## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the hours worked by Dale, Mary, Jean, and Peggy as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the given problem into a symbolic representation
The objective function to maximize is: $3.77x_0 + 4.37x_1 + 2.05x_2 + 7.17x_3$.

## 3: List the constraints in symbolic notation
Constraints:
1. $15x_0 \geq 0$ (Implicit, as $x_0$ is non-negative)
2. $13x_0 \geq 0$ (Implicit, as $x_0$ is non-negative)
3. $23x_1 \geq 0$ (Implicit, as $x_1$ is non-negative)
4. $11x_1 \geq 0$ (Implicit, as $x_1$ is non-negative)
5. $6x_2 \geq 0$ (Implicit, as $x_2$ is non-negative)
6. $20x_2 \geq 0$ (Implicit, as $x_2$ is non-negative)
7. $19x_3 \geq 0$ (Implicit, as $x_3$ is non-negative)
8. $2x_3 \geq 0$ (Implicit, as $x_3$ is non-negative)
9. $15x_0 + 23x_1 + 19x_3 \geq 96$
10. $15x_0 + 6x_2 + 19x_3 \geq 96$
11. $23x_1 + 6x_2 + 19x_3 \geq 96$
12. $15x_0 + 23x_1 + 19x_3 \geq 93$
13. $15x_0 + 6x_2 + 19x_3 \geq 93$
14. $23x_1 + 6x_2 + 19x_3 \geq 93$
15. $15x_0 + 23x_1 + 19x_3 \geq 105$
16. $15x_0 + 6x_2 + 19x_3 \geq 105$
17. $23x_1 + 6x_2 + 19x_3 \geq 105$
18. $13x_0 + 2x_3 \geq 92$
19. $13x_0 + 11x_1 \geq 141$
20. $15x_0 + 23x_1 \leq 349$
21. $6x_2 + 19x_3 \leq 464$
22. $23x_1 + 19x_3 \leq 446$
23. $15x_0 + 23x_1 + 19x_3 \leq 393$
24. $15x_0 + 23x_1 + 6x_2 \leq 484$
25. $23x_1 + 6x_2 + 19x_3 \leq 176$
26. $15x_0 + 6x_2 + 19x_3 \leq 215$
27. $15x_0 + 23x_1 + 6x_2 + 19x_3 \leq 215$
28. $13x_0 + 20x_2 \leq 533$
29. $20x_2 + 2x_3 \leq 396$
30. $11x_1 + 2x_3 \leq 320$
31. $13x_0 + 11x_1 + 2x_3 \leq 522$
32. $13x_0 + 11x_1 + 20x_2 + 2x_3 \leq 522$

## 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Dale'),
        ('x1', 'hours worked by Mary'),
        ('x2', 'hours worked by Jean'),
        ('x3', 'hours worked by Peggy')
    ],
    'objective_function': '3.77*x0 + 4.37*x1 + 2.05*x2 + 7.17*x3',
    'constraints': [
        '15*x0 + 23*x1 + 19*x3 >= 96',
        '15*x0 + 6*x2 + 19*x3 >= 96',
        '23*x1 + 6*x2 + 19*x3 >= 96',
        '15*x0 + 23*x1 + 19*x3 >= 93',
        '15*x0 + 6*x2 + 19*x3 >= 93',
        '23*x1 + 6*x2 + 19*x3 >= 93',
        '15*x0 + 23*x1 + 19*x3 >= 105',
        '15*x0 + 6*x2 + 19*x3 >= 105',
        '23*x1 + 6*x2 + 19*x3 >= 105',
        '13*x0 + 2*x3 >= 92',
        '13*x0 + 11*x1 >= 141',
        '15*x0 + 23*x1 <= 349',
        '6*x2 + 19*x3 <= 464',
        '23*x1 + 19*x3 <= 446',
        '15*x0 + 23*x1 + 19*x3 <= 393',
        '15*x0 + 23*x1 + 6*x2 <= 484',
        '23*x1 + 6*x2 + 19*x3 <= 176',
        '15*x0 + 6*x2 + 19*x3 <= 215',
        '15*x0 + 23*x1 + 6*x2 + 19*x3 <= 215',
        '13*x0 + 20*x2 <= 533',
        '20*x2 + 2*x3 <= 396',
        '11*x1 + 2*x3 <= 320',
        '13*x0 + 11*x1 + 2*x3 <= 522',
        '13*x0 + 11*x1 + 20*x2 + 2*x3 <= 522'
    ]
}
```

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

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name='x0', lb=0)  # hours worked by Dale
x1 = m.addVar(name='x1', lb=0, integrality=gurobi.GRB.INTEGER)  # hours worked by Mary
x2 = m.addVar(name='x2', lb=0, integrality=gurobi.GRB.INTEGER)  # hours worked by Jean
x3 = m.addVar(name='x3', lb=0, integrality=gurobi.GRB.INTEGER)  # hours worked by Peggy

# Define the objective function
m.setObjective(3.77 * x0 + 4.37 * x1 + 2.05 * x2 + 7.17 * x3, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(15 * x0 + 23 * x1 + 19 * x3 >= 96)
m.addConstr(15 * x0 + 6 * x2 + 19 * x3 >= 96)
m.addConstr(23 * x1 + 6 * x2 + 19 * x3 >= 96)
m.addConstr(15 * x0 + 23 * x1 + 19 * x3 >= 93)
m.addConstr(15 * x0 + 6 * x2 + 19 * x3 >= 93)
m.addConstr(23 * x1 + 6 * x2 + 19 * x3 >= 93)
m.addConstr(15 * x0 + 23 * x1 + 19 * x3 >= 105)
m.addConstr(15 * x0 + 6 * x2 + 19 * x3 >= 105)
m.addConstr(23 * x1 + 6 * x2 + 19 * x3 >= 105)
m.addConstr(13 * x0 + 2 * x3 >= 92)
m.addConstr(13 * x0 + 11 * x1 >= 141)
m.addConstr(15 * x0 + 23 * x1 <= 349)
m.addConstr(6 * x2 + 19 * x3 <= 464)
m.addConstr(23 * x1 + 19 * x3 <= 446)
m.addConstr(15 * x0 + 23 * x1 + 19 * x3 <= 393)
m.addConstr(15 * x0 + 23 * x1 + 6 * x2 <= 484)
m.addConstr(23 * x1 + 6 * x2 + 19 * x3 <= 176)
m.addConstr(15 * x0 + 6 * x2 + 19 * x3 <= 215)
m.addConstr(15 * x0 + 23 * x1 + 6 * x2 + 19 * x3 <= 215)
m.addConstr(13 * x0 + 20 * x2 <= 533)
m.addConstr(20 * x2 + 2 * x3 <= 396)
m.addConstr(11 * x1 + 2 * x3 <= 320)
m.addConstr(13 * x0 + 11 * x1 + 2 * x3 <= 522)
m.addConstr(13 * x0 + 11 * x1 + 20 * x2 + 2 * x3 <= 522)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', m.objVal)
    print('Hours worked by Dale: ', x0.varValue)
    print('Hours worked by Mary: ', x1.varValue)
    print('Hours worked by Jean: ', x2.varValue)
    print('Hours worked by Peggy: ', x3.varValue)
else:
    print('No optimal solution found')
```