To solve the given optimization problem, we first need to understand the objective function and the constraints. The goal is to maximize the function `4.03 * hours_worked_by_Bill + 7.68 * hours_worked_by_John` subject to several constraints related to organization score, productivity rating, computer competence rating, and specific linear combinations of hours worked by Bill and John.

The constraints can be summarized as follows:
1. Total combined organization score ≥ 5 and ≤ 13.
2. Total combined productivity rating ≥ 39 and ≤ 78.
3. Total combined computer competence rating ≥ 7 and ≤ 15.
4. A specific linear constraint: `4 * hours_worked_by_Bill - 7 * hours_worked_by_John ≥ 0`.

Given the problem's nature, we'll use Gurobi to model and solve this linear programming problem.

```python
from gurobi import *

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

# Add variables: hours worked by Bill and John
hours_worked_by_Bill = m.addVar(lb=0, name="hours_worked_by_Bill")
hours_worked_by_John = m.addVar(lb=0, name="hours_worked_by_John")

# Set the objective function
m.setObjective(4.03 * hours_worked_by_Bill + 7.68 * hours_worked_by_John, GRB.MAXIMIZE)

# Add constraints
# Organization score constraint: total combined ≥ 5 and ≤ 13
m.addConstr(6.44 * hours_worked_by_Bill + 5.84 * hours_worked_by_John >= 5, name="org_score_min")
m.addConstr(6.44 * hours_worked_by_Bill + 5.84 * hours_worked_by_John <= 13, name="org_score_max")

# Productivity rating constraint: total combined ≥ 39 and ≤ 78
m.addConstr(2.16 * hours_worked_by_Bill + 2.06 * hours_worked_by_John >= 39, name="prod_rating_min")
m.addConstr(2.16 * hours_worked_by_Bill + 2.06 * hours_worked_by_John <= 78, name="prod_rating_max")

# Computer competence rating constraint: total combined ≥ 7 and ≤ 15
m.addConstr(4.71 * hours_worked_by_Bill + 2.29 * hours_worked_by_John >= 7, name="comp_comp_min")
m.addConstr(4.71 * hours_worked_by_Bill + 2.29 * hours_worked_by_John <= 15, name="comp_comp_max")

# Specific linear constraint
m.addConstr(4 * hours_worked_by_Bill - 7 * hours_worked_by_John >= 0, name="linear_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Bill: {hours_worked_by_Bill.x}")
    print(f"Hours worked by John: {hours_worked_by_John.x}")
else:
    print("No optimal solution found")
```