## Problem Description and Formulation

The problem is an optimization problem with the goal of minimizing an objective function subject to several constraints. The variables are the hours worked by Laura, Hank, and Mary. The objective function to minimize is \(2.81 \times \text{hours worked by Laura} + 5.39 \times \text{hours worked by Hank} + 8.18 \times \text{hours worked by Mary}\).

The constraints include:

1. The likelihood to quit index and organization score for each individual are given but do not directly constrain the model as they are constants.
2. The total combined likelihood to quit index from hours worked by Laura and Hank must be at least 20.
3. The total combined likelihood to quit index from all hours worked must be at least 20.
4. The total combined organization score from hours worked by Laura and Hank must be at least 23.
5. The total combined organization score from hours worked by Laura and Mary must be at least 44.
6. The total combined organization score from all hours worked must be at least 44 and at most 150.
7. \(-10 \times \text{hours worked by Hank} + 5 \times \text{hours worked by Mary} \geq 0\).
8. \(-8 \times \text{hours worked by Laura} + 1 \times \text{hours worked by Hank} \geq 0\).

## Gurobi Code Formulation

```python
import gurobi

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

    # Define variables
    laura_hours = model.addVar(name="laura_hours", lb=0)  # Assuming hours cannot be negative
    hank_hours = model.addVar(name="hank_hours", lb=0)
    mary_hours = model.addVar(name="mary_hours", lb=0)

    # Define parameters
    r0_x0, r0_x1, r0_x2 = 23.08, 9.62, 8.92
    r1_x0, r1_x1, r1_x2 = 20.84, 17.17, 20.23

    # Objective function
    model.setObjective(2.81 * laura_hours + 5.39 * hank_hours + 8.18 * mary_hours, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(r0_x0 * laura_hours + r0_x1 * hank_hours >= 20, name="likelihood_laura_hank")
    model.addConstr(r0_x0 * laura_hours + r0_x1 * hank_hours + r0_x2 * mary_hours >= 20, name="likelihood_all")
    model.addConstr(r1_x0 * laura_hours + r1_x1 * hank_hours >= 23, name="organization_laura_hank")
    model.addConstr(r1_x0 * laura_hours + r1_x2 * mary_hours >= 44, name="organization_laura_mary")
    model.addConstr(r1_x0 * laura_hours + r1_x1 * hank_hours + r1_x2 * mary_hours >= 44, name="organization_all_min")
    model.addConstr(r1_x0 * laura_hours + r1_x1 * hank_hours + r1_x2 * mary_hours <= 150, name="organization_all_max")
    model.addConstr(-10 * hank_hours + 5 * mary_hours >= 0, name="hank_mary_constraint")
    model.addConstr(-8 * laura_hours + 1 * hank_hours >= 0, name="laura_hank_constraint")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Laura hours: {laura_hours.varValue}")
        print(f"Hank hours: {hank_hours.varValue}")
        print(f"Mary hours: {mary_hours.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

optimization_problem()
```