## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bobby', 'hours worked by Jean', 'hours worked by Hank', and 'hours worked by George', which can be represented symbolically as $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $9x_1 + 8x_2 + 2x_3 + 6x_4$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
1. $15x_2 + 17x_4 \geq 40$
2. $3x_1 + 15x_2 + 17x_4 \geq 38$
3. $3x_1 + 15x_2 + 12x_3 + 17x_4 \geq 38$
4. $-3x_3 + 10x_4 \geq 0$
5. $3x_1 + 17x_4 \leq 113$
6. $3x_1 + 12x_3 + 17x_4 \leq 52$

## 4: Consider the bounds and integrality constraints
- $x_1$ can be non-integer.
- $x_2$ can be non-integer.
- $x_3$ must be an integer.
- $x_4$ can be non-integer.

## 5: Represent the problem in the required JSON format
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Bobby'), 
        ('x2', 'hours worked by Jean'), 
        ('x3', 'hours worked by Hank'), 
        ('x4', 'hours worked by George')
    ], 
    'objective_function': '9*x1 + 8*x2 + 2*x3 + 6*x4', 
    'constraints': [
        '15*x2 + 17*x4 >= 40', 
        '3*x1 + 15*x2 + 17*x4 >= 38', 
        '3*x1 + 15*x2 + 12*x3 + 17*x4 >= 38', 
        '-3*x3 + 10*x4 >= 0', 
        '3*x1 + 17*x4 <= 113', 
        '3*x1 + 12*x3 + 17*x4 <= 52'
    ]
}
```

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

def solve_optimization_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 Jean
    x3 = model.addVar(name="x3", lb=0, integrality=1)  # hours worked by Hank
    x4 = model.addVar(name="x4", lb=0)  # hours worked by George

    # Define the objective function
    model.setObjective(9 * x1 + 8 * x2 + 2 * x3 + 6 * x4, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(15 * x2 + 17 * x4 >= 40)
    model.addConstr(3 * x1 + 15 * x2 + 17 * x4 >= 38)
    model.addConstr(3 * x1 + 15 * x2 + 12 * x3 + 17 * x4 >= 38)
    model.addConstr(-3 * x3 + 10 * x4 >= 0)
    model.addConstr(3 * x1 + 17 * x4 <= 113)
    model.addConstr(3 * x1 + 12 * x3 + 17 * x4 <= 52)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Bobby: {x1.varValue}")
        print(f"Hours worked by Jean: {x2.varValue}")
        print(f"Hours worked by Hank: {x3.varValue}")
        print(f"Hours worked by George: {x4.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```