To solve the optimization problem described, we will use the Gurobi Python API. The problem involves maximizing an objective function subject to several constraints related to the hours worked by Ringo and Paul.

### Problem Description:

- Variables: `Ringo_hours`, `Paul_hours`
- Objective Function: Maximize `9 * Ringo_hours + 3 * Paul_hours`
- Constraints:
  - Total combined computer competence rating >= 22
  - Total combined productivity rating >= 9
  - Total combined paperwork competence rating >= 28
  - Total combined likelihood to quit index >= 10
  - `-5 * Ringo_hours + 10 * Paul_hours >= 0`
  - Total combined computer competence rating <= 36
  - Total combined productivity rating <= 35
  - Total combined paperwork competence rating <= 38
  - Total combined likelihood to quit index <= 34

Given the descriptions:
- Ringo's ratings: computer=4, productivity=9, paperwork=5, quit index=3
- Paul's ratings: computer=1, productivity=1, paperwork=3, quit index=4

### Gurobi Code:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define the variables
Ringo_hours = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="Ringo_hours")
Paul_hours = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="Paul_hours")

# Define the objective function
m.setObjective(9 * Ringo_hours + 3 * Paul_hours, GRB.MAXIMIZE)

# Add constraints
m.addConstr(4 * Ringo_hours + 1 * Paul_hours >= 22, "Computer_Rating_Min")
m.addConstr(9 * Ringo_hours + 1 * Paul_hours >= 9, "Productivity_Rating_Min")
m.addConstr(5 * Ringo_hours + 3 * Paul_hours >= 28, "Paperwork_Rating_Min")
m.addConstr(3 * Ringo_hours + 4 * Paul_hours >= 10, "Quit_Index_Min")
m.addConstr(-5 * Ringo_hours + 10 * Paul_hours >= 0, "Custom_Constraint")

m.addConstr(4 * Ringo_hours + 1 * Paul_hours <= 36, "Computer_Rating_Max")
m.addConstr(9 * Ringo_hours + 1 * Paul_hours <= 35, "Productivity_Rating_Max")
m.addConstr(5 * Ringo_hours + 3 * Paul_hours <= 38, "Paperwork_Rating_Max")
m.addConstr(3 * Ringo_hours + 4 * Paul_hours <= 34, "Quit_Index_Max")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Ringo hours: {Ringo_hours.x}")
    print(f"Paul hours: {Paul_hours.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```