## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill', 'hours worked by Bobby', 'hours worked by John', and 'hours worked by Jean', which can be represented symbolically as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $1 \cdot x_0 + 7 \cdot x_1 + 7 \cdot x_2 + 2 \cdot x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $6.03x_0 \geq 0$ (not a constraint as it's about Bill's rating, but $6.03x_0$ is part of other constraints)
- $9.43x_0 \geq 0$ (not a constraint as it's about Bill's cost)
- $7.15x_1 \geq 0$ (not a constraint as it's about Bobby's rating)
- $11.64x_1 \geq 0$ (not a constraint as it's about Bobby's cost)
- $3.18x_2 \geq 0$ (not a constraint as it's about John's rating)
- $9.03x_2 \geq 0$ (not a constraint as it's about John's cost)
- $10.35x_3 \geq 0$ (not a constraint as it's about Jean's rating)
- $4.78x_3 \geq 0$ (not a constraint as it's about Jean's cost)
- $6.03x_0 + 7.15x_1 \geq 27$
- $3.18x_2 + 10.35x_3 \geq 53$
- $6.03x_0 + 7.15x_1 + 10.35x_3 \geq 49$
- $6.03x_0 + 7.15x_1 + 3.18x_2 + 10.35x_3 \geq 49$
- $11.64x_1 + 4.78x_3 \geq 30$
- $11.64x_1 + 9.03x_2 \geq 20$
- $9.03x_2 + 4.78x_3 \geq 31$
- $11.64x_1 + 9.03x_2 + 4.78x_3 \geq 29$
- $9.43x_0 + 11.64x_1 + 9.03x_2 + 4.78x_3 \geq 29$
- $7x_1 - x_2 \geq 0$
- $-7x_1 + 7x_3 \geq 0$
- $4x_0 - 10x_3 \geq 0$
- $6.03x_0 + 7.15x_1 + 3.18x_2 \leq 212$
- $9.43x_0 + 9.03x_2 + 4.78x_3 \leq 152$

## 4: Represent the problem in JSON format
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Bill'),
        ('x1', 'hours worked by Bobby'),
        ('x2', 'hours worked by John'),
        ('x3', 'hours worked by Jean')
    ],
    'objective_function': '1*x0 + 7*x1 + 7*x2 + 2*x3',
    'constraints': [
        '6.03*x0 + 7.15*x1 >= 27',
        '3.18*x2 + 10.35*x3 >= 53',
        '6.03*x0 + 7.15*x1 + 10.35*x3 >= 49',
        '6.03*x0 + 7.15*x1 + 3.18*x2 + 10.35*x3 >= 49',
        '11.64*x1 + 4.78*x3 >= 30',
        '11.64*x1 + 9.03*x2 >= 20',
        '9.03*x2 + 4.78*x3 >= 31',
        '11.64*x1 + 9.03*x2 + 4.78*x3 >= 29',
        '9.43*x0 + 11.64*x1 + 9.03*x2 + 4.78*x3 >= 29',
        '7*x1 - x2 >= 0',
        '-7*x1 + 7*x3 >= 0',
        '4*x0 - 10*x3 >= 0',
        '6.03*x0 + 7.15*x1 + 3.18*x2 <= 212',
        '9.43*x0 + 9.03*x2 + 4.78*x3 <= 152'
    ]
}
```

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

def optimize_problem():
    model = gurobi.Model()
    
    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Bill
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Bobby
    x2 = model.addVar(name="x2", lb=0)  # hours worked by John
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Jean

    # Objective function
    model.setObjective(1*x0 + 7*x1 + 7*x2 + 2*x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(6.03*x0 + 7.15*x1 >= 27)
    model.addConstr(3.18*x2 + 10.35*x3 >= 53)
    model.addConstr(6.03*x0 + 7.15*x1 + 10.35*x3 >= 49)
    model.addConstr(6.03*x0 + 7.15*x1 + 3.18*x2 + 10.35*x3 >= 49)
    model.addConstr(11.64*x1 + 4.78*x3 >= 30)
    model.addConstr(11.64*x1 + 9.03*x2 >= 20)
    model.addConstr(9.03*x2 + 4.78*x3 >= 31)
    model.addConstr(11.64*x1 + 9.03*x2 + 4.78*x3 >= 29)
    model.addConstr(9.43*x0 + 11.64*x1 + 9.03*x2 + 4.78*x3 >= 29)
    model.addConstr(7*x1 - x2 >= 0)
    model.addConstr(-7*x1 + 7*x3 >= 0)
    model.addConstr(4*x0 - 10*x3 >= 0)
    model.addConstr(6.03*x0 + 7.15*x1 + 3.18*x2 <= 212)
    model.addConstr(9.43*x0 + 9.03*x2 + 4.78*x3 <= 152)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Hours worked by Bill: ", x0.varValue)
        print("Hours worked by Bobby: ", x1.varValue)
        print("Hours worked by John: ", x2.varValue)
        print("Hours worked by Jean: ", x3.varValue)
    else:
        print("The model is infeasible")

optimize_problem()
```