## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. The variables are:
- $x_0$: hours worked by John
- $x_1$: hours worked by Bobby
- $x_2$: hours worked by Peggy
- $x_3$: hours worked by Ringo

The objective function to maximize is: $9.56x_0 + 4.75x_1 + 9.42x_2 + 4.98x_3$

## Step 2: List the constraints
The constraints are as follows:
- Productivity ratings:
  - $1.25x_0$
  - $1.57x_1$
  - $6.67x_2$
  - $5.68x_3$
- Dollar cost per hour:
  - $7.88x_0$
  - $8.32x_1$
  - $6.01x_2$
  - $1.31x_3$
- Likelihood to quit index:
  - $5.1x_0$
  - $7.48x_1$
  - $0.37x_2$
  - $0.75x_3$

## 3: Translate constraints into algebraic form
1. $1.57x_1 + 5.68x_3 \geq 45$
2. $1.25x_0 + 1.57x_1 \geq 28$
3. $1.57x_1 + 6.67x_2 \geq 28$
4. $7.88x_0 + 8.32x_1 + 1.31x_3 \geq 31$
5. $7.88x_0 + 8.32x_1 + 6.01x_2 \geq 31$
6. $8.32x_1 + 6.01x_2 + 1.31x_3 \geq 31$
7. $7.88x_0 + 8.32x_1 + 1.31x_3 \geq 24$
8. $7.88x_0 + 8.32x_1 + 6.01x_2 \geq 24$
9. $8.32x_1 + 6.01x_2 + 1.31x_3 \geq 24$
10. $7.88x_0 + 8.32x_1 + 1.31x_3 \geq 31$
11. $7.88x_0 + 8.32x_1 + 6.01x_2 \geq 31$
12. $8.32x_1 + 6.01x_2 + 1.31x_3 \geq 31$
13. $5.1x_0 + 7.48x_1 \geq 41$
14. $5.1x_0 + 0.75x_3 \geq 80$
15. $1.25x_0 + 5.68x_3 \leq 181$
16. $1.57x_1 + 5.68x_3 \leq 118$
17. $1.57x_1 + 6.67x_2 \leq 310$
18. $6.67x_2 + 5.68x_3 \leq 177$
19. $1.25x_0 + 1.57x_1 + 6.67x_2 + 5.68x_3 \leq 177$
20. $7.88x_0 + 1.31x_3 \leq 121$
21. $6.01x_2 + 1.31x_3 \leq 43$
22. $8.32x_1 + 1.31x_3 \leq 36$
23. $7.88x_0 + 8.32x_1 \leq 51$
24. $7.88x_0 + 8.32x_1 + 6.01x_2 \leq 117$
25. $7.88x_0 + 8.32x_1 + 6.01x_2 + 1.31x_3 \leq 117$
26. $5.1x_0 + 0.75x_3 \leq 167$
27. $0.37x_2 + 0.75x_3 \leq 327$
28. $7.48x_1 + 0.37x_2 \leq 176$
29. $5.1x_0 + 0.37x_2 \leq 114$
30. $7.48x_1 + 0.75x_3 \leq 326$
31. $5.1x_0 + 7.48x_1 + 0.37x_2 + 0.75x_3 \leq 326$

## 4: Symbolic representation
```json
{
    'sym_variables': [
        ('x0', 'hours worked by John'),
        ('x1', 'hours worked by Bobby'),
        ('x2', 'hours worked by Peggy'),
        ('x3', 'hours worked by Ringo')
    ],
    'objective_function': '9.56x0 + 4.75x1 + 9.42x2 + 4.98x3',
    'constraints': [
        '1.57x1 + 5.68x3 >= 45',
        '1.25x0 + 1.57x1 >= 28',
        '1.57x1 + 6.67x2 >= 28',
        '7.88x0 + 8.32x1 + 1.31x3 >= 31',
        '7.88x0 + 8.32x1 + 6.01x2 >= 31',
        '8.32x1 + 6.01x2 + 1.31x3 >= 31',
        '7.88x0 + 8.32x1 + 1.31x3 >= 24',
        '7.88x0 + 8.32x1 + 6.01x2 >= 24',
        '8.32x1 + 6.01x2 + 1.31x3 >= 24',
        '7.88x0 + 8.32x1 + 1.31x3 >= 31',
        '7.88x0 + 8.32x1 + 6.01x2 >= 31',
        '8.32x1 + 6.01x2 + 1.31x3 >= 31',
        '5.1x0 + 7.48x1 >= 41',
        '5.1x0 + 0.75x3 >= 80',
        '1.25x0 + 5.68x3 <= 181',
        '1.57x1 + 5.68x3 <= 118',
        '1.57x1 + 6.67x2 <= 310',
        '6.67x2 + 5.68x3 <= 177',
        '1.25x0 + 1.57x1 + 6.67x2 + 5.68x3 <= 177',
        '7.88x0 + 1.31x3 <= 121',
        '6.01x2 + 1.31x3 <= 43',
        '8.32x1 + 1.31x3 <= 36',
        '7.88x0 + 8.32x1 <= 51',
        '7.88x0 + 8.32x1 + 6.01x2 <= 117',
        '7.88x0 + 8.32x1 + 6.01x2 + 1.31x3 <= 117',
        '5.1x0 + 0.75x3 <= 167',
        '0.37x2 + 0.75x3 <= 327',
        '7.48x1 + 0.37x2 <= 176',
        '5.1x0 + 0.37x2 <= 114',
        '7.48x1 + 0.75x3 <= 326',
        '5.1x0 + 7.48x1 + 0.37x2 + 0.75x3 <= 326',
        'x0 >= 0', 'x0 is integer',
        'x1 >= 0', 'x1 is integer',
        'x2 >= 0', 'x2 is integer',
        'x3 >= 0', 'x3 is integer'
    ]
}
```

## 5: Gurobi code
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="x0", vtype=gurobi.GRB.INTEGER)  # hours worked by John
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Bobby
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy
    x3 = model.addVar(name="x3", vtype=gurobi.GRB.INTEGER)  # hours worked by Ringo

    # Objective function
    model.setObjective(9.56 * x0 + 4.75 * x1 + 9.42 * x2 + 4.98 * x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(1.57 * x1 + 5.68 * x3 >= 45)
    model.addConstr(1.25 * x0 + 1.57 * x1 >= 28)
    model.addConstr(1.57 * x1 + 6.67 * x2 >= 28)
    model.addConstr(7.88 * x0 + 8.32 * x1 + 1.31 * x3 >= 31)
    model.addConstr(7.88 * x0 + 8.32 * x1 + 6.01 * x2 >= 31)
    model.addConstr(8.32 * x1 + 6.01 * x2 + 1.31 * x3 >= 31)
    model.addConstr(7.88 * x0 + 8.32 * x1 + 1.31 * x3 >= 24)
    model.addConstr(7.88 * x0 + 8.32 * x1 + 6.01 * x2 >= 24)
    model.addConstr(8.32 * x1 + 6.01 * x2 + 1.31 * x3 >= 24)
    model.addConstr(7.88 * x0 + 8.32 * x1 + 1.31 * x3 >= 31)
    model.addConstr(7.88 * x0 + 8.32 * x1 + 6.01 * x2 >= 31)
    model.addConstr(8.32 * x1 + 6.01 * x2 + 1.31 * x3 >= 31)
    model.addConstr(5.1 * x0 + 7.48 * x1 >= 41)
    model.addConstr(5.1 * x0 + 0.75 * x3 >= 80)
    model.addConstr(1.25 * x0 + 5.68 * x3 <= 181)
    model.addConstr(1.57 * x1 + 5.68 * x3 <= 118)
    model.addConstr(1.57 * x1 + 6.67 * x2 <= 310)
    model.addConstr(6.67 * x2 + 5.68 * x3 <= 177)
    model.addConstr(1.25 * x0 + 1.57 * x1 + 6.67 * x2 + 5.68 * x3 <= 177)
    model.addConstr(7.88 * x0 + 1.31 * x3 <= 121)
    model.addConstr(6.01 * x2 + 1.31 * x3 <= 43)
    model.addConstr(8.32 * x1 + 1.31 * x3 <= 36)
    model.addConstr(7.88 * x0 + 8.32 * x1 <= 51)
    model.addConstr(7.88 * x0 + 8.32 * x1 + 6.01 * x2 <= 117)
    model.addConstr(7.88 * x0 + 8.32 * x1 + 6.01 * x2 + 1.31 * x3 <= 117)
    model.addConstr(5.1 * x0 + 0.75 * x3 <= 167)
    model.addConstr(0.37 * x2 + 0.75 * x3 <= 327)
    model.addConstr(7.48 * x1 + 0.37 * x2 <= 176)
    model.addConstr(5.1 * x0 + 0.37 * x2 <= 114)
    model.addConstr(7.48 * x1 + 0.75 * x3 <= 326)
    model.addConstr(5.1 * x0 + 7.48 * x1 + 0.37 * x2 + 0.75 * x3 <= 326)

    # Non-negativity constraints
    model.addConstr(x0 >= 0)
    model.addConstr(x1 >= 0)
    model.addConstr(x2 >= 0)
    model.addConstr(x3 >= 0)

    # Optimize
    model.optimize()

    # Print 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("No solution found")

solve_optimization_problem()
```