## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' corresponds to 'hours worked by Bobby'
- 'x2' corresponds to 'hours worked by Peggy'
- 'x3' corresponds to 'hours worked by Dale'
- 'x4' corresponds to 'hours worked by Paul'
- 'x5' corresponds to 'hours worked by Mary'

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is: 
\[ 5.99x_1 + 7.61x_2 + 6.69x_3 + 4.64x_4 + 6.21x_5 \]

## 3: List the constraints in symbolic notation
The constraints are:
- \( 18x_1 \geq 0 \) (Implicit, as x1 cannot be negative)
- \( 9x_2 \geq 0 \) (Implicit, as x2 cannot be negative)
- \( 11x_3 \geq 0 \) (Implicit, as x3 cannot be negative)
- \( 11x_4 \geq 0 \) (Implicit, as x4 cannot be negative)
- \( 6x_5 \geq 0 \) (Implicit, as x5 cannot be negative)
- \( 18x_1 + 11x_3 + 6x_5 \geq 54 \)
- \( 18x_1 + 9x_2 + 6x_5 \geq 54 \)
- \( 11x_3 + 11x_4 + 6x_5 \geq 54 \)
- \( 18x_1 + 11x_3 + 6x_5 \geq 37 \)
- \( 18x_1 + 9x_2 + 6x_5 \geq 37 \)
- \( 11x_3 + 11x_4 + 6x_5 \geq 37 \)
- \( 18x_1 + 11x_3 + 6x_5 \geq 39 \)
- \( 18x_1 + 9x_2 + 6x_5 \geq 39 \)
- \( 11x_3 + 11x_4 + 6x_5 \geq 39 \)
- \( -4x_1 + x_4 \geq 0 \)
- \( 18x_1 + 9x_2 \leq 240 \)
- \( 11x_3 + 6x_5 \leq 201 \)
- \( 9x_2 + 11x_4 \leq 170 \)
- \( 9x_2 + 6x_5 \leq 138 \)
- \( 18x_1 + 11x_4 \leq 184 \)
- \( 9x_2 + 11x_3 + 6x_5 \leq 213 \)
- \( 18x_1 + 11x_3 + 11x_4 \leq 208 \)
- \( 11x_3 + 11x_4 + 6x_5 \leq 113 \)
- \( 18x_1 + 9x_2 + 11x_3 + 11x_4 + 6x_5 \leq 113 \)

## 4: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Bobby'), 
        ('x2', 'hours worked by Peggy'), 
        ('x3', 'hours worked by Dale'), 
        ('x4', 'hours worked by Paul'), 
        ('x5', 'hours worked by Mary')
    ], 
    'objective_function': '5.99*x1 + 7.61*x2 + 6.69*x3 + 4.64*x4 + 6.21*x5', 
    'constraints': [
        '18*x1 + 11*x3 + 6*x5 >= 54',
        '18*x1 + 9*x2 + 6*x5 >= 54',
        '11*x3 + 11*x4 + 6*x5 >= 54',
        '18*x1 + 11*x3 + 6*x5 >= 37',
        '18*x1 + 9*x2 + 6*x5 >= 37',
        '11*x3 + 11*x4 + 6*x5 >= 37',
        '18*x1 + 11*x3 + 6*x5 >= 39',
        '18*x1 + 9*x2 + 6*x5 >= 39',
        '11*x3 + 11*x4 + 6*x5 >= 39',
        '-4*x1 + x4 >= 0',
        '18*x1 + 9*x2 <= 240',
        '11*x3 + 6*x5 <= 201',
        '9*x2 + 11*x4 <= 170',
        '9*x2 + 6*x5 <= 138',
        '18*x1 + 11*x4 <= 184',
        '9*x2 + 11*x3 + 6*x5 <= 213',
        '18*x1 + 11*x3 + 11*x4 <= 208',
        '11*x3 + 11*x4 + 6*x5 <= 113',
        '18*x1 + 9*x2 + 11*x3 + 11*x4 + 6*x5 <= 113'
    ]
}
```

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

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

    # Define the variables
    x1 = model.addVar(name='x1', lb=0)  # hours worked by Bobby
    x2 = model.addVar(name='x2', lb=0)  # hours worked by Peggy
    x3 = model.addVar(name='x3', lb=0)  # hours worked by Dale
    x4 = model.addVar(name='x4', lb=0)  # hours worked by Paul
    x5 = model.addVar(name='x5', lb=0)  # hours worked by Mary

    # Define the objective function
    model.setObjective(5.99 * x1 + 7.61 * x2 + 6.69 * x3 + 4.64 * x4 + 6.21 * x5, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(18 * x1 + 11 * x3 + 6 * x5 >= 54)
    model.addConstr(18 * x1 + 9 * x2 + 6 * x5 >= 54)
    model.addConstr(11 * x3 + 11 * x4 + 6 * x5 >= 54)
    model.addConstr(18 * x1 + 11 * x3 + 6 * x5 >= 37)
    model.addConstr(18 * x1 + 9 * x2 + 6 * x5 >= 37)
    model.addConstr(11 * x3 + 11 * x4 + 6 * x5 >= 37)
    model.addConstr(18 * x1 + 11 * x3 + 6 * x5 >= 39)
    model.addConstr(18 * x1 + 9 * x2 + 6 * x5 >= 39)
    model.addConstr(11 * x3 + 11 * x4 + 6 * x5 >= 39)
    model.addConstr(-4 * x1 + x4 >= 0)
    model.addConstr(18 * x1 + 9 * x2 <= 240)
    model.addConstr(11 * x3 + 6 * x5 <= 201)
    model.addConstr(9 * x2 + 11 * x4 <= 170)
    model.addConstr(9 * x2 + 6 * x5 <= 138)
    model.addConstr(18 * x1 + 11 * x4 <= 184)
    model.addConstr(9 * x2 + 11 * x3 + 6 * x5 <= 213)
    model.addConstr(18 * x1 + 11 * x3 + 11 * x4 <= 208)
    model.addConstr(11 * x3 + 11 * x4 + 6 * x5 <= 113)
    model.addConstr(18 * x1 + 9 * x2 + 11 * x3 + 11 * x4 + 6 * x5 <= 113)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objval)
        print('Hours worked by Bobby: ', x1.varValue)
        print('Hours worked by Peggy: ', x2.varValue)
        print('Hours worked by Dale: ', x3.varValue)
        print('Hours worked by Paul: ', x4.varValue)
        print('Hours worked by Mary: ', x5.varValue)
    else:
        print('No optimal solution found')

optimize_problem()
```