## Step 1: Define the optimization problem and the objective function
The objective function to maximize is: $9.2x_0 + 1.4x_1 + 1.15x_2 + 2.34x_3$, where $x_0$ represents the hours worked by John, $x_1$ by Jean, $x_2$ by Ringo, and $x_3$ by Bill.

## 2: List all the constraints
1. $8x_0 \leq 105$ (Implicitly, $x_0 \geq 0$ and $8x_0 \leq 105$ translates to $x_0 \leq 13.125$ but we focus on given constraints directly)
2. $x_0$ has a likelihood to quit index of 8, which is directly given.
3. $x_0$ has a work quality rating of 1.
4. $9x_1 \leq 105$
5. $x_1$ has a likelihood to quit index of 9.
6. $x_1$ has a work quality rating of 16.
7. $18x_2 \leq 105$
8. $x_2$ has a likelihood to quit index of 18.
9. $x_2$ has a work quality rating of 10.
10. $12x_3 \leq 105$
11. $x_3$ has a likelihood to quit index of 12.
12. $x_3$ has a work quality rating of 9.
13. $8x_0 + 9x_1 \geq 22$
14. $9x_1 + 18x_2 \geq 18$
15. $16x_1 + 9x_3 \geq 33$
16. $1x_0 + 9x_3 \geq 35$
17. $1x_0 + 16x_1 \geq 42$
18. $1x_0 + 10x_2 \geq 36$
19. $16x_1 + 10x_2 \geq 38$
20. $1x_0 + 10x_2 + 9x_3 \geq 31$
21. $16x_1 + 10x_2 + 9x_3 \geq 31$
22. $1x_0 + 10x_2 + 9x_3 \geq 39$
23. $16x_1 + 10x_2 + 9x_3 \geq 39$
24. $8x_0 + 9x_1 \leq 47$
25. $8x_0 + 9x_1 + 18x_2 \leq 37$
26. $8x_0 + 9x_1 + 18x_2 + 12x_3 \leq 37$
27. $1x_0 + 10x_2 \leq 143$
28. $1x_0 + 9x_3 \leq 105$
29. $1x_0 + 16x_1 + 9x_3 \leq 93$
30. $1x_0 + 10x_2 + 9x_3 \leq 93$
31. $1x_0 + 16x_1 + 10x_2 + 9x_3 \leq 93$

## 3: Convert the problem into Gurobi code
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="hours_worked_by_John", lb=0)
    x1 = model.addVar(name="hours_worked_by_Jean", lb=0)
    x2 = model.addVar(name="hours_worked_by_Ringo", lb=0)
    x3 = model.addVar(name="hours_worked_by_Bill", lb=0)

    # Objective function
    model.setObjective(9.2*x0 + 1.4*x1 + 1.15*x2 + 2.34*x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(8*x0 <= 105)
    model.addConstr(9*x1 <= 105)
    model.addConstr(18*x2 <= 105)
    model.addConstr(12*x3 <= 105)

    model.addConstr(8*x0 + 9*x1 >= 22)
    model.addConstr(9*x1 + 18*x2 >= 18)
    model.addConstr(16*x1 + 9*x3 >= 33)
    model.addConstr(x0 + 9*x3 >= 35)
    model.addConstr(x0 + 16*x1 >= 42)
    model.addConstr(x0 + 10*x2 >= 36)
    model.addConstr(16*x1 + 10*x2 >= 38)
    model.addConstr(x0 + 10*x2 + 9*x3 >= 31)
    model.addConstr(16*x1 + 10*x2 + 9*x3 >= 31)
    model.addConstr(x0 + 10*x2 + 9*x3 >= 39)
    model.addConstr(16*x1 + 10*x2 + 9*x3 >= 39)

    model.addConstr(8*x0 + 9*x1 <= 47)
    model.addConstr(8*x0 + 9*x1 + 18*x2 <= 37)
    model.addConstr(8*x0 + 9*x1 + 18*x2 + 12*x3 <= 37)

    model.addConstr(x0 + 10*x2 <= 143)
    model.addConstr(x0 + 9*x3 <= 105)
    model.addConstr(x0 + 16*x1 + 9*x3 <= 93)
    model.addConstr(x0 + 10*x2 + 9*x3 <= 93)
    model.addConstr(x0 + 16*x1 + 10*x2 + 9*x3 <= 93)

    # Solve the model
    model.optimize()

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

optimize_problem()
```