## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Ringo', 'hours worked by Jean', and 'hours worked by Paul', 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 $6x_0 + 1x_1 + 6x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
- $7x_1 + 3x_2 \geq 15$
- $18x_0 + 7x_1 + 3x_2 \geq 24$
- $2x_1 + 11x_2 \geq 36$
- $6x_0 + 2x_1 + 11x_2 \geq 36$
- $-x_1 + x_2 \geq 0$
- $2x_1 + 11x_2 \leq 111$
- $x_0, x_1, x_2$ can be non-integer.

## 4: Represent the problem in the required JSON format
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Ringo'),
        ('x1', 'hours worked by Jean'),
        ('x2', 'hours worked by Paul')
    ],
    'objective_function': '6*x0 + 1*x1 + 6*x2',
    'constraints': [
        '7*x1 + 3*x2 >= 15',
        '18*x0 + 7*x1 + 3*x2 >= 24',
        '2*x1 + 11*x2 >= 36',
        '6*x0 + 2*x1 + 11*x2 >= 36',
        '-x1 + x2 >= 0',
        '2*x1 + 11*x2 <= 111'
    ]
}
```

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

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

    # Define the variables
    x0 = model.addVar(name='hours_worked_by_Ringo', lb=0)
    x1 = model.addVar(name='hours_worked_by_Jean', lb=0)
    x2 = model.addVar(name='hours_worked_by_Paul', lb=0)

    # Define the objective function
    model.setObjective(6*x0 + x1 + 6*x2, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(7*x1 + 3*x2 >= 15)
    model.addConstr(18*x0 + 7*x1 + 3*x2 >= 24)
    model.addConstr(2*x1 + 11*x2 >= 36)
    model.addConstr(6*x0 + 2*x1 + 11*x2 >= 36)
    model.addConstr(-x1 + x2 >= 0)
    model.addConstr(2*x1 + 11*x2 <= 111)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Hours worked by Ringo: ', x0.varValue)
        print('Hours worked by Jean: ', x1.varValue)
        print('Hours worked by Paul: ', x2.varValue)
    else:
        print('The model is infeasible')

solve_optimization_problem()
```