## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' or 'hours worked by Paul'
- 'x2' or 'hours worked by Hank'
- 'x3' or 'hours worked by Bobby'
- 'x4' or 'hours worked by John'
- 'x5' or 'hours worked by Dale'
- 'x6' or 'hours worked by George'

## Step 2: Define the coefficients for each variable in the objective function
The objective function to maximize is: 
\[ 4x_1 + 6x_2 + 6x_3 + 3x_4 + 9x_5 + 6x_6 \]

## Step 3: List all the constraints
Constraints are as follows:
- \( 14x_1 \leq 275 \)
- \( 1x_2 \leq 275 \)
- \( 18x_3 \leq 275 \)
- \( 1x_4 \leq 275 \)
- \( 6x_5 \leq 275 \)
- \( 4x_6 \leq 275 \)
- \( 1x_2 + 4x_6 \geq 23 \)
- \( 18x_3 + 4x_6 \geq 19 \)
- \( 18x_3 + 6x_5 \geq 34 \)
- \( 14x_1 + 6x_5 \geq 32 \)
- \( 1x_4 + 4x_6 \geq 45 \)
- \( 1x_2 + 6x_5 + 4x_6 \geq 28 \)
- \( 1x_2 + 1x_4 + 4x_6 \geq 28 \)
- \( 18x_3 + 1x_4 + 4x_6 \geq 28 \)
- \( 1x_2 + 18x_3 + 4x_6 \geq 28 \)
- \( 1x_2 + 6x_5 + 4x_6 \geq 33 \)
- \( 1x_2 + 1x_4 + 4x_6 \geq 33 \)
- \( 18x_3 + 1x_4 + 4x_6 \geq 33 \)
- \( 1x_2 + 18x_3 + 1x_4 \geq 33 \)
- \( 1x_2 + 18x_3 + 4x_6 \geq 33 \)
- \( 1x_2 + 6x_5 + 4x_6 \geq 25 \)
- \( 1x_2 + 1x_4 + 4x_6 \geq 25 \)
- \( 18x_3 + 1x_4 + 4x_6 \geq 25 \)
- \( 1x_2 + 18x_3 + 1x_4 \geq 25 \)
- \( 1x_2 + 18x_3 + 4x_6 \geq 25 \)
- \( 1x_2 + 6x_5 + 4x_6 \geq 37 \)
- \( 1x_2 + 1x_4 + 4x_6 \geq 37 \)
- \( 18x_3 + 1x_4 + 4x_6 \geq 37 \)
- \( 1x_2 + 18x_3 + 1x_4 \geq 37 \)
- \( 1x_2 + 18x_3 + 4x_6 \geq 37 \)
- \( 1x_2 + 6x_5 + 4x_6 \geq 45 \)
- \( 1x_2 + 1x_4 + 4x_6 \geq 45 \)
- \( 18x_3 + 1x_4 + 4x_6 \geq 45 \)
- \( 1x_2 + 18x_3 + 1x_4 \geq 45 \)
- \( 1x_2 + 18x_3 + 4x_6 \geq 45 \)
- \( 1x_4 + 4x_6 \leq 228 \)
- \( 14x_1 + 1x_2 \leq 258 \)
- \( 6x_5 + 4x_6 \leq 70 \)
- \( 1x_2 + 18x_3 \leq 166 \)
- \( 14x_1 + 1x_4 \leq 93 \)
- \( 14x_1 + 6x_5 \leq 77 \)
- \( 14x_1 + 4x_6 \leq 128 \)
- \( 14x_1 + 1x_2 + 18x_3 + 1x_4 + 6x_5 + 4x_6 \leq 128 \)

## 4: Write down the problem in a symbolic representation
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Paul'),
        ('x2', 'hours worked by Hank'),
        ('x3', 'hours worked by Bobby'),
        ('x4', 'hours worked by John'),
        ('x5', 'hours worked by Dale'),
        ('x6', 'hours worked by George')
    ],
    'objective_function': '4*x1 + 6*x2 + 6*x3 + 3*x4 + 9*x5 + 6*x6',
    'constraints': [
        '14*x1 <= 275',
        '1*x2 <= 275',
        '18*x3 <= 275',
        '1*x4 <= 275',
        '6*x5 <= 275',
        '4*x6 <= 275',
        '1*x2 + 4*x6 >= 23',
        '18*x3 + 4*x6 >= 19',
        '18*x3 + 6*x5 >= 34',
        '14*x1 + 6*x5 >= 32',
        '1*x4 + 4*x6 >= 45',
        '1*x2 + 6*x5 + 4*x6 >= 28',
        '1*x2 + 1*x4 + 4*x6 >= 28',
        '18*x3 + 1*x4 + 4*x6 >= 28',
        '1*x2 + 18*x3 + 4*x6 >= 28',
        '1*x2 + 6*x5 + 4*x6 >= 33',
        '1*x2 + 1*x4 + 4*x6 >= 33',
        '18*x3 + 1*x4 + 4*x6 >= 33',
        '1*x2 + 18*x3 + 1*x4 >= 33',
        '1*x2 + 18*x3 + 4*x6 >= 33',
        '1*x2 + 6*x5 + 4*x6 >= 25',
        '1*x2 + 1*x4 + 4*x6 >= 25',
        '18*x3 + 1*x4 + 4*x6 >= 25',
        '1*x2 + 18*x3 + 1*x4 >= 25',
        '1*x2 + 18*x3 + 4*x6 >= 25',
        '1*x2 + 6*x5 + 4*x6 >= 37',
        '1*x2 + 1*x4 + 4*x6 >= 37',
        '18*x3 + 1*x4 + 4*x6 >= 37',
        '1*x2 + 18*x3 + 1*x4 >= 37',
        '1*x2 + 18*x3 + 4*x6 >= 37',
        '1*x2 + 6*x5 + 4*x6 >= 45',
        '1*x2 + 1*x4 + 4*x6 >= 45',
        '18*x3 + 1*x4 + 4*x6 >= 45',
        '1*x2 + 18*x3 + 1*x4 >= 45',
        '1*x2 + 18*x3 + 4*x6 >= 45',
        '1*x4 + 4*x6 <= 228',
        '14*x1 + 1*x2 <= 258',
        '6*x5 + 4*x6 <= 70',
        '1*x2 + 18*x3 <= 166',
        '14*x1 + 1*x4 <= 93',
        '14*x1 + 6*x5 <= 77',
        '14*x1 + 4*x6 <= 128',
        '14*x1 + 1*x2 + 18*x3 + 1*x4 + 6*x5 + 4*x6 <= 128'
    ]
}
```

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

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(name="hours_worked_by_Paul", lb=0)
x2 = m.addVar(name="hours_worked_by_Hank", lb=0)
x3 = m.addVar(name="hours_worked_by_Bobby", lb=0)
x4 = m.addVar(name="hours_worked_by_John", lb=0)
x5 = m.addVar(name="hours_worked_by_Dale", lb=0)
x6 = m.addVar(name="hours_worked_by_George", lb=0)

# Define the objective function
m.setObjective(4*x1 + 6*x2 + 6*x3 + 3*x4 + 9*x5 + 6*x6, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(14*x1 <= 275)
m.addConstr(1*x2 <= 275)
m.addConstr(18*x3 <= 275)
m.addConstr(1*x4 <= 275)
m.addConstr(6*x5 <= 275)
m.addConstr(4*x6 <= 275)
m.addConstr(1*x2 + 4*x6 >= 23)
m.addConstr(18*x3 + 4*x6 >= 19)
m.addConstr(18*x3 + 6*x5 >= 34)
m.addConstr(14*x1 + 6*x5 >= 32)
m.addConstr(1*x4 + 4*x6 >= 45)
m.addConstr(1*x2 + 6*x5 + 4*x6 >= 28)
m.addConstr(1*x2 + 1*x4 + 4*x6 >= 28)
m.addConstr(18*x3 + 1*x4 + 4*x6 >= 28)
m.addConstr(1*x2 + 18*x3 + 4*x6 >= 28)
m.addConstr(1*x2 + 6*x5 + 4*x6 >= 33)
m.addConstr(1*x2 + 1*x4 + 4*x6 >= 33)
m.addConstr(18*x3 + 1*x4 + 4*x6 >= 33)
m.addConstr(1*x2 + 18*x3 + 1*x4 >= 33)
m.addConstr(1*x2 + 18*x3 + 4*x6 >= 33)
m.addConstr(1*x2 + 6*x5 + 4*x6 >= 25)
m.addConstr(1*x2 + 1*x4 + 4*x6 >= 25)
m.addConstr(18*x3 + 1*x4 + 4*x6 >= 25)
m.addConstr(1*x2 + 18*x3 + 1*x4 >= 25)
m.addConstr(1*x2 + 18*x3 + 4*x6 >= 25)
m.addConstr(1*x2 + 6*x5 + 4*x6 >= 37)
m.addConstr(1*x2 + 1*x4 + 4*x6 >= 37)
m.addConstr(18*x3 + 1*x4 + 4*x6 >= 37)
m.addConstr(1*x2 + 18*x3 + 1*x4 >= 37)
m.addConstr(1*x2 + 18*x3 + 4*x6 >= 37)
m.addConstr(1*x2 + 6*x5 + 4*x6 >= 45)
m.addConstr(1*x2 + 1*x4 + 4*x6 >= 45)
m.addConstr(18*x3 + 1*x4 + 4*x6 >= 45)
m.addConstr(1*x2 + 18*x3 + 1*x4 >= 45)
m.addConstr(1*x2 + 18*x3 + 4*x6 >= 45)
m.addConstr(1*x4 + 4*x6 <= 228)
m.addConstr(14*x1 + 1*x2 <= 258)
m.addConstr(6*x5 + 4*x6 <= 70)
m.addConstr(1*x2 + 18*x3 <= 166)
m.addConstr(14*x1 + 1*x4 <= 93)
m.addConstr(14*x1 + 6*x5 <= 77)
m.addConstr(14*x1 + 4*x6 <= 128)
m.addConstr(14*x1 + 1*x2 + 18*x3 + 1*x4 + 6*x5 + 4*x6 <= 128)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Paul: ", x1.varValue)
    print("Hours worked by Hank: ", x2.varValue)
    print("Hours worked by Bobby: ", x3.varValue)
    print("Hours worked by John: ", x4.varValue)
    print("Hours worked by Dale: ", x5.varValue)
    print("Hours worked by George: ", x6.varValue)
else:
    print("The model is infeasible")
```