x_0 + 2x_1 + 6x_2 + 11x_3 \geq 45$
13. $-10x_0 + 3x_3 \geq 0$
14. $28x_0 + 6x_3 \leq 261$
15. $27x_2 + 6x_3 \leq 273$
16. $28x_0 + 8x_1 \leq 203$
17. $2x_1 + 6x_2 \leq 93$
18. $4x_0 + 11x_3 \leq 103$

## 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Laura'),
        ('x1', 'hours worked by John'),
        ('x2', 'hours worked by George'),
        ('x3', 'hours worked by Hank')
    ],
    'objective_function': '3.4*x0 + 3.75*x1 + 9.58*x2 + 9.08*x3',
    'constraints': [
        '8*x1 + 6*x3 >= 45',
        '28*x0 + 8*x1 >= 55',
        '28*x0 + 8*x1 + 27*x2 + 6*x3 >= 55',
        '4*x0 + 2*x1 >= 30',
        '4*x0 + 11*x3 >= 44',
        '6*x2 + 11*x3 >= 31',
        '4*x0 + 6*x2 >= 30',
        '4*x0 + 6*x2 + 11*x3 >= 32',
        '4*x0 + 2*x1 + 11*x3 >= 32',
        '4*x0 + 6*x2 + 11*x3 >= 45',
        '4*x0 + 2*x1 + 11*x3 >= 45',
        '4*x0 + 2*x1 + 6*x2 + 11*x3 >= 45',
        '-10*x0 + 3*x3 >= 0',
        '28*x0 + 6*x3 <= 261',
        '27*x2 + 6*x3 <= 273',
        '28*x0 + 8*x1 <= 203',
        '2*x1 + 6*x2 <= 93',
        '4*x0 + 11*x3 <= 103'
    ]
}
```

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

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

    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Laura
    x1 = model.addVar(name="x1", lb=0, integrality=gurobi.GRB.INTEGER)  # hours worked by John
    x2 = model.addVar(name="x2", lb=0)  # hours worked by George
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Hank

    # Objective function
    model.setObjective(3.4 * x0 + 3.75 * x1 + 9.58 * x2 + 9.08 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(8 * x1 + 6 * x3 >= 45)
    model.addConstr(28 * x0 + 8 * x1 >= 55)
    model.addConstr(28 * x0 + 8 * x1 + 27 * x2 + 6 * x3 >= 55)
    model.addConstr(4 * x0 + 2 * x1 >= 30)
    model.addConstr(4 * x0 + 11 * x3 >= 44)
    model.addConstr(6 * x2 + 11 * x3 >= 31)
    model.addConstr(4 * x0 + 6 * x2 >= 30)
    model.addConstr(4 * x0 + 6 * x2 + 11 * x3 >= 32)
    model.addConstr(4 * x0 + 2 * x1 + 11 * x3 >= 32)
    model.addConstr(4 * x0 + 6 * x2 + 11 * x3 >= 45)
    model.addConstr(4 * x0 + 2 * x1 + 11 * x3 >= 45)
    model.addConstr(4 * x0 + 2 * x1 + 6 * x2 + 11 * x3 >= 45)
    model.addConstr(-10 * x0 + 3 * x3 >= 0)
    model.addConstr(28 * x0 + 6 * x3 <= 261)
    model.addConstr(27 * x2 + 6 * x3 <= 273)
    model.addConstr(28 * x0 + 8 * x1 <= 203)
    model.addConstr(2 * x1 + 6 * x2 <= 93)
    model.addConstr(4 * x0 + 11 * x3 <= 103)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Laura: {x0.varValue}")
        print(f"Hours worked by John: {x1.varValue}")
        print(f"Hours worked by George: {x2.varValue}")
        print(f"Hours worked by Hank: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```