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

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $4x_0 + 3x_1 + 8x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
- $7.97x_1 + 10.3x_2 \geq 19$
- $4.21x_0 + 10.3x_2 \geq 28$
- $4.21x_0 + 7.97x_1 + 10.3x_2 \geq 31$
- $4.21x_0 + 7.97x_1 + 10.3x_2 \geq 31$ (same as the previous one, redundant)
- $16.79x_1 + 5.13x_2 \geq 61$
- $2.1x_0 + 5.13x_2 \geq 61$
- $2.1x_0 + 16.79x_1 + 5.13x_2 \geq 32$
- $2.1x_0 + 16.79x_1 + 5.13x_2 \geq 32$ (same as the previous one, redundant)
- $-2x_0 + x_2 \geq 0$
- $4.21x_0 + 10.3x_2 \leq 49$
- $2.1x_0 + 5.13x_2 \leq 113$
- $2.1x_0 + 16.79x_1 \leq 151$
- $x_0$ can be non-integer, $x_1$ must be integer, $x_2$ must be integer.

## 4: Correcting and Simplifying Constraints
Upon review, notice that some constraints are redundant or can be simplified. The correct list of unique constraints to consider is:
- $7.97x_1 + 10.3x_2 \geq 19$
- $4.21x_0 + 10.3x_2 \geq 28$
- $4.21x_0 + 7.97x_1 + 10.3x_2 \geq 31$
- $16.79x_1 + 5.13x_2 \geq 61$
- $2.1x_0 + 5.13x_2 \geq 61$
- $-2x_0 + x_2 \geq 0$
- $4.21x_0 + 10.3x_2 \leq 49$
- $2.1x_0 + 5.13x_2 \leq 113$
- $2.1x_0 + 16.79x_1 \leq 151$

## 5: Symbolic Representation
The symbolic representation of the problem is:
```json
{
'sym_variables': [('x0', 'hours worked by Paul'), ('x1', 'hours worked by Jean'), ('x2', 'hours worked by Laura')],
'objective_function': '4*x0 + 3*x1 + 8*x2',
'constraints': [
    '7.97*x1 + 10.3*x2 >= 19',
    '4.21*x0 + 10.3*x2 >= 28',
    '4.21*x0 + 7.97*x1 + 10.3*x2 >= 31',
    '16.79*x1 + 5.13*x2 >= 61',
    '2.1*x0 + 5.13*x2 >= 61',
    '-2*x0 + x2 >= 0',
    '4.21*x0 + 10.3*x2 <= 49',
    '2.1*x0 + 5.13*x2 <= 113',
    '2.1*x0 + 16.79*x1 <= 151'
]
}
```

## 6: Gurobi Code
Now, let's implement this in Gurobi:
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="hours_worked_by_Paul", lb=-float('inf'), ub=float('inf'))
    x1 = model.addVar(name="hours_worked_by_Jean", lb=0, ub=float('inf'), integrality=1)
    x2 = model.addVar(name="hours_worked_by_Laura", lb=0, ub=float('inf'), integrality=1)

    # Objective function
    model.setObjective(4*x0 + 3*x1 + 8*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(7.97*x1 + 10.3*x2 >= 19)
    model.addConstr(4.21*x0 + 10.3*x2 >= 28)
    model.addConstr(4.21*x0 + 7.97*x1 + 10.3*x2 >= 31)
    model.addConstr(16.79*x1 + 5.13*x2 >= 61)
    model.addConstr(2.1*x0 + 5.13*x2 >= 61)
    model.addConstr(-2*x0 + x2 >= 0)
    model.addConstr(4.21*x0 + 10.3*x2 <= 49)
    model.addConstr(2.1*x0 + 5.13*x2 <= 113)
    model.addConstr(2.1*x0 + 16.79*x1 <= 151)

    # Optimize
    model.optimize()

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

solve_optimization_problem()
```