To capture the optimization problem described and provide a solution using Gurobi, we need to translate the natural language description into mathematical formulations. The objective function is given as \(4 \times \text{hours worked by Mary} + 7 \times \text{hours worked by Paul} + 3 \times \text{hours worked by John}\), subject to various constraints related to computer competence ratings, organization scores, and specific combinations of hours worked.

Let's define the variables:
- \(M\): Hours worked by Mary
- \(P\): Hours worked by Paul
- \(J\): Hours worked by John

The objective function can be written as: Minimize \(4M + 7P + 3J\)

Constraints based on the given information:
1. Computer competence rating constraints:
   - \(14M + 8J \geq 22\) (Combined computer competence from Mary and John)
   - \(14M + 3P \geq 16\) (Combined computer competence from Mary and Paul)
   - \(14M + 3P + 8J \geq 16\) (Combined computer competence from all)
2. Organization score constraints:
   - \(9M + 9P \geq 14\) (Combined organization score from Mary and Paul)
   - \(9M + 9J \geq 8\) (Combined organization score from Mary and John)
   - \(9P + 9J \geq 21\) (Combined organization score from Paul and John)
   - \(9M + 9P + 9J \geq 21\) (Combined organization score from all)
3. Additional constraints:
   - \(5M - 8J \geq 0\)
4. Variable type constraints:
   - \(M, P, J \in \mathbb{Z}^+\) (All hours worked are positive integers)

Given these formulations, the Gurobi code can be written as follows:

```python
from gurobipy import *

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

# Define variables
M = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Mary")
P = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Paul")
J = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_John")

# Set objective function
m.setObjective(4*M + 7*P + 3*J, GRB.MINIMIZE)

# Add constraints
m.addConstr(14*M + 8*J >= 22, "computer_competence_Mary_John")
m.addConstr(14*M + 3*P >= 16, "computer_competence_Mary_Paul")
m.addConstr(14*M + 3*P + 8*J >= 16, "computer_competence_all")
m.addConstr(9*M + 9*P >= 14, "organization_score_Mary_Paul")
m.addConstr(9*M + 9*J >= 8, "organization_score_Mary_John")
m.addConstr(9*P + 9*J >= 21, "organization_score_Paul_John")
m.addConstr(9*M + 9*P + 9*J >= 21, "organization_score_all")
m.addConstr(5*M - 8*J >= 0, "additional_constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print("Hours worked by Mary:", M.x)
    print("Hours worked by Paul:", P.x)
    print("Hours worked by John:", J.x)
else:
    print("No optimal solution found")
```