## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the variables as follows:
- $x_1$ = patches per day
- $x_2$ = intrusion analysts
- $x_3$ = security engineers
- $x_4$ = system administrators

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

## 3: Define the constraints
Based on the problem description, the constraints are:
- $20x_1 + 14x_2 + 2x_3 + 14x_4 \leq 313$ (network latency impact)
- $5x_1 + 5x_2 + 21x_3 + 8x_4 \leq 343$ (data confidentiality impact)
- $18x_1 + 15x_2 + 14x_3 + 5x_4 \leq 259$ (power consumption)
- $20x_1 + 14x_2 \geq 40$ (network latency from patches per day and intrusion analysts)
- $20x_1 + 14x_4 \geq 58$ (network latency from patches per day and system administrators)
- $14x_2 + 2x_3 + 14x_4 \geq 57$ (network latency from intrusion analysts, security engineers, and system administrators)
- $20x_1 + 14x_2 + 2x_3 + 14x_4 \geq 57$ (network latency from all sources)
- $5x_2 + 8x_4 \geq 51$ (data confidentiality impact from intrusion analysts and system administrators)
- $5x_1 + 5x_2 + 21x_3 + 8x_4 \geq 51$ (data confidentiality impact from all sources)
- $14x_3 + 5x_4 \geq 63$ (power consumption from security engineers and system administrators)
- $15x_2 + 14x_3 \geq 60$ (power consumption from intrusion analysts and security engineers)
- $18x_1 + 5x_4 \geq 62$ (power consumption from patches per day and system administrators)
- $15x_2 + 5x_4 \geq 61$ (power consumption from intrusion analysts and system administrators)
- $18x_1 + 14x_3 \geq 51$ (power consumption from patches per day and security engineers)
- $18x_1 + 15x_2 \geq 49$ (power consumption from patches per day and intrusion analysts)
- $18x_1 + 15x_2 + 14x_3 + 5x_4 \geq 49$ (power consumption from all sources)
- $10x_1 - 5x_3 \geq 0$ (linear constraint)
- $-x_3 + 9x_4 \geq 0$ (linear constraint)
- $20x_1 + 2x_3 \leq 265$ (network latency from patches per day and security engineers)
- $14x_2 + 2x_3 \leq 145$ (network latency from intrusion analysts and security engineers)
- $20x_1 + 14x_2 \leq 79$ (network latency from patches per day and intrusion analysts)
- $20x_1 + 14x_4 \leq 97$ (network latency from patches per day and system administrators)
- $20x_1 + 14x_2 + 14x_4 \leq 262$ (network latency from patches per day, intrusion analysts, and system administrators)
- $5x_2 + 21x_3 + 8x_4 \leq 279$ (data confidentiality impact from intrusion analysts, security engineers, and system administrators)
- $15x_2 + 5x_4 \leq 96$ (power consumption from intrusion analysts and system administrators)

## 4: Symbolic Representation
The symbolic representation of the problem is:
```json
{
    'sym_variables': [
        ('x1', 'patches per day'),
        ('x2', 'intrusion analysts'),
        ('x3', 'security engineers'),
        ('x4', 'system administrators')
    ],
    'objective_function': '7*x1 + 6*x2 + 5*x3 + 4*x4',
    'constraints': [
        '20*x1 + 14*x2 + 2*x3 + 14*x4 <= 313',
        '5*x1 + 5*x2 + 21*x3 + 8*x4 <= 343',
        '18*x1 + 15*x2 + 14*x3 + 5*x4 <= 259',
        '20*x1 + 14*x2 >= 40',
        '20*x1 + 14*x4 >= 58',
        '14*x2 + 2*x3 + 14*x4 >= 57',
        '20*x1 + 14*x2 + 2*x3 + 14*x4 >= 57',
        '5*x2 + 8*x4 >= 51',
        '5*x1 + 5*x2 + 21*x3 + 8*x4 >= 51',
        '14*x3 + 5*x4 >= 63',
        '15*x2 + 14*x3 >= 60',
        '18*x1 + 5*x4 >= 62',
        '15*x2 + 5*x4 >= 61',
        '18*x1 + 14*x3 >= 51',
        '18*x1 + 15*x2 >= 49',
        '18*x1 + 15*x2 + 14*x3 + 5*x4 >= 49',
        '10*x1 - 5*x3 >= 0',
        '-x3 + 9*x4 >= 0',
        '20*x1 + 2*x3 <= 265',
        '14*x2 + 2*x3 <= 145',
        '20*x1 + 14*x2 <= 79',
        '20*x1 + 14*x4 <= 97',
        '20*x1 + 14*x2 + 14*x4 <= 262',
        '5*x2 + 21*x3 + 8*x4 <= 279',
        '15*x2 + 5*x4 <= 96'
    ]
}
```

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

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

    # Define variables
    x1 = model.addVar(name='patches_per_day', vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name='intrusion_analysts', vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name='security_engineers', vtype=gurobi.GRB.INTEGER)
    x4 = model.addVar(name='system_administrators', vtype=gurobi.GRB.INTEGER)

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

    # Constraints
    model.addConstr(20 * x1 + 14 * x2 + 2 * x3 + 14 * x4 <= 313)
    model.addConstr(5 * x1 + 5 * x2 + 21 * x3 + 8 * x4 <= 343)
    model.addConstr(18 * x1 + 15 * x2 + 14 * x3 + 5 * x4 <= 259)
    model.addConstr(20 * x1 + 14 * x2 >= 40)
    model.addConstr(20 * x1 + 14 * x4 >= 58)
    model.addConstr(14 * x2 + 2 * x3 + 14 * x4 >= 57)
    model.addConstr(20 * x1 + 14 * x2 + 2 * x3 + 14 * x4 >= 57)
    model.addConstr(5 * x2 + 8 * x4 >= 51)
    model.addConstr(5 * x1 + 5 * x2 + 21 * x3 + 8 * x4 >= 51)
    model.addConstr(14 * x3 + 5 * x4 >= 63)
    model.addConstr(15 * x2 + 14 * x3 >= 60)
    model.addConstr(18 * x1 + 5 * x4 >= 62)
    model.addConstr(15 * x2 + 5 * x4 >= 61)
    model.addConstr(18 * x1 + 14 * x3 >= 51)
    model.addConstr(18 * x1 + 15 * x2 >= 49)
    model.addConstr(18 * x1 + 15 * x2 + 14 * x3 + 5 * x4 >= 49)
    model.addConstr(10 * x1 - 5 * x3 >= 0)
    model.addConstr(-x3 + 9 * x4 >= 0)
    model.addConstr(20 * x1 + 2 * x3 <= 265)
    model.addConstr(14 * x2 + 2 * x3 <= 145)
    model.addConstr(20 * x1 + 14 * x2 <= 79)
    model.addConstr(20 * x1 + 14 * x4 <= 97)
    model.addConstr(20 * x1 + 14 * x2 + 14 * x4 <= 262)
    model.addConstr(5 * x2 + 21 * x3 + 8 * x4 <= 279)
    model.addConstr(15 * x2 + 5 * x4 <= 96)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('patches_per_day: ', x1.varValue)
        print('intrusion_analysts: ', x2.varValue)
        print('security_engineers: ', x3.varValue)
        print('system_administrators: ', x4.varValue)
    else:
        print('No optimal solution found')

solve_optimization_problem()
```