## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Mary', 'hours worked by Paul', 'hours worked by Jean', 'hours worked by Ringo', 'hours worked by Hank']. Let's denote them as $x_0, x_1, x_2, x_3, x_4$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $5x_0 + 5x_1 + 2x_2 + 9x_3 + 5x_4$.

## Step 3: List the constraints
The constraints are:
- $13x_0 \leq 405$
- $23x_0 \leq 454$
- $15x_1 \leq 405$
- $4x_1 \leq 454$
- $20x_2 \leq 405$
- $6x_2 \leq 454$
- $12x_3 \leq 405$
- $12x_3 \leq 454$
- $2x_4 \leq 405$
- $18x_4 \leq 454$
- $12x_3 + 2x_4 \geq 78$
- $13x_0 + 15x_1 \geq 45$
- $15x_1 + 12x_3 \geq 49$
- $13x_0 + 2x_4 \geq 48$
- $15x_1 + 20x_2 \geq 34$
- $20x_2 + 12x_3 \geq 40$
- $13x_0 + 20x_2 \geq 59$
- $13x_0 + 15x_1 + 2x_4 \geq 80$
- $15x_1 + 20x_2 + 2x_4 \geq 80$
- $13x_0 + 15x_1 + 2x_4 \geq 52$
- $15x_1 + 20x_2 + 2x_4 \geq 52$
- $4x_1 + 6x_2 \geq 56$
- $12x_3 + 18x_4 \geq 68$
- $23x_0 + 6x_2 + 12x_3 \geq 52$
- $13x_0 + 20x_2 \leq 120$
- $12x_3 + 2x_4 \leq 161$
- $13x_0 + 15x_1 \leq 247$
- $13x_0 + 15x_1 + 20x_2 + 12x_3 + 2x_4 \leq 247$
- $6x_2 + 12x_3 \leq 205$
- $23x_0 + 18x_4 \leq 294$
- $4x_1 + 6x_2 \leq 320$
- $4x_1 + 12x_3 + 18x_4 \leq 330$
- $23x_0 + 6x_2 + 12x_3 \leq 175$
- $4x_1 + 6x_2 + 18x_4 \leq 309$
- $23x_0 + 6x_2 + 18x_4 \leq 293$
- $23x_0 + 4x_1 + 6x_2 + 12x_3 + 18x_4 \leq 293$
- $x_0, x_1, x_2, x_3, x_4$ are integers.

## 4: Write down the problem in a symbolic representation
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Mary'), 
        ('x1', 'hours worked by Paul'), 
        ('x2', 'hours worked by Jean'), 
        ('x3', 'hours worked by Ringo'), 
        ('x4', 'hours worked by Hank')
    ], 
    'objective_function': '5*x0 + 5*x1 + 2*x2 + 9*x3 + 5*x4', 
    'constraints': [
        '13*x0 <= 405',
        '23*x0 <= 454',
        '15*x1 <= 405',
        '4*x1 <= 454',
        '20*x2 <= 405',
        '6*x2 <= 454',
        '12*x3 <= 405',
        '12*x3 <= 454',
        '2*x4 <= 405',
        '18*x4 <= 454',
        '12*x3 + 2*x4 >= 78',
        '13*x0 + 15*x1 >= 45',
        '15*x1 + 12*x3 >= 49',
        '13*x0 + 2*x4 >= 48',
        '15*x1 + 20*x2 >= 34',
        '20*x2 + 12*x3 >= 40',
        '13*x0 + 20*x2 >= 59',
        '13*x0 + 15*x1 + 2*x4 >= 80',
        '15*x1 + 20*x2 + 2*x4 >= 80',
        '13*x0 + 15*x1 + 2*x4 >= 52',
        '15*x1 + 20*x2 + 2*x4 >= 52',
        '4*x1 + 6*x2 >= 56',
        '12*x3 + 18*x4 >= 68',
        '23*x0 + 6*x2 + 12*x3 >= 52',
        '13*x0 + 20*x2 <= 120',
        '12*x3 + 2*x4 <= 161',
        '13*x0 + 15*x1 <= 247',
        '13*x0 + 15*x1 + 20*x2 + 12*x3 + 2*x4 <= 247',
        '6*x2 + 12*x3 <= 205',
        '23*x0 + 18*x4 <= 294',
        '4*x1 + 6*x2 <= 320',
        '4*x1 + 12*x3 + 18*x4 <= 330',
        '23*x0 + 6*x2 + 12*x3 <= 175',
        '4*x1 + 6*x2 + 18*x4 <= 309',
        '23*x0 + 6*x2 + 18*x4 <= 293',
        '23*x0 + 4*x1 + 6*x2 + 12*x3 + 18*x4 <= 293'
    ]
}
```

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

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name='x0', vtype=gurobi.GRB.INTEGER)  # hours worked by Mary
    x1 = model.addVar(name='x1', vtype=gurobi.GRB.INTEGER)  # hours worked by Paul
    x2 = model.addVar(name='x2', vtype=gurobi.GRB.INTEGER)  # hours worked by Jean
    x3 = model.addVar(name='x3', vtype=gurobi.GRB.INTEGER)  # hours worked by Ringo
    x4 = model.addVar(name='x4', vtype=gurobi.GRB.INTEGER)  # hours worked by Hank

    # Objective function
    model.setObjective(5 * x0 + 5 * x1 + 2 * x2 + 9 * x3 + 5 * x4, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(13 * x0 <= 405)
    model.addConstr(23 * x0 <= 454)
    model.addConstr(15 * x1 <= 405)
    model.addConstr(4 * x1 <= 454)
    model.addConstr(20 * x2 <= 405)
    model.addConstr(6 * x2 <= 454)
    model.addConstr(12 * x3 <= 405)
    model.addConstr(12 * x3 <= 454)
    model.addConstr(2 * x4 <= 405)
    model.addConstr(18 * x4 <= 454)
    model.addConstr(12 * x3 + 2 * x4 >= 78)
    model.addConstr(13 * x0 + 15 * x1 >= 45)
    model.addConstr(15 * x1 + 12 * x3 >= 49)
    model.addConstr(13 * x0 + 2 * x4 >= 48)
    model.addConstr(15 * x1 + 20 * x2 >= 34)
    model.addConstr(20 * x2 + 12 * x3 >= 40)
    model.addConstr(13 * x0 + 20 * x2 >= 59)
    model.addConstr(13 * x0 + 15 * x1 + 2 * x4 >= 80)
    model.addConstr(15 * x1 + 20 * x2 + 2 * x4 >= 80)
    model.addConstr(13 * x0 + 15 * x1 + 2 * x4 >= 52)
    model.addConstr(15 * x1 + 20 * x2 + 2 * x4 >= 52)
    model.addConstr(4 * x1 + 6 * x2 >= 56)
    model.addConstr(12 * x3 + 18 * x4 >= 68)
    model.addConstr(23 * x0 + 6 * x2 + 12 * x3 >= 52)
    model.addConstr(13 * x0 + 20 * x2 <= 120)
    model.addConstr(12 * x3 + 2 * x4 <= 161)
    model.addConstr(13 * x0 + 15 * x1 <= 247)
    model.addConstr(13 * x0 + 15 * x1 + 20 * x2 + 12 * x3 + 2 * x4 <= 247)
    model.addConstr(6 * x2 + 12 * x3 <= 205)
    model.addConstr(23 * x0 + 18 * x4 <= 294)
    model.addConstr(4 * x1 + 6 * x2 <= 320)
    model.addConstr(4 * x1 + 12 * x3 + 18 * x4 <= 330)
    model.addConstr(23 * x0 + 6 * x2 + 12 * x3 <= 175)
    model.addConstr(4 * x1 + 6 * x2 + 18 * x4 <= 309)
    model.addConstr(23 * x0 + 6 * x2 + 18 * x4 <= 293)
    model.addConstr(23 * x0 + 4 * x1 + 6 * x2 + 12 * x3 + 18 * x4 <= 293)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Mary: ", x0.varValue)
        print("Hours worked by Paul: ", x1.varValue)
        print("Hours worked by Jean: ", x2.varValue)
        print("Hours worked by Ringo: ", x3.varValue)
        print("Hours worked by Hank: ", x4.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```