## Step 1: Define the symbolic representation of the variables
The variables are defined as follows:
- $x_0$ : pen testers
- $x_1$ : system administrators
- $x_2$ : deployed decoys
- $x_3$ : automatic alerts
- $x_4$ : security engineers

## Step 2: Convert the objective function into symbolic notation
The objective function to minimize is: $6x_0 + 1 \cdot (x_1 + 6) \cdot x_2 + 6x_3 + 7x_4$

## 3: List the constraints in symbolic notation
Constraints:
1. $6x_0 \leq 219$
2. $11x_0 \leq 218$
3. $4x_0 \leq 249$
4. $8x_1 \leq 219$
5. $8x_1 \leq 218$
6. $14x_1 \leq 249$
7. $1x_2 \leq 219$
8. $9x_2 \leq 218$
9. $1x_2 \leq 249$
10. $8x_3 \leq 219$
11. $11x_3 \leq 218$
12. $3x_3 \leq 249$
13. $5x_4 \leq 219$
14. $9x_4 \leq 218$
15. $3x_4 \leq 249$
16. $6x_0 + 8x_1 + 5x_4 \geq 18$
17. $1x_2 + 5x_4 \geq 18$
18. $8x_1 + 8x_3 \geq 39$
19. $1x_2 + 8x_3 \geq 43$
20. $6x_0 + 8x_1 + 1x_2 + 8x_3 + 5x_4 \geq 43$
21. $8x_1 + 11x_3 \geq 40$
22. $9x_2 + 9x_4 \geq 28$
23. $11x_3 + 9x_4 \geq 15$
24. $8x_1 + 9x_4 \geq 14$
25. $11x_0 + 9x_2 \geq 26$
26. $9x_2 + 11x_3 \geq 14$
27. $11x_0 + 8x_1 + 9x_2 + 11x_3 + 9x_4 \geq 14$
28. $4x_0 + 3x_4 \geq 30$
29. $1x_2 + 3x_3 \geq 42$
30. $4x_0 + 3x_3 \geq 17$
31. $1x_2 + 3x_4 \geq 33$
32. $3x_3 + 3x_4 \geq 40$
33. $4x_0 + 14x_1 + 1x_2 + 3x_3 + 3x_4 \geq 40$
34. $6x_0 + 8x_1 + 5x_4 \leq 160$
35. $6x_0 + 8x_3 + 5x_4 \leq 129$
36. $8x_1 + 1x_2 + 5x_4 \leq 98$
37. $11x_0 + 8x_1 + 9x_2 \leq 157$
38. $11x_0 + 8x_1 + 11x_3 \leq 65$
39. $11x_0 + 11x_3 + 9x_4 \leq 215$
40. $14x_1 + 1x_2 \leq 190$
41. $14x_1 + 3x_4 \leq 164$
42. $4x_0 + 14x_1 \leq 221$
43. $14x_1 + 3x_3 \leq 248$
44. $4x_0 + 3x_3 \leq 226$
45. $4x_0 + 3x_4 \leq 219$
46. $4x_0 + 1x_2 + 3x_3 \leq 196$

## 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'pen testers'),
        ('x1', 'system administrators'),
        ('x2', 'deployed decoys'),
        ('x3', 'automatic alerts'),
        ('x4', 'security engineers')
    ],
    'objective_function': '6*x0 + 1*(x1 + 6)*x2 + 6*x3 + 7*x4',
    'constraints': [
        '6*x0 <= 219',
        '11*x0 <= 218',
        '4*x0 <= 249',
        '8*x1 <= 219',
        '8*x1 <= 218',
        '14*x1 <= 249',
        '1*x2 <= 219',
        '9*x2 <= 218',
        '1*x2 <= 249',
        '8*x3 <= 219',
        '11*x3 <= 218',
        '3*x3 <= 249',
        '5*x4 <= 219',
        '9*x4 <= 218',
        '3*x4 <= 249',
        '6*x0 + 8*x1 + 5*x4 >= 18',
        '1*x2 + 5*x4 >= 18',
        '8*x1 + 8*x3 >= 39',
        '1*x2 + 8*x3 >= 43',
        '6*x0 + 8*x1 + 1*x2 + 8*x3 + 5*x4 >= 43',
        '8*x1 + 11*x3 >= 40',
        '9*x2 + 9*x4 >= 28',
        '11*x3 + 9*x4 >= 15',
        '8*x1 + 9*x4 >= 14',
        '11*x0 + 9*x2 >= 26',
        '9*x2 + 11*x3 >= 14',
        '11*x0 + 8*x1 + 9*x2 + 11*x3 + 9*x4 >= 14',
        '4*x0 + 3*x4 >= 30',
        '1*x2 + 3*x3 >= 42',
        '4*x0 + 3*x3 >= 17',
        '1*x2 + 3*x4 >= 33',
        '3*x3 + 3*x4 >= 40',
        '4*x0 + 14*x1 + 1*x2 + 3*x3 + 3*x4 >= 40',
        '6*x0 + 8*x1 + 5*x4 <= 160',
        '6*x0 + 8*x3 + 5*x4 <= 129',
        '8*x1 + 1*x2 + 5*x4 <= 98',
        '11*x0 + 8*x1 + 9*x2 <= 157',
        '11*x0 + 8*x1 + 11*x3 <= 65',
        '11*x0 + 11*x3 + 9*x4 <= 215',
        '14*x1 + 1*x2 <= 190',
        '14*x1 + 3*x4 <= 164',
        '4*x0 + 14*x1 <= 221',
        '14*x1 + 3*x3 <= 248',
        '4*x0 + 3*x3 <= 226',
        '4*x0 + 3*x4 <= 219',
        '4*x0 + 1*x2 + 3*x3 <= 196'
    ]
}
```

## 5: Write the Gurobi code
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="x0", vtype=gurobi.GRB.INTEGER)  # pen testers
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # system administrators
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # deployed decoys
    x3 = model.addVar(name="x3", vtype=gurobi.GRB.INTEGER)  # automatic alerts
    x4 = model.addVar(name="x4", vtype=gurobi.GRB.INTEGER)  # security engineers

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

    # Constraints
    model.addConstr(6 * x0 <= 219)
    model.addConstr(11 * x0 <= 218)
    model.addConstr(4 * x0 <= 249)
    model.addConstr(8 * x1 <= 219)
    model.addConstr(8 * x1 <= 218)
    model.addConstr(14 * x1 <= 249)
    model.addConstr(x2 <= 219)
    model.addConstr(9 * x2 <= 218)
    model.addConstr(x2 <= 249)
    model.addConstr(8 * x3 <= 219)
    model.addConstr(11 * x3 <= 218)
    model.addConstr(3 * x3 <= 249)
    model.addConstr(5 * x4 <= 219)
    model.addConstr(9 * x4 <= 218)
    model.addConstr(3 * x4 <= 249)
    model.addConstr(6 * x0 + 8 * x1 + 5 * x4 >= 18)
    model.addConstr(x2 + 5 * x4 >= 18)
    model.addConstr(8 * x1 + 8 * x3 >= 39)
    model.addConstr(x2 + 8 * x3 >= 43)
    model.addConstr(6 * x0 + 8 * x1 + x2 + 8 * x3 + 5 * x4 >= 43)
    model.addConstr(8 * x1 + 11 * x3 >= 40)
    model.addConstr(9 * x2 + 9 * x4 >= 28)
    model.addConstr(11 * x3 + 9 * x4 >= 15)
    model.addConstr(8 * x1 + 9 * x4 >= 14)
    model.addConstr(11 * x0 + 9 * x2 >= 26)
    model.addConstr(9 * x2 + 11 * x3 >= 14)
    model.addConstr(11 * x0 + 8 * x1 + 9 * x2 + 11 * x3 + 9 * x4 >= 14)
    model.addConstr(4 * x0 + 3 * x4 >= 30)
    model.addConstr(x2 + 3 * x3 >= 42)
    model.addConstr(4 * x0 + 3 * x3 >= 17)
    model.addConstr(x2 + 3 * x4 >= 33)
    model.addConstr(3 * x3 + 3 * x4 >= 40)
    model.addConstr(4 * x0 + 14 * x1 + x2 + 3 * x3 + 3 * x4 >= 40)
    model.addConstr(6 * x0 + 8 * x1 + 5 * x4 <= 160)
    model.addConstr(6 * x0 + 8 * x3 + 5 * x4 <= 129)
    model.addConstr(8 * x1 + x2 + 5 * x4 <= 98)
    model.addConstr(11 * x0 + 8 * x1 + 9 * x2 <= 157)
    model.addConstr(11 * x0 + 8 * x1 + 11 * x3 <= 65)
    model.addConstr(11 * x0 + 11 * x3 + 9 * x4 <= 215)
    model.addConstr(14 * x1 + x2 <= 190)
    model.addConstr(14 * x1 + 3 * x4 <= 164)
    model.addConstr(4 * x0 + 14 * x1 <= 221)
    model.addConstr(14 * x1 + 3 * x3 <= 248)
    model.addConstr(4 * x0 + 3 * x3 <= 226)
    model.addConstr(4 * x0 + 3 * x4 <= 219)
    model.addConstr(4 * x0 + x2 + 3 * x3 <= 196)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("x0: ", x0.varValue)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
        print("x3: ", x3.varValue)
        print("x4: ", x4.varValue)
    else:
        print("The model is infeasible")

optimization_problem()
```