## Step 1: Define the optimization problem and the objective function
The objective function to maximize is $3.77B + 5.17G + 8.77J$, where $B$, $G$, and $J$ represent the hours worked by Bill, George, and Jean, respectively.

## 2: List all the constraints
1. $3B + 10G + 6J \geq 11$ (total combined productivity rating)
2. $13G + 5J \geq 10$ (total combined organization score from George and Jean)
3. $8B + 5J \geq 18$ (total combined organization score from Bill and Jean)
4. $8B + 13G \geq 14$ (total combined organization score from Bill and George)
5. $3B + 10G \leq 14$ (total combined productivity rating from Bill and George)
6. $3B + 6J \leq 30$ (total combined productivity rating from Bill and Jean)
7. $3B + 10G + 6J \leq 30$ (total combined productivity rating from all)
8. $13G + 5J \leq 54$ (total combined organization score from George and Jean)
9. $8B + 13G \leq 20$ (total combined organization score from Bill and George)
10. $8B + 5J \leq 29$ (total combined organization score from Bill and Jean)
11. $8B + 13G + 5J \leq 38$ (total combined organization score from all)
12. $B$ must be an integer
13. $G$ must be an integer
14. $J$ must be an integer

## 3: Convert the problem into Gurobi code
```python
import gurobi

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

    # Define the variables
    B = model.addVar(name="B", lowBound=0, integrality=1)  # Bill's hours
    G = model.addVar(name="G", lowBound=0, integrality=1)  # George's hours
    J = model.addVar(name="J", lowBound=0, integrality=1)  # Jean's hours

    # Define the objective function
    model.setObjective(3.77 * B + 5.17 * G + 8.77 * J, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(3 * B + 10 * G + 6 * J >= 11)  # Productivity rating constraint
    model.addConstr(13 * G + 5 * J >= 10)  # Organization score from George and Jean
    model.addConstr(8 * B + 5 * J >= 18)  # Organization score from Bill and Jean
    model.addConstr(8 * B + 13 * G >= 14)  # Organization score from Bill and George
    model.addConstr(3 * B + 10 * G <= 14)  # Productivity rating from Bill and George
    model.addConstr(3 * B + 6 * J <= 30)  # Productivity rating from Bill and Jean
    model.addConstr(3 * B + 10 * G + 6 * J <= 30)  # Productivity rating from all
    model.addConstr(13 * G + 5 * J <= 54)  # Organization score from George and Jean
    model.addConstr(8 * B + 13 * G <= 20)  # Organization score from Bill and George
    model.addConstr(8 * B + 5 * J <= 29)  # Organization score from Bill and Jean
    model.addConstr(8 * B + 13 * G + 5 * J <= 38)  # Organization score from all

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print(f"Bill's hours: {B.varValue}")
        print(f"George's hours: {G.varValue}")
        print(f"Jean's hours: {J.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```