## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the variables as follows:
- $x_0$ : security engineers
- $x_1$ : pen testers
- $x_2$ : automatic alerts
- $x_3$ : network administrators
- $x_4$ : system administrators

## Step 2: Formulate the objective function
The objective function to minimize is: $4x_0 + 6x_1 + 5x_2 + 7x_3 + 2x_4$

## 3: Define the constraints
Constraints based on the problem description:
1. $22x_0 \leq 802$
2. $15x_0 + 22x_1 + 14x_2 + 23x_3 + 8x_4 \leq 403$
3. $2x_1 + 22x_2 \geq 153$
4. $22x_2 + 9x_4 \geq 133$
5. $2x_1 + 9x_4 \geq 83$
6. $22x_0 + 9x_4 \geq 92$
7. $22x_2 + 5x_3 + 9x_4 \geq 118$
8. $22x_0 + 22x_2 + 5x_3 \geq 118$
9. $2x_1 + 22x_2 + 5x_3 \geq 118$
10. $2x_1 + 22x_2 + 9x_4 \geq 118$
11. $22x_2 + 5x_3 + 9x_4 \geq 85$
12. $22x_0 + 22x_2 + 5x_3 \geq 85$
13. $2x_1 + 22x_2 + 5x_3 \geq 85$
14. $2x_1 + 22x_2 + 9x_4 \geq 85$
15. $22x_2 + 5x_3 + 9x_4 \geq 95$
16. $22x_0 + 22x_2 + 5x_3 \geq 95$
17. $2x_1 + 22x_2 + 5x_3 \geq 95$
18. $2x_1 + 22x_2 + 9x_4 \geq 95$
19. $22x_2 + 5x_3 + 9x_4 \geq 150$
20. $22x_0 + 22x_2 + 5x_3 \geq 150$
21. $2x_1 + 22x_2 + 5x_3 \geq 150$
22. $2x_1 + 22x_2 + 9x_4 \geq 150$
23. $22x_0 + 2x_1 + 22x_2 + 5x_3 + 9x_4 \geq 150$
24. $14x_2 + 23x_3 \geq 51$
25. $14x_2 + 8x_4 \geq 53$
26. $15x_0 + 14x_2 \geq 32$
27. $22x_1 + 14x_2 \geq 46$
28. $15x_0 + 8x_4 \geq 52$
29. $22x_1 + 14x_2 + 23x_3 \geq 46$
30. $15x_0 + 22x_1 + 14x_2 + 23x_3 + 8x_4 \geq 46$
31. $-x_1 + 5x_2 \geq 0$
32. $7x_0 - 6x_2 \geq 0$
33. $-4x_2 + 9x_4 \geq 0$
34. $22x_0 + 2x_1 + 9x_4 \leq 560$
35. $22x_0 + 2x_1 + 5x_3 \leq 343$
36. $22x_0 + 22x_2 + 9x_4 \leq 285$
37. $22x_0 + 5x_3 + 9x_4 \leq 263$
38. $2x_1 + 5x_3 + 9x_4 \leq 168$
39. $22x_0 + 22x_2 + 5x_3 \leq 670$
40. $23x_3 + 8x_4 \leq 133$
41. $15x_0 + 22x_1 + 8x_4 \leq 381$

## 4: Symbolic Representation
```json
{
    'sym_variables': [
        ('x0', 'security engineers'),
        ('x1', 'pen testers'),
        ('x2', 'automatic alerts'),
        ('x3', 'network administrators'),
        ('x4', 'system administrators')
    ],
    'objective_function': '4*x0 + 6*x1 + 5*x2 + 7*x3 + 2*x4',
    'constraints': [
        '22*x0 <= 802',
        '15*x0 + 22*x1 + 14*x2 + 23*x3 + 8*x4 <= 403',
        '2*x1 + 22*x2 >= 153',
        '22*x2 + 9*x4 >= 133',
        '2*x1 + 9*x4 >= 83',
        '22*x0 + 9*x4 >= 92',
        '22*x2 + 5*x3 + 9*x4 >= 118',
        '22*x0 + 22*x2 + 5*x3 >= 118',
        '2*x1 + 22*x2 + 5*x3 >= 118',
        '2*x1 + 22*x2 + 9*x4 >= 118',
        '22*x2 + 5*x3 + 9*x4 >= 85',
        '22*x0 + 22*x2 + 5*x3 >= 85',
        '2*x1 + 22*x2 + 5*x3 >= 85',
        '2*x1 + 22*x2 + 9*x4 >= 85',
        '22*x2 + 5*x3 + 9*x4 >= 95',
        '22*x0 + 22*x2 + 5*x3 >= 95',
        '2*x1 + 22*x2 + 5*x3 >= 95',
        '2*x1 + 22*x2 + 9*x4 >= 95',
        '22*x2 + 5*x3 + 9*x4 >= 150',
        '22*x0 + 22*x2 + 5*x3 >= 150',
        '2*x1 + 22*x2 + 5*x3 >= 150',
        '2*x1 + 22*x2 + 9*x4 >= 150',
        '22*x0 + 2*x1 + 22*x2 + 5*x3 + 9*x4 >= 150',
        '14*x2 + 23*x3 >= 51',
        '14*x2 + 8*x4 >= 53',
        '15*x0 + 14*x2 >= 32',
        '22*x1 + 14*x2 >= 46',
        '15*x0 + 8*x4 >= 52',
        '22*x1 + 14*x2 + 23*x3 >= 46',
        '15*x0 + 22*x1 + 14*x2 + 23*x3 + 8*x4 >= 46',
        '-x1 + 5*x2 >= 0',
        '7*x0 - 6*x2 >= 0',
        '-4*x2 + 9*x4 >= 0',
        '22*x0 + 2*x1 + 9*x4 <= 560',
        '22*x0 + 2*x1 + 5*x3 <= 343',
        '22*x0 + 22*x2 + 9*x4 <= 285',
        '22*x0 + 5*x3 + 9*x4 <= 263',
        '2*x1 + 5*x3 + 9*x4 <= 168',
        '22*x0 + 22*x2 + 5*x3 <= 670',
        '23*x3 + 8*x4 <= 133',
        '15*x0 + 22*x1 + 8*x4 <= 381'
    ]
}
```

## 5: Gurobi Code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="security_engineers", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="pen_testers", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="automatic_alerts", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="network_administrators", vtype=gurobi.GRB.INTEGER)
    x4 = model.addVar(name="system_administrators", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(4 * x0 + 6 * x1 + 5 * x2 + 7 * x3 + 2 * x4, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(22 * x0 <= 802)
    model.addConstr(15 * x0 + 22 * x1 + 14 * x2 + 23 * x3 + 8 * x4 <= 403)
    model.addConstr(2 * x1 + 22 * x2 >= 153)
    model.addConstr(22 * x2 + 9 * x4 >= 133)
    model.addConstr(2 * x1 + 9 * x4 >= 83)
    model.addConstr(22 * x0 + 9 * x4 >= 92)
    model.addConstr(22 * x2 + 5 * x3 + 9 * x4 >= 118)
    model.addConstr(22 * x0 + 22 * x2 + 5 * x3 >= 118)
    model.addConstr(2 * x1 + 22 * x2 + 5 * x3 >= 118)
    model.addConstr(2 * x1 + 22 * x2 + 9 * x4 >= 118)
    model.addConstr(22 * x2 + 5 * x3 + 9 * x4 >= 85)
    model.addConstr(22 * x0 + 22 * x2 + 5 * x3 >= 85)
    model.addConstr(2 * x1 + 22 * x2 + 5 * x3 >= 85)
    model.addConstr(2 * x1 + 22 * x2 + 9 * x4 >= 85)
    model.addConstr(22 * x2 + 5 * x3 + 9 * x4 >= 95)
    model.addConstr(22 * x0 + 22 * x2 + 5 * x3 >= 95)
    model.addConstr(2 * x1 + 22 * x2 + 5 * x3 >= 95)
    model.addConstr(2 * x1 + 22 * x2 + 9 * x4 >= 95)
    model.addConstr(22 * x2 + 5 * x3 + 9 * x4 >= 150)
    model.addConstr(22 * x0 + 22 * x2 + 5 * x3 >= 150)
    model.addConstr(2 * x1 + 22 * x2 + 5 * x3 >= 150)
    model.addConstr(2 * x1 + 22 * x2 + 9 * x4 >= 150)
    model.addConstr(22 * x0 + 2 * x1 + 22 * x2 + 5 * x3 + 9 * x4 >= 150)
    model.addConstr(14 * x2 + 23 * x3 >= 51)
    model.addConstr(14 * x2 + 8 * x4 >= 53)
    model.addConstr(15 * x0 + 14 * x2 >= 32)
    model.addConstr(22 * x1 + 14 * x2 >= 46)
    model.addConstr(15 * x0 + 8 * x4 >= 52)
    model.addConstr(22 * x1 + 14 * x2 + 23 * x3 >= 46)
    model.addConstr(15 * x0 + 22 * x1 + 14 * x2 + 23 * x3 + 8 * x4 >= 46)
    model.addConstr(-x1 + 5 * x2 >= 0)
    model.addConstr(7 * x0 - 6 * x2 >= 0)
    model.addConstr(-4 * x2 + 9 * x4 >= 0)
    model.addConstr(22 * x0 + 2 * x1 + 9 * x4 <= 560)
    model.addConstr(22 * x0 + 2 * x1 + 5 * x3 <= 343)
    model.addConstr(22 * x0 + 22 * x2 + 9 * x4 <= 285)
    model.addConstr(22 * x0 + 5 * x3 + 9 * x4 <= 263)
    model.addConstr(2 * x1 + 5 * x3 + 9 * x4 <= 168)
    model.addConstr(22 * x0 + 22 * x2 + 5 * x3 <= 670)
    model.addConstr(23 * x3 + 8 * x4 <= 133)
    model.addConstr(15 * x0 + 22 * x1 + 8 * x4 <= 381)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Security Engineers: ", x0.varValue)
        print("Pen Testers: ", x1.varValue)
        print("Automatic Alerts: ", x2.varValue)
        print("Network Administrators: ", x3.varValue)
        print("System Administrators: ", x4.varValue)
    else:
        print("No optimal solution found")

solve_optimization_problem()
```