## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Bill', 'hours worked by Hank', 'hours worked by Bobby', 'hours worked by John'] which can be represented symbolically as 'x0', 'x1', 'x2', 'x3' respectively.

## Step 2: Define the symbolic representation of the problem
The symbolic representation of the variables and their corresponding natural language objects is:
- 'x0' : 'hours worked by Bill'
- 'x1' : 'hours worked by Hank'
- 'x2' : 'hours worked by Bobby'
- 'x3' : 'hours worked by John'

## Step 3: Formulate the objective function
The objective function to maximize is: $9x_0 + 3x_1 + 2x_2 + 2x_3$

## 4: List the constraints
The constraints are:
- $6x_0 \geq 31$ 
- $6x_0 + 7x_2 \geq 31$ 
- $6x_0 + 6x_1 + 11x_3 \geq 37$ 
- $6x_0 + 6x_1 + 7x_2 \geq 37$ 
- $9x_1 + 18x_2 \geq 17$ 
- $9x_1 + 12x_3 \geq 20$ 
- $8x_0 + 9x_1 \geq 27$ 
- $18x_2 + 12x_3 \geq 19$ 
- $7x_0 + x_2 + 13x_3 \geq 26$ 
- $7x_0 + 4x_1 + x_2 \geq 26$ 
- $7x_0 + x_2 + 13x_3 \geq 27$ 
- $7x_0 + 4x_1 + x_2 \geq 27$ 
- $7x_2 + 11x_3 \leq 99$ 
- $6x_0 + 11x_3 \leq 52$ 
- $6x_0 + 7x_2 \leq 118$ 
- $6x_0 + 7x_2 + 11x_3 \leq 60$ 
- $6x_0 + 6x_1 + 7x_2 + 11x_3 \leq 60$ 
- $8x_0 + 18x_2 \leq 104$ 
- $9x_1 + 18x_2 \leq 63$ 
- $8x_0 + 12x_3 \leq 34$ 
- $8x_0 + 9x_1 \leq 72$ 
- $8x_0 + 9x_1 + 18x_2 + 12x_3 \leq 72$ 
- $7x_0 + 4x_1 \leq 64$ 
- $7x_0 + x_2 \leq 71$ 
- $7x_0 + 13x_3 \leq 106$ 
- $4x_1 + 13x_3 \leq 29$ 
- $7x_0 + 4x_1 + x_2 + 13x_3 \leq 29$

## 5: Write the Gurobi code
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Bill
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Hank
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Bobby
    x3 = model.addVar(name="x3", lb=0)  # hours worked by John

    # Define the objective function
    model.setObjective(9 * x0 + 3 * x1 + 2 * x2 + 2 * x3, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(6 * x0 >= 31)  
    model.addConstr(6 * x0 + 7 * x2 >= 31)  
    model.addConstr(6 * x0 + 6 * x1 + 11 * x3 >= 37)  
    model.addConstr(6 * x0 + 6 * x1 + 7 * x2 >= 37)  
    model.addConstr(9 * x1 + 18 * x2 >= 17)  
    model.addConstr(9 * x1 + 12 * x3 >= 20)  
    model.addConstr(8 * x0 + 9 * x1 >= 27)  
    model.addConstr(18 * x2 + 12 * x3 >= 19)  
    model.addConstr(7 * x0 + x2 + 13 * x3 >= 26)  
    model.addConstr(7 * x0 + 4 * x1 + x2 >= 26)  
    model.addConstr(7 * x0 + x2 + 13 * x3 >= 27)  
    model.addConstr(7 * x0 + 4 * x1 + x2 >= 27)  
    model.addConstr(7 * x2 + 11 * x3 <= 99)  
    model.addConstr(6 * x0 + 11 * x3 <= 52)  
    model.addConstr(6 * x0 + 7 * x2 <= 118)  
    model.addConstr(6 * x0 + 7 * x2 + 11 * x3 <= 60)  
    model.addConstr(6 * x0 + 6 * x1 + 7 * x2 + 11 * x3 <= 60)  
    model.addConstr(8 * x0 + 18 * x2 <= 104)  
    model.addConstr(9 * x1 + 18 * x2 <= 63)  
    model.addConstr(8 * x0 + 12 * x3 <= 34)  
    model.addConstr(8 * x0 + 9 * x1 <= 72)  
    model.addConstr(8 * x0 + 9 * x1 + 18 * x2 + 12 * x3 <= 72)  
    model.addConstr(7 * x0 + 4 * x1 <= 64)  
    model.addConstr(7 * x0 + x2 <= 71)  
    model.addConstr(7 * x0 + 13 * x3 <= 106)  
    model.addConstr(4 * x1 + 13 * x3 <= 29)  
    model.addConstr(7 * x0 + 4 * x1 + x2 + 13 * x3 <= 29)

    # 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 Hank: ", x1.varValue)
        print("Hours worked by Bobby: ", x2.varValue)
        print("Hours worked by John: ", x3.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'hours worked by Bill'), ('x1', 'hours worked by Hank'), ('x2', 'hours worked by Bobby'), ('x3', 'hours worked by John')],
    'objective_function': '9*x0 + 3*x1 + 2*x2 + 2*x3',
    'constraints': [
        '6*x0 >= 31', 
        '6*x0 + 7*x2 >= 31', 
        '6*x0 + 6*x1 + 11*x3 >= 37', 
        '6*x0 + 6*x1 + 7*x2 >= 37', 
        '9*x1 + 18*x2 >= 17', 
        '9*x1 + 12*x3 >= 20', 
        '8*x0 + 9*x1 >= 27', 
        '18*x2 + 12*x3 >= 19', 
        '7*x0 + x2 + 13*x3 >= 26', 
        '7*x0 + 4*x1 + x2 >= 26', 
        '7*x0 + x2 + 13*x3 >= 27', 
        '7*x0 + 4*x1 + x2 >= 27', 
        '7*x2 + 11*x3 <= 99', 
        '6*x0 + 11*x3 <= 52', 
        '6*x0 + 7*x2 <= 118', 
        '6*x0 + 7*x2 + 11*x3 <= 60', 
        '6*x0 + 6*x1 + 7*x2 + 11*x3 <= 60', 
        '8*x0 + 18*x2 <= 104', 
        '9*x1 + 18*x2 <= 63', 
        '8*x0 + 12*x3 <= 34', 
        '8*x0 + 9*x1 <= 72', 
        '8*x0 + 9*x1 + 18*x2 + 12*x3 <= 72', 
        '7*x0 + 4*x1 <= 64', 
        '7*x0 + x2 <= 71', 
        '7*x0 + 13*x3 <= 106', 
        '4*x1 + 13*x3 <= 29', 
        '7*x0 + 4*x1 + x2 + 13*x3 <= 29'
    ]
}
```