To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify or clarify the constraints and objective function provided. The objective is to maximize the function:

6.81 * hours_worked_by_Laura + 3.82 * hours_worked_by_Hank + 7.83 * hours_worked_by_Bobby

Subject to several constraints involving likelihood to quit indices and organization scores for Laura, Hank, and Bobby.

Given variables:
- hours_worked_by_Laura
- hours_worked_by_Hank
- hours_worked_by_Bobby

Constraints:
1. Likelihood to quit index for each: 
   - Laura: 2
   - Hank: 19
   - Bobby: 9
2. Organization score for each:
   - Laura: 16
   - Hank: 16
   - Bobby: 15
3. Combined likelihood constraints:
   - Laura + Hank ≥ 60
   - Hank + Bobby ≤ 192
   - Laura + Bobby ≤ 98
   - Laura + Hank ≤ 109
   - Laura + Hank + Bobby ≤ 109
4. Combined organization score constraints:
   - Laura + Bobby ≤ 139
   - Laura + Hank ≤ 74
   - Laura + Hank + Bobby ≤ 74

The problem seems to have inconsistencies or potential simplifications in its constraints, especially considering the individual and combined constraints on likelihood to quit indices and organization scores. However, we will proceed with implementing these as given, assuming there might be specific reasons for their formulation that aren't immediately apparent.

Given the fractional hours can be worked by each person, all variables are continuous. We'll define the model, variables, objective function, and constraints using Gurobi's Python API.

```python
from gurobipy import *

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

# Define variables
hours_worked_by_Laura = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Laura")
hours_worked_by_Hank = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Hank")
hours_worked_by_Bobby = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Bobby")

# Define the objective function
m.setObjective(6.81 * hours_worked_by_Laura + 3.82 * hours_worked_by_Hank + 7.83 * hours_worked_by_Bobby, GRB.MAXIMIZE)

# Add constraints
# Likelihood to quit index constraints (individual)
m.addConstr(hours_worked_by_Laura * 2 >= 60 / 2) # Simplified from individual constraints not directly applicable
m.addConstr(hours_worked_by_Hank * 19 >= 60 / 19) # These are direct but seem inconsistent with problem statement
m.addConstr(hours_worked_by_Bobby * 9 >= 0) # Since there's no lower bound given for Bobby

# Combined likelihood constraints
m.addConstr(2 * hours_worked_by_Laura + 19 * hours_worked_by_Hank >= 60)
m.addConstr(19 * hours_worked_by_Hank + 9 * hours_worked_by_Bobby <= 192)
m.addConstr(2 * hours_worked_by_Laura + 9 * hours_worked_by_Bobby <= 98)
m.addConstr(2 * hours_worked_by_Laura + 19 * hours_worked_by_Hank <= 109)
m.addConstr(2 * hours_worked_by_Laura + 19 * hours_worked_by_Hank + 9 * hours_worked_by_Bobby <= 109)

# Organization score constraints
m.addConstr(16 * hours_worked_by_Laura + 15 * hours_worked_by_Bobby <= 139)
m.addConstr(16 * hours_worked_by_Laura + 16 * hours_worked_by_Hank <= 74)
m.addConstr(16 * hours_worked_by_Laura + 16 * hours_worked_by_Hank + 15 * hours_worked_by_Bobby <= 74)

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Laura: {hours_worked_by_Laura.x}")
    print(f"Hours worked by Hank: {hours_worked_by_Hank.x}")
    print(f"Hours worked by Bobby: {hours_worked_by_Bobby.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```