## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Paul', 'hours worked by Hank', 'hours worked by John', and 'hours worked by Mary'. Let's denote them as $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $3.68x_1 + 2.07x_2 + 5.86x_3 + 3.14x_4$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $8x_1 + 4x_2 \geq 12$
- $8x_1 + 13x_3 \geq 11$
- $13x_3 + 7x_4 \geq 13$
- $4x_2 + 7x_4 \geq 30$
- $4x_2 + 7x_4 \leq 127$
- $13x_3 + 7x_4 \leq 34$
- $8x_1 + 4x_2 \leq 96$
- $4x_2 + 13x_3 + 7x_4 \leq 38$
- $8x_1 + 4x_2 + 13x_3 + 7x_4 \leq 38$

## 4: Create the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Paul'), 
        ('x2', 'hours worked by Hank'), 
        ('x3', 'hours worked by John'), 
        ('x4', 'hours worked by Mary')
    ], 
    'objective_function': '3.68*x1 + 2.07*x2 + 5.86*x3 + 3.14*x4', 
    'constraints': [
        '8*x1 + 4*x2 >= 12',
        '8*x1 + 13*x3 >= 11',
        '13*x3 + 7*x4 >= 13',
        '4*x2 + 7*x4 >= 30',
        '4*x2 + 7*x4 <= 127',
        '13*x3 + 7*x4 <= 34',
        '8*x1 + 4*x2 <= 96',
        '4*x2 + 13*x3 + 7*x4 <= 38',
        '8*x1 + 4*x2 + 13*x3 + 7*x4 <= 38'
    ]
}
```

## 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='hours_worked_by_Paul', lb=0)  # No upper bound given
    x2 = model.addVar(name='hours_worked_by_Hank', lb=0)   # No upper bound given
    x3 = model.addVar(name='hours_worked_by_John', lb=0)   # No upper bound given
    x4 = model.addVar(name='hours_worked_by_Mary', lb=0)   # No upper bound given

    # Define the objective function
    model.setObjective(3.68*x1 + 2.07*x2 + 5.86*x3 + 3.14*x4, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(8*x1 + 4*x2 >= 12)
    model.addConstr(8*x1 + 13*x3 >= 11)
    model.addConstr(13*x3 + 7*x4 >= 13)
    model.addConstr(4*x2 + 7*x4 >= 30)
    model.addConstr(4*x2 + 7*x4 <= 127)
    model.addConstr(13*x3 + 7*x4 <= 34)
    model.addConstr(8*x1 + 4*x2 <= 96)
    model.addConstr(4*x2 + 13*x3 + 7*x4 <= 38)
    model.addConstr(8*x1 + 4*x2 + 13*x3 + 7*x4 <= 38)

    # Optimize the model
    model.optimize()

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

optimize_problem()
```