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

## Step 2: Translate the given attributes into symbolic notation
The work quality ratings are given as: $r_0 = 15x_0 + 10x_1 + 4x_2 + 6x_3$.
The organization scores are given as: $r_1 = 6x_0 + 17x_1 + x_2 + 13x_3$.

## 3: Express the objective function in symbolic notation
The objective function to maximize is $3x_0 + 5x_1 + 8x_2 + 9x_3$.

## 4: List all constraints in symbolic notation
1. $15x_0 = 15$ (Ringo's work quality rating)
2. $6x_0 = 6$ (Ringo's organization score)
3. $10x_1 = 10$ (Jean's work quality rating)
4. $17x_1 = 17$ (Jean's organization score)
5. $4x_2 = 4$ (Hank's work quality rating)
6. $x_2 = 1$ (Hank's organization score)
7. $6x_3 = 6$ (Peggy's work quality rating)
8. $13x_3 = 13$ (Peggy's organization score)
9. $10x_1 + 6x_3 \geq 51$
10. $15x_0 + 10x_1 + 4x_2 \geq 58$
11. $15x_0 + 10x_1 + 6x_3 \geq 58$
12. $10x_1 + 4x_2 + 6x_3 \geq 58$
13. $15x_0 + 10x_1 + 4x_2 \geq 69$
14. $15x_0 + 10x_1 + 6x_3 \geq 69$
15. $10x_1 + 4x_2 + 6x_3 \geq 69$
16. $15x_0 + 10x_1 + 4x_2 \geq 57$
17. $15x_0 + 10x_1 + 6x_3 \geq 57$
18. $10x_1 + 4x_2 + 6x_3 \geq 57$
19. $17x_1 + 13x_3 \geq 62$
20. $x_2 + 13x_3 \geq 59$
21. $17x_1 + x_2 \geq 80$
22. $6x_0 + 17x_1 + 13x_3 \geq 77$
23. $17x_1 + x_2 + 13x_3 \geq 77$
24. $6x_0 + 17x_1 + 13x_3 \geq 59$
25. $17x_1 + x_2 + 13x_3 \geq 59$
26. $6x_1 + 9x_2 - 9x_3 \geq 0$
27. $10x_1 + 4x_2 \leq 165$
28. $15x_0 + 10x_1 \leq 447$
29. $15x_0 + 4x_2 \leq 380$
30. $10x_1 + 6x_3 \leq 153$
31. $15x_0 + 6x_3 \leq 120$
32. $4x_2 + 6x_3 \leq 258$
33. $15x_0 + 10x_1 + 4x_2 + 6x_3 \leq 447$
34. $6x_0 + 17x_1 \leq 243$
35. $6x_0 + 13x_3 \leq 220$
36. $17x_1 + 13x_3 \leq 112$
37. $x_2 + 13x_3 \leq 201$
38. $6x_0 + 17x_1 + 13x_3 \leq 159$
39. $17x_1 + x_2 + 13x_3 \leq 303$
40. $6x_0 + 17x_1 + x_2 + 13x_3 \leq 303$

## 5: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Ringo'),
        ('x1', 'hours worked by Jean'),
        ('x2', 'hours worked by Hank'),
        ('x3', 'hours worked by Peggy')
    ],
    'objective_function': '3*x0 + 5*x1 + 8*x2 + 9*x3',
    'constraints': [
        '15*x0 == 15',
        '6*x0 == 6',
        '10*x1 == 10',
        '17*x1 == 17',
        '4*x2 == 4',
        'x2 == 1',
        '6*x3 == 6',
        '13*x3 == 13',
        '10*x1 + 6*x3 >= 51',
        '15*x0 + 10*x1 + 4*x2 >= 58',
        '15*x0 + 10*x1 + 6*x3 >= 58',
        '10*x1 + 4*x2 + 6*x3 >= 58',
        '15*x0 + 10*x1 + 4*x2 >= 69',
        '15*x0 + 10*x1 + 6*x3 >= 69',
        '10*x1 + 4*x2 + 6*x3 >= 69',
        '15*x0 + 10*x1 + 4*x2 >= 57',
        '15*x0 + 10*x1 + 6*x3 >= 57',
        '10*x1 + 4*x2 + 6*x3 >= 57',
        '17*x1 + 13*x3 >= 62',
        'x2 + 13*x3 >= 59',
        '17*x1 + x2 >= 80',
        '6*x0 + 17*x1 + 13*x3 >= 77',
        '17*x1 + x2 + 13*x3 >= 77',
        '6*x0 + 17*x1 + 13*x3 >= 59',
        '17*x1 + x2 + 13*x3 >= 59',
        '6*x1 + 9*x2 - 9*x3 >= 0',
        '10*x1 + 4*x2 <= 165',
        '15*x0 + 10*x1 <= 447',
        '15*x0 + 4*x2 <= 380',
        '10*x1 + 6*x3 <= 153',
        '15*x0 + 6*x3 <= 120',
        '4*x2 + 6*x3 <= 258',
        '15*x0 + 10*x1 + 4*x2 + 6*x3 <= 447',
        '6*x0 + 17*x1 <= 243',
        '6*x0 + 13*x3 <= 220',
        '17*x1 + 13*x3 <= 112',
        'x2 + 13*x3 <= 201',
        '6*x0 + 17*x1 + 13*x3 <= 159',
        '17*x1 + x2 + 13*x3 <= 303',
        '6*x0 + 17*x1 + x2 + 13*x3 <= 303',
        'x3 % 1 == 0' # Ensuring x3 is an integer
    ]
}
```

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

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

    # Define variables
    x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x0")
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x1")
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x2")
    x3 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x3", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(3*x0 + 5*x1 + 8*x2 + 9*x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(15*x0 == 15)
    model.addConstr(6*x0 == 6)
    model.addConstr(10*x1 == 10)
    model.addConstr(17*x1 == 17)
    model.addConstr(4*x2 == 4)
    model.addConstr(x2 == 1)
    model.addConstr(6*x3 == 6)
    model.addConstr(13*x3 == 13)
    model.addConstr(10*x1 + 6*x3 >= 51)
    model.addConstr(15*x0 + 10*x1 + 4*x2 >= 58)
    model.addConstr(15*x0 + 10*x1 + 6*x3 >= 58)
    model.addConstr(10*x1 + 4*x2 + 6*x3 >= 58)
    model.addConstr(15*x0 + 10*x1 + 4*x2 >= 69)
    model.addConstr(15*x0 + 10*x1 + 6*x3 >= 69)
    model.addConstr(10*x1 + 4*x2 + 6*x3 >= 69)
    model.addConstr(15*x0 + 10*x1 + 4*x2 >= 57)
    model.addConstr(15*x0 + 10*x1 + 6*x3 >= 57)
    model.addConstr(10*x1 + 4*x2 + 6*x3 >= 57)
    model.addConstr(17*x1 + 13*x3 >= 62)
    model.addConstr(x2 + 13*x3 >= 59)
    model.addConstr(17*x1 + x2 >= 80)
    model.addConstr(6*x0 + 17*x1 + 13*x3 >= 77)
    model.addConstr(17*x1 + x2 + 13*x3 >= 77)
    model.addConstr(6*x0 + 17*x1 + 13*x3 >= 59)
    model.addConstr(17*x1 + x2 + 13*x3 >= 59)
    model.addConstr(6*x1 + 9*x2 - 9*x3 >= 0)
    model.addConstr(10*x1 + 4*x2 <= 165)
    model.addConstr(15*x0 + 10*x1 <= 447)
    model.addConstr(15*x0 + 4*x2 <= 380)
    model.addConstr(10*x1 + 6*x3 <= 153)
    model.addConstr(15*x0 + 6*x3 <= 120)
    model.addConstr(4*x2 + 6*x3 <= 258)
    model.addConstr(15*x0 + 10*x1 + 4*x2 + 6*x3 <= 447)
    model.addConstr(6*x0 + 17*x1 <= 243)
    model.addConstr(6*x0 + 13*x3 <= 220)
    model.addConstr(17*x1 + 13*x3 <= 112)
    model.addConstr(x2 + 13*x3 <= 201)
    model.addConstr(6*x0 + 17*x1 + 13*x3 <= 159)
    model.addConstr(17*x1 + x2 + 13*x3 <= 303)
    model.addConstr(6*x0 + 17*x1 + x2 + 13*x3 <= 303)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("x0: ", x0.varValue)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
        print("x3: ", x3.varValue)
    else:
        print("The model is infeasible")

optimize_problem()
```