## Problem Description and Formulation

The problem is an optimization problem where we need to maximize an objective function subject to several constraints. The objective function is:

Maximize: 9.01 * (hours worked by Hank) + 2.2 * (hours worked by Ringo) + 5.01 * (hours worked by Paul) + 4.0 * (hours worked by Laura)

The variables are:

- hours worked by Hank
- hours worked by Ringo
- hours worked by Paul
- hours worked by Laura

The constraints are:

- The likelihood to quit index for each individual is given and fixed: Hank = 2, Ringo = 13, Paul = 13, Laura = 8.
- The total combined likelihood to quit index from hours worked by Ringo and Laura should be >= 23.
- The total combined likelihood to quit index from hours worked by Hank and Paul should be >= 22.
- The total combined likelihood to quit index from hours worked by Hank and Laura should be >= 10.
- The total combined likelihood to quit index from hours worked by Ringo, Paul, and Laura should be >= 19.
- The total combined likelihood to quit index from hours worked by Hank and Laura should be <= 101.
- The total combined likelihood to quit index from hours worked by Paul and Laura should be <= 74.
- The total combined likelihood to quit index from hours worked by Hank and Paul should be <= 86.
- The total combined likelihood to quit index from hours worked by Hank and Ringo should be <= 120.
- The total combined likelihood to quit index from hours worked by Ringo and Laura should be <= 91.
- The total combined likelihood to quit index from hours worked by Ringo and Paul should be <= 119.
- The total combined likelihood to quit index from hours worked by Hank, Ringo, and Laura should be <= 47.
- The total combined likelihood to quit index from hours worked by Ringo, Paul, and Laura should be <= 90.
- The total combined likelihood to quit index from hours worked by all should be <= 90.
- All hours worked must be integers.

## Gurobi Code Formulation

```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define the variables
    hours_worked_by_Hank = model.addVar(name="hours_worked_by_Hank", vtype=gurobi.GRB.INTEGER)
    hours_worked_by_Ringo = model.addVar(name="hours_worked_by_Ringo", vtype=gurobi.GRB.INTEGER)
    hours_worked_by_Paul = model.addVar(name="hours_worked_by_Paul", vtype=gurobi.GRB.INTEGER)
    hours_worked_by_Laura = model.addVar(name="hours_worked_by_Laura", vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(9.01 * hours_worked_by_Hank + 2.2 * hours_worked_by_Ringo + 5.01 * hours_worked_by_Paul + 4.0 * hours_worked_by_Laura, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(13 * hours_worked_by_Ringo + 8 * hours_worked_by_Laura >= 23)
    model.addConstr(2 * hours_worked_by_Hank + 13 * hours_worked_by_Paul >= 22)
    model.addConstr(2 * hours_worked_by_Hank + 8 * hours_worked_by_Laura >= 10)
    model.addConstr(13 * hours_worked_by_Ringo + 13 * hours_worked_by_Paul + 8 * hours_worked_by_Laura >= 19)
    model.addConstr(2 * hours_worked_by_Hank + 8 * hours_worked_by_Laura <= 101)
    model.addConstr(13 * hours_worked_by_Paul + 8 * hours_worked_by_Laura <= 74)
    model.addConstr(2 * hours_worked_by_Hank + 13 * hours_worked_by_Paul <= 86)
    model.addConstr(2 * hours_worked_by_Hank + 13 * hours_worked_by_Ringo <= 120)
    model.addConstr(13 * hours_worked_by_Ringo + 8 * hours_worked_by_Laura <= 91)
    model.addConstr(13 * hours_worked_by_Ringo + 13 * hours_worked_by_Paul <= 119)
    model.addConstr(2 * hours_worked_by_Hank + 13 * hours_worked_by_Ringo + 8 * hours_worked_by_Laura <= 47)
    model.addConstr(13 * hours_worked_by_Ringo + 13 * hours_worked_by_Paul + 8 * hours_worked_by_Laura <= 90)
    model.addConstr(2 * hours_worked_by_Hank + 13 * hours_worked_by_Ringo + 13 * hours_worked_by_Paul + 8 * hours_worked_by_Laura <= 90)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Hank: {hours_worked_by_Hank.varValue}")
        print(f"Hours worked by Ringo: {hours_worked_by_Ringo.varValue}")
        print(f"Hours worked by Paul: {hours_worked_by_Paul.varValue}")
        print(f"Hours worked by Laura: {hours_worked_by_Laura.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```