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

## Step 2: Define the symbolic representation of the problem
The objective function to minimize is $7x_0 + 4x_1 + x_2$.

## Step 3: List the constraints
The constraints given are:
- $4x_0 \leq 60$
- $7x_0 \leq 63$
- $3x_0 \leq 59$
- $7x_1 \leq 60$
- $2x_1 \leq 63$
- $5x_1 \leq 59$
- $2x_2 \leq 60$
- $1x_2 \leq 63$
- $5x_2 \leq 59$
- $4x_0 + 7x_1 \geq 19$
- $4x_0 + 2x_2 \geq 18$
- $7x_1 + 2x_2 \geq 8$
- $4x_0 + 7x_1 + 2x_2 \geq 8$
- $7x_0 + 2x_1 \geq 15$
- $7x_0 + x_2 \geq 8$
- $7x_0 + 2x_1 + x_2 \geq 8$
- $3x_0 + 5x_1 \geq 17$
- $5x_1 + 5x_2 \geq 6$
- $3x_0 + 5x_1 + 5x_2 \geq 6$
- $5x_1 - 9x_2 \geq 0$
- $-6x_0 + 2x_1 \geq 0$
- $4x_0 + 2x_2 \leq 46$
- $3x_0 + 5x_1 \leq 20$

## 4: Write down the problem in JSON format
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Ringo'),
        ('x1', 'hours worked by Laura'),
        ('x2', 'hours worked by Mary')
    ],
    'objective_function': '7*x0 + 4*x1 + x2',
    'constraints': [
        '4*x0 <= 60',
        '7*x0 <= 63',
        '3*x0 <= 59',
        '7*x1 <= 60',
        '2*x1 <= 63',
        '5*x1 <= 59',
        '2*x2 <= 60',
        'x2 <= 63',
        '5*x2 <= 59',
        '4*x0 + 7*x1 >= 19',
        '4*x0 + 2*x2 >= 18',
        '7*x1 + 2*x2 >= 8',
        '4*x0 + 7*x1 + 2*x2 >= 8',
        '7*x0 + 2*x1 >= 15',
        '7*x0 + x2 >= 8',
        '7*x0 + 2*x1 + x2 >= 8',
        '3*x0 + 5*x1 >= 17',
        '5*x1 + 5*x2 >= 6',
        '3*x0 + 5*x1 + 5*x2 >= 6',
        '5*x1 - 9*x2 >= 0',
        '-6*x0 + 2*x1 >= 0',
        '4*x0 + 2*x2 <= 46',
        '3*x0 + 5*x1 <= 20'
    ]
}
```

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

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

    # Define the variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Ringo
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Laura
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Mary

    # Define the objective function
    model.setObjective(7 * x0 + 4 * x1 + x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(4 * x0 <= 60)
    model.addConstr(7 * x0 <= 63)
    model.addConstr(3 * x0 <= 59)
    model.addConstr(7 * x1 <= 60)
    model.addConstr(2 * x1 <= 63)
    model.addConstr(5 * x1 <= 59)
    model.addConstr(2 * x2 <= 60)
    model.addConstr(x2 <= 63)
    model.addConstr(5 * x2 <= 59)
    model.addConstr(4 * x0 + 7 * x1 >= 19)
    model.addConstr(4 * x0 + 2 * x2 >= 18)
    model.addConstr(7 * x1 + 2 * x2 >= 8)
    model.addConstr(4 * x0 + 7 * x1 + 2 * x2 >= 8)
    model.addConstr(7 * x0 + 2 * x1 >= 15)
    model.addConstr(7 * x0 + x2 >= 8)
    model.addConstr(7 * x0 + 2 * x1 + x2 >= 8)
    model.addConstr(3 * x0 + 5 * x1 >= 17)
    model.addConstr(5 * x1 + 5 * x2 >= 6)
    model.addConstr(3 * x0 + 5 * x1 + 5 * x2 >= 6)
    model.addConstr(5 * x1 - 9 * x2 >= 0)
    model.addConstr(-6 * x0 + 2 * x1 >= 0)
    model.addConstr(4 * x0 + 2 * x2 <= 46)
    model.addConstr(3 * x0 + 5 * x1 <= 20)

    # 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 Laura: ", x1.varValue)
        print("Hours worked by Mary: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```