## Problem Description and Formulation

The problem is an optimization problem with two variables: 'hours worked by Bill' and 'hours worked by Bobby'. The objective is to minimize the function: $4.11 \times \text{hours worked by Bill} + 4.49 \times \text{hours worked by Bobby}$.

The problem has several constraints:

1. The dollar cost per hour for Bill is 4, and for Bobby is 9.
2. The likelihood to quit index for Bill is 1, and for Bobby is 3.
3. The total combined dollar cost per hour must be at least 38.
4. The total combined likelihood to quit index must be at least 46.
5. $-2 \times \text{hours worked by Bill} + 2 \times \text{hours worked by Bobby} \geq 0$.
6. The total combined dollar cost per hour must be at most 87.
7. The total combined likelihood to quit index must be at most 94.

## Gurobi Code

```python
import gurobi

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

    # Define the variables
    bill_hours = model.addVar(name="bill_hours", lb=0)  # Assuming hours cannot be negative
    bobby_hours = model.addVar(name="bobby_hours", lb=0)  # Assuming hours cannot be negative

    # Define the objective function
    model.setObjective(4.11 * bill_hours + 4.49 * bobby_hours, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(4 * bill_hours + 9 * bobby_hours >= 38, name="dollar_cost_constraint")
    model.addConstr(bill_hours + 3 * bobby_hours >= 46, name="likelihood_to_quit_constraint")
    model.addConstr(-2 * bill_hours + 2 * bobby_hours >= 0, name="hours_worked_constraint")
    model.addConstr(4 * bill_hours + 9 * bobby_hours <= 87, name="dollar_cost_upper_bound")
    model.addConstr(bill_hours + 3 * bobby_hours <= 94, name="likelihood_to_quit_upper_bound")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Bill: {bill_hours.varValue}")
        print(f"Hours worked by Bobby: {bobby_hours.varValue}")
        print(f"Objective function value: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("No feasible solution found.")
    else:
        print("Optimization failed.")

solve_optimization_problem()
```