To solve the given optimization problem, we will use the Gurobi Python library. The objective is to maximize the weighted sum of hours worked by Paul, Hank, John, and Mary, subject to several constraints related to their computer competence ratings.

First, let's analyze the constraints:

- Minimum combined computer competence rating for Paul and Hank: 12
- Minimum combined computer competence rating for Paul and John: 11
- Minimum combined computer competence rating for John and Mary: 13
- Minimum combined computer competence rating for Hank and Mary: 30
- Maximum combined computer competence rating for Hank and Mary: 127
- Maximum combined computer competence rating for John and Mary: 34
- Maximum combined computer competence rating for Paul and Hank: 96
- Maximum combined computer competence rating for Hank, John, and Mary: 38
- Maximum combined computer competence rating for all (Paul, Hank, John, and Mary): 38

The objective function coefficients are:
- Hours worked by Paul: 3.68
- Hours worked by Hank: 2.07
- Hours worked by John: 5.86
- Hours worked by Mary: 3.14

Given that all variables can be fractional, we don't need to specify any integer constraints.

Here is the Gurobi code to solve this problem:

```python
from gurobipy import *

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

# Define the variables
Paul = m.addVar(name="hours_worked_by_Paul", lb=0)
Hank = m.addVar(name="hours_worked_by_Hank", lb=0)
John = m.addVar(name="hours_worked_by_John", lb=0)
Mary = m.addVar(name="hours_worked_by_Mary", lb=0)

# Define the objective function
m.setObjective(3.68 * Paul + 2.07 * Hank + 5.86 * John + 3.14 * Mary, GRB.MAXIMIZE)

# Add constraints
m.addConstr(8 * Paul + 4 * Hank >= 12, name="Paul_Hank_min")
m.addConstr(8 * Paul + 13 * John >= 11, name="Paul_John_min")
m.addConstr(13 * John + 7 * Mary >= 13, name="John_Mary_min")
m.addConstr(4 * Hank + 7 * Mary >= 30, name="Hank_Mary_min")
m.addConstr(4 * Hank + 7 * Mary <= 127, name="Hank_Mary_max")
m.addConstr(13 * John + 7 * Mary <= 34, name="John_Mary_max")
m.addConstr(8 * Paul + 4 * Hank <= 96, name="Paul_Hank_max")
m.addConstr(4 * Hank + 13 * John + 7 * Mary <= 38, name="Hank_John_Mary_max")
m.addConstr(8 * Paul + 4 * Hank + 13 * John + 7 * Mary <= 38, name="All_max")

# Optimize the model
m.optimize()

# Print the solution
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print("Objective:", m.objVal)
```