## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the hours worked by Hank, Laura, John, Dale, and Peggy as $x_0$, $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Translate the objective function into algebraic terms using symbolic variables
The objective function to maximize is $2x_0 + 5x_1 + 8x_2 + 4x_3 + 8x_4$.

## 3: List all constraints in semi-algebraic form using symbolic variables
Constraints:
1. $11x_0 \geq 11$ (Hank's likelihood to quit index is 11, but this seems to be an equality constraint, so $11x_0 = 11$ or simply $x_0 = 1$)
2. $x_0 \geq 0$ and $x_1 \geq 0$ and $x_2 \geq 0$ and $x_3 \geq 0$ and $x_4 \geq 0$ (non-negativity constraints, though not explicitly stated, they are implied for hours worked)
3. $11x_0 + 2x_1 + 10x_4 \geq 18$ (total combined likelihood to quit index)
4. $10x_2 + 5x_3 \geq 11$ (total combined dollar cost per hour from John and Dale)
5. $x_0 + 11x_1 \geq 30$ (total combined dollar cost per hour from Hank and Laura)
6. $11x_1 + 5x_3 \geq 26$ (total combined dollar cost per hour from Laura and Dale)
7. $x_0 + 10x_2 \geq 17$ (total combined dollar cost per hour from Hank and John)
8. $x_0 + 5x_3 \geq 12$ (total combined dollar cost per hour from Hank and Dale)
9. $x_0 + 11x_1 + 10x_2 \geq 25$ (total combined dollar cost per hour from Hank, Laura, and John)
10. $10x_2 + 5x_3 + 3x_4 \geq 25$ (total combined dollar cost per hour from John, Dale, and Peggy)
11. $x_0 + 10x_2 + 5x_3 \geq 25$ (total combined dollar cost per hour from Hank, John, and Dale)
12. $11x_1 + 5x_3 + 3x_4 \geq 25$ (total combined dollar cost per hour from Laura, Dale, and Peggy)
13. $x_0 + 11x_1 + 10x_2 \geq 21$ (similar to constraint 9 but with a different RHS)
14. $10x_2 + 5x_3 + 3x_4 \geq 21$ (similar to constraint 10)
15. $x_0 + 10x_2 + 5x_3 \geq 21$ (similar to constraint 11)
16. $11x_1 + 5x_3 + 3x_4 \geq 21$ (similar to constraint 12)
17. $x_0 + 11x_1 + 10x_2 \geq 22$ 
18. $10x_2 + 5x_3 + 3x_4 \geq 22$ 
19. $x_0 + 10x_2 + 5x_3 \geq 22$ 
20. $11x_1 + 5x_3 + 3x_4 \geq 22$ 
21. $3x_3 + 10x_4 \leq 76$ (total combined likelihood to quit index from Dale and Peggy)
22. $2x_1 + 3x_3 \leq 87$ (total combined likelihood to quit index from Laura and Dale)
23. $11x_0 + 2x_1 \leq 72$ (total combined likelihood to quit index from Hank and Laura)
24. $x_2 + 3x_3 \leq 92$ (total combined likelihood to quit index from John and Dale)
25. $11x_0 + 3x_3 \leq 69$ (total combined likelihood to quit index from Hank and Dale)
26. $11x_0 + 3x_3 + 10x_4 \leq 33$ 
27. $11x_0 + 2x_1 + 10x_4 \leq 129$ 
28. $11x_0 + 2x_1 + x_2 + 3x_3 + 10x_4 \leq 129$ 
29. $x_0 + 3x_4 \leq 49$ (total combined dollar cost per hour from Hank and Peggy)
30. $11x_1 + 10x_2 \leq 101$ (total combined dollar cost per hour from Laura and John)
31. $11x_1 + 3x_4 \leq 159$ (total combined dollar cost per hour from Laura and Peggy)
32. $x_0 + 11x_1 + 5x_3 \leq 132$ 
33. $x_0 + 5x_3 + 3x_4 \leq 160$ 
34. $x_0 + 10x_2 + 5x_3 \leq 109$ 
35. $x_0 + 11x_1 + 10x_2 \leq 153$ 
36. $11x_1 + 5x_3 + 3x_4 \leq 113$ 
37. $x_0 + 10x_2 + 3x_4 \leq 85$ 
38. $10x_2 + 5x_3 + 3x_4 \leq 85$ 
39. $11x_1 + 10x_2 + 5x_3 \leq 85$ 
40. $x_0 + 11x_1 + 10x_2 + 5x_3 + 3x_4 \leq 85$

## 4: Create a symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Hank'),
        ('x1', 'hours worked by Laura'),
        ('x2', 'hours worked by John'),
        ('x3', 'hours worked by Dale'),
        ('x4', 'hours worked by Peggy')
    ],
    'objective_function': '2*x0 + 5*x1 + 8*x2 + 4*x3 + 8*x4',
    'constraints': [
        'x0 = 1',
        '11*x0 + 2*x1 + 10*x4 >= 18',
        '10*x2 + 5*x3 >= 11',
        'x0 + 11*x1 >= 30',
        '11*x1 + 5*x3 >= 26',
        'x0 + 10*x2 >= 17',
        'x0 + 5*x3 >= 12',
        'x0 + 11*x1 + 10*x2 >= 25',
        '10*x2 + 5*x3 + 3*x4 >= 25',
        'x0 + 10*x2 + 5*x3 >= 25',
        '11*x1 + 5*x3 + 3*x4 >= 25',
        'x0 + 11*x1 + 10*x2 >= 21',
        '10*x2 + 5*x3 + 3*x4 >= 21',
        'x0 + 10*x2 + 5*x3 >= 21',
        '11*x1 + 5*x3 + 3*x4 >= 21',
        'x0 + 11*x1 + 10*x2 >= 22',
        '10*x2 + 5*x3 + 3*x4 >= 22',
        'x0 + 10*x2 + 5*x3 >= 22',
        '11*x1 + 5*x3 + 3*x4 >= 22',
        '3*x3 + 10*x4 <= 76',
        '2*x1 + 3*x3 <= 87',
        '11*x0 + 2*x1 <= 72',
        'x2 + 3*x3 <= 92',
        '11*x0 + 3*x3 <= 69',
        '11*x0 + 3*x3 + 10*x4 <= 33',
        '11*x0 + 2*x1 + 10*x4 <= 129',
        '11*x0 + 2*x1 + x2 + 3*x3 + 10*x4 <= 129',
        'x0 + 3*x4 <= 49',
        '11*x1 + 10*x2 <= 101',
        '11*x1 + 3*x4 <= 159',
        'x0 + 11*x1 + 5*x3 <= 132',
        'x0 + 5*x3 + 3*x4 <= 160',
        'x0 + 10*x2 + 5*x3 <= 109',
        'x0 + 11*x1 + 10*x2 <= 153',
        '11*x1 + 5*x3 + 3*x4 <= 113',
        'x0 + 10*x2 + 3*x4 <= 85',
        '10*x2 + 5*x3 + 3*x4 <= 85',
        '11*x1 + 10*x2 + 5*x3 <= 85',
        'x0 + 11*x1 + 10*x2 + 5*x3 + 3*x4 <= 85'
    ]
}
```

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

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

    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Hank
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Laura
    x2 = model.addVar(name="x2", lb=0)  # hours worked by John
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Dale
    x4 = model.addVar(name="x4", lb=0)  # hours worked by Peggy

    # Set x0 = 1 (from constraint 1)
    model.addConstr(x0 == 1)

    # Add constraints
    model.addConstr(11*x0 + 2*x1 + 10*x4 >= 18)
    model.addConstr(10*x2 + 5*x3 >= 11)
    model.addConstr(x0 + 11*x1 >= 30)
    model.addConstr(11*x1 + 5*x3 >= 26)
    model.addConstr(x0 + 10*x2 >= 17)
    model.addConstr(x0 + 5*x3 >= 12)
    model.addConstr(x0 + 11*x1 + 10*x2 >= 25)
    model.addConstr(10*x2 + 5*x3 + 3*x4 >= 25)
    model.addConstr(x0 + 10*x2 + 5*x3 >= 25)
    model.addConstr(11*x1 + 5*x3 + 3*x4 >= 25)
    model.addConstr(x0 + 11*x1 + 10*x2 >= 21)
    model.addConstr(10*x2 + 5*x3 + 3*x4 >= 21)
    model.addConstr(x0 + 10*x2 + 5*x3 >= 21)
    model.addConstr(11*x1 + 5*x3 + 3*x4 >= 21)
    model.addConstr(x0 + 11*x1 + 10*x2 >= 22)
    model.addConstr(10*x2 + 5*x3 + 3*x4 >= 22)
    model.addConstr(x0 + 10*x2 + 5*x3 >= 22)
    model.addConstr(11*x1 + 5*x3 + 3*x4 >= 22)
    model.addConstr(3*x3 + 10*x4 <= 76)
    model.addConstr(2*x1 + 3*x3 <= 87)
    model.addConstr(11*x0 + 2*x1 <= 72)
    model.addConstr(x2 + 3*x3 <= 92)
    model.addConstr(11*x0 + 3*x3 <= 69)
    model.addConstr(11*x0 + 3*x3 + 10*x4 <= 33)
    model.addConstr(11*x0 + 2*x1 + 10*x4 <= 129)
    model.addConstr(11*x0 + 2*x1 + x2 + 3*x3 + 10*x4 <= 129)
    model.addConstr(x0 + 3*x4 <= 49)
    model.addConstr(11*x1 + 10*x2 <= 101)
    model.addConstr(11*x1 + 3*x4 <= 159)
    model.addConstr(x0 + 11*x1 + 5*x3 <= 132)
    model.addConstr(x0 + 5*x3 + 3*x4 <= 160)
    model.addConstr(x0 + 10*x2 + 5*x3 <= 109)
    model.addConstr(x0 + 11*x1 + 10*x2 <= 153)
    model.addConstr(11*x1 + 5*x3 + 3*x4 <= 113)
    model.addConstr(x0 + 10*x2 + 3*x4 <= 85)
    model.addConstr(10*x2 + 5*x3 + 3*x4 <= 85)
    model.addConstr(11*x1 + 10*x2 + 5*x3 <= 85)
    model.addConstr(x0 + 11*x1 + 10*x2 + 5*x3 + 3*x4 <= 85)

    # Objective function
    model.setObjective(2*x0 + 5*x1 + 8*x2 + 4*x3 + 8*x4, gurobi.GRB.MAXIMIZE)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Hank: ", x0.varValue)
        print("Hours worked by Laura: ", x1.varValue)
        print("Hours worked by John: ", x2.varValue)
        print("Hours worked by Dale: ", x3.varValue)
        print("Hours worked by Peggy: ", x4.varValue)
    else:
        print("No optimal solution found")

solve_optimization_problem()
```